Chapter 1 :Digital Modulation techniques techniques 1.1 WHAT IS THE MODULATION? Modulation is the process of encoding information from a message source in a manner suitable for t ransmission. ransmission. It is generally involves translating a baseband baseband message signal (called the source) sour ce) to a bandpass signal at frequencies that are very high when compared to the base band frequency. The bandpass signal is called the modulated signal, and the baseband message signal is called the modulating signal. Modulation may be done by varying the amplitude, phase or frequency of a high frequency carrier in accordance with the amplitude of the message signal. Demodulation is the process of extracting the baseband message from the carrier so that it may be processed by the intended receiver. 1.1.1
Why we modulate signals? In order to ease propagation process and use an antenna of a suitable length. Since the effective radiation of EM waves requires antenna dimensions comparable with the wavelength: e.g. -Antenna for 3 kHz would be ~100 km long. long. -Antenna for 3 GHz carrier is 10 cm long. Sharing the access to the telecomm t elecommunication unication channel resources: This is done by using FDM using FDM (Frequency (Frequency division multiplexing) technique. technique. In order to transmit larger power for wide area: If we amplify the data power using power amplifiers, it will be distorted, so we perform modulation and amplify the carrier power. In order to reduce noise effects in case of non-white Gaussian noise.
1.1.2 Why Digital? (Analog versus Digital): Modern mobile communication systems use digital modulation techniques. Advancements in very large-scale integration (VLSI) and digital signal processing (DSP) technology have made digital modulation more cost effective than analog transmission systems. Digital modulation offers many advantages over analog modulation. Some advantages include greater noise immunity and robustness to channel impairments, easier multiplexing of various forms of information (e.g., voice, data, and video), and greater security. Furthermore, digital transmissions accommodate digital error-control codes which detect and/or correct transmi tr ansmission ssion errors, error s, and support complex signal conditioning and processing techniques such as source coding, encryption, and equalization to improve the performance of the overall communication link. New multipurpose programmable digital digital signal processors have made it possible to implement digital modulators and demodulators completely in software. Instead of having a particular modem design permanently frozen as hardware, embedded
software implementations now allow alterations and improvements without having to redesign or replace the modem. We introduce here in table(1.1) a comparison between analog and digital modulation schemes to conclude the assessment of both modulation schemes usage in Wireless communication systems
Analog
Digital
Less bandwidth( Advantage)
Large bandwidth( Disadvantage)
More accurate ( Advantage)
Less accurate due to the Quantization error that can not be avoided or corrected. (Disadvantage)
Low noise immunity ( Disadvantage).
High noise immunity as the amplitude of the digital has two levels only and channel coding(error correcting correcting codes) can be used. ( Advantage)
Low level of security. (Disadvantage)
High level of security as you can use Encryption (Ciphering) and Authentication. Authentication. (Advantage)
No signal conditioning and processing are used (Disadvantage)
Support complex signal conditioning and processing techniques such as source coding, encryption, and equalization((Advantage)
Low QOS. ( Disadvantage)
High QOS. (Advantage)
You can use FDM only( Disadvantage)
You can use FDM, TDM, CDM, OFDM multiplexing t echniques. echniques. (Advantage)
In mobile communications, analog supports voice service only. (Disadvantage)
In mobile communications, digital supports voice, SMS, data (you can access the internet), images and video call. (Advantage) Easily designed using software (Advantage).
More difficult to design design than Digital. D igital. (Disadvantage)
Table (1.1) comparisons between analog and digital modulation schemes
1.1.3 Factors that influence the choice of digital modulation: A desirable modulation scheme should provide:
Low bit error rates at low receive r eceived d signal to noise ratio. Performs well in multi-path and fading conditions, and in interference environment. Occupies a minimum bandwidth. Easy and cost-effective to implement. Cost and complexity of the receiver subscribers must be minimized. Modulation which is simple to detect is most attractive. Note That: That: There is no modulation scheme that satisfies all these requirements, so trade-offs are made when selecting a modulation scheme.
1.1.4 The performance of a modulation scheme : We assess the performance of the modulation scheme by measuring the Power efficiency (ηP). Bandwidth efficiency e fficiency((ηB). Power spectral density. System complexity.
1.1.4.1 Power efficiency ηP: The power The power efficiency is defined as the required E b /No (Ratio of the signal energy per bit to noise power spectral density) at the input of the receiver for a certain bit bit error probability probability Pb over an AWGN channel. Power efficiency describes the ability of a modulation technique to preserve the bit error probability of digital message at low power levels. In digital modulation systems, in order to increase the noise immunity, it is necessary to increase the signal power, so there is a trade-off between the signal power and the bit error probability. The power efficiency is a measure of how favorably this trade off is made.
1.1.4.2 Bandwidth efficiency (Spectral efficiency) ηB: Bandwidth efficiency describes the ability of a modulation scheme to accommodate data within a limited bandwidth. As the data rate increases, pulse width of the digital symbols decreases and hence the bandwidth increases.
=
/
eqn (1.1)
The system capacity capacity of a digital mobile communication system system is directly d irectly related to the bandwidth efficiency for a modulation scheme. So a modulation scheme with greater value of ηB will transmit more data in a given spectrum allocation. Note that the maximum possible bandwidth efficiency is limited by the noise in the channel according to Shannon's Theorem as:
=
=
2
1+
Where C is the channel capacity in bps , and S/N is the signal to noise ratio .
eqn (1.2)
1.1.4.3 Bandwidth efficiency, Power efficiency Trade-off:
Adding error control coding to message increases the required bandwidth, then decreases, decreases, but the required received power for a particular bit error rate decreases and hence increases. On the other hand using high levels M'ary modulation schemes (except ( except in M’ary FSK modulation which isn’t bandwidth limited modulation scheme), scheme) , decreases the bandwidth occupancy, increases, but the required received power for a particular bit error rate increases and hence decreases.
1.1.4.4 System Complexity System complexity complexity refers to the amount of o f circuits involved and the t he technical difficulty of the system. Associated with the system complexity is the cost of manufacturing, which is of course a major concern in choosing a modulation technique. Usually the demodulator is more complex than the modulator. Coherent demodulator is much more complex than no coherent demodulator since carrier recovery is required. For some demodulation methods, sophisticated algorithms like the Viterbi algorithm are required. Also note that, for all personal communication systems which serve a large user community, the cost and complexity of the subscriber receiver must be minimized, and a modulation which is simple to detection is most attractive All these are basis for complexity comparison. Since power efficiency, bandwidth efficiency, and system complexity are the main criteria of choosing a modulation technique, technique, we will always pay attention to them in the analysis of modulation techniques.
1.1.4.5 Other considerations While power and bandwidth efficiency considerations are very important, other factors also affect the choice of a digital modulation scheme. For example The performance of the modulation scheme under various types of channel impair ments such as Rayleigh and Rician fading and multipath time dispersion, given a particular demodulator implementation, is another key factor in selecting a modulation. In cellular systems where interference is a major issue, the performance of a modulation scheme in an interference environment is extremely important. Sensitivity Sensitivity to detection detection of timing jitter, caused by time-varying channels, channels, is also an important consideration in choosing a particular modulation modulation scheme. In general, ge neral, the modulation, interference, interference, and implementation implementation of the t he time varying effects of the channel as well as the performance of the specific demodulator are analyzed as a complete system using simulation to determine relative performance performance and ultimate u ltimate selection.
1.1.5 Hierarchy of Digital modulation schemes Digital modulation techniques may be classified into coherent and noncoherent techniques depending on whether the receiver is equipped with a phase-recovery circuit or not. The phase recovery circuit ensures that the oscillator supplying the locally generated carrier wave in the receiver is synchronized (in both frequency and phase) to the transmitter oscillator.
Digital modulation schemes (according to receiver)
coherent demodulation
noncoherent demodulation
(All types of modulation )
(All types of modulation except PSK)
Fig.(1.1) Digital modulation according to demodulation type
The modulation schemes listed in the fig.(1.2) and the tree are classified into two large categories: constant envelope and nonconstant envelope. Under constant envelope class, there are three subclasses: FSK and PSK. Under nonconstant envelope class, there are three subclasses: ASK and QAM. Digital Modulation schemes constant Envelope
nonconstant envelope
FSK
PSK
-BFSK
-BPSK
-M'ary FSK
-DPSK
-MSK
-M'ary PSK.
-GMSK
-QPSK.
ASK
M'ary QAM
-On-Off keying.
-Rectangular QAM.
-M'ary ASK
-OQPSK. π / 4 – QPSK
Fig.(1.2) Digital modulation hierarchy
-circular QAM
1.1.6 Types of modulation schemes in different advanced digital communication systems: In the table (1.2) we give examples of the used modulation schemes in different wireless modern communication systems
Communication system
Used modulation scheme
GSM(Global System for Mobile communications) 2G. GPRS(General Packet Radio Service) 2.5G. EDGE (Enhanced Data Rates for GSM Evolution) 2.75G. CDMA 2000 ( Code Division Multiple Access)
GMSK
UMTS (Universal Mobile Telecommunications System) 3G HSDPA (High-Speed Downlink Packet Access). 3.5G
Wi Fi (Wireless Fidelity) WiMAX (the Worldwide Interoperability for Microwave Access) , Fixed and mobile
8PSK -QPSK in the forward channel (From BTS to MS). -OQPSK in the reverse channel QPSK -Adaptive modulation: depending on signal quality and cell usage. - QPSK , data rate: 1.8 Mbit/s - 16QAM , data rate: 3.6 Mbit/s in good radio conditions. BPSK , QPSK , 16 QAM , 64 QAM Adaptive Modulation: QPSK, 16 QAM, 64 QAM
Table (1.2) Modulation schemes used in advanced communication systems
1.1.7 Geometric representation of Modulated signal(Constellation diagram). To proceed with the analysis of the digital modulation schemes we introduce the constellation diagram as we can see the Digital modulation means choosing particular signals from a finite set of a possible signal waveforms (symbols) based on the information bits applied to modulator. If there are total of M possible signals ., S= 1 , 2 , For binary information bit S will contain two signals and For signal size of M It is possible to transmit log 2M bits to represent a symbol.(ex. M=8 3 bits/symbol) Vector space analysis provides valuable insight into the performance of particular modulation scheme. The idea is any realizable waveforms in a vector space can be expressed as a linear combination of “N” orthonormal waveforms (called a basis signal).Once the basis signal is determined we can express any signal as a linear combination of them.
……
1.1.7.1 The Basis signal conditions = =1 ( ) (1)
eqn (1.3)
that
means that any signal can be represented by linear combination of basis functions (2) Basis signals are orthogonal to each other in time
− ≠ − ∞
=0 (3) Basis signals are normalized to unit energy ∞
∞
i
2
=1 i.e. basis signals forms a coordinate system for the vector space ∞
eqn(1.4) eqn (1.5)
Note that:
no. of basis signals is less than or equal the signal set
No of basis signals is called dimension
1.1.7.2Constellation diagram interpretation The constellation diagram provides graphical representation of the complex e nvelope of each possible symbol state. The X-axis of the diagram is called in-phase component and the y-axis represents the quadrature component The distance between signals on constellation diagram relates to how different t he modulation waveforms are and how well the receiver can differentiate between all possible symbols when random noise is present. Some of properties of the modulation scheme can be inferred from the constellation diagram: BW occupied by the modulation signals decreases as no. of points increases i.e. if modulation scheme has a densely packed constellation it would be more bandwidth efficient.
Pe is proportional to the distance between the closest points in constellation densely packed modulation scheme is less energy efficient than the modulation scheme that has sparse constellation
High Power efficiency low Power efficiency Low Bw efficiency high Bw efficiency ____________________________________________________________ Fig.(1.3) comparison between constellation diagram interpretation on power and BW efficiencies.
1.1.7.3 Probability of error and constellation diagram The constellation diagram can also be employed to find the upper bound for symbol error rate in AWGN channel with PSD=N o Is
≤ ≠ ∞ − ( | )
Where the Q-function is
=
=1,
1
2
exp (
2
2
2)
eqn (1.6)
eqn (1.7)
And dij is Euclidean distance between ith and the jth points.
1.2
LINE CODES
Line codes ( Baseband modulation) is defined as a direct transmission without Frequency transform. It is the technology of representing digital sequences by pulse waveforms suitable for baseband transmission. A variety of waveforms have been proposed in an effort to find ones with some desirable properties, such as good bandwidth and power efficiency, and adequate timing information. These baseband modulation waveforms are variably called line codes, baseband formats (or waveforms), PCM waveforms (or formats, or codes). Any of several line codes can be used for the electrical representation of a binary data stream. Figure (1.4) displays the waveforms of five important line codes for the example data stream 01101001. Figure (1.5) displays their individual power spectra (for positive frequencies) for randomly generated binary data, Assuming that
symbols 0 and 1 are equiprobable, the average power is normalized to unity, and The frequency f is normalized with respect to the bit rate 1/T b. The five line codes illustrated in Figure (1.4) are described here:
1.2.1 Unipolar nonreturn-to-zero (NRZ) signaling In this line code, symbol 1 is represented by transmitting a pulse A for the duration of the symbol, and symbol 0 is represented by switching off the pulse, as in Figure (1.4) (a).This line code is also referred to as on-off signaling. Disadvantages of on-off signaling are the waste of power due to transmitted DC level and the fact that the power spectrum of the transmitt ed signal does not approach zero at zero frequency.
1.2.2 Polar nonreturn-to-zero (NRZ) signaling In this second line code, symbol 1 and 0 are represented by transmitting pulse of amplitudes +A and – A, respectively, as illustrated in Figure (1.4) (b). This line code is relatively easy to generate but disadvantage is that the power spectrum of the signal is large near zero frequency.
1.2.3 Unipolar return-to-zero (RZ) signaling In this other line code, symbol 1 is represented by a rectangular pulse of amplitude A and half-symbol 0 width, and symbol 0 is represented by transmitting no pulse, as illustrated in Figure (1.4) (c). An attractive feature of this line code is the presence of delta functions at f = 1/Tb in the power spectrum of the transmitted signal, which can be used for bit-timing recovery at the receiver. However, its disadvantage is that it requires 3db more power than polar return-to-zero signaling for the same probability of symbol error.
____________________________________________________________________ Figure (1.4) Line codes for the electrical representation of binary data: (a) Unipolar NRZ signaling. (b) Polar NRZ signaling. (c) Unipolar RZ signaling. (d) Bipolar RZ signaling. (e) Split-phase or Manchester code.
_____________________________________________________________________ Figure(1.5) Power spectra of line codes: (a) Unipolar NRZ signal. (b) Polar NRZ signal. (c) Unipolar RZ signal. (d) Bipolar RZ signal. (e) Ma nchester-encoded signal. The frequency is normalized with respect to the bit rate 1/Tb and the average power is normalized to unity.
1.2.4 Bipolar return-to-zero (BRZ) signaling This line code uses three amplitude level as indicated in Figure (1.4) (d). Specifically, positive and negative pulses of equal amplitude (i.e., +A and – A) are used alternately for symbol 1, with each pulse having a half-symbol width; no pulse is always used for symbol 0. A useful property of the BRZ signaling is that the power spectrum of the transmitted signal has no DC component and relatively insignificant low-frequency components for the case when symbols 1 and 0 occur with equal probability. This line code is also called alternate mark inversion (AMI) signaling .
1.2.5 Split-phase (Manchester code) In this method of signaling, illustrated in Figure (1.4) (e). symbol 1 is represented by a positive pulse of amplitude A followed by a negative pulse of amplitude – A, with both pulses being half-symbol wide. For symbol 0, the polarities of these two pulses are reversed. The Manchester code suppresses the DC component and has relatively insignificant low-frequency components, regardless of the signal statistics. This property is essential in some applications.
1.2.6 Differential encoding This method is used to encode information in terms of signal transitions. In particular, a transition is used to designate symbol 0 in the incoming binary data stream, while no transition is used to designate symbol l, as illustrated in Figure (1.6). In Figure (1.6)(b).we show the differentially encoded data stream for the example data specified in Figure (1.6)(a) .The original binary data stream used here is the same that used in Figure (1.4). The waveform of the differentially encoded data is shown in Figure (1.6)(c)., assuming the use of unipolar nonreturn-to-zero signaling. From Figure (1.6) it is apparent that a differentially encoded signal may be inverted without affecting its interpretation. The original binary information is recovered simply by comparing the polarity of adjacent binary symbols to establish whether or not a transition has occurred. Note that differential encoding requires the use of a reference bit before initiating the encoding process. In Figure (1.6), symbol 1 is used as the reference bit.
_____________________________________________________________________ Figure (1.6)(a) Original binary data. (b) Differentially encoded data, assuming reference bit 1. (c) Waveform of differentially encoded data using unipolar NRZ signaling.
Distorted PCM wave
Amplifierequalizer
Decisionmaking device
Regenerated PCM wave
Timing circuit
_____________________________________________________________________ Figure (1.7). Block diagram of regenerative repeater.
1.3 PULSE SHAPING TECHNIQUES When rectangular pulses are passed through a bandlimited channel, the pulses will spread in time, and the pulse for each symbol will smear into the time intervals of succeeding symbols. This causes intersymbol interference (ISI) and leads to an increased probability of the receiver making an error in detecting a symbol. One obvious way to minimize intersymbol interference is to increase the channel bandwidth. However, mobile communication systems operate with minimal bandwidth, and techniques that reduce the modulation bandwidth and suppress out-ofband radiation, while reducing intersymbol interference, are highly desirable. Out-ofband radiation in the adjacent channel in a mobile radio system should generally be 40 dB to 80 dB below that in the desired passband. Since it is difficult to directly manipulate the transmitter spectrum at RF frequencies, spectral shaping is done through baseband or IF processing. There are a number of well known pulse shaping techniques which are used to simultaneously reduce the intersymbol effects and the spectral width of a modulated digital signal.
1.3.1 Intersymbol Interference (ISI) Intersymbol interference (ISI) is a source of bit errors in a baseband-pulse transmission system. It arises when the channel is dispersive. Consider this baseband binary transmission system as shown in figure
____________________________________________________________________ Figure (1.8) Baseband binary data transmission system
The output of the receiver would be
− − −− =
+
( )
eqn (1.8)
Input binary data bk consists of symbols 1 and 0 each of duration T b. PAM modifies this binary sequences into a new sequence of short pulses.
+1 1
ak =
s t =
y t = μ
if symbol bk is 1 if symbol bk is 0
k ak g
t
eqn (1.9)
kTb
ak p t
eqn(1.10)
kTb + n(t)
where is a scaling factor and p(t) is to be defined and normalized i.e p(0) = 1 P(t) = g(t) * h(t) * c(t)
eqn (1.11)
* denotes convolution Convolution in time domain
multiplication in (f) domain
P(f) = G(f) H(f) C(f) )
eqn (1.12)
− −− − ≠
Receive filter output y(t) is sampled at time t i = iTb.
=
=
∞
=
+
+
∞
∞
=
+
∞
eqn(1.13)
ai is the contribution of the ith transmitted bit BUT Second term represents the ISI
[Residual effect due to the occurrence of pulse before and after the sampling time instant t i is called ISI] Note that:
Under normal (ideal) conditions the ith transmitted bit is decoded correctly. ISI and noise in system introduce errors in decision device at the receiver. We want to minimize these effects to reach good decoding. We will neglect noise now to concentrate on ISI only.
1.3.2 Nyquist’s criterion for Distortion less Base Band Binary Transmission Typically The frequency response of the channel and the transmission pulse shape are specified, the problem is to determine the frequency responses of the transmit and receive filters to reconstruct the original binary data sequence (b k ). Extraction involves sampling the o/p y(t) at time t=iTb. The decoding requires that the weighted pulse contribution ak P(iT b – kTb) for k=i be free from
− ≠
ISI due to overlapping tails of all other weighted pulse contributions represented by k i We control pulse p(t) such that
=
If p(t) satisfies this ISI will vanish.
1 0
=
How to design this? Converting to frequency domain considering sampling process in time and frequency domain and periodicity in (f) domain.
F.T of infinite periodic sequence of delta function of period T b whose individual areas are weighted by the respective sample value of p(t) that is given P (f) is given by
− − − − − − − − − − ∞
=
=
=
∞
∞
∞
(
∞
=
)
2
∞
eqn(1.14)
Let m = i – k i = k corresponds to m = 0
=
i k corresponds to m 0 ∞
∞
0
2
( )
=
0 = 1
eqn(1.15)
Condition of zero ISI is ∞
=
=
∞
eqn(1.16)
Nyquist criterion for distortion less baseband transmission in the absence of noise
=
Ideal Nyquist channel
=
2
=
Note:
1
2
2
=
1
2
(2
)
eqn(1.17)
Rb = 2w is called Nyquist rate. W is called Nyquist bandwidth
_____________________________________________________________________ Figure (1.9) Nyquist criterion for ISI cancellation (ideal Nyquist channel)
(a) Ideal magnitude. (b) Ideal basic pulse shape
This transfer function corresponds to a rectangular "brick-wall" filter with absolute bandwidth=Rb/2 where Rb is the bit rate. While this transfer function satisfies the zero ISI criterion with a minimum of bandwidth, there are practical difficulties in implementing it, since it corresponds to a noncausal system (h(t) exists for t< 0) and is thus difficult to approximate. Also, the (sin t) /t pulse has a waveform slope that is 1/t at each zero crossing, and is zero only at exact multiples of 7's, thus any error in the sampling time of zerocrossings will cause significant ISI due to overlapping from adjacent symbols (A slope of 1/t2 or 1/t3 is more desirable to minimize the ISI due to timing jitter in adjacent samples).
1.3.3 Raised Cosine Filter To overcome the practical difficulties encountered with ideal Nyquist channel by extending the B.W from the minimum value w = Rb/2 to an adjustable value between w and 2w we use the overall frequency response p(f) to satisfy a condition more elaborate than that for the ideal Nyquist channel
− ≤ −≤≤ ≤ − − − ≤ ≥ −≤ − − +
2
+
+2
=
1
2
1
2
=
1
4
(
1
2
0
1
1
2
)
2 1
0
Where
=1
2
1
eqn(1.18)
eqn(1.19)
1
1
is called roll off factor which indicates the excess bandwidth over the ideal solution w. Transmission B.W BT = 2w – f 1 = (1+) W. This transfer function is plotted in Figure 1.10 for various values of a. When = 0. the raised cosine rolloff filter corresponds to a rectangular filter of minimum bandwidth. The corresponding impulse response of the filter can be obtained by taking the inverse Fourier transform of the transfer function, and is given by
− απα
p t = sinc 2wt
cos 2
wt
1 16 2 w 2 t 2
eqn (1.20)
Notice that the impulse response decays much faster at the zero-crossings (approximately as 1/t3 for t>> when compared to the 'brick-wall" filter (=0). The rapid time rolloff allows it to be truncated in time with little deviation in performance from theory. As seen from Figure 1.10, as the rolloff factor a increases, the bandwidth of the filter also increases, and the time side lobe levels decrease in adjacent symbol slots. This implies that increasing a decreases the sensitivity to timing jitter, but increases the occupied bandwidth. The spectral efficiency offered by a raised cosine filter only occurs if the exact pulse shape is preserved at the carrier. This becomes difficult if nonlinear RF
amplifiers are used. Small distortions in the baseband pulse shape can dramatically change the spectral occupancy of the transmitted signal. If not properly controlled, this can cause serious adjacent channel interference in mobile communication systems. A dilemma for mobile, communication designers is that the reduced bandwidth offered by Nyquist pulse shaping requires linear amplifiers which are not power efficient. An obvious solution to this problem would be to develop linear amplifiers which use real time feedback to offer more power efficiency, and this is currently an active research thrust for mobile communications.
_______________________________________________________________ Figure (1.10) Responses for different rolloff factors of raised cosine filter.
(a) Frequency response. (b) Time response.
1.3.4 Gaussian Filter It is also possible to use non-Nyquist techniques for pulse shaping. Prominent among such techniques is the use of a Gaussian pulse-shaping filter which is particularly effective when used in conjunction with Minimum Shift Keying (MSK) modulation, or other modulations which are well suited for power efficient nonlinear amplifiers. Unlike Nyquist filters which have zero-crossings at adjacent symbol peaks and a truncated transfer function, the Gaussian filter has a smooth transfer function with no zero-crossings. The impulse response of the Gaussian filter gives rise to a transfer function that is highly dependent upon the 3-dB bandwidth. The Gaussian Iowpass filter has a transfer function given By 2 2 = exp ( ) eqn(1.21) The parameter α is related to bandwidth , the 3-dB bandwidth of the baseband Gaussian shaping filter is given by,
−
=
0.5887
eqn(1.22)
As a increases, the spectral occupancy of the Gaussian filter decreases and time dispersion of the applied signal increases. The impulse response of the Gaussian filter is given by
− =
exp
2
2
2
eqn(1.23)
Figure 1.11 shows the impulse response of the baseband Gaussian filter for various values of 3-dB bandwidth-symbol time product (BTS). The Gaussian filter has a narrow absolute bandwidth (although not as narrow as a raised cosine rolloff filter), and has sharp cut-off, low overshoot, and pulse area preservation properties which make it very attractive for use in modulation techniques that use nonlinear RF amplifiers and do not accurately preserve the transmitted pulse shape . It should be noted that since the Gaussian pulse-shaping filter does not satisfy the Nyquist criterion for ISI cancellation, reducing the spectral occupancy creates degradation in performance due to increased ISI. Thus, a trade-off is made between the desired RF bandwidth and the irreducible error due to ISI of adjacent symbols when Gaussian pulse shaping is used. Gaussian pulses are used when cost is a major factor and the bit error rates due to ISI are deemed to be lower than what is nominally required.
Figure (1.11) impulse response of Gaussian shaping filter
1.4 AMPLITUDE-SHIFT KEYING (ASK) MODULATION 1.4.1 Introduction Amplitude shift keying (ASK) is nonconstant modulation scheme where the amplitude of the carrier frequency is changed with respect to the message signal. When the amplitude is altered between “A” and zero volt the modulation is considered on-off keying .Also the ASK modulation can ne extended t o M’ary modulation scheme with Multi-level signal. The ASK can be coherently or noncoherently demodulated.
1.4.2 Binary Amplitude-Shift Keying (BASK)
≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ∅ π
A binary amplitude-shift keying (BASK) signal can be defined by
=
cos2
0
eqn (1.23)
where A is a constant, m(t) = 1 or 0, fc is the carrier frequency, and T is the bit duration. It has a power P =
=
A2
2 cos2 2
=
2
=
2
, so that A =
0
,
cos2
cos2
2P . Thus equation (1) can be written as
,
,
0 0
eqn (1.24)
where E = P T is the energy contained in a bit duration. If we take
1
t =
2
T
cos2 f c t as the orthonormal basis function, the applicable
signal space or constellation diagram of the BASK signals is shown in Figure (1.11).
Figure (1.11) BASK signal constellation diagram.
Figure (1.12) shows the BASK signal sequence generated by the binary sequence 0 1 0 1 0 0 1. The amplitude of a carrier is switched or keyed by the binary signal m(t). This is sometimes called on-off keying (OOK).
_____________________________________________________________________ Figure (1.12) (a) Binary modulating signal and (b) BASK signal
The Fourier transform of the BASK signal s(t ) is
− −−− − − =
∞
2
2
∞
2
=
2
2
+
∞
2
2
∞
+
2
+
eqn (1.25)
The effect of multiplication by the carrier signal Acos 2πfct is simply to shift the spectrum of the modulating signal m (t) to fc. Figure 1.13 shows the amplitude spectrum of the BASK signals when m(t) is a periodic pulse train. Since we define the bandwidth as the range occupied by the baseband signal m(t) from 0 Hz to the first zero-crossing point, we have B Hz of bandwidth for the baseband signal and 2B Hz for the BASK signal.
_____________________________________________________________________ Figure (1.13) (a) Modulating signal, (b) spectrum of (a), and (c) spectrum of BASK signals.
Figure (1.14) shows the modulator and a possible implementation of the coherent demodulator for BASK signals.
_____________________________________________________________________ Figure (1.14) (a) BASK modulator and (b) coherent demodulator.
1.4.3 M-ary Amplitude-Shift Keying ( M -ASK)
≤ ≤
An M-ary amplitude-shift keying ( M -ASK) signal can be defined by
=
2
0
0,
eqn (1.26) where
Ai = A[2i - ( M - 1)]
eqn (1.27)
for i = 0, 1, ..., M - 1 and M > 4. Here, A is a constant, f c is the carrier frequency, and T 2
is the symbol duration. The signal has a power Pi = ,, so that Ai = 2 2
≤ ≤ ≤ ≤ ≤ ≤
Thus equation (4) can be written as
=
2
cos2 2
=
=
2
0
,
cos2
cos2
.
,
,
0
0
eqn(1.28)
where E i = PiT is the energy of s(t ) contained in a symbol duration for i = 0, 1, ..., M -1.
Figure (1.15) shows the signal constellation diagrams of M -ASK and 4-ASK signals.
_____________________________________________________________________ Figure (1.15) (a) M-ASK and (b) 4-ASK signal constellation diagrams.
Figure (1.16) shows the 4-ASK signal sequence generated by the binary sequence 00 01 10 11.
____________________________________________________________________ Figure (1.16) 4-ASK modulation: (a) binary sequence, (b) 4-ary signal, and (b) 4-ASK signal.
Figure (1.17) shows the modulator and a possible implementation of the coherent demodulator for M -ASK signals.
_____________________________________________________________________ Figure 1.17 (a) M-ASK modulator and (b) coherent demodulator.
1.4.4 Probability of error: For binary ASK (or as special case OOK signal) the probability of error would be
− − =
2
eqn(1.29)
0
And For M-ary ASK (MAM) the probability of error would be
=
2(
1)
6(
2
)
2
1
eqn(1.30)
1.5 PHASE SHIFT KEYING MODULATION TECHNIQUES Phase shift keying is constant envelope modulation technique where the phase of the carrier is switched according to the message signal and normally cannot be noncoherently demodulated . We begin this section with binary PSK(BPSK) followed by the differential PSK (DPSK) as a brilliant solution of noncoherent demodulation of the PSK, Then we introduce t he M’ary PSK followed by a common and robust special case modulation scheme the later which is quadrature PSK (QPSK) and its modified versions offset QPSK(OQPSK) and (π/4 QPSK)
1.5.1 Binary phase shift keying (BPSK):Here the phase of constant amplitude carrier signal is switched between two values according to the possible signals m 1, m2 which corresponds to 1, 0. Normally m1, m2 phases are separated by 180 phase shift and amplitude of A c and
≤≤ − ≤≤ energy per bit (E b=
1 2
2
Tb)
1.5.1.1 BPSK Signal equation: =
2
cos 2
+
OR: The signal is shifted by
=
2
cos 2
0
(for binary 1)
eqn(1.31)
when transmitting binary zero which means
+
0
(for binary 0)
eqn(1.32)
These signals are referred to as antipodal signals and is normalized to unit energy The reason that they are chosen is that they have a correlation coefficient of -1, which leads to the minimum error probability for the same E b /No, as we will see shortly. If m(t) represents binary data which takes on one of two possible pulse shapes(1,-1) as general case
≤≤ =
( )
2
cos 2
+
0
eqn(1.33)
Therefore The BPSK signal is equivalent to a double sideband suppressed carrier amplitude modulated waveform, where cos ( 2 ) is applied as the carrier, and the data signal in m(t) is applied as the modulating waveform. Hence a BPSK signal can be generated using a balanced modulator.
1.5.1.2 Time domain For the binary data {10110} the modulated carrier would be
Figure 1.18 BPSK signal in time domain
1.5.1.3 Spectrum & Bandwidth
The power spectral density (PSD) of the complex envelope can be shown to be: 2 =2 eqn(1.34) Where Eb is bit energy and T b is bit duration
− − −−−−
That is equivalent to PSD at RF
=
sin ( (
2
(
)
)
2
+
sin ( ( (
)
2
)
eqn(1.35)
Which result in Null to null BW=twice bit rate
=2
eqn(1.36)
From figure (1.19) we conclude that 90% of BPSK energy is contained within an approximately equal to 1.6 R b and we can also find that with using a raised cosine = 0.5 all energy are contained within 1.5 R b filter of
Figure (1.19) BPSK spectrum with rectangular and raised cosine filter with roll of factor=0.5
1.5.1.4 Constellation diagram Let,
1
=
2
cos 2
+
is the basis signal then we will have two
constellation points separated by 180 degree phase shift Therefore A coherent binary PSK system is characterized by having a signal space that is one dimensional (i.e. N=1), with a signal constellation consisting of two message points
Figure (1.20) BPSK constellation diagram
1.5.1.5 Modulator Using a balanced modulator after putting the binary data on the form of polar NRZ (non return to zero) (-1,+1) (- 1,+1) we can generate generate the BPSK signal note that t hat the carrier frequency must satisfy that = for satisfying synchronization i.e ensure that each transmitted bit contains an integral number of cycles of the carrier wave.
Figure (1.21) BPSK modulator
1.5.1.6 Demodulator:As we pointed out before the PSK modulation modulation must be coherently demodulated so a carrier recovery circuit (Costas loop-phase locked loop) must be employed to obtain the carrier. To detect the original binary sequence of 1’s and zero’s we apply the noisy PSK signal to a correlator which is supplied with the locally generated generated carrier the correlator output is compared with a threshold of zero volts if the output exceeds zero the receiver decides in favor of symbol 1 otherwise the receiver decides in favor of zero.
Figure (1.22) BPSK demodulator
0
=
2
2
2
+
=
2
1 2
1
+ cos (2(2 2
+ )
eqn(1.37)
When no pilot signal is transmitted transm itted a Costas loop or squaring loop may be used to synthesize the carrier phase and frequency from the received BPSK signal. Figure (1.23) shows the block diagram diagram of a BPSK receiver along with w ith the carrier recovery circuits.
Figure (1.23) shows the block diagram of a BPSK receiver along with the carrier
recovery circuits.
The received signal is squared to generate a dc signal and an amplitude varying sinusoid at twice the carrier frequency. The de signal is filtered out using a bandpass filter with center frequency tuned to A frequency divider is then used to recreate the waveform.
1.5.1.7 Power sufficiency & bandwidth efficiency:Since we have only two constellation points hence we have High power efficiency
Low bandwidth efficiency: the symbol is represented by 1 bit
=
= 0.5
eqn(1.38)
−
1.5.1.8 Probability of error:-
Since that Distance between constellation points =2
.
Then the probability probability of o f error is derived from the general probability of error equation equation of the matched filter (correlator) receiver
With
− =
1+ 2
=
2 12 2
1 2
eqn(1.39)
1 and E1=E2=Eb in the BPSK modulation therefore =
2
=
1 2
eqn(1.40)
1.5.2 Differential phase shift keying (DPSK):As we have seen in BPSK modulation that the demodulator must be coherent i.e. it needs a reference signal to be demodulated which will increase the complexity of the demodulator demodulator by the synchronization circuits and the reason of t his that the demodulator must preserve the phase of the carrier which includes the message. From here a noncoherent version of BPSK is needed. the idea here is to equip the receiver with storage capability so as it can measure the relative phase difference between the waveforms received during two successive bit intervals provided that the unknown phase varies slowly (slow enough to be considered constant over over the two bit intervals) intervals) That is we consider the differential PSK (DPSK) as Noncohe No ncoherent rent form of PSK. which will result in many advantages such as: no need for coherent reference signal and the receivers are cheap to build. This would be done by differential encoding i.e. The input binary sequence is first differentially encoded & then modulated using BPSK modulator.
1.5.2.1 Differential encoding procedure: Here we encode the baseband data before modulating it onto carrier.The encoded output bit is determined from the input bit and the previous output bit. Let ak : original binary data. And dk : encoded binary data sequence. Encoding: = eqn(1.41) 1 Decoding:
=
⨁− ⨁−
1
eqn(1.42)
The effect:to leave symbol d k unchanged from the previous symbol if a k =1 & toggle if else. Example of differential encoding: m k d k-1 d k
1
1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 Table (1.3) Example of differential encoding
1 0 0
0 0 1
1.5.2.2 Modulator:
Figure (1.24) DPSK modulator
It consists of a one bit delay element and a logic circuit i nterconnected nterconnected so as to generate the differentially encoded sequence from the input binary sequence. The output is passed through a product modulator to obtain the DPSK signal i.e. output bit is delayed by 1 bit duration and XNORed with newer i/p bit,Then the o/p sequence is transformed to polar NRZ and then it will be like BPSK.
1.5.2.3 Demodulator:(1) Suboptimum receiver: At the receiver, the t he original sequence is recovered from the demodulated differentially encoded signal through a complementary process,
Figure (1.25) Suboptimum receiver of DPSK modulation
(2) Optimum receiver: The demodulator does not require phase synchronization between the reference signals and the received signal. But it does require the reference frequency be the same as the received signal this can be maintained by using stable oscillators, such as crystal oscillators, in both transmitter and receiver. However, in the case where Doppler shift exists in the carrier frequency, such as in mobile communications, frequency tracking is needed to maintain the same frequency Therefore the suboptimum receiver is more practical, and indeed it is the usual-sense DBPSK receiver. Its error performance is slightly inferior to that of t he optimum
Figure (1.26) Optimum receiver of DPSK modulation
1.5.2.4 Example: A complete example of differential PSK (DPSK) is shown in Table (1.4)
Modulation
⨁−
Message ak = Encoding Signal phase Demodulation Output of correlator
Demodulator output
1
1 0
1 1 0
0 0
1 0
ref 1 0 0 1 0
1
-1
1
1
-1
-1
-1
1
1
1
0
0
0
1
1
0 0
0 1 0
1 1 0
1 1 0
1 0 1 Table(1.4) DPSK example
1.5.2.5 Advantages & disadvantages: Advantage.: reduce the receiver complexity. Disadvantage.: energy efficiency is less than coherent PSK by 3 dB
1.5.2.6 Power spectral density: The same as BPSK Since the difference of differentially encoded BPSK from BPSK is differential encoding, which always produces an asymptotically equally likely data sequence the PSD ofthe differentially encoded BPSK is the same as BPSK which we assume is equally likely
−
1.5.2.7 Probability of error:-
=
1
/
2
eqn (1.43)
Which provides a gain of 3 dB over noncoherent FSK for same E b /No
Figure (1.27) Performance comparison between coherent BPSK,coherent DPSK
,optimum and suboptimum DPSK
1.5.3 M-ary phase shift keying(M’ary PSK/MPSK) The motivation behind MPSK is to increase the bandwidth efficiency of the PSK modulation schemes. In BPSK, a data bit is represented by a symbol. In MPSK, n = log2 M data bits are represented by a symbol, thus the bandwidth efficiency is increased to n times. Among all MPSK schemes, QPSK is the most-often-used scheme since it does not suffer from BER degradation while the bandwidth efficiency is increased. We will see this in Section 4.6. Other MPSK schemes increase bandwidth efficiency at the expenses of BER performance. Here carrier phase takes on one of M possible values namely
Where i=1,2,3,….M
=
2(
− 1)
eqn(1.44)
1.5.3.1 Signal Equation:-
− ≤≤ 2
=
cos 2
+
2
1
0
eqn (1.45)
i=1,2,…..,M & Ts: is symbol time=(log2M)Tb . And Es=symbol energy=(log 2M)Eb Using trigonometric identities:-
− − − − − − =
Let
1(
2
)=
[cos(( 2
1)
2
cos 2
=
[cos((
)cos (2 ,
2(
1)
)
2
)=
2
)
sin((
1(
1)
sin 2
)
sin((
2
)sin (2
)]
eqn(1.46)
are the basis signals
1)
2
)
2(
)]
eqn(1.47)
1.5.3.2 Constellation diagram:(1) Since we have two basis signals two dimensional diagram (2) From equation the envelope is constant (when no pulse shaping is employed) while the phase is varying that can be represented by equally spaced message points on a circle of radius
(3) Gray coding is usually used in signal assignment in MPSK to make only one bit difference to two adjacent signals1 bit error An example of 8-ary PSK with gray coding is as shown:-
Figure (1.28) 8PSK modulation with gray coding as signment
1.5.3.3 Probability of error:
From the geometry of the constellation we will find that the distance between adjacent symbols is equal to 2
sin
Figure (1.29) Formulation of probability of error expression for MPSK signal
And hence using eqn(1.39) we will find that average symbol error probability equal
≤ ≈ 2
2
≥
& For M 4:-
2
2
eqn(1.48)
4
eqn(1.49)
2
1.5.3.4 Power spectra of M-ary PSK:The first null BW decrease as M increases while bit rate is held constant
=2
=2
2
2
2
(
2
)
eqn (1.50)
Figure (1.30) Spectrum and the bandwidth of MPSK signal
1.5.3.5 Power & BW efficiency:-
As the value of M increases, the bandwidth efficiency increases. That is, for fixed Rb, increases and Bandwidth decreases as M is increased. At the same time, increasing M implies that the constellation is more densely packed, and hence the power efficiency (noise tolerance) is decreased so As M increases (a) Bandwidth efficiency increases (b) Power efficiency decreases. Where
=
2
=
2
log 2
eqn(1.51)
Therefore,
=
log 2 2
And To ensure that there is no degradation in error performance (BER) the ratio Eb /No must increase. Table (1.5) gives a values of both the bandwidth and power efficiencies of M-ary PSK signals
2 4 8 16 32 64 M 0.5 1 1.5 2 2.5 3 = / -6 10.5 10.5 14 18.5 23.4 28.5 Eb /No for BER =10 Table (1.5) bandwidth and power efficiencies of M-ary PSK signals
The relation between symbol error & Eb /No is as following:
Figure(1.31) symbol error rate versus signal to noise ratio for various modulation PSK schemes
≥
1.5.3.6 Modulator:
4we can use a quadrature modulator. The only difference for different values of M is the level generator The level generator gives two signals corresponding to each n bits of the input sequence(symbol) by changing the levels of these signals we can vary the phase. Note that the M-ary can be directly modulated or differentially encoded to provide noncoherent detection For M
Figure (1.32) MPSK modulator
1.5.3.7 Demodulator:-
Figure (1.33) MPSK demodulator
1.5.4 Quadrature phase shift keying (QPSK) QPSK has the twice bandwidth efficiency of BPSK, since 2 bits are transmitted in a single modulation symbol. The phase of the carrier takes on 1 of 4 equally spaced value such as 0, π/2, π, 3π/2, where each value of phase corresponds to a unique pair of message bits. For example:
Message
Phase
00 0 π/2 01 11 π 10 3π/2 Table (1.6) QPSK output phases
Note that : it is better to arrange the states with Gray Coding , this makes each adjacent symbol only differs by one bit to minimize the bit error rate (BER).
1.5.4.1 Signal Equation The QPSK signal for this set of symbol states may be defined as:
− ≤≤ =
2
cos[2
+
1
2
]
0
= 1,2,3,4.
eqn (1.52)
Where TS is the symbol duration and is equal to twice the bit period T b. Using trigonometric identities:
cos(x+y) = cos x cos y – sin x sin y
− − − − – − =
2
cos[
1
2
] cos(2
2
)
sin[
1
2
] sin(2
)
eqn (1.53)
If the basis functions are: 1
2
=
cos(2
)
,
2
=
2
sin(2
)
Then the 4 signals in the set can be expressed in the terms of the basis functions as:
=
cos
1
2
1
sin
1
2
2
eqn (1.54)
= 1,2,3,4
1.5.4.2 Constellation Diagram and probability of error Based on this representation the QPSK signal can be depicted using a two dimensional constellation diagram with four points as shown:
Figure (1.34) (a) QPSK constellation where the carrier phases are 0, π/2 , π,3π/2
(b) QPSK constellation where the carrier phases are π/4, 3π/4 ,5π/4,7π/4
From the constellation diagram, it can be seen that the distance between two adjacent points in the constellation is
2
.
Since each symbol corresponds to two bits, t hen ES=2E b, then the distance between two adjacent points in the constellation is 2
.
Then the average probability of bit error in AWGN channel:
=
Note that
2
=
1 2
eqn (1.55)
QPSK has the same probability of bit error as BPSK, but twice as much data can be sent in the same bandwidth. Thus compared to BPSK, QPSK provides twice the spectral efficiency with exactly the same power efficiency. Similar to BPSK, QPSK can also be differentially encoded to allow noncoherent detection.
1.5.4.3 Spectrum and bandwidth of QPSK signal: The Null to null RF bandwidth is equal to the bit rate. BW of QPSK= R b =Half BW of BPSK
Figure (1.35) QPSK spectrum and bandwidth
1.5.4.4 QPSK Transmitter:
Figure (1.36) QPSK modulator
The unipolar binary message stream has bit rate Rb and is first converted into a bipolar non return to zero (NRZ) sequence using a unipolar to bipolar converter. The data sequence is separated by the serial-to-parallel converter (S/P) to form the odd numbered bit sequence for I-channel (cosine) and the even numbered bit sequence for Q-channel (sine). Next the odd-numbered- bit pulse train is multiplied to cos 2π f ct and the evennumbered- bit pulse train is multiplied to sin 2π f ct. It is clear that the I-channel and Q-channel signals are BPSK signals with symbol duration of 2T b. Finally a summer adds these two waveforms together to produce the final QPSK signal. The BPF at the output of the modulator confines the power spectrum of the QPSK signal within the allocated band, this prevents spill-over of signal energy into adjacent channels.
1.5.4.5 QPSK Receiver:
Figure (1.37) QPSK demodulator
The frontend bandpass filter removes out -of -band noise and adjacent channel interference. The filtered output is split into two parts , each part is coherently demodulated using the in-phase and quadrature carriers which are recovered from the received signal using carrier recovery circuit. The outputs of the demodulators are passed through decision circuits which generate the in-phase and quadrature binary streams. The two components are then multiplexed to r eproduce the original binary sequence.
1.5.5 Offset Quadrature phase shift keying (OQPSK) Offset Quadrature phase-shift keying (OQPSK) is a variant of phase-shift keying modulation using 4 different values of t he phase to transmit as QPSK. Taking four values of the phase (two bits) at a time to construct a QPSK symbol can allow the phase of the signal to jump by as much as 180° at a time. The amplitude of a QPSK signal is ideally constant. However, when QPSK signals are pulse shaped, they lose the constant envelope property. The occasional phase shift of π radians can cause the signal envelope to pass through zero for just an instant. Any kind of hard limiting or nonlinear amplification of the zerocrossings brings back the filtered side lobes since the fidelity of the signal at small voltage levels is lost in transmission. The prevent the regeneration of side lobes and spectral widening; it is imperative that QPSK signals be amplified only using linear amplifiers, which are less efficient. A modified form o f QPSK, called offset QPSK (OQPSK) or staggered QPSK is less susceptible to these deleterious effects and supports more efficient amplification. By offsetting the timing of the odd and even bits by one bit-period, or half a symbol-period, the in-phase and quadrature components will never change at the same time. This will limit the phase-shift to no more than 90° at a time, this yields much lower amplitude fluctuations than non-offset QPSK and is sometimes preferred in practice.
Figure (1.38) QPSK and OQPSK phase transitions
The above figure shows the difference in the behavior of the phase between ordinary QPSK and OQPSK. It can be seen that in the first plot (ordinary QPSK) the phase can change by 180° at once, while in OQPSK the changes are never greater than 90°. The following figure shows the even and odd bit streams, m I (t) and mQ(t) and the offset in their relative alignment by one bit period (half-symbol period):
Figure (1.39) OQPSK generation
Due to the time alignment of mI (t) and mQ (t) in standard QPSK, phase transitions occur only once every Ts = 2Tb s, and will be a maximum of 180 degree if there is a change in the value of both mI (t) and mQ (t) However, in OQPSK signaling, bit transitions (and hence phase transitions) occur every T b s. Since the transitions instants of mI (t) and mQ (t) are offset, at any given time only one of the two bit streams can change values. This implies that the maximum phase shift of the transmitted signal at any given time is limited to ±90°. Hence by switching phases more frequently (i.e., every Tb s instead of 2Tbs) OQPSK signaling eliminates 180° phase transitions. Since 180° phase transitions have been eliminated, bandlimiting of (i.e., pulse shaping) OQPSK signals does not cause the signal envelope to go to zero. Obviously, there will be some amount of ISI caused by the bandlimiting process, especially at the 90 degree phase transition points. But the envelope variations are considerably less, and hence hard limiting or nonlinear amplification of OQPSK signals does not regenerate the high frequency side lobes as much as in QPSK. Thus, spectral occupancy is significantly reduced, while permitting more efficient RF amplification.
The modulated signal is shown in the figure below for a short segment of a random binary data-stream:
Figure (1.40) OQPSK modulated signal
Note that half symbol-period offset between the two component waves. The spectrum of an OQPSK signal is identical to that of a QPSK signal, hence both signals occupy the same bandwidth. The staggered alignment of the even and odd bit streams does not change the nature of the spectrum. OQPSK retains its band limited nature even after nonlinear amplification, and therefore is very attractive for mobile communication systems where bandwidth efficiency and efficient nonlinear amplifiers are critical for low power drain. Further, OQPSK signals also appear to perform better than QPSK in the presence of phase jitter due to noisy reference signals at the receiver
1.5.6 π / 4 – QPSK The π/4 shifted QPSK modulation is a quadrature phase shift keying technique which offers a compromise between OQPSK and QPSK in terms of the allowed maximum phase transitions. It may be demodulated in a coherent or noncoherent fashion. In π/4 QPSK, the maximum phase ch ange is limited to ± 135° as o compared to 180° for QPSK and 90 for OQPSK. Hence, the bandlimited π/4 QPSK signal preserves the constant envelope property better t han bandlimited QPSK, but is more susceptible to envelope variations than OQPSK. An extremely attractive feature of π/4 QPSK is that it can be noncoherently detected, which greatly simplifies receiver design. Further, it has been found that in the presence of in multipath spread and fading, π/4 QPSK performs better than OQPSK . Very often, π/4 QPSK signals are differentially encoded to facilitate easier implementation of differential detection or coherent demodulation with phase ambiguity in the recovered carrier. When differentially encoded π/4 QPSK is called π/4 DQPSK.
π / 4–QPSK uses two identical constellations which are rotated by 45° ( π / 4
radians, hence the name) with respect to one another. Usually, either the even or odd data bits are used to select points from one of the constellations or t he other bits select points from the other constellation. This also reduces the phase-shifts from a maximum of 180°, but only to a maximum of 135° and so the amplitude fluctuations of π / 4 – QPSK are between OQPSK and non-offset QPSK.One property this modulation scheme possesses is that if the modulated signal is represented in the complex domain, it does not have any paths through the origin. In other words, the signal does not pass through the origin. This lowers the dynamical range of fluctuations in the signal which is desirable in communications. π/4 QPSK modulator, signaling points of the modulated signal are se lected from two QPSK constellations which are shifted by π/4 with respect to each other. The figure shows the two constellations along with the combined constellation where the links between two signal points indicate the possible phase transitions. Switching between two constellations, every successive bit ensures that there is at least a phase shift which is an integer multiple of π/4 radians between successive symbols. This ensures that there is a phase transition for every symbol, which enables a receiver to perform timing recovery and synchronization. Information bits mI,mQ Phase π /4 11 01 3π/4 00 -3π /4 10 -π/4 Table (1.7): Carrier phase shifts corresponding to various input bit pairs.
_____________________________________________________________________ Figure (1.41) Constellation diagram of π/4 QPSK signal (a) possible states of wken /4 (b) possible states when /2 (c) all possible states 1 = 1 =
−
−
1.5.6.1 Example Sketch the modulated symbols for the input bit stream: 11000110
_____________________________________________________________________ Figure (1.42) constellation diagram of π/4 QPSK The modulated signal is shown below for a short segment of a random binary datastream:
Figure (1.43) modulated signal when 11000110 is transmitted
Note that: Successive symbols are taken from the two constellations shown in the diagram. Thus, the first symbol (1 1) is taken from the 'blue' constellation and the second symbol (0 0) is taken from the 'green' constellation.
1.5.6.2 π /4 QPSK Transmission Techniques A block diagram of a gener ic π/4 QPSK transmitter is shown in Figure.
Figure (1.44) π/4 QPSK transmitter
The input bit stream is partitioned by a serial-to-parallel (S/P) converter into two parallel data streams mIk and mQk each with a symbol rate equal to half that of the th incoming bit rate. The K in-phase and quadrature pulses, I k and Qk are produced at the output of the signal mapping circuit over time kT t (k + 1)T and are determined by their previous values, I k -1 and Qk -1 as well as θk which itself is a function of k which is a function of the current input symbols mIk and mQk . Ik and Qk represent rectangular pulses over one symbol duration having amplitudes given by:
≤≤
ϕ
− −− − − − = cos
=
= sin
=
=
Where
1
cos
1
1
sin
+
1
sin
eqn (1.56)
1
cos
eqn (1.57)
+
eqn(1.58)
Just as in a QPSK modulator, the in-phase and quadrature bit streams I k and Qk are then separately modulated by two carriers which are in quadrature with one another, to produce the π/4 QPSK waveform given by:
− − −− − − −− − − − − − − =
cos
( ) sin
4
Where
=
1 =0
=
1 =0
2
2
=
=
1 =0 cos
1 =0 sin
2
2
eqn(1.59) eqn(1.60)
Both Ik and Qk are usually passed through raised cosine roll off pulse shaping filters before modulation, in order to reduce the bandwidth occupancy. The function P(t) in equations (1.59),(1.60) corresponds to the pulse shape, and Ts is the symbol period. Pulse shaping also reduces the spectral restor ation problem which may be significant in fully saturated, nonlinear amplified systems. It should be noted that the values of I k and Qk and the peak amplitude of the waveforms I(t) and Q(t) can take one of the five possible values 0, +1, -1,
+1/ 2 , -1/ 2 . From the above discussion it is clear that the information in a π/4 QPSK signal is completely contained in the phase difference φk of the carrier between two adjacent symbols. Since the information is completely contained in the phase difference, it is possible to use noncoherent differential detection even in the absence of differential encoding.
1.5.6.3 π/4 QPSK Detection Techniques Due to ease of hardware implementation, differential detection is often employed to demodulate π/4 QPSK signals. In an AWGN channel, the BER pe rformance of a differentially detected π/4 QPSK is about 3 dB inferior to QPSK, while coherently detected π/4 QPSK has the same error performance as QPSK. In low bit rate, fast Rayleigh fading channels, differential detection offers a lower error floor since it does not rely on phase synchronization. There are various types of detection techniques that are used for the detection of π/4QPSK signals. They include baseband differential detection, IF differential detection, and FM discriminator detection. While both the baseband and IF differential detector determines the cosine and sine functions of the phase difference, and then decides on the phase difference accordingly, the FM discriminator detects the phase difference directly in a noncoherent manner. Interestingly, simulations have shown that all 3 receiver structures offer very similar bit error rate performances, although there are implementation issues which are specific to each technique.
1.5.6.3.1 Baseband Differential Detection Figure (1.45) shows a block diagram of a baseband differential detector. The Incoming π/4 QPSK signal is quadrature demodulated using two local oscillator signals that have the same frequency as the unmodulated carrier at the transmitter,
but not necessarily the same phase
ϕ
k
−
= tan
1 Q k is I k
the phase of the carrier due to
the kth data bit, the output wk and zk from the two low pass filters in the in-phase and quadrature arms of the demodulator can be expressed as: = cos eqn (1.61)
− − = sin
Figure (1.45) Block diagram of a baseband differential detector.
eqn(1.62)
where γ is a phase shift due to noise, propagation, and interference. The phase γ is assumed to change much slower than φ k so it is essentially a constant. The two sequences w k and zk are passed through a differential decoder which operates on t he following rule: = eqn(1.63) 1 + 1 = eqn(1.64) 1 + 1
−− −− −− −− −− −− −− −− −−−− The output of the differential decoder can be expressed as = cos cos + sin sin 1 = sin cos + cos sin 1
1
1
=cos =sin
1
1
eqn (1.65)
The output of the differential decoder is applied to the decision circuit, which uses Table (1.7) to determine:
= 1, = 1,
>0 >0
= 0, = 0,
<0 <0
Where SI and SQ are the detected bits in the in-phase and quadrature arms, respectively.
1.5.6.3.2 IF Differential Detector The IF differential detector shown in Figure (1.46) avoids the need for a local oscillator by using a delay line and two phase detectors. The received signal is converted to IF and is bandpass filtered. The bandpass filter is designed to match the transmitted pulse shape, so that the carrier phase is preserved and noise power is minimized. To minimize the effect of ISI and noise, the bandwidth of the filters are chosen to be 0.57/ Ts .The received IF signal is differentially decoded using a delay line and two mixers. The bandwidth of the signal at the output of the differential detector is twice that of the baseband signal at the transmitter end.
Figure (1.46) Block diagram of an IF differential detector for π/4 QPSK.
1.5.6.3.3 FM Discriminator Figure (1.47) shows a block diagram of an FM discriminator detector for π/4QPSK. The input signal is first filtered using a bandpass filter that is matched to the transmitted signal. The filtered signal is then hard limited to remove any envelope fluctuations. Hard limiting preserves the phase changes in the input signal and hence no information is lost. The FM discriminator extracts the instantaneous frequency deviation of the received signal which, when integrated over each symbol period gives the phase difference between two sampling instants. The phase difference is then detected by a four level threshold comparator to obtain the original signal. The phase difference can also be detected using a modulo-2π phase detector. The modulo-2π phase detector improves the BER performance and reduces the effect of click noise.
Figure(1.47) FM discriminator detector for π/4 DQPSK demodulation.
1.6 FREQUENCY SHIFT KEYING FSK FSK (Frequency Shift Keying) is also known as frequency shift modulation and frequency shift signaling. Frequency Shift Keying is a data signal converted into a specific frequency or tone in order to transmit it over wire, cable, optical fiber or wireless media to a destination point. The history of FSK dates back to the early 1900s, when this technique was discovered and then used to work alongside teleprinters to transmit messages by radio (RTTY). But FSK, with some modifications, is still effective in many instances including the digital world where it is commonly used in conjunction with computers and low speed modems. In fact, the contributions of FSK are much more far reaching. For example, the principle of FSK has laid the path to the development of other similar techniques such as the Audio Frequency Shift Keying (AFSK) and Multiple Frequency Shift Keying (MFSK) just to name a few. In Frequency Shift Keying, the modulating signals shift the output frequency between predetermined levels.
Technically FSK has two classifications, the non-coherent and coherent FSK. In non-coherent FSK, the instantaneous frequency is shifted between two discrete values named mark and space frequency, respectively. On the other hand, in coherent Frequency Shift Keying or binary FSK, there is no phase discontinuity in the output signal. In this digital era, the modulation of signals are carried out by a computer, which converts the binary data to FSK signals for transmission, and in turn receives the incoming FSK signals and converts it to corresponding digital low and high, the language the computer understands best. The basic principle of Frequency Shift Keying is at least a century old. Despite its age, FSK has successfully maintained its use during more modern times and has adapted well to the digital domain, and continues to serve those that need to transfer data via computer, cable, or wire. There is no doubt that FSK will be around as long as there is a need to transmit information in a highly effective and affordable manner.
1.6.1Binary phase shift keying (BFSK) In binary frequency shift keying ( BFSK), the frequency of a constant amplitude carrier signal is switched between two values according to the two possible message states (High and Low), corresponding to a binary 1 or 0. + /2 , and 1 is transmitted by a A 0 is transmitted by a pulse of frequency /2 such a waveform may be considered to be two pulse of frequency interleaved ASK waves. An FSK signal described as mentioned may be represented as:
− ω ωω ≤≤ ω − ≤≤ 0
1
=
=
2
2
cos
cos (
c
+
0
2
c
2
)
(
0
0)
(
eqn(1.66)
1)
eqn(1.67)
Where Δω is a constant offset from the nominal carrier frequency. The most important factor to keep in mind when designing FSK is to keep the frequency of the different symbols orthogonal to minimize the correlation between the two symbols to the zero assuming perfect synchronization of receiver oscillators. To achieve this we must do the correlation function between to transmitted symbols and get the conditions to achieve the orthogonality
=
ω ω ω
ω − ω ω 0
1
0
=
2
cos
c +
0
=
cos
t
2
+
0
=
cos
cos 2
c
2
c t
0
+
2
2
eqn (1.68)
In practice therefore
≪
1, and the second term on the right hand side can be ignored
=
eqn(1.69)
in order for E = 0 from the previous equation: Δf = n/2 Tb where n is any integer.
eqn(1.70)
Larger Δf means wider separat ion between signaling frequencies. Thus binary FSK system is characterized by having a signal space that is two dimensional with two message point as shown in figure (1.48) Decision
Forward hint We can think in the different orthogonal carriers of the FSK signal as a multidimensional system with each carrier represents an axis in this system. In binary FSK we only have two dimensions. The M-ary FSK is built on this idea
boundary
Region
Region
Figure 1.48 signal space diagram for binary FSK system
1.6.1.1 Binary FSK Modulator To generate a binary FSK signal we may use the scheme shown in fig 1.49.the input binary sequence is represented in its on-off form, with symbol 1 represented by constant amplitude of volts and symbol 0 represented by zero volts. By using an inverter in the lower channel in fig 1.49, we in fact make sure that when we have symbol 1 at the input, the oscillator with frequency 1 in the upper channel is switched on while the oscillator with frequency 2 in the lower channel is switched off, with the result that frequency 1 is transmitted. Conversely, when we have symbol 0 at the input, the oscillator in the upper channel is switched off, and the oscillator in the lower channel is switched on, with the result that frequency 2 is transmitted. The two frequencies 1 and 2 are chosen integer multiple of the bit rate 1/ which we previously proved to be orthogonal. In this transmitter we assume that the two oscillators are synchronized, so that their outputs satisfy the requirements of t he two orthogonal basis functions 1 & 0 . We may use a single keyed (voltage controlled) oscillator. In either case, the frequency of the modulated wave is shifted with a continues phase, in accordance with the input binary wave that is to say, phase continuity is always maintained, including the inter-bit switching time. We refer to this digital modulation as continues-phase frequency-shift keying (CPFSK).
m (t)
Binary wave
+
(on-off
Binary FSK
signaling
+
form)
wave
Inverter
_____________________________________________________________________ Fig 1.49 Block diagram for binary FSK transmitter
Figure (1.50) shows generating a FSK signal, there are four signals first the binary one second the 1 signal with lower frequency third the 0 signal with the higher frequency and last the final FSK signal.
_______________________________________________________ Figure 1.50 FSK signal
1.6.1.2 Power Spectral Density Now we proceed to find the power spectrum of the FSK signal. We expand the FSK signal as following: 1 = cos2 + 2
− −
=
cos
cos 2
=
sin
sin 2
cos( )cos 2
sin( )sin 2
eqn(1.71)
Where the last expression is derived using the fact that ak = ±1. The in phase component
π
π
t
t
A cos( ) is independent of the data. The quadrature component A ak sin ( ) is T
T
directly related to data. The in phase and quadrature components are independent of each other.
ℱ − =
+
( )
eqn(1.72)
Where I f is the in phase component and Q f is the quadrature phase component. S f can be found easily since the in phase component is independent of data. It is defined on the entire time axis. Thus:
=
Where
( )
2
=
1
2
+ ( +
2
1
2
)
eqn(1.73)
stands for Fourier transform. It is seen that the spectrum of the in phase part
of the FSK signal are two delta functions.
ℱ 1
=
=
(
2
2
)
0
− 1
2
2
cos
1
2
≤≤ eqn (1.74)
2
The complete baseband PSD of the binary FSK signal is the sum of I(f) & Q(f) :
− ( )=
2
1
2
+ ( +
−
1 2
) +
1
2
2
cos
1
2
2
eqn(1.75)
1.6.1.3 Coherent demodulation and error performance In order to detect the original binary sequence given the noisy received wave x(t), we may use the receiver shown in Fig 1.5.5, it consist of two correlators with common input, which are supplies with local generated coherent reference signal 1 t & 2 t . The correlator outputs are then subtracted, one from the other, and the resulting difference, l, is compared with a threshold of zero volts. If l > 0, the receiver decides in favor of 1. On the other hand, if l < 0, it decides in favor of 0.
Φ Φ
Decision device
Fig 1.51 Block diagram for coherent binary FSK receiver
To study the coherent demodulator error performance of the transmitted FSK signal we need to look at fig 1.51 where the distance between the two message points is
equal to
2Eb and the error probelity is driven from the relation
=
>0
0
)+
<0
1
)
We can drive that Pe 0 & Pe (1) have the same value and it is equal to :
(0 1 =
1 2
(
2
)
eqn(1.76)
0
Averaging Pe 0 & Pe (1), we find that the average probability of symbol error for coherent binary FSK is:
=
1 2
2
eqn (1.77)
0
1.6.1.4 Noncoherent demodulation and error performance
For the noncoherant detection, the receiver consists of a pair of matched filters followed by envelope detectors, as in Fig 1.52 the filter in the upper path of the receiver is matched to the first symbol signal with frequency 1 and the filter in the upper path of the receiver is matched to the first symbol signal with frequency 2 . The resulting envelope detector outputs are sampled at = and their values are compared. The envelope sample of the upper and lower paths are shown as 1 & 2 respectively; then, if 1 > 2 , the receiver decides in favor of symbol 1, and if 1 < 2 , the receiver decides in favor of symbol 0.
Filter matched to
Envelope detector
Comparison device Filter matched to
Envelope detector
Fig 1.52 Noncoherant
receiver for detection of binary FDK signals
The noncoherant binary FSK described is a special case of noncoherant orthogonal modulation with:
− =
And
=
Where is the bit duration and proven to be
is the signal energy per bit. Hence, the
=
1 2
exp (
2 0
)
can be
eqn(1.78)
1.6.2 M-ARY FSK In an M-ary FSK scheme, the transmitted signals are defined by
≤≤ … ≠ =
2
cos
+
eqn(1.79)
/2 for some fixed integer . Where = 1,2, , and the carrier frequency = The transmitted signals are equal duration T and have equal energy E. Since the individual signal frequencies are separated by 1/2T hertz, the signals describes above are as proved orthogonal, that is: =0 eqn(1.80) For coherent M-ary FSK, the optimum receiver consists of band of M correlators or matched filters, with the wanted signals providing the pertinent references. At the sampling times t=kT, the receiver makes decisions based on the largest matched filter output. 0
The M-ary FSK can be considered multidimensional system with M orthogonal axis. Giving the probability of error parameters of the M-ary FSK as following:
Average symbol error probability: eqn(1.81) Average bit error probability:
eqn(1.82) Asymptotic power efficiency: eqn(1.83)
Shannon bandwidth: eqn(1.84) Bandwidth efficiency: eqn(1.86)
Fig 1.53
The bit error rate of different M’s for the M-ary FSK is shown in Fig 1.53, it can be shown from the Fig that the more M t he lower level of probability o f error for the same SNR Coherent detection of M-ary FSK requires the use of exact phase references, the provision for which at the receiver can be costly and difficult to maintain. We may avoid the need for such a provision by using noncoherant detection, which result in a slightly inferior performance. In a noncoherant receiver, the individual matched filters are followed by envelope detectors that destroy the phase information. The probability of symbol error of the noncoherant detection of M-ary FSK:
−
=
1
2
exp (
−
2 0
)
eqn(1.86)
1.6.3 Minimum shift keying MSK In the coherent detection of binary FSK signal described before, the phase information contained in the receiver signal was not fully exploited, other than to provide for synchronization of the receiver to the transmitter. We now show that by proper utilization of the phase when performing detection, it is possible to improve the noise performance of the receiver significantly. This improvement is, however, achieved at the expense of increasing receiver complexity. Consider a continues-phase frequency-shift keying (CPFSK) signal, which is defined for interval 0 , as follows:
≤≤ 2
=
2
cos 2
+
1
cos 2
0
+
2
0
1 eqn(1.87)
0
Where is the transmitted signal energy per bit, and is the bit duration. The phase (0), denoting the value of phase at time = 0, depends on the past history of the modulation process. The frequency 1 & 2 are sent in response to binary symbol 1 and 0 appearing at the modulation input, respectively. Another useful way of representing the CPFSK signal s(t) is to express it in the conventional form of an angle-modulation waves follows
=
2
cos 2
+
eqn(1.88)
The phase is a continues function of time, this leads to the modulated wave s(t) itself to be continues all the time including the inter -bit switching times.
The phase bit period of
≤≤ 1
= ( 2
1
+
2)
eqn(1.89)
of CPFSK signal increases or decreases linearly with time during each seconds, as shown by:
=
0 ±
0
eqn(1.90)
Where the plus sign corresponds to sending symbol 1, and mines sign corresponds to sending symbol 0. The parameter h is defined by:
− − −
= (1 eqn(1.91) 2) We refer to h as the deviation ratio, measured with respect to the bit rate 1/ . At time t=
0 =
1 0
eqn(1.92)
That is to say, the sending of symbol 1 increases the phase of CPFSK s(t) by πh , whereas the sending of symbol 0 reduces it by an equal amount This can be cleared using the phase trellis method
Fig 1.54
Using some mathematical operations we can express the CPFSK s(t) in terms of its inphase and quadrature components as follows:
eqn(1.93) So we have the following four cases:
= π/2, corresponding to transmission of symbol 1. The phase 0 = 0 and 1. = π/2, corresponding to transmission of symbol 2. The phase 0 = 0 and 0. 3. The phase 0 = 0 and = -π /2 (or, equivalently, 3 π /2, modulo 2 π), corresponding to transmission of symbol 1. 4. The phase 0 = 0 and = -π /2 (or, equivalently, 3 π /2, modulo 2 π), corresponding to transmission of symbol 0. 5. This in turn means that the MSK signal itself can take one of four possible forms, depending on the values 0 and
∅ ∅ ∅ − ≤≤ ∅ ≤≤ ∅ ∅ ≤≤
This, in turn, means that the MSK signal itself may assume any one of four possible forms, depending on the value of 0 & The appropriate form for the orthogonal basis function 1 ( ) and 2 ( ): 1
2
=
=
2
Tb
2
cos
sin
2
2
cos 2
sin 2
0
2
Correspondingly, we may express the MSK signal in the form
= 1 1 + 2 2 0 Accordingly, the signal constellation for an MSK signal is two-dimensional, with four message points, as shown in Fig 1.5.9
Figure (1.55) Constellation diagram for MSK signal If we made a comparison between
the constellation of MSK and the QPSK signals we would notice that they have identical format. Note, however, that the coordinates of the message points for the QPSK signal are expressed in terms of signal energy per symbol, E, whereas for the MSK signal they are expressed in terms of the signal energy per bit.
∅ ∅
∅ ∅
The basic difference between QPSK & MSK signals is in the choice of orthogonal signals 1 ( ) and 2 ( ). For QPSK 1 ( ) and 2 ( ) are represented by a pair of quadrature carriers, whereas for an MSK signal, they are represented by a pair of sinusoidally modulated quadrature carriers.
Fig 1.56 Sequence and waveforms for MSK signal
To generate the signal described before of MSK we can use the following MSK transmitter shown in Fig 1.57, the advantage of this modulator is that the signal coherence and deviation ratio are largely unaffected by variation in the input data rate.
Fig 1.57 MSK modulator
Fig 1.58 shows the block diagram of typical MSK receiver.
Fig 1.58 MSK Demodulator
Earlier we remarked that the MSK and QPSK signals have similar signal space diagram. It follows, therefore, that for the case of AWGN channel, they have the same forela for their average probability of error:
=
(
)
0
eqn(1.94)
This is much better than the ordinary FSK.
1.6.4 Gaussian minimum shift keying GMSK Gaussian Minimum Shift Keying (GMSK) is a modification of MSK (i.e. CPFSK with h = 1/2). A filter used to reduce the bandwidth of a baseband pulse train prior to modulation is called a pre-modulation filter. The Gaussian pre-modulation filter smoothes the phase trajectory of the MSK signal thus limiting the instantaneous frequency variations. The result is an FM modulated signal with a much narrower bandwidth. This bandwidth reduction does not come for free since the pre-modulation filter smears the individual pulses in pulse train. As a consequence of this smearing in time, adjacent pulses interfere with each other generating what is commonly called inter-symbol interference or ISI. In the applications where GMSK is used, the tradeoff between power efficiency and bandwidth efficiency is well worth the cost. There are two methods to generate GMSK, one is frequency shift keyed modulation, the other is quadrature phase shift keyed modulation.
Fig 1.59 GMSK implemented by Frequency Shift keying
Fig 1.60 GMSK implemented by a quadrature baseband
The shaded areas in the two above figures have the same function. The GMSK VCOmodulator architecture as shown in the first is simple but is not however, suitable for coherent demodulation due to component tolerance problems. This method requires that the frequency deviation factor of the VCO exactly equals 0.5, but the modulation index of conventional VCO based transmitters drifts over t ime and temperature. The implementation in the second employs a quadrature baseband process followed by a quadrature modulator. With this implementation, the modulation index can be maintained at exactly 0.5. This method is also cheaper to implement. Both methods lead to the same GMSK modulated signal. We are going to be looking at the second of these two methods that is we shall be looking at a quadrature baseband processor followed by a quadrature modulator as shown in the second. The Gaussian low-pass filter has an impulse response given by the following equation
π
g t =
1
2T
Q 2 Bb
− − t
T 2
ln 2
π
Q 2 Bb
t+
T 2
ln 2
eqn(1.95)
For
0
≤ ≤∞ Bb
Fig 1.61 The truncated and scaled impulse response of the Gaussian low-pass
Recall the probability of error for plain MSK is given by
=
(
2
0
)
eqn(1.96)
GMSK > Pe FSK this arises from the trade off By comparing we can conclude that Pe between power and bandwidth efficient: GMSK achieves better bandwidth efficiency than MSK at the expense of power efficiency.
1.7 QUADRATURE AMPLITUDE MODULATION (QAM) M-Ary PSK systems are consisted of fixed step phase shifts with constant envelope. In a try to increase such system capacity, the constellation points will get closer to each other increasing the bit error rate. A simple solution is to increase the radius of the constellation points, but of course it’ll also increase the power used. A new technique was developed to overcome that problem by making use of available space inside the constellation circle.
(b) (a)
Figure (1.62) showing 16-Ary PSK (a) crowded on the constellation circle and equivalent average power 16-Ary QAM (b) with constellation points distributed to make use of the same space
This technique is called quadrature amplitude modulation as it combines with or make use of both ASK and PSK. As in Fig(1.62)-b is a constellation diagram showing 16-Ary QAM, it can be represented as two quadrature carriers each is modulated with 4-level ASK.
1.7.1 Types of QAM: 1.7.1.1 Circular QAM: Simply this type of QAM is considered as multi-level PSK, with phase off-set to maximize the minimum Euclidian distance to obtain minimum average energy per symbol.
Where
=
2 0
is the normalized level,
.
j(2
+
)
,
0 ≤ t ≤ Tb
eqn(1.97)
is the symbol’s phase.
_____________________________________________________________________ Figure (1.63) circular QAM
1.7.1.2 Rectangular QAM: The general form of M-Ary QAM is defined by the transmitted signal:
− − − − ⋯ − − −− −− ⋮ −− −− ⋯⋱ −− −⋮− ≤≤ ≤≤ − − −−− −− −−− −− −− −− 2 0
=
cos 2
+
2 0
sin 2
,
0 ≤ t ≤ T b eqn(1.98)
E0 is the energy of the signal with the lowest amplitude, a i and bi are a pair of independent integers chosen to specify a certain constellation point, i є [-L+1 L-1], where
,
=
=
( (
+ 1, + 1,
1) 3)
( (
+ 3, + 3,
1) 3)
( (
1, 1,
1) 3)
eqn (1.99)
( + 1, + 1) ( + 3, + 1) ( 1, + 1) The rectangular QAM signal could be represented in terms of 2 independent basis functions:
,
1
=
2
=
2
2
cos 2
0
eqn(1.100)
sin 2
0
eqn(1.101)
The coordinates of the ith message point are
for example, for the 16-QAM with L = 4: 3,+3 3,+1 , = 3, 1 3, 3
1,+3 1,+1 1, 1 1, 3
0
and
+1, +3 +1, +1 +1, 1 +1, 3
0,
+3, +3 +3, +1 +3, 1 +3, 3
eqn (1.102)
The first rectangular QAM constellation usually encountered is 16-QAM, the constellation diagram for which is shown here. A Gray coded bit-assignment is also given. The reason that 16-QAM is usually the first is that a brief consideration reveals that 2-QAM and 4-QAM are in fact binary phase-shift keying (BPSK) and quadrature phase-shift keying (QPSK), respectively. Also, the error-rate performance of 8-QAM is close to that of 16-QAM (only about 0.5dB better), but its data rate is only threequarters that of 16-QAM. 1.7.1.3 Circular or Rectangular 8-QAM? When dealing with 8-Ary constellation, APK is preferable as it is space efficient rather than QAM. In 16-Ary constellation, it’s more advisable to go to QAM constellation as its more energy efficient, even some standards actually uses 16 APK like V29.bis telephone standard to maximize the phase difference between points having the same energy in the expense of increasing the amplitude levels. Also the symmetry of rectangular QAM sometimes doesn’t suit the channel characteristics or the detection process, and more, the designer wishes to put the constellation points anywhere, from here came the APK.
_____________________________________________________________________ Figure (1.64) rectangular QAM versus APK
1.7.2 Probability of symbol error calculations: As both in-phase and quadrature components are independent, probability of correct detection is:
− − − − − ≅ − − −
′ 2 = (1 ) , where ′ is the probability of symbol error for one of the components. Referring to PAM symbol error eqn, ′ could be written as: ′
1
= 1
0
(
)
eqn(1.103)
eqn(1.104)
0
The probability of symbol error for QAM is:
=1
=2 1
, so
1
=1
(1
0
)2
′
2
′
, but
0
=
2(
−
1) 0 3
So finally
=2 1
1
3
2(
1) 0
eqn(1.105)
APK (amplitude-phase keying): its constellation is simply multi level of amplitudes. 8-QAM is considered the optimal constellation as it requires least mean energy.
− 2
<
1
(
2 0
) eqn(1.106)
The separation between each point having the same magnitude is 45 degrees.
Figure (1.65) formulation of
probability of symbol error in circular QAM
1.7.3 QAM modulation
___________________________________________________________ Figure (1.66) M-Ary QAM Modulator
Binary data are split into 2 parallel paths, in each path a number of bits = is amplitude shift keyed to L levels then phase shift keyed using the 2 independent carriers. Then the paths are combined again to form the M-Ary QAM signal.
1.7.4 QAM demodulation: In QAM modulation, coherent and differentially coherent detection could be used as for PSK systems, we shall concern with coherent detection.
Figure (1.67) M-Ary QAM Demodulator
As in PSK, the i/p signal is multiplicated by both in-phase and quadrature carriers then integrated over the symbol period to get a multi-level baseband symbol set. The decision circuit translates those levels to bits which are then combined using the S/P converter to get the modulated binary data. Example for 16-Ary QAM symbols amp=4.2426, ph=-135
amp=3.1623, ph=-161.5651
amp=3.1623, ph=161.5651
amp=4.2426, ph=135
5
5
5
5
0
0
0
0
-5
0
50
100
-5
0
amp=3.1623, ph=-108.4349
50
100
-5
0
amp=1.4142, ph=-135
50
100
-5
amp=1.4142, ph=135
5
5
5
0
0
0
0
0
50
100
-5
0
amp=3.1623, ph=-71.5651
50
100
-5
0
amp=1.4142, ph=-45
50
100
-5
amp=1.4142, ph=45
5
5
5
0
0
0
0
0
50
100
-5
amp=4.2426, ph=-45
0
50
100
-5
amp=3.1623, ph=-18.4349
0
50
100
-5
5
5
0
0
0
0
50
100
-5
0
50
100
-5
0
50
50
100
0
100
50
100
amp=4.2426, ph=45
5
0
0
amp=3.1623, ph=18.4349
5
-5
100
amp=3.1623, ph=71.5651
5
-5
50
amp=3.1623, ph=108.4349
5
-5
0
-5
0
50
100
_____________________________________________________________________ Figure (1.68) All possible QAM signals
1.7.5 BW efficiency:
It’s identical to M-Ary PSK where
=
=
2
2
eqn (1.107)
1.8 SYNCHRONIZATION The coherent detection of a digitally modulated signal , irrespective of its form, requires that the receiver be synchronous to the transmitter. We say that two sequences of events (representing a transmitter and a receiver) are synchronous relative to each other when the events in one sequence and the corresponding in the other occur simultaneously. The process of making situation synchronous, and maintaining in this situation is called synchronization. From the discussion presented on the operation of d igital modulation techniques, we recognize the need for two basic modes of synchronization: When coherent detection is used , knowledge of both the frequency and the phase of the carrier is necessary. The estimation of the carrier phase and frequency is called carrier recovery or carrier synchronization.
To perform demodulation , the receiver has to know the instants of time at which the modulation can change its state. That is, it has to know the starting and finishing times of individual symbols , so that it may determine when to sample and when to quench the product-integrators. The estimation of these times is called clock recovery or symbol synchronization. We have observed that in a digital communication system, the output of the demodulator must be sampled periodically, once per symbol interval, in order to recover the transmitted information. Since the propagation delay from t he transmitter to the receiver is generally unknown at the receiver, symbol timing must be derived from the received signal in order to synchronously sample the output of the demodulator. The propagation delay in the transmitted signal also results in a carrier offset, which must be estimated at the receiver if the detector is phase coherent.
1.8.1 Carrier Recovery and Symbol Synchronization in Signal Demodulation Symbol synchronization is required in every digital communication system which transmits information synchronously. Carrier recovery is required if the signal is detected coherently. Figure (1.69) illustrates the block diagram of a binary PSK (or binary PAM) signal
π ϕ
demodulator and detector. As shown, the carrier phase estimate generating the reference signal g t cos 2 f c t +
ϕ
is used in
for the correlator.
The symbol synchronizer controls the sampler and the output of the signal pulse generator. If the signal pulse is rectangular then the signal generator can be eliminated.
Figure (1.69) Block digram of binary PSK receiver
The block diagram of an M-ary PSK demodulator is shown in Fig(1.70). In this case, two correlators (or matched filters) are required to correlate the received
π ϕ ϕ
π ϕ
signal with the two quadrature carrier signals g t cos 2 f c t +
g t sin 2 f c t +
. where
and
is the carrier phase estimate. The detector is now
a phase detector, which compares the received signal phases with the possible transmitted signal phases.
Figure(1.70) Block diagram of M-ary PSK receiver
Finally, we illustrate the block diagram of a QAM demodulator in Fig( ). An AGC is required to maintain a constant average power signal at the input to the demodulator. We observe that the demodulator is similar to a PSK demodulator, in that both generate in-phase and quadrature signal samples (X, Y) for the detector. In the case of QAM the detector computes the euclidean distance between the received noise corrupted signal point and the M possible transmitted points, and selects the signal closest to the received point.
Figure(1.71) Block diagram of QAM receiver.
1.8.2 Carrier Recovery: The PSK signals have no spectral line at carrier frequency. Therefore a device is needed in the carrier recovery circuit to generate such a line spectrum. There are two main types of carrier synchronizers, the Mth power loop, and the Costas loop. Figure (1.71 ) is the Mth power loop for carrier recovery for M-ary PSK. For BPSK (or DBPSK), M = 2, thus it is a squaring loop. For QPSK (or OQPSK,DQPSK), M = 4, it is a quadrupling loop, and so on. It is the Mth power device that produces the spectral line at Mfc. The phase lock loop consisting of the phase detector, the LPF, and the VCO, tracks and locks onto the frequency and phase of the M fc component. The divide-by-M device divides the frequency of this component to produce the desired carrier at frequency fc and with almost the same phase of the received signal. Before locking, there is a phase difference in the received signal relative to the VCO output signal. We denote the phase of the received
θ
signal as θ and the phase of the VCO output as M .
Figure(1.71) Mth power synchronizer for carrier recovery.
A difficulty in circuit implementation of the Mth power loop is the Mth power device, especially at high frequencies. Costas loop design avoids this device. Figure (1.72) is the Costas loop for carrier recovery for BPSK. Initially the VCO generates a sinusoid with a frequency close to the carrier frequency fc and some initial phase. The frequency difference and the initial phase are accounted for by the phase
θ θ −θ θ −θ θ −θ
. The multipliers in the I and Q-channels produce 2 fc terms and zero frequency terms. The LPFs attenuate the 2fc terms and their outputs are proportional to
a t cos 1
or a t sin
term a2 t sin2 2
. Then these two terms multiply again to give the
which is low-pass filtered one more time to get rid of any
amplitude fluctuation in a2 t , thus the control signal to the VCO is proportional to
θ −θ
sin 2
θ −θ
which drives the VCO such that the difference
and smaller. For sufficiently small signal.
θ −θ
becomes smaller
, the I-channel output is the demodulated
Figure(1.72) Costas loop for carrier recovery for BPSK.
The Costas loop for QPSK is shown in Figure (1.73). The figure is selfexplanatory and its working principle is similar to that of BPSK. The limiters are bipolar, which are used to control the amplitude of the two channels' signal to maintain balance.
ϕ θ −θ
When the phase difference = is sufficiently small, the I- and Q-channel outputs are the demodulated signals. A difficulty in Costas loop implementation is to maintain the balance between the I- and Q-channel. The two multipliers and low-pass filters in these two channels must be perfectly matched in order to achieve the theoretical performance. Although the appearance of the Mth power loop and the Costas loop are quite different, their performance can be shown to be the same.
Figure(1.73) Costas loop for carrier recovery for QPSK.
1.8.3 Clock Recovery The clock or symbol timing recovery can be classified into two basic groups. One group is the open loop synchronizer which uses nonlinear devices. These circuits recover the clock signal directly from the data stream by nonlinear operations on the received data stream. Another group is the closed-loop synchronizers which attempt to lock a local clock signal onto the received data stream by use of comparative measurements on the local and received signals. Two examples of the open-loop synchronizer are shown in Figure (1.74). The data stream that we use in the phase shift keying modulation is NRZ waveform. We know that this waveform has no spectral energy at the clock Frequency. Thus in the open-loop synchronizers in Figure (1.74), the first thing that one needs to do is to create spectral energy at the clock frequency. In the first example, a Fourier component at the data clock frequency is generated by the delay-and-multiply operation on the demodulated signal m(t). This frequency component is then extracted by the BPF that follows and shaped into square wave by the final stage. The second example generates the clock frequency component by using the differentiator-rectifier combination. The differentiator is very sensitive to wideband noise, therefore a low-pass filter is placed in the front end of the synchronizer.
Figure(1.74 ) Two types of open-loop symbol synchronizers.
An early/late-gate circuit shown in Figure (1.75) is an example of the class of closed-loop synchronizers. The working principle is easily understood by referencing Figure (1.75). The time zero point is set by the square wave clock locally generated by the VCO. If the VCO square wave clock is in perfect synchronism with the demodulated signal m(t), the early-gate integrator and the late-gate integrator will accumulate the same amount of signal energy so that the error signal e = 0. If the VCO frequency is higher than that of m(t),then m(t) is late by Δ < d, relative to the VCO clock. Thus the integration time in the early-gate integrator will be T - d - Δ, while the integration time in the late-gate integrator is still the entire T - d. The error
signal will be proportional to - Δ. This error signal will reduce the VCO frequency and retard the VCO timing to bring it back toward the timing of m(t). If the VCO frequency had been lower and the timing had been late, the error signal would be proportional to +Δ, and the reverse process would happen, that is, the VCO frequency would be increased and its timing would be advanced toward that of the incoming signal.
Figure(1.75) Early/late-gate clock synchronizer.
Figure(1.76) Early-late-gate timing illustration.
1.8 COMPARISON BETWEEN DIGITAL MODULATION SCHEMES As we introduce the main digital modulation schemes in the previous section we here introduce a comparison between them as a conclusion Table (1.8) presents a comparison between the previously introduced modulation schemes from the probability of error point of view Modulation scheme
Probability of error
ASK
M ary ASK
BFSK
M-ary FSK BPSK
DPSK M-ary PSK
− ≤ =
MSK
/
2
2
2
QPSK
1
2
− − =
(
)
0
GMSK
=
(
2
)
0
QAM
=2 1
1
3
2(
1)
0
Table (1.8) Error probabilities for various modulation schemes
Table (1.8) introduces a comparison between relevant modulation schemes from the Bandwidth efficiency and power efficiency and the error free E b /No.
Table(1.8) Error free Eb/No for relevant modulation schemes
1.9 DISCUSSION OF THE ABOVE MODULATION SCHEMES:
Coherent reception provides better performance than differential, but requires a more complex receiver. The above table shows that bandwidth efficiency is traded off against power efficiency. MFSK is power efficient, but not bandwidth efficient (because the probability of error decreases by increasing M; however that would increase the transmission bandwidth). MPSK and QAM are bandwidth efficient but not power efficient. , therefore PSK is more Mobile radio systems are bandwidth limited suited. Phase Shift Keying is often used, as it provides a highly bandwidth efficient modulation scheme. The constant envelope class is generally suitable for communication systems whose power amplifiers must operate in the nonlinear region of the inputoutput characteristic in order to achieve maximum amplifier efficiency. An example is the TWTA (traveling wave tube amplifier) in satellite communications. QPSK, modulation is very robust, but requires some form of linear amplification. OQPSK and π /4-QPSK can be implemented, and reduce the envelope variations of the signal. o The π/4-QPSK is worth special attention due to its ability to avoid 180 abrupt phase shift and to enable differential demodulation. It has been
used in digital mobile cellular systems, such as the United States digital cellular (USDC) system. The PSK schemes have constant envelope but discontinuous phase transitions from symbol to symbol. The CPM schemes have not only constant envelope, but also continuous phase transitions. Thus they have less side lobe energy in their spectra in comparison with the PSK schemes. The CPM class includes, GMSK, and MSK. MSK is a special case of CPFSK, but it also can be derived from OQPSK with extra sinusoidal pulse-shaping. MSK has excellent power and bandwidth efficiency. Its modulator and demodulator are also not too complex. MSK has been used in NASA's Advanced Communication Technology Satellite (ACTS). GMSK has a Gaussian frequency pulse. Thus it can achieve even better bandwidth efficiency than MSK. GMSK is used in the US cellular digital packet data (CDPD) system and European GSM (global system for mobile communication) system. Constant envelope schemes (such as GMSK) can be employed since an efficient, non-linear amplifier can be used.
The generic nonconstant envelope schemes, such as ASK and QAM, are generally not suitable for systems with nonlinear power amplifiers. However QAM, with a large signal constellation, can achieve extremely high bandwidth efficiency. QAM has been widely used in modems used in telephone networks, such as computer modems. QAM can even be considered for satellite systems. In this case, however, back-off in TWWs input and output power must be provided to ensure the linearity of the power amplifier. High level M-ary schemes (such as 64-QAM) are very bandwidth efficient, but more susceptible to noise and require linear amplification.
1.10 SIMULATION RESULTS USING MATLAB: We now proceed to simulate and assess the above introduced modulation schemes we used in simulation four methods which are:
BER tool of communication blockset. SIMULINK models. M-file commands. And GUI(graphical user interface) of our own.
1.10.1 BER tool of communication blockset:
Using the Bit error rate tool located in the communication toolbox . The tool gives the capability of: Managing a series of simulations with different values of Eb/N0. Collecting the results of bit error rate & importing to workspace.
Creating a plot & Produce a comparison curves between different types of modulation. The simulation can be theoretical, semi-analytic or Monte carlo analysis. The simulation can be achieved in AWGN & Rayleigh fading channel. The tool supports: Channel coded sequence(convolution & block) Coherent and noncoherent detection Differential encoded sequence. Synchronization errors addition.
Figure (1.77) BER tool
By using the above tool in the theoretical mode we obtain the next results: 1.10.1.1 Phase shift keying (PSK) modulation scheme simulation
Figure (1.78) Simulation of PSK modulation schemes
By simulating PSK modulation schemes from BPSK to 64 PSK in the signal to noise ratio range of 0-20 dB(for 32PSK and 64PSK we extend the range to 30 dB) we will find that:
BPSK and QPSK have the same probability of error but QPSK has higher spectral efficiency. As M increases the probability of error increases which represents a power /bandwidth efficiency trade off. From the figure we deduct that the QPSK is robust modulation scheme that we can employ in noisy channels in WiMAX From the figure we conclude the power efficiency and spectral efficiency BPSK QPSK 8PSK 16PSK 64PSK 0.5 1 1.5 2 3 Spectral efficiency (log2 M/2) -6 23.2dB 28.5dB Power efficiency (for BER=10 ) 10.5dB 10.5dB 18.5dB Table (1.9) PSK power and spectral efficiencies simulation results
Figure (1.79) QPSK, OQPSK, DQPSK simulation
Figure (1.79) shows a simulation for QPSK,OQPSK and DQPSK simulation from 0:20 dB We found that
OQPSK and QPSK has the same BER but with less phase abrupt changes DQPSK is inferior to both by 3dB.
Figure (1.80) shows a simulation for BPSK versus DPSK simulation in the range of 015 dB We found that DPSK is slightly inferior to BPSK but in the expense on increasing the complexity of the coherent demodulator employed to demodulate PSK signal
Figure (1.80) BPSK and DPSK simulation
1.10.1.2 Frequency shift keying simulation (FSK) simulation Figure (1.81) shows the effect of increasing the order of FSK modulation from the BFSK into 32FSK.
Figure (1.81) M-ary FSK simulation
From the figure we conclude that:
As the order of modulation increases (M increases) the BER decreases. That increase in power efficiency is traded off by the required transmission bandwidth Table (1.10) shows the power efficiencies of M-ary FSK
-6
Power efficiency (for BER=10 )
BFSK 13.5 dB
4FSK 10.7dB
8FSK 9.2 dB
16FSK 8.2 dB
32PSK 7.5 dB
Table(1.10) M-ary FSK power efficiency
Figure (1.82) BFSK, MSK simulation
Figure (1.82) shows a simulation of BFSK and MSK and we can here assess that the MSK has a better BER performance than BFSK and hence better more power efficient.
Figure (1.83) coherent and noncoherent FSK
Figure (1.83) shows that noncoherent FSK is inferior to coherent FSK 1.10.1.3 QAM simulation
Figure (1.84) QAM simulation
Figure (1.84) shows QAM simulation for various modulation order (16,32, 64,128,256,512,1024) in the signal to noise ratio of 0-30 dB and we found that:
As M increases the BER increases. 2QAM and 4QAM are BPSK and QPSK which are previously simulated As M increases the spectral efficiency increases. 16QAM and 64QAM are suitable modulation scheme to be employed in good channel conditions in WiMAX Table (1.11) shows the power efficiencies /bandwidth efficiency of QAM modulation QAM order 8 16 32 64 128 256 512 1024 1.5 2 2.5 3 3.5 4 4.5 5 Spectral efficiency (log2 M/2) 13.5 14.5 17.5 18.7 22 23.5 27 28.5 Power efficiency (for -6 BER=10 ) in dB Table (1.9) QAM power and spectral efficiencies simulation results
1.10.1.4 Comparative simulation of various modulation schemes
Figure (1.85) modulation schemes used in WiMAX simulation
Figure (1.85) shows the three modulation schemes used in WiMAX system, the results instruct an adaptive modulation to be employed in WiMAX system according to channel conditions (signal to noise ratio)
Figure (1.86) comparative simulation for 16FSK, 16PSK, 16QAM
Figure (1.86) shows a comparative simulation between different modu lation schemes each having same modulation order 16 so we found that:
16 FSK is the best modulation scheme which is pointed out before (since it trades the better performance by the excessive transmission bandwidth). 16 QAM is better than 16PSK (since the symbols in 16QAM cover all the spaces in the constellation diagram and not confined to a densely packed circle). Hence when it is required to achieve same spectral efficiency square QAM is used instead PSK. However PSK is used when the linear amplification is considered.
Figure (1.87) shows a comparative simulation between all i ntroduced modulation schemes
Figure (1.87) Comparative simulation between all modulation schemes
As we found from figure we can arrange the modulation schemes descendingly from the most power efficient scheme to the least as following: 32FSK16FSK8FSKBPSK/QPSK4FSKBFSK8PSK16QAM 32 QAM16PSK64QAM256QAM.
1.10.2 SIMULINK simulation & constellation diagram
The simulink library has several of uesful blocks that can be in the performance analysis of modulation schemes such that: Binary generators for creating a random sequence of bits. Channels (AWGN-Rayleigh fading). Scatter plot scope and eye diagram scope. Dynamic error rate calculation. Various modulation schemes.
1.10.2.1 The SIMULINK model used in evaluating BER Figure (1.88) shows the SIMULINK model used for simulation and evaluating BER The model includes: Bernoulli generator for generating bits. Block of desired modulation & demodulation technique. Channel (ex:AWGN). Scatter plot for plotting the constellation diagram. Error rate calculator to calculate BER. NOTE: by changing the blocks of modulation and demodulation we can study different schemes.
Figure (1.89) BER calculation using SIMULINK
1.10.3.2 Sample runs with different Signal to noise ratios (a) With SNR=5dB.
_____________________________________________________________________ Figure (1.90) Simulation results with SNR=5dB.
Figure (1.90) shows a simulation results when the signal to noise ratio is very low (5 dB) we can see that:
The constellation points moves away from its designated points due to large noise power. The receiver cannot differentiate between symbols correctly and hence that will lead to 0.004 bit error rate.
(b) with SNR=10dB
Figure(1.91) Simulation results with 10 dB
As shown in figure (1.91) that although the constellation points moves away around its transmitted places but the movement is confined to the decision region and hence no error occurs
(c) With SNR=100dB(no noise approximately)
_____________________________________________________________________ Figure (1.92) Simulation results when SNR=100dB
In figure (1.92) we can see that when no noise is added (approximately) the constellation points will lay in its correct places.
1.10.4 M-files commands The communication toolbox presents a rich library of commands that cover all modulation schemes and will lead to similar results that we obtained before and here we present them without displaying the output which was introduced before •
For generating the random bits: randint(n,1). Converting to symbols: xsym = bi2de(reshape(x,k,length(x)/k).','left-msb'); Channels: y = awgn(x,snr) Constellation: h = scatterplot(yrx(1:nsamp*5e3),nsamp,0,'g.'); Bit error rate: [number_of_errors,bit_error_rate] = biterr(x,z) •
•
•
•
•
•
•
•
•
And the modulation commands:
Figure (1.93) Modulation commands in MATLAB
And also MODEM objects:
Figure (1.94) MODEM objects in MATLAB
1.10.3 Our GUI for modulation: Last we introduce a GUI for modulation. The program was designed to illustrate different modulation schemes in time and frequency domain
The program asks the user for the message bits and the frequency of the carrier and the user can choose the appropriate modulation scheme from the set {ASK,BFSK,BPSK,QPSK} and the program will draw tha modulated signal in time and frequency domain. Sample Run: QPSK of {1 0 0 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0}
