Properties of the Fourier Transform: x(t t0 ) x( t) x(t) e j 2πf t
− −
X (f ) f )e−j 2πf t X ( f ) f ) X (f f 0 ), if f f 0 is real 1 [X (f + + f 0 ) + X (f f 0 )] 2 j [X (f + + f 0 ) X (f f 0 )] 2
↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔
0
x(t) cos(2πf cos(2πf 0 t) x(t) sin(2πf sin(2πf 0 t) d x(t) dt x(t) v(t) x(t)v(t)
∗
0
− − −
−
− − − −
j 2πf X (f ) f ) X (f ) f )V ( V (f ) f ) X (f ) f ) V ( V (f ) f )
∗
Common Fourier Transform Pairs: δ (t) rect(t/T rect(t/T ))
1 T sinc( T sinc(ff T ) T ) 1 cos(2πf cos(2πf 0 t) [δ (f + + f 0 ) + δ (f f 0 )] 2 j sin(2πf sin(2πf 0 t) [δ (f + + f 0 ) δ (f f 0 )] 2 Autocorrelation function and spectral density for deterministic signals:
↔ ↔ ↔ ↔
− − − − −
∞
Rx (τ ) τ ) =
x(t)x(t + τ ) τ ) dt = dt = x x((τ ) τ ) x(τ ), τ ), for an energy signal
∗
−∞
Ψx (f ) f ) = Rx (τ ) τ ) = Gx (f ) f ) = Random Variables:
|X (f )|2 1 T 0
T 0
| | 2
−
T 0
x(t)x(t + τ ) τ ) dt, dt , for a power signal with period T 0
2
cn 2 δ (f
− − nf 0) x
F X (x) =
p( p(λ) dλ
−∞ ∞
mX
=
x p(x) dx
−∞ ∞
P X
=
x2 p( p(x) dx
−∞
2 σX
− mX2
= P X
WSS Random Processes: mY = mX H (0), (0), when X when X ((t) is filtered to get Y ( Y (t) RY (τ (τ )) = RX (τ ) τ ) Rh (τ ) τ ), when X when X ((t) is filtered to get Y ( Y (t) GY (f (f ) = GX (f ) f ) H (f ) f ) 2 , when X when X ((t) is filtered to get Y ( Y (t)
∗ |
|
Sampling and Quantization: X s (f ) = SNR peak l
1 T s
∞
X f
n=−∞ 2
= 3L
≥
− −
log2
1 2 p
n T s
for p for p% % peak distortion distortion
Baseband Modulation: R W (i) bi-NRZ: zeroes are represented by negative ( V ) and ones are represented by positive (+ V ) pulses (ii) uni-NRZ: zeroes are represented by no pulse (0) and ones are represented by positive (+ V ) pulses (iii) bi/uni-RZ: same as for NRZ, but, pulses return to zero level at half-bit ( T 2 ) duration (iv) NRZ-M: zeroes maintain the previous voltage and ones toggle it (v) Manchester coding (bi-φ-L): ones are represented by half-bit wide pulses in the first half of the bit duration and zeroes are represented by half-bit wide pulses in the second half of the bit duration (vi) Duobinary & precoding: bandwidth efficiency =
−
yk wk
= xk + xk−1 , for duobinary signaling with x k being pulse ( 1) voltages = dk wk−1 , for precoding with d k being binary (‘0’ or ‘1’) data
±
⊕
Baseband Demodulation:
ψi2 (t) dt
= 1, and,
ψj (t)ψk (t) dt = 0 for an orthonormal basis
s(t) = a1 ψ1 (t) + . . . + aN ψN (t), with an N -dim basis, where, s(t)ψj (t) dt aj = , and, K j = ψ j2 (t) dt K j E s
=
N
s2 (t) dt =
a2j is the waveform energy
1
P (a ) 2 a1 + a2 σ0 ln( P (a ) ) γ = + is the decision threshold; here, a i = s i (t) h(t) t=T 2 (a1 a2 ) 2 1
∗
−
P B
=
Q − Q − Q − Q E d 2N 0
P B
= P (0)
P B
=
P B
=
, where, E d is the energy in s 1 (t)
γ
a2
σ
a1
Q
+ P (1)
a2
a1
|
− s2(t)
− γ
σ
2σ
E b (1 ρ) N 0
ρ =
s 1 (t)s2 (t) dt is the correlation coefficient E b
h(t) = s 1 (T
− t) − s2(T − t) for the matched filter
The raised cosine (RC) spectrum with bandwidth W and symbol rate R s = 2W 0 has the characteristic: H (f ) =
r
=
W =
1 for f < 2W 0 W 2 π |f |+W −2W cos 4 W −W for 2W 0 W < f < W 0 for f > W W W 0 , is the roll-off factor W 0 1 (1 + r)Rs 2
−
0
0
| | | |
−
−
||
The Range Equation: EIRP = P t Gt Ae = ηA p for an antenna with efficiency η
4πd λ
Ls =
P r = G = M =
2
is the free space loss at distance d
EIRP G r is the power received Ls Lo
4πA e relates antenna directional gain to effective area λ2
EIRP Gr is the link margin (k = 1.38 (E b /N 0 )reqd RkT o Ls Lo
× 10−23J/K)
Noise Analysis: o o N = kT eq W relates noise power N to noise equivalent temperature T eq and bandwidth W
T o , and, 290 = (F 1) 290 relate noise figure to noise temperature
F = 1 + T o
− ×
F = L for a lossy line with loss factor L
Gcomp
= G1
× G2 × . . . × Gn F 2 − 1 F 3 − 1 F 1 + + + ...
F comp
=
F n 1 G1 G1 G2 G1 G2 . . . Gn−1 o o T T 3 T no = T 1o + 2 + +... G1 G1 G2 G1 G2 . . . G n−1
o T comp
−
o T sys = T Ao + T Lo + LT Ro is the system noise temperature
Satellite Repeaters: P B
β =
P r N 0
1 , where, P T = P T + kT s W
= ij
≈ P u + P d for a regenerative repeater
k Ak P k is
the total received power, and β EIRPs is the variable gain
EIRPs γ j βA i P i for a non-regenerative repeater (Ai and γ j are U/L and D/L attenuations, resp.) EIRPs γ j βN s + N g −1
−1
E b N 0
=
ov
E b N s
+
u
E b N g
−1
relates the overall SNR to U/L and D/L SNRs d
M -PSK
Q Q − 2E S 2πi cos 2πf c t + T M
si (t) =
2E S π sin N 0 M
P E = 2 P B
=
P B
=
P B
=
2E b N 0
for BPSK and QPSK
1 E b exp for DPSK 2 N 0 P E for all other PSK k 2 . T
W = M -FSK si (t) =
2E S cos(2πf i t) T
Q −
P E = (M P E = P B
=
W
≈ ≈
W
− 1) M − 1 exp
E S for coherent reception N 0 E S 2N 0
for non-coherent reception 2 M/2 P E M 1 M for coherent reception 2T M for non-coherent reception. T
−
M -ASK si (t) = W =
2E i cos(2πf c t) T
2 . T
Random Access ρ = G e −2G , for Aloha with G = λ t τ ρ = G e −G , for S-Aloha with G = λ t τ . Spread Spectrum M = J 0 2 ρ0 P B,max
= = =
γG P GV GA , where, G A = G V = 2.5, γ = 1.5, H 0 = 1.55 (E b /I 0 )reqd H 0 J = Jammer PSD when jamming bandwidth W with power J 2W 2 for (E b /J 0 ) > 2 (E b /J ) for optimum partial band jamming 1 for (E b /J 0 ) 2
0
e−1 (E b /J 0 ) E 1 − 2J b0 2e
≤
for (E b /J 0 ) > 2 for (E b /J 0 )
≤2
with optimum partial band jammer
The PDF of a Gaussian random variable with mean m and variance σ 2 is p(x) =
1 exp 2πσ
√
−
(x
− m)2
2σ 2
Bayes’ Rule:
P (A) =
P (A Bi )P (Bi ).
|
i
Jensen’s Inequality: If a i are non-negative numbers with
N 1 ai =
1, then,
N
N
ai log(zi )
i=1
x2 x1
If a(x) is a non-negative function of x with x2
≤ log
ai zi
i=1
a(x)dx = 1, then, x2
a(x)log(b(x)) dx
≤ log
x1
a(x)b(x) dx .
x1
Entropy and Information : H (X ) =
p(xi )log
xi
H (X, Y ) =
xi
H (X Y ) =
|
yj
1 p(xi , yj )log p(xi , yj )
p(yj )H (X Y = yj )
|
yj
=
1 p(xi )
p(yj )
yj
| xi
1 p(xi yj )log p(xi yj )
|
1 dx p(x) I (X ; Y ) = H (X ) H (X Y ) = h(X ) h(X Y ) = H (Y ) H (Y X ) = h(Y ) h(Y X ) D = E [X Y ] for a binary source h(X ) =
D
=
p(x)log
− | − | − | − | ⊕ M b 2 E [(X − Y ) ] = (x − yi )2 p(x) dx for a continuous source i=1 b Hb( p) − Hb(D) if D < p for a binary source.
i
i−1
R(D) = R(D) =
1 log 2 2
σ2 D
if D < σ 2 for a Gaussian source.
1 P P C = max I (X ; Y ) = log 2 1 + = W log 2 1 + 2 N N 0 W p(x) P C ∞ = log2 e N 0 p = Pr[C C p ], where, C p is the p% outage capacity. 100
≤
EE 641: RF Wireless Communication Systems List of Commonly Needed Expressions and Relationships I. Fundamentals of RF Wireless Communication Systems 1. Spectral Efficiency !spec = Data rate R b (bits/sec) / Transmission bandwidth B (Hz) 2. Power Efficiency of modulation scheme, ! pow = Data rate R b (bits/sec) / Radiated power P rad (Watts)
2. Power Efficiency of radiating antenna,
! pow = Radiated power Prad (Watts) / Power drawn from source PDC (watts) S # & 3. Shannon’s channel capacity C = B log 2 $1 + ! bits/sec % N " II. Electromagnetic Waves and Radiators
1. Maxwell’s Equations # ( µ H ) $ % E
= &
#t
,
$% H
=
J
+
# (µ E ) #t
,
$.E
=
!
,
$.H
=
"
2. Wave Equation 2
# E
2 + ! " µ E =
0,
2
# H
2 + ! " µ H =
0
3. Wave impedance of the medium !
=
µ / "
=
120# µ rel / " rel
In free space, ! = 120# $ = 377 $ 4. Propagation constant of the medium "
% = & + j ' = j ! ((2)µ) , " if ) = )r + j )i, then % = & + j ' = j( [ µ)r (1 – j )i/)r ) ]
5. Phase velocity of the electromagnetic waves in the medium v p (ohms) = 1/! (µ)) = c / * (µrel)rel) where c = 3+10 m/sec 6. Poynting Vector 8
S = E + H 7. Radiation Intensity due to a source at the origin 2
U(r, ,, -) = r . S (r, ,, -) 8. Power radiated from a source at the origin "
P rad
=
2"
! ! U (# , $ ) sin # d # d $ # 0 =
$ 0 =
9. Far Field condition Conditions of Far Field :
2 R . 2D / /
0
R >> D R >> / 10. Fields due to a Hertzian Dipole of length 0z and current I, placed at origin along z axis. E(r, ,, -) = Er (r, ,, -) ar + E (r, ,, -) a + 0 a ,
2
Er (r ," , #)
I )z ! =
2%
$ e
& j ! r
E" (r ," , # ) = H" (r ,# , " )
j I (z ! =
, ! r
2
e
4$
Directivity
D (! , " )
Radiation resistance
' & sin "
$ e& j ! r *
4%
% j ! r
3 =
2
Rrad
-
' cos " cos " ( & j * ( ! r )2 + ( ! r )3 ,
2
j I )z !
,
j sin "
+
( ! r )2
+
sin " (
+
( ! r )3 -
& sin # sin # ' ) ! r % ( ! r )2 * + ,
sin 2 ! 80!
=
11. Dipoles: Short (Hertzian):
2
$ # z % & ' ( " )
2
D = 1.5 = 1.76 dB D = 1.64 = 2.15 dB and R rad = 73 $
Half-wave:
III. Receiving Antenna Characteristics
1. Friis Equation
Pr
PG t t
=
4" d
2
Aeff
2. Effective Area
Aeff (! , " ) !
3. Reciprocity Theorem
G
4! =
2
"
=
# ! 2 PG t t 2
4" d
% ' 4"
S inc (! , " )
Aeff
Pn,av = kTn,antB
5. Noise temperature of antenna Tn , ant
=
2!
!
4! " $0 =
sin " d"
(
P av
4. Noise power available from a resistor
1
$
Gr &
$ d# D(" ,# )T
B
(" ,# )
# 0 =
6. G/T Ratio G/T = [ 10 log10 Gant ] / Tant dB/K IV. Physical Model of Wave Propagation 1. Reflection Coefficient
For E field parallel to ground
$! %
For E field in plane of incidence
E ref ! E inc!
%# &
E ref # E inc #
=
! 2 cos "trans
# !1 cos " inc
!2 cos "trans
+
=
!1 cos " inc
! 2 cos "inc
$ !1 cos " trans
!2 cos "inc
+
!1 cos "trans
2. Transmission Coefficient For E field parallel to ground
For E field in plane of incidence
T !
#
E trans! E inc!
T # $
=
E trans # E inc #
2! 2 cos " trans
!2 cos "trans =
+
!1 cos " inc
2!2 cos" inc
!2 cos"inc
3. Power received under free-space propagation (Friis equation) :
+
!1 cos "trans P rec Ptr
=
Gt Gr
# ! $ % & ' 4" R (
2
4. Power received due to perfectly reflecting ground with antennas at heights ht and hr : 2
P rec P tr
=
& ' # & ht hr # Gt Gr $ ! ! $ % 4( R " % 4( R' "
2
5. Normalized diffraction parameter "
=
h
2(d 1 + d 2 ) ! d 1 d 2
6. Excess Path Loss due to diffraction from single knife edge
V. Empirical Models of Wave propagation 1. Delisle model of path loss in urban environment
4 2 " !17 r ( f /1MHz ) $4.27 #10 hb2shmo b $ L = % 4 2 $4.27 #10!16 r ( f /1MHz ) $& hb2shmo b
for hmob
<
10m
for hmob
>
10m
2. Ikegami’s model of excess path loss between two edges separated by ds :
L
=
( f /1MHz) {(ho ! hmob ) /1m}2 186 ( d s /1m)
3. Okamura – Hata model for VHF/UHF (150 MHz to 1 GHz)
L(dB) = 69.55 + 26.16 log10 ( f c /1MHz) !13.82log10 hbs ! a(hmob ) + (44.9 ! 6.55log10 hbs ) log10 r ! C Where
8.29 [log10 (1.54 hmob /1m) ! 1.1 for l arg e city and fc " 300 MHz # $ a(hmob ) = % 3.2 [log10 (11.75 hmob /1m) ! 4.97 for l arg e city and fc > 300 MHz $1.1 log [( f /1MHz) ! 0.7] ( h /1m) ! 1.56 log [( f /1MHz) ! 0.8] for small city 10 c mob 10 c & 0 ! " and C = # 5.4 + 2 [log10 ( f c / 28MHz )]2 "40.94 + 4.78 [log ( f /1MHz)]2 $ 18.33 log ( f /1MHz) 10 c 10 c %
for Urban area for Sunurban area for Open area
VI. Statistical Model of Wave Propagation 1. Rayleigh density function for received signal amplitude
$ 0 ! * + y 2 ' f Y ( y ) = # y % exp(( 2 % !, 2 2 , ) & "
y < 0 y>0
2. Exponential density function for received signal power 1 f P ( p) exp " p / 2! 2 u ( p) 2 2! 3. Rician distribution in the presence of a strong signal
(
=
)
2
2
2
2
2
f X(x) = ( x / 1R ) exp [ - (x + A )/ 21R ] . Io ( x A / 1R ) 4. Error Probability in the absence of Fading
P E
1 =
2
! S " $ N % &
erfc #
1 =
2
! E S " $ N % O&
erfc #
5. Error Probability in the presence of fading for Rayleigh-distributed signal
P E
=
1& ( S / N ) # 1 ' $ ! 2% 1 + (S / N ) "
6. Diversity Gain
G
( S / N ) for diversity system =
( S / N ) for sin gle channel
7. Doppler Shift Frequency
f D
=
f carr
v cos! c
VII. Channel Characterization
1. Given the power delay profile P( 2), Average delay
<
T D
)
!
>=
0
P (! ) "
# $ % ) P(! )d ! & '0 (
% [
!
Power delay spread
"
D
(t )
=
d !
2
(t ) $ T D (t )]
P (! ) d!
0
#
% P(
!
) d !
0
2. R.M.S. Delay spread
!
rms
or, Multipath spread T MUL
"
=
D
3. Coherence time – Doppler spread relationship:
=
2
!
D
Tcoh 3 1 / 2f D
4. Coherence Bandwidth – r.m.s. Delay Spread Relationship
Bcor
"
1 2! rms
VIII. Multiple Access and Cellular Systems
1. Minimum signal-to-interference Power Ratio (for hexagonal cells with 4-th power law)
( S % & # ' I $ min
R =
"!
( N c " 1)
[3 N R]"
!
/2
c
2. Erlang’s B formula for the probability of call blocking with N available duplex channels, as a function of total caller traffic intensity of U erlangs:
P[ Blocking ]
U N =
! U m " N ! ' # $ m 0 % m! & N
=
$ S
sig
ACI
=
( f ) | H BP ( f " #f ) |2 df
"!
!
3. Adjacent channel interference ratio :
$ S
sig
( f ) | H BP ( f ) |2 df
"!
IX. Noise and Interference
1. Power spectral density of noise at the output of a noiseless linear filter with frequency response H(jf) excited at its input with a random signal of power spectral density Sx(jf) : Sy(f) = |H(jf)| 2 Sx(f) 2. Noise bandwidth (or noise-equivalent bandwidth) of a filter with frequency response H(jf): 2
# | H ( jf ) | Beq
=
No df
"!
N o
2. Noise figure of a linear system:
! Sin / N in " $ % Sout / N out & T
F # =
n , source
3. Equivalent noise temperature Teq = (F – 1) Tref
T ref
=
or, Noise figure F = 1 + (Teq/Tref )
!
" T # 1% p & Gav ' T ref
4. Noise figure of a passive filter at temperature T p: F = 1 + $
1
5. Combined noise temperature of a cascade of n linear systems,
Or, combined noise figure,
Ftot
=
F 1 +
F 2
!1
Gav ,1
+
F 3
!1
Gav,1Gav,2
+
Ttot
....
X. Nonlinear and Intermodulation Distortion
For a memory-less nonlinear system with transfer characteristic 2
3
y = a0 + a1 x + a2 x + a3 x + ignorable higher-order terms
= T + 1
T 2 Gav,1
+
T 3 Gav,1Gav ,2
+
....
1. When excited with a harmonic signal of amplitude Vamp, Gain compression
Gnonlin Glin
=
3 a1 + 34 a3V amp
a1 2
Vamp|@2f = " a1 Vamp
Second-harmonic generation
2. When excited with two harmonic signals of frequencies f 1 and f 2, with equal amplitudes Vamp Amplitude of intermodulation signal (at each of 2f 2 ± f 1 and 2f 1 ± f 2) : VIMD = # a3 Vamp3 2 2 2 4 Intermodulation power ratio: IMPR 5 VIMD / (a1Vamp) = # (a3 / a1 ) Vamp
Third-order intercept (TOI) point, referred to the input: PTOI = Input Power (" a1Vamp) |@IMPR=1 = 2a1 / 3a3 2
3. Dynamic range
DR = [ PTOI / No ]
2/3
3
