SS Lab # 1
OBJECTIVES OF THE LAB
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Matlab will be used extensively in all the succeeding labs. The goal of this of this first lab is to gain familiarity with Matlab and build some basic skills in the Matlab language. Some specific topics covered in this lab are: • • • • • • • •
Introduction to Matlab Matlab Environment Matlab Help Variable arithmetic Built in Built in Mathematical Functions Mathematical Functions Input and Input and display display Timing functions Timing functions Introduction to M‐ files
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
1.1 WHAT IS MATLAB? MATLAB is a commercial "MATrix MATrix LABoratory" LABoratory" package, by MathWorks, MathWorks, which operates as an interactive programming environment with graphical output. The MATLAB programming language is exceptionally straightforward since almost every data object is assumed to be an array. Hence, for some areas of engineering MATLAB is displacing popular programming languages, due to its interactive interface, reliable algorithmic foundation, fully extensible environment, and computational speed.
1.2 ENTERING AND RUNNING MATLAB Double click on the MATLAB icon to launch and a command window will appear with the prompt: >> You are now in MATLAB. From this point on, individual MATLAB commands may be given at the program prompt. They will be processed when you hit the
key. The following figure shows the screenshot of matlab. of matlab.
1.3 LEAVING MATLAB A MATLAB session may be terminated by simply typing >> quit or by typing >> exit at the MATLAB prompt.
1.4 MATLAB HELP Online help is available from the MATLAB prompt, both generally (listing all available commands). >> help [a long list of help topics follows] and for specific commands: >> help command_name
If you want to search for all the commands related to some particular functionality, use the keyword lookfor followed by a keyword that explains the functionality. >> lookfor convolution will return a number of commands that perform convolution related tasks.
1.5 VARIABLES MATLAB has built‐in variables like pi, eps, and ans. You can learn their values from the MATLAB interpreter. >> eps eps = 2.2204e‐16 >> pi ans = 3.1416
1.5.1
Variable Assignment
The equality sign is used to assign values to variables: >> x = 3 x= 3 >> y = x^2 y= 9
Variables in MATLAB are case sensitive. Hence, the variables "x" and "y" are distinct from "X" and "Y" (at this point, the latter are in fact, undefined). Output can be suppressed by appending a semicolon to the command lines. >> x = 3; >> y = x^2; >> y y= 9
1.5.2
Active Variables
At any time you want to know the active variables you can use who: >> who Your variables are: ans x y
1.5.3
Removing a Variable
To remove a variable, try this: >> clear x To remove all the variables from workspace, use clear >> clear
1.5.4
Saving and Restoring Variables
To save the value of the variable "x" to a plain text file named "x.value" use >> save x.value x ‐ascii To save all variables in a file named mysession.mat, in reloadable format, use >> save mysession To restore the session, use >> load mysession
1.6 VARIABLE ARITHMETIC MATLAB uses some fairly standard notation. More than one command may be entered on a single line, if they are separated by commas. >> 2+3; >> 3*4, 4^2; Powers are performed before division and multiplication, which are done before subtraction and addition. For example >> 2+3*4^2; generates ans = 50. That is: 2+3*4^2 ==> 2 + 3*4^2 <== exponent has the highest precedence ==> 2 + 3*16 <== then multiplication operator ==> 2 + 48 <== then addition operator ==> 50
1.6.1
Double Precision Arithmetic
All arithmetic is done to double precision, which for 32‐bit machines means to about 16 decimal digits of accuracy. Normally the results will be displayed in a shorter form. >> a = sqrt(2) a= 1.4142 >> format long, b=sqrt(2) b=
1.41421356237310 >> format short
1.6.2
Command-Line Editing
The arrow keys allow "command‐line editing," which cuts down on the amount of typing required, and allows easy error correction. Press the "up" arrow, and add "/2." What will this produce? >> 2+3*4^2/2 Parentheses may be used to group terms, or to make them more readable. For example: >> (2 + 3*4^2)/2 generates ans = 25.
1.6.3
Built-In Mathematical Functions
MATLAB has a platter of built‐in functions for mathematical and scientific computations. Here is a summary of relevant functions.
Function Meaning Example ====================================================== sin si ne si n( pi ) = 0. 0 cos cosi ne cos( pi ) = 1. 0 tan t angent t an( pi / 4) = 1. 0 asin ar csi ne asi n( pi / 2) = 1. 0 acos ar ccosi ne acos( pi / 2) = 0. 0 atan ar ct angent at an( pi / 4) = 1. 0 exp exponent i al exp( 1. 0) = 2. 7183 log nat ur al l ogar i t hm l og( 2. 7183) = 1. 0 log10 l ogar i t hm base 10 l og10( 100. 0) = 2. 0 ======================================================
The arguments to trigonometric functions are given in radians.
Example: Let's verify that sin(x)^2 + cos(x)^2 = 1.0 for arbitrary x. The MATLAB code is: >> x = pi/3; >> sin(x)^2 + cos(x)^2 ‐ 1.0 ans = 0
1.7 TIMING COMMANDS Timing functions may be required to determine the time taken by a command to execute or an operation to complete. Several commands are available to accomplish it:
1.7.1
Clock
CLOCK returns Current date and time as date vector. CLOCK returns a six element date vector containing the current time and date in decimal form: CLOCK = [year month day hour minute seconds] The first five elements are integers. The second’s element is accurate to several digits beyond the decimal point. FIX(CLOCK) rounds to integer display format.
1.7.2
Etime
ETIME Elapsed time. ETIME(T1,T0) returns the time in seconds that has elapsed between vectors T1 and T0. The two vectors must be six elements long, in the format returned by CLOCK: T = [Year Month Day Hour Minute Second] Time differences over many orders of magnitude are computed accurately. The result can be thousands of seconds if T1 and T0 differ in their first five components or small fractions of seconds if the first five components are equal. t0 = clock; operation etime(clock,t0)
1.7.3
Tic Toc
TIC Start a stopwatch timer. The sequence of commands TIC, operation, TOC prints the number of seconds required for the operation.
1.8 INPUT & DISPLAY 1.8.1
INPUT
INPUT prompts for user input. R = INPUT('How many apples') gives the user the prompt in the text string and then waits for input from the keyboard. The input can be any MATLAB expression, which is evaluated, using the variables in the current workspace, and the result returned in R. If the user presses the return key without
entering anything,
INPUT returns an empty matrix.
Example: Entering a single variable >> x=input('Enter a variable: ') Enter a variable: 4 x= 4
Example: Entering a vector A vector is entered by specifying [] and elements are inserted inside these brackets, separated by space. >> x=input('Enter a vector: ') Enter a vector: [3 4 1] x= 3
4
1
Example: A \n entered after the string results in starting a new line. >> x=input('Enter a value\n') Enter a value 5 x= 5
1.8.2
DISP
DISP Display array. DISP(X) displays the array, without printing the array name. In all other ways it's the same as leaving the semicolon off an expression except that empty arrays don't display. DISP(‘string’) is another variation of the same function that is used to display a string on the command prompt. Example: >> disp('I am using MATLAB 7.0') I am using MATLAB 7.0
1.9 M-Files Typing errors are time‐consuming to fix if you are working in the command window because you need to retype all or part of the program. Even if you do not make any mistakes, all of your work may be lost if you inadvertently quit MATLAB. To preserve large sets of commands, you can store them in a special type of file called an M‐ file. MATLAB supports two types of M‐files: script and function M‐ files. To hold a large collection of commands, we use a script M‐file. The function M‐file is discussed in coming lab. The script file has a ' . m'
extension and is referred to as an M‐file (for example, myf i l e. m
myf unci on. m, etc.). The commands in the script file can then be executed by typing the file name without its extension in the command window. Commands in a script utilize and modify the contents of the current workspace. It is possible to embed comments in a script file.
To make a script M‐file, you need to open a file using the built‐in MATLAB editor. There are two ways to accomplish it: 1. From file menu, click NEW 2. Type edit on command line A new window appears like one shown in the figure below.
When you are finished with typing in this new window, click File‐>Save to save this file. The extension of this file be .m. In order to execute this program, 1. Write the name of file on command window (excluding the .m) or 2. Click Debug‐>Run
-------------------------TASK 01-------------------------Matlab stores numeric data as double‐precision floating point by default. To store data as an 8‐ bit integer, int8 (a conversion function) can be used. Type the sample code in matlab command window: >> x = 26
>> whos >> y = int8(x) >> whos What difference do you see? State your findings.
-------------------------TASK 02-------------------------Take your name in command window e.g. name = ‘Ali’. Convert it into 8‐bit integer format using int8 function.
-------------------------TASK 03-------------------------Create logical array “x” of 5 elements by entering a true (1) or false (0) value for each element.
-------------------------TASK 04-------------------------Create an m‐file to get 10 numbers from user and generate the square of those numbers.
-------------------------TASK 05-------------------------Writ a Calculator program: Take two variables i.e. x and y from user. Find their sum, difference, product, multiplication, and modulus. Show the results.
Lab # 2
OBJECTIVES OF THE LAB
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ In this lab, we will cover the following topics: • Built in Matrix Functions • Indexing Matrices • Sub Matrices • Matrix element level operations • Round Floating Point numbers to Integers
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
2.1 MATRICES MATLAB works with essentially only one kind of object, a rectangular numerical matrix possibly, with complex entries. Every MATLAB variable refers to a matrix [a number is a 1 by 1 matrix]. In some situations, 1‐by‐1 matrices are interpreted as scalars, and matrices with only one row or one column are interpreted as vectors. A matrix is a rectangular array of numbers. For example:
⎡3 ⎢1 ⎢ ⎢2 ⎢ ⎣1
6
9
2⎤
⎥ ⎥ 8 7 5⎥ ⎥ 4 2 3⎦ 4 8
5
defines a matrix with 4 rows, 4 columns, and 16 elements.
Example: Consider the following three equations:
This family of equations can be written in the form
, where
X=
Depending on the specific values of coefficients in matrices A and B, there may be: (a) no solutions to (b) a unique solution to
, or , or
(c) an infinite number of solutions to In above example, the solution matrix is
2.1.1
Defining Matrices In Matlab
MATLAB is designed to make definition of matrices and matrix manipulation as simple as possible.
Matrices can be introduced into MATLAB in several different ways. For example, either of the statements >> A = [1 2 3; 4 5 6; 7 8 9]; and >> A = [ 1 2 3 456 789] creates the obvious 3‐by‐3 matrix and assigns it to a variable A.
Note that:
•
The elements within a row of a matrix may be separated by commas as well as a blank.
•
The elements of a matrix being entered are enclosed by brackets;
•
A matrix is entered in "row‐major order" [i.e. all of the first row, then all of the second row, etc];
•
Rows are separated by a semicolon [or a newline], and the elements of the row may be separated by either a comma or a space. [Caution: Watch out for extra spaces!]
The matrix element located in the ith row and jth column of A is referred to in the usual way: >> A(1,2), A(2,3) ans = 2 ans = 6 It's very easy to modify matrices: >> A(2,3) = 10;
2.1.2
Building Matrices from a Block
Large matrices can be assembled from smaller matrix blocks. For example, with matrix A in hand, we can enter the following commands: >> C = [A; 10 11 12];
<== generates a (4x3) matrix
>> D = [A; A; A];
<== generates a (9x3) matrix
>> E = [A, A, A];
<== generates a (3x9) matrix
As with variables, use of a semicolon with matrices suppresses output. This feature can be especially useful when large matrices are being generated.
2.1.3
Built-in matrix functions
MATLAB has many types of matrices which are built into its system. For example, Function Description =============================================== diag r et ur ns di agonal M. E. as vect or eye i dent i t y mat r i x hilb Hi l ber t mat r i x magic magi c squar e ones mat r i x of ones rand r andoml y gener at ed mat r i x upper t r i angul ar par t of a mat r i x triu tril l ower t r i angul ar par t of a mat r i x zeros mat r i x of zer os ===============================================
Here are some examples: i.
Matrices of Random Entries: A 3 by 3 matrix with random entries is produced by typing >> rand(3) ans = 0.0470
0.9347
0.8310
0.6789
0.3835
0.0346
0.6793
0.5194
0.0535
General m‐by‐n matrices of random entries are generated with >> rand(m,n);
ii.
Magic Squares: A magic square is a square matrix which has equal sums along all its rows and columns. For example: >> magic(4) ans = 16
2
3
13
5
11
10
8
9
7
6
12
4
14
15
1
The elements of each of each row and column sum to 34.
iii.
iv.
Matrices of Ones: The functions
•
eye (m,n) produces an m‐by‐n matrix of ones. of ones.
•
eye (n) produces an n‐by‐n matrix of ones. of ones.
Matrices of Zeros: The commands
•
zeros (m,n) produces an m‐by‐n matrix of zeros. of zeros.
•
zeros (n) produces an n‐by‐n one;
If A If A is a matrix, then zeros (A) produces a matrix of zeros of zeros of the of the same size as A.
v.
Diagonal Matrices: If x is a vector, diag(x) is the diagonal matrix with x down the diagonal. If A If A is a square matrix, then diag(A) is a vector consisting of the of the diagonal of A. of A. What is diag(diag(A))? Try it.
2.2 MATRIX OPERATIONS The following matrix operations are available in MATLAB: Operator Description Operator Description ================================================
+ * ^
addi t i on s ub ubt r ac a c t i on mul t i pl i c at i on power power
' t r anspose spose \ l ef t di vi s i on / r i ght di vi s i on
================================================
These matrix operations apply, of course, of course, to scalars (1‐by‐1 matrices) as well. If the If the sizes of the of the matrices are incompatible for the matrix operation, an error message will result, except in the case of scalar‐ matrix operations (for addition, subtraction, and division as well as for multiplication) in which case each entry of the of the matrix is operated on by the scalar.
2.2.1
Matrix Transpose
The transpose of a matrix is the result of interchanging rows and columns. MATLAB denotes the [conjugate] transpose by following the matrix with the single‐quote [apostrophe]. For example: >> A' ans = 1
4
7
2
5
8
3
6
9
>> B = [1+i
2 + 2*i
3 ‐ 3*i]'
B= 1.0000 ‐ 1.0000i 2.0000 ‐ 2.0000i 3.0000 + 3.0000i
2.2.2
Matrix Addition/Subtraction
Let matrix "A" have m rows and n columns, and matrix "B" have p rows and q columns. The matrix sum "A + B" is defined only when m equals p and n equals q, the result is n‐by‐m matrix having the element‐by‐element sum of components of components in A and B. For example: >> E = [ 2 3; 4 5.0; 6 7]; >> F = [ 1 ‐2; 3 6.5; 10 ‐45]; >> E+F ans = 3.0000
1.0000
7.0000
11.5000
16.0000
‐38.0000
2.2.3
Matrix Multiplication
Matrix multiplication requires that the sizes match. If they If they don't, an error message is generated. >> A*B, B*A; >> B'*A; >> A*A', A'*A; >> B'*B, B*B'; Scalars multiply matrices as expected, and matrices may be added in the usual way (both are done "element by element”): >> 2*A, A/4; >> A + [b,b,b];
Example: We can use matrix multiplication to check the "magic" property of magic of magic squares. >> A = magic(5); >> b = ones(5,1); >> A*b;
<== (5x1) matrix containing row sums.
>> v = ones(1,5); >> v*A;
2.2.4
<== (1x5) matrix containing column sum.
Matrix Functions "any" and "all”
There is a function to determine if a matrix has at least one nonzero entry, any, as well as a function to determine if all if all the entries are nonzero, all. >> A = zeros(1,4) >> any(A) >> D = ones(1,4) >> any(D) >> all(A)
2.2.5
Returning more than One Value
Some MATLAB functions can return more than one value. In the case of max the interpreter returns the maximum value and also the column index where the maximum value occurs.
>> [m, i] = max(B) >> min(A) >> b = 2*ones(A) >> A*B >> A
2.2.6
Size of Matrix
Size of a matrix can be calculate by using function ‘size ‘. >> x = [1 2 3 ;1 2 3]; >> s = size(x) s= 2
2.2.7
3
Length of Array
Length of an array can be found using function ‘length’. >> n = [‐3:1:3]; >> l = length(n) l= 7
2.2.8
Finding an element in a matrix
This function can be used to find index of any particular value. Say given array is x= [0 2 4 6 8]; To find the indices of all values that are greater than 4, following is used >> y = find(x>4) y= 4
5
-------------------------TASK 01-------------------------Write a program to generate a new matrix B from the matrix A given below such that each column in the new matrix except the first one is the result of subtraction of that column from
the previous one i.e. 2nd new column is the result of subtraction of 2nd column and 1st column and so on. Copy the first column as it is in the new matrix.
⎡3 6 9 ⎤ ⎢ ⎥ A = 1 4 8 ⎢ ⎥ ⎢⎣2 8 7 ⎥⎦ -------------------------TASK 02-------------------------Generate two 5000 sampled random discrete time signals (1 dimensional) using rand() function i.e. rand(1, 5000). Write a program to add the two signals together using simple vector addition.
2.3 SUB-MATRICES 2.3.1
A note on colon notation
A central part of the MATLAB language syntax is the "colon operator," which produces a list. For example: >> x = ‐3:3 x=
‐3 ‐2 ‐1 0 1 2 3
The default increment is by 1, but that can be changed. For example: >> x = ‐3 : .5 : 3 x= Columns 1 through 7
‐3.0000 ‐2.5000 ‐2.0000 ‐1.5000 ‐1.0000 ‐0.5000
0
Columns 8 through 13 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000
This can be read: "x is the name of the list, which begins at ‐3, and whose entries increase by .5, until 3 is surpassed." You may think of x as a list, a vector, or a matrix, whichever you like.
In our third example, the following statements generate a table of sine. >> x = [0.0:0.1:2.0]'; >> y = sin(x); >> [x y] Try it. Note that since sin operates entry‐wise, it produces a vector y from the vector x.
The colon notation can also be combined with the earlier method of constructing matrices. >> a = [1:6 ; 2:7 ; 4:9]
-------------------------TASK 03-------------------------Using colon notation, generate the following sequence:
‐120, ‐116, ‐112, . . . , ‐4, 0, 4, 8, . . ., 112, 116, 120.
2.3.2
Sub-matrix extraction using colon notation
Colon notation can be used to generate vectors. A very common use of the colon notation is to extract rows, or columns, as a sort of "wild‐card" operator which produces a default list. For example,
•
A(1:4,3)
is the column vector consisting of the first four entries of the third column of A
•
A(:,3)
is the third column of A. A colon by itself denotes an entire row or column
•
A(1:4,:)
is the first four rows of A.
An arbitrary integral vector can be used as subscripts. The statement A(:,[2 4]) contains as columns, columns 2 and 4 of matrix A.
This subscripting scheme can be used on both sides of an assignment statement: A(:,[2 4 5]) = B(:,1:3) It replaces columns 2, 4, and 5 of matrix A with the first three columns of matrix B. Note that the "entire" altered matrix A is printed and assigned. Try it.
-------------------------TASK 04-------------------------Given the matrices:
⎡− 7 16⎤ A = ⎢ , ⎥ ⎣ 4 9⎦
⎡ 6 − 5⎤ B = ⎢ ⎥ − 12 2 ⎣ ⎦
Find the following: 1) Array addition; store the result in matrix C 2) Array subtraction; store the result in matrix D 3) Array multiplication using .* operator; store the result in matrix E 4) Array division using ./ operator; store the result in matrix F 5) Array exponentiation using .^ operator; store the result in matrix G
-------------------------TASK 05-------------------------Type the given matrix in matlab:
⎡ 3 ⎢− 5 A = ⎢ ⎢ 6 ⎢ ⎣15
7 − 4 12 ⎤ 9 10 13 8 5
4
⎥ ⎥ 11⎥ ⎥ 1 ⎦ 2
Find the following: 1) Create 4x3 array B consisting of all elements in the second through fourth columns of A 2) Create 3x4 array C consisting of all elements in the second through fourth rows of A 3) Create 2x3 array D consisting of all elements in the first two rows and the last three columns of A
-------------------------TASK 06-------------------------MATLAB has functions to round floating point numbers to integers. These are round, fix, ceil, and floor. Test how these functions work. Determine the output of the following: >> f = [‐.5 .1 .5]; >> round(f) >> fix(f) >> ceil(f)
>> floor(f)
-------------------------TASK 07-------------------------Given the following matrix:
⎡− 3 5⎤ A = ⎢ ⎥ ⎣ 4 8⎦ Find the following: 1) Column‐wise sum of all elements of A using sum function; for information about sum function, type help sum in matlab 2) Column‐wise product of all elements of A using prod function; for information about prod function, type help prod in matlab 3) Length of matrix A 4) Size of matrix A
Lab # 3
OBJECTIVES OF THE LAB ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ In this lab, we will get an understanding of the following topics:
• • • • • •
Making Functions Control Structures Relational Constructs Logical Constructs Branching Constructs Looping constructs
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
3.1 MAKING FUNCTIONS A function can be created by the following syntax: function [output1,output2,...] = function_name(input1,input2,...)
A function is a reusable piece of code that can be called from program to accomplish some specified functionality. A function takes some input arguments and returns some output. To create a function that adds two numbers and stores the result in a third variable, type in it the following code:
function add x = 3; y = 5; z = x + y Save the file by the name of add (in work folder, which is chosen by default), go back to the command window and write >> add z= 8 You see that the sum z is displayed in the command window. Now go back to the editor/debugger and modify the program as follows
function addv(x,y) z = x + y Save the above program with a new name addv, go back to the command window and type the following: >> addv(3, 5) z= 8 >> addv(5, 5) z= 10 We have actually created a function of our own and called it in the main program and gave values to the variables (x, y).
Now go back to the editor/debugger and modify the program as follows
function adv(x, y)
%------------------------------------------------% This function takes two values as input, % finds its sum, & displays the result. % inputs: x & y % output: z % Example: addv(3,6) % Result: z=9 %-------------------------------------------------z = x + y
Save the program with the same name addv, go back to command window, type the following >> help addv
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ This function takes two values as input, finds its sum, & displays the result. inputs: x & y output: z Example: addv(3, 6) Result: z=9
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 3.1.1
Script Vs Function
1) A script is simply a collection of Matlab commands in an m‐file. Upon typing the name of the file (without the extension), those commands are executed as if they had been entered at the keyboard. Functions are used to create user‐defined matlab commands. 2) A script can have any name. A function file is stored with the name specified after keyword function. 3) The commands in the script can refer to the variables already defined in Matlab, which are said to be in the global workspace. When a function is invoked, Matlab creates a local workspace. The commands in the function cannot refer to variables from the global (interactive) workspace unless they are passed as inputs. By the same token, variables created as the function executes are erased when the execution of the function ends, unless they are passed back as outputs.
-------------------------TASK 01-------------------------Write a function that accepts temperature in degrees F and computes the corresponding value in degrees C. The relation between the two is
T c =
5 9
(T c − 32 )
Be sure to test your function.
3.2 CONTROL STRUCTURES Control‐of ‐flow in MATLAB programs is achieved with logical/relational constructs, branching constructs, and a variety of looping constructs.
3.2.1
Relational and logical constructs
The relational operators in MATLAB are Operator Description ===================================
< > <= >= == ~=
l ess t han gr eat er t han l ess t han or equal gr eat er t han or equal equal not equal
===================================
Note that ''='' is used in an assignment statement while ''=='' is used in a relation. Relations may be connected or quantified by the logical operators Operator Description ===================================
& | ~
and or not
===================================
When applied to scalars, a relation is actually the scalar 1 or 0 depending on whether the relation is true or false (indeed, throughout this section you should think of 1 as true and 0 as false). For example >> 3 < 5 ans = 1
>> a = (3 == 5) a= 0
When logical operands are applied to matrices of the same size, a relation is a matrix of 0's and 1's giving the value of the relation between corresponding entries. For example: >> A = [ 1 2; 3 4 ]; >> B = [ 6 7; 8 9 ]; >> A == B ans = 0
0
0
0 >> A < B
ans = 1
1
1
1
To see how the other logical operators work, you should also try >> ~A >> A&B >> A & ~B >> A | B >> A | ~A
-------------------------TASK 02-------------------------For the arrays x and y given below, write matlab code to find all the elements in x that are greater than the corresponding elements in y. x = [‐3, 0, 0, 2, 6, 8] y = [‐5, ‐2, 0, 3, 4, 10]
3.2.2
Branching constructs
MATLAB provides a number of language constructs for branching a program's control of flow. i.
if ‐end Construct : The most basic construct is:
if end Here the condition is a logical expression that will evaluate to either true or false (i.e., with values 1 or 0). When a logical expression evaluates to 0, program control moves on to the next program construction. You should keep in mind that MATLAB regards A==B and A<=B as functions with values 0 or 1.
Example: >> a = 1; >> b = 2; >> if a < b c = 3; end >> c c= 3
ii.
If ‐else‐end Construct: Frequently, this construction is elaborated with
if else end In this case if condition is 0, then program2 is executed. iii.
If-elseif-end Construct: Another variation is if elseif end
Now if condition1 is not 0, then program1 is executed, if condition1 is 0 and if condition2 is not 0, then program2 is executed, and otherwise control is passed on to the next construction.
-------------------------TASK 03-------------------------For 0 < a
≤ 16,
C=
find the values of C defined as follows: 4ab
for
1
≤a ≤8
ab
for
8
≤ 16
and b = 12.
-------------------------TASK 04-------------------------For the values of integer a going from 1 to 10, using separately the methods of if syntax and the Boolean alternative expressions, find the values of C if: 2
C=
a a+3 a
for for for
a<3 3 ≤a < 7 a>7
-------------------------TASK 05-------------------------Rewrite the following statements to use only one if statement. if x < y if z < 5 w = x*y*z end end
3.2.3 i.
Looping constructs For Loop : A for loop is a construction of the form for i= 1 : n end
This will repeat once for each index value i = 1, 2 .... n. Here are some examples of MATLAB's for loop capabilities:
Example: The basic for loop >> for i = 1 : 5 c = 2*i end c= 2 ..... lines of output removed ... c= 10 computes and prints "c = 2*i" for i = 1, 2, ... 5.
Example: For looping constructs may be nested. Here is an example of creating a matrix content inside a nested for loop: >> for i = 1:10 for j = 1:10 A(i, j) = i / j; end end There are actually two loops here, with one nested inside the other; they define A(1,1), A(1,2), A(1,3) ... A(1,10), A(2,1), A(2,2) ... A(2,10), ... A(10,1), A(10,2) ... A(10,10) in that order.
Example: MATLAB will allow you to put any vector in place of the vector 1:n in this construction. Thus the construction >> for i = [2,4,5,6,10] end
is perfectly legitimate. In this case program will execute 5 times and the values for the variable i during execution are successively, 2, 4, 5, 6, 10.
-------------------------TASK 06-------------------------Using for loop, generate the cube of the first ten integers.
-------------------------TASK 07-------------------------Add the following two matrices using for loop.
⎡5 ⎢ A = 9 ⎢ ⎢⎣2
12 6 2
⎡2 ⎢ B = 10 ⎢ ⎢⎣ 3
3⎤
⎥ 5 ⎥ 1⎥ ⎦
1
9⎤
5
6
4
⎥ ⎥ 2⎥ ⎦
-------------------------TASK 08-------------------------Write matlab function that creates a special square matrix that has ones in the first row and first column, and whose remaining elements are the sum of two elements i.e. the element above and the element to the left, if the sum is less than 20. Otherwise, the element is the maximum of those two element values. Name the function specmat with single input defining the matrix dimension. A sample program run is: >>specmat(3) ans =
⎡1 ⎢1 ⎢ ⎢⎣1
ii.
1
1⎤
2
3
3
⎥ ⎥ 6⎥ ⎦
While Loops
A while loop is a construction of the form
while end where condition is a MATLAB function, as with the branching construction. The program will execute successively as long as the value of condition is not 0. While loops carry an implicit danger in that there is no guarantee in general that you will exit a while loop. Here is a sample program using a while loop.
Example: function l = twolog(n) % l = twolog(n). l is the floor of the base 2 % logarithm of n. l = 0;
m = 2; while m<=n l=l+1; m=2*m; end
-------------------------TASK 09-------------------------Consider the following script file. Fill in the lines of the following table with the values that would be displayed immediately after the while statement if you ran the script file. Write in the values the variables have each time the while statement is executed. You might need more or fewer lines in the table. Then type in the file, and run it to check your answers. k = 1; b = -2; x = -1; y = -2; while k <= 3 k, b, x, y
Pass
y = x^2 -3;
First
if y < b b = y; end x = x + 1; k = k + 1;
k
b
x
y
Second Third Fourth Fifth
end
-------------------------TASK 10-------------------------Create an m‐file that inputs a number from user and then finds out the factorial of that number.
-------------------------TASK 11-------------------------Create an m‐file that takes two vectors from user. Make sure that the second vector taken is of the same size as the first vector (Hint: use while loop). In a while loop, generate a third vector that contains the sum of the squares of corresponding entries of both the vectors.
Lab # 4
OBJECTIVES OF THE LAB
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ This lab will help you grasp the following concepts: Discrete Signal representation in Matlab Matlab Graphics Two Dimensional Plots Plot and subplot Different Plotting Functions Used in Matlab • • • • •
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
4.1 DISCRETE-TIME SIGNAL REPRESENTATION IN MATLAB In MATLAB, finite‐duration sequence (or discrete time signal) is represented by row matrix/vector of appropriate values. Such representation does not have any information about sample position n. Therefore, for correct representation, two vectors are required, one for x and other for n. Consider the following finite duration sequence & its implementation: x(n) = { 1 ‐1 0 2 1 4 6 } ↑ >> n = [‐3:1:3] n= ‐3
‐2
‐1
0
1
2
3
>> x = [1 ‐1 0 2 1 4 6] x= 1
‐1
0
2
1
4
6
NOTE # 01: When the sequence begins at n=0, x‐vector representation alone is enough. NOTE # 02: An arbitrary infinite‐sequence can’t be represented in MATLAB due to limited memory.
-------------------------TASK 01-------------------------Given the signals: x1[n] = [2 5 8 4 3] x2[n] = [4 3 2] a) Write a Matlab program that adds these two signals. Use vector addition and multiplication. b) Instead of using vector addition and multiplication, use for loop to add and multiply the
signals.
-------------------------TASK 02-------------------------Amplitude scaling by a factor β causes each sample to get multiplied by β. Write a user‐defined function that has two input arguments: (i) a signal to be scaled and (ii) scaling factor β. The function should return the scaled output to the calling program. In the calling program, get the discrete time signal as well as the scaling factor from user and then call the above‐mentioned function.
-------------------------TASK 03-------------------------X2[n] X1[n]
2
2
1
2 1
0
1
2
3
3
2
1
4
1
0 1
2
3
4
Write a Matlab program to compare the signals x1[n] and x2[n]. Determine the index where a sample of x1[n] has smaller amplitude as compared to the corresponding sample of x2[n]. Use for loop.
4.2 GRAPHICS Two‐ and three‐dimensional MATLAB graphs can be given titles, have their axes labeled, and have text placed within the graph. The basic functions are: Funct i on Descr i pt i on ============================================================================ pl ot ( x, y) pl ot s y vs x pl ot ( x, y1, x, y2, x, y3) pl ot s y1, y2 and y3 vs x on t he same gr aph st em( x) pl ot s x and dr aws a ver t i cal l i ne at each dat apoi nt t o t he hor i zont al axi s xl abel ( ' x axi s l abel ' ) l abel s x axi s yl abel ( ' y axi s l abel ' ) l abel s y axi s t i t l e ( ' t i t l e of pl ot ' ) put s a t i t l e on t he pl ot gt ext ( ' t ext ' ) act i vat es t he use of t he mouse t o posi t i on a cr osshai r on t he gr aph, at whi ch poi nt t he ' t ext' wi l l be pl aced when any key i s pr essed. zoom al l ows zoom I N/ OUT usi ng t he mouse curs or gr i d dr aws a gr i d on t he graph area pr i nt f i l ename. ps saves t he pl ot as a bl ack and whi t e post scr i pt fi l e Shg br i ngs t he cur r ent f i gur e wi ndow f orwar d. CLF cl ear s cur r ent f i gur e. ============================================================================
4.2.1
Two-dimensional plots
The plot command creates linear x‐y plots; if x and y are vectors of the same length, the command plot(x,y) opens a graphics window and draws an x‐y plot of the elements of x versus the elements of y.
Example: Let's draw the graph of the sine function over the interval ‐4 to 4 with the following
commands:
>> x = ‐4:.01:4; y = sin(x); plot(x,y) >> grid >> xlabel('x') >> ylabel('sin(x)') The vector x is a partition of the domain with meshsize 0.01 while y is a vector giving the values of sine at the nodes of this partition (recall that sin operates entrywise). Following figure shows the result.
4.2.1.1 MULTIPLE PLOTS ON SAME FIGURE WINDOW Two ways to make multiple plots on a single graph are: i.
Single plot command
x = 0:.01:2*pi; y1=sin(x); y2=sin(2*x); y3 = sin(4*x); plot(x,y1,x,y2,x,y3) ii.
Multiple plot commands
Another way is with hold. The command hold freezes the current graphics screen so that subsequent plots are superimposed on it. Entering hold again releases the ``hold.''
x = 0:.01:2*pi;
y1=sin(x); y2=sin(2*x); y3 = sin(4*x); plot(x,y1); hold on; plot(x, y2); plot(x,y3); 4.2.1.2 OVERRIDING THE DEFAULT PLOT SETTINGS One can override the default linetypes and pointtypes. For example, the command sequence
x = 0:.01:2*pi; y1=sin(x); y2=sin(2*x); y3=sin(4*x); plot(x,y1,'--',x,y2,':',x,y3,'+') grid title ('Dashed line and dotted line graph')
The line‐type and mark‐type are ============================================================= L i net ypes : s ol i d ( - ) , das hed ( - ) . dot t ed ( : ) , das hdot ( - . ) Mar k t ypes : poi nt ( . ) , pl us ( +) , s t ar ( * , ci r c l e ( o) , x- mar k ( x) =============================================================
-------------------------TASK 04-------------------------Plot the two curves y1 = 2x + 3 and y2 = 4x + 3 on the same graph using different plot styles.
4.2.1.3 AXES COMMANDS (MANUAL ZOOMING) MATLAB automatically adjusts the scale on a graph to accommodate the coordinates of the points being plotted. The axis scaling can be manually enforced by using the command axis([xmin xmax ymin ymax]). A signal can be zoomed out by specifying the axis coordinates by
user himself. Example:
x = -5*pi:.01:5*pi; y1=sin(x); plot(x,y1, 'r') The plot is shown in the figure below.
In order to see only one cycle of this signal from 0 to 2π, the signal is zoomed using axis command. Here we have specified xmin and xmax as 0 and 2π respectively.
x = -5*pi:0.01:5*pi; y1=sin(x); plot(x,y1, 'r') axis([0 2*pi -1 1]) The magnified plot is shown in the figure below.
Similarly the y‐axis can be adjusted according to requirements.
x = -5*pi:0.01:5*pi; y1=sin(x); plot(x,y1, 'r') axis([0 2*pi -2 2])
4.2.1.4 LABELING A GRAPH To add labels to your graph, the functions xlabel, ylabel, and title can be used as follows: xlabel('x‐ xlabel('x‐axis') ylabel('y‐ ylabel('y‐axis') title('points in a plane')
4.2.1.5 SUBPLOT SUBPLOT Create axes in tiled positions. MATLAB graphics windows will contain one plot by default. The command subplot can be used to partition the screen so that up to four plots can be viewed simultaneously. A single figure can be divided into a number of plotting areas where different graphs can be plotted. This can be accomplished by using the command subplot(m, n, p) where m, n specifies the total number of rows and columns respectively in the figure window and p specifies the specific cell to plot into.
x=0:1:10; y=x.^2; z=10*x;
Now type the following code
figure subplot (1,2,1) plot(x,y) subplot (1,2,2) plot(x,z)
In the above case subplot(m,n,p) command was used, in our case subplot (1,2,1) and subplot (1,2,2). Here m=1 means that divide the figure into 1 row, n=2 means to divide the figure into 2 columns. This gives us a total of 2 subplots in one figure. Where p=1 means the window on the left (starting from row 1 and counting p=1 subplots to the right) and p=2 means the subplot on the right (starting from row 1 and counting p=2 subplots to the right). Example: Performing operations on signals entered by user
clear x=input('Enter the first discrete time signal\n'); len_x=length(x); y=input('Enter the second discrete time signal\n'); len_y=length(y);
while(len_y~=len_x) disp('Error: Length of signals must match. Enter the 2nd signal again') y=input(''); len_y=length(y); end z=x+y; subplot(3,1,1) stem(x,'filled') title('Signal 1') xlabel('Sample number') ylabel('Signal Amplitude') subplot(3,1,2) stem(y,'filled') title('Signal 2') xlabel('Sample number') ylabel('Signal Amplitude') subplot(3,1,3) stem(z,'filled') title('Resultant title('Resultant Signal') xlabel('Sample number') ylabel('Signal Amplitude')
output:
Enter the first discrete time signal [3 5 1 0 2] Enter the second discrete time signal [1 1 3 2 1]
-------------------------TASK 05-------------------------Make two separate functions for signal addition and multiplication. The functions should take the signals as input arguments and return the resultant signal. In the main program, get the signals from user, call the functions for signal addition and multiplication, and plot the original signals as well as the resultant signals.
-------------------------TASK 06-------------------------Given the signals: X1[n] = 2δ[n] + 5δ[n‐1] + 8δ[n‐2] + 4δ[n‐3] + 3δ[n‐4] X2[n] = δ[n‐4] + 4δ[n‐5] +3δ[n‐6] + 2δ[n‐7] Write a Matlab program that adds these two signals. Plot the original signals as well as the final result.
-------------------------TASK 07-------------------------Take a discrete‐time signal from user. Count the number of samples with amplitude greater than a threshold of 3 and less than a threshold of ‐3 (use for loop).
-------------------------TASK 08-------------------------Write your own function to downsample a signal i.e. retain odd numbered samples of the original signal and discard the even‐numbered (downsampling by 2). The function must take a signal as input and return the downsampeled version of that signal. See Fig for example. Call this function from a matlab file. Verify your result by using the command “downsample ”. Plot
the original signal, downsampled signal determined by your program, and downsampled signal obtained by the command downsample.
Fig. DownSampling
-------------------------TASK 09-------------------------Write your own function to upsample a signal i.e. copy the 1st sample of original signal in the new signal and then place an extra sample of 0, copy the 2nd sample of original signal and then place a 0, and so on. See Fig for example. Call this function from a matlab file. Verify your result by using the command “upsample ”. Plot the original signal, upsampled signal determined by your program, and upsampled signal obtained by the command upsample .
Fig. Upsampling
Lab # 5
OBJECTIVES OF THE LAB ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ In this lab, we will cover the following topics: • • •
Gain familiarity with Complex Numbers and plot them Complex exponential signals Real exponential signals
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
5.1 COMPLEX NUMBERS A complex number z is an ordered pair (x, y) of real numbers. Complex numbers can be represented in rectangular form as z = x + iy, which is the vector in two‐dimensional plane. The horizontal coordinate x is called the real part of z and can be represented as x = Re {z}, while the vertical coordinate y is called the imaginary part of z and represented as y = Imag {z}. That is: z = (x, y) = x + iy = Re {x} + i Imag {x} Another way to represent a complex number is in polar form. In polar form, the vector is defined by its length (r) or magnitude (|z|) and its direction (θ). A rectangular form can be converted into polar form using formulas: |z| = r = (x2 + y2)½ θ
= arctan (y/x)
z = r e j
θ
where e j = cos θ
θ+
i sin θ, and known as the Euler’s formula.
5.2 BUILT-IN MATRIX FUNCTIONS Function Description ================================================== real returns the real part x of z returns the imaginary part y of z imag abs returns the length r of z angle returns the direction θ of z returns the complex conjugate ẕ of z conj ==================================================
Here are some examples: Example To define the complex number, for instance, z = (3, 4) in matlab write in matlab editor >> z = 3 + 4i z= 3.0000 + 4.0000i
Example To find the real and imaginary parts of the complex number, write >> x = real(z) x= 3 >> y = imag(z) y= 4
Example To find the length and direction of z, write >> r = abs(z) r= 5 >> θ
θ=
angle(z)
= 0.9273
Example To find the conjugate of z, write >> zx = conj(z) zx = 3.0000 – 4.0000i
-------------------------TASK 01-------------------------Write matlab function zprint, which takes a complex number and returns it real part, imaginary part, magnitude, phase in radians, and phase in degrees. A sample run of program is: >> zprint(z) Z=
X 3
+
jY 4
Magnitude Phase Ph(deg) 5
0.927
53.13
-------------------------TASK 02-------------------------Compute the conjugate
ź (i.e.
z_conj [give variable name]) and the inverse 1/z (i.e. z_inv [give
variable name]) for any complex number z. Display the results numerically with zprint.
-------------------------TASK 03-------------------------Take two complex number and compute z1 +z2 and display the results numerically using zprint.
-------------------------TASK 04-------------------------Take two complex numbers and compute z1z2 and z1/z2. Use zprint to display the results numerically.
5.3 COMPLEX EXPONENTIAL SIGNALS The complex exponential signal is defined as x’(t) = A e j(w0t + ø) which is a complex‐valued function of t, where the magnitude of x’(t) is |x’(t)| = A arg x’(t) = (w0t + ø)
magnitude or length of x’(t) angle or direction of x’(t)
Using Euler’s formula, it can be expressed in rectangular or Cartesian form, i.e. x’(t) = A e j(w0t + ø) = A cos (w0t + ø) + j A sin (w0t + ø) where A = amplitude, ø=phase shift w0 = frequency in rad/sec
Example clear, close all, clc n=0:1/10:10; k=5; a=pi/2; x=k * exp(a*n*i); % plot the real part subplot(2,1,1) stem(n, real(x), 'filled')
title('Real part of complex exp') xlabel('sample #') ylabel('signal amplitude') grid % plot the imaginary part subplot(2,1,2) stem(n, imag(x), 'filled') title('Imaginary part of complex exp') xlabel('sample #') ylabel('signal amplitude') grid
-------------------------TASK 05-------------------------Determine the complex conjugate of the exponential signal given in above example and plot its real and imaginary portions.
-------------------------TASK 06-------------------------Generate the complex valued signal y(n) = exp
(-0.2 + j0.5)n
,
‐10≤n≤10
and plot its magnitude, phase, the real part, and the imaginary part in separate subplots.
-------------------------TASK 07-------------------------n
a) Generate a real‐exponential x=a for a=0.7 and n ranging from 0‐10. Find the discrete time as well as the continuous time version of this signal. Plot the two signals on same graph (holding both the graphs). b) Repeat the same program with value of a=1.3.
-------------------------TASK 08-------------------------(i*n*pi/4)
n
Multiply the two discrete signals x1=5exp and x2= a (use point‐by‐point multiplication of the two signals). Plot the real as well as the exponential parts for 01.
-------------------------TASK 09-------------------------Plot the discrete signal x=a^|n| for n ranging from ‐10 to 10. Draw two subplots for 01.
-------------------------TASK 10------------------------- j(ωt + π)
a) Generate the signal x(t) = Ae for A = 3, π= ‐0.4, and ω = 2π(1250). Take a range for t that will cover 2 or 3 periods. b) Plot the real part versus t and the imaginary part versus t. Use subplot(2,1,i) to put both plots in the same window. c) Verify that the real and imaginary parts are sinusoids and that they have the correct frequency, phase, and amplitude.
Lab # 6
OBJECTIVES OF THE LAB ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ In this lab, we will cover the following topics: • • •
Generating Sinusoids Addition of Sinusoids with Variation in Parameters and their Plots Linear Phase Shift Concept When Dealing With Sum of Sinusoids
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
6.1
GENERATING SINUSOIDS
Sinusoidal sequences are implemented using sin() & cos() functions.
Example: Continuous ‐Time Sinusoid clc; clear all; close all; f0 = 3; A = 5; t = -1:0.005:1; y = A*cos(2*pi*f0*t); figure, plot(t, y,'*:'); xlabel('Time, sec'), ylabel('Amplitude'); title('Graph of sinusoid');
Program: Discrete‐Time Sinusoid clc; clear all; close all; M=10;
%samples/sec
n=-3:1/M:3; A=2; phase=0; f=1; x=A * sin(2*pi*f*n + phase); stem(n,x,'linewidth', 2) title('Discrete-Time Sine Wave: A sin(2*\pi*f*n + \phi)') xlabel('Time Index') ylabel('Signal Amplitude') axis([n(1) n(end) -A A]) grid
6.2
CREATING PHASE SHIFT
Phase shift can be created by adding an angle to 2πft for a sinusoid.
Example clc; clear all; close all; fs=1000; t=-3:1/fs:3; A=2; phase=0; f=1; x=A * sin(2*pi*f*t + phase); plot(t,x, 'linewidth', 2) title('Continuous-Time Sine Wave: A sin(2*\pi*f*t + \phi)') xlabel('Time Index') ylabel('Signal Amplitude') axis([t(1) t(end) -A A]) grid
-------------------------TASK 01-------------------------Generate the 1x10 row vector v whose i ‐th component is cos (iπ/4).
-------------------------TASK 02-------------------------Write matlab code that draw graphs of sin (nπx) on the interval ‐1≤x≤1 for n = 1, 2, 3, …, 8. (Hint: Use for loop)
-------------------------TASK 03-------------------------Given the signal exp(‐x)sin(8x) for 0≤x≤2π, plot its continuous‐time and discrete‐time representations. Use subplot and label properly.
-------------------------TASK 04-------------------------Modify the example given in topic 6.2 to generate a sine wave with phase shift of +pi/2. Then plot a cosine wave of same frequency, amplitude, and phase shift of 0 in another subplot. Compare both the signals and determine the relationship between the two.
-------------------------TASK 05-------------------------Write a program to generate a continuous ‐time sine wave of frequency 3 Hz, positive phase shift of pi/2, and amplitude of 5. Also generate a continuous‐time cosine wave of frequency 3 Hz, amplitude of 5, and phase shift of 0. Plot the two signals on separate subplots and properly label them. Determine the relationship between the two signals.
6.3
ADDITION OF SINUSOIDS 6.3.1
CASE 1: When Frequency, Phases, and amplitude of the sinusoids are same clc ; clear all; close all; t=-2:0.01:2; x1=cos(2*pi*0.5*t); x2=cos(2*pi*0.5*t); x3=x1+x2; subplot(3,1,1); plot(t,x1,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time');
title('COS WAVE , AMPLITUDE = 1, FREQ = 0.5 HZ, Phase = 0 RADIAN'); subplot(3,1,2); plot(t,x2,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('COS WAVE , AMPLITUDE = 1, FREQ = 0.5 HZ, Phase= 0 RADIAN'); subplot(3,1,3); plot(t,x3,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('SUM OF THE ABOVE TWO COSINE SIGNALS');
6.3.2 CASE 2: When Frequencies and Phases of the sinusoids are same but Amplitudes are different. t=-2:0.01:2;
x1=2*cos(2*pi*0.5*t); x2=cos(2*pi*0.5*t); x3=x1+x2; subplot(3,1,1); plot(t,x1,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('COS WAVE , AMPLITUDE = 2, FREQ = 0.5 HZ, Phase = 0 RADIAN'); subplot(3,1,2); plot(t,x2,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('COS WAVE , AMPLITUDE = 1, FREQ = 0.5 HZ, Phase= 0 RADIAN'); subplot(3,1,3); plot(t,x3,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('SUM OF THE ABOVE TWO COSINE SIGNALS');
6.3.3 CASE 3: When Amplitudes and Phases of the sinusoids are the same but Frequencies are different. t=-2:0.01:2; x1=cos(2*pi*0.5*t); x2=cos(2*pi*1*t); x3=x1+x2; subplot(3,1,1); plot(t,x1,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('COS WAVE , AMPLITUDE = 1, FREQ = 0.5 HZ, Phase = 0 RADIAN'); subplot(3,1,2); plot(t,x2,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time');
title('COS WAVE , AMPLITUDE = 1, FREQ = 1 HZ, Phase = 0 RADIAN'); subplot(3,1,3); plot(t,x3,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('SUM OF THE ABOVE TWO COSINE SIGNALS');
6.3.4 CASE 4: When Amplitudes and Frequencies of the sinusoids are the same but Phases are different t=-2:0.01:2; x1=cos(2*pi*0.5*t); x2=cos((2*pi*0.5*t)+1); x3=x1+x2; subplot(3,1,1); plot(t,x1,'linewidth',3); grid; ylabel('Amplitude');
xlabel('Time'); title('COS WAVE , AMPLITUDE = 1, FREQ = 0.5 HZ, Phase = 0 RADIAN'); subplot(3,1,2); plot(t,x2,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('COS WAVE , AMPLITUDE = 1, FREQ = 0.5 HZ, Phase = 1 RADIAN'); subplot(3,1,3); plot(t,x3,'linewidth',3); grid; ylabel('Amplitude'); xlabel('Time'); title('SUM OF THE ABOVE TWO COSINE SIGNALS');
-------------------------TASK 06-------------------------Write a general program that takes ‘n’ sinusoids from user of same frequency, amplitude, and phase. Plot the individual sinusoids & the resultant using subplot function on same figure. Do perform proper labeling. Note: Take the amplitude, frequency, and phase given in example of case 1. Run the code for different values of n and state the result on paper.
-------------------------TASK 07-------------------------Write a general program that takes ‘n’ sinusoids from user of same frequency and phase with varying amplitudes. Take amplitude from user on run time. Plot the individual sinusoids & the resultant using subplot function on same figure. Do perform proper labeling. Note: Take the amplitude and frequency given in example of case 2. Run the code for different values of n and state the result on paper.
-------------------------TASK 08-------------------------Write a general program that takes ‘n’ sinusoids from user of same amplitude and phase with varying frequencies. Take each frequency from user on run time. Plot the individual sinusoids & the resultant using subplot function on same figure. Do perform proper labeling. Note: Take the amplitude and phase given in example of case 3. Run the code for different values of n and state the result on paper.
-------------------------TASK 09-------------------------Write a general program that takes ‘n’ sinusoids from user of same amplitude and frequency with varying phases. Take each phase from user on run time. Plot the individual sinusoids & the resultant using subplot function on same figure. Do perform proper labeling. Note: Take the amplitude and frequency given in example of case 4. Run the code for different values of n and state the result on paper.
Lab # 7
OBJECTIVES OF THE LAB -----------------------------------------------------------------------------------------------------------This lab aims at the understanding of:
Generating unit impulse and unit step sequences Basic signal operations
------------------------------------------------------------------------------------------------------------
7.1 GENERATING UNIT IMPULSE AND UNIT STEP SEQUENCES Use matlab commands zeros and ones to generate unit impulse and unit step sequences. Example: Unit Impulse Sequence n=‐10:10; % unit impulse x1=[zeros(1,10) 1 zeros(1,10)]; stem(n,x1,'filled'); xlabel('sample #'); ylabel('signal amplitude'); title('Unit impulse'); axis([‐10 10 ‐1 2]);
Example: Unit Step Sequence n= ‐10:10; %unit step x1=[zeros(1,10) ones(1,11)]; stem(n,x1,'filled'); xlabel('sample #'); ylabel('signal amplitude'); title('Unit step'); axis([‐10 10 ‐1 2]);
Using ones function, plot the signum sequence over interval ‐10≤n≤10. It can be defined as:
1, 0 1,0, 00
Prove the following:
1
7.2 BASIC SIGNAL OPERATIONS 1) Signal Shifting clc clear all close all n=0:0.002:4; x=sin(2*pi*1*n); subplot(2,1,1); plot(n,x,'linewidth',2); title('Original Signal'); xlabel('Time'); ylabel('Signal Amplitude'); axis([‐3 4 ‐1 1]); grid; subplot(2,1,2); plot(n‐1,x,'linewidth',2); title('Advanced signal'); xlabel('Time'); ylabel('Signal Amplitude');
axis([‐3 4 ‐1 1]); grid;
Delay the original signal given in above example by 1 sec. Plot both the delayed & original signal on the same figure.
2) Signal Flipping clear n=‐1:1/1000:1; x1=5*sin(2*pi*1*n); subplot(2,1,1); plot(n,x1, 'g', 'linewidth',2); axis([‐1 1 ‐5 5]); xlabel('time'); ylabel('signal amplitude');
title('Original sine wave'); grid; subplot(2,1,2); plot(‐n,x1, 'r', 'linewidth',2); axis([‐1 1 ‐5 5]); xlabel('time'); ylabel('signal amplitude'); title('Flipped sine wave'); grid;
Flip the following signal:
5 exp ∗ ∗4
Plot the original signal as well as the flipped one in the same figure.
Flip the following signal: x[n]= 2δ[n]+ 5δ[n‐1] + 8δ[n‐2] + 4δ[n‐3] + 3δ[n‐4] Plot the original signal as well as the flipped one in the same figure.
3) Amplitude Scaling clear n=1:7; x=[1 2 2 3 2 2 1];
subplot(2,1,1); stem(n,x, 'filled'); title('Original signal'); xlabel('Time index'); ylabel('Signal Amplitude'); axis([1 7 0 4]); grid;
S=2; subplot(2,1,2); stem(n,S*x, 'filled'); title('Amplitude Scaled signal'); xlabel('Time index'); ylabel('Signal Amplitude'); axis([1 7 0 8]); grid;
Scale the continuous‐time sinusoid used in signal shifting example by a factor of 2.
4) Time Scaling %Decimation (down‐sampling) clear n=‐2:1/1000:2; x1=sin(2*pi*2*n); x2=decimate(x1,2); subplot(2,1,1); plot(x1); title('Original signal'); xlabel('Sample Number'); ylabel('Signal Amplitude');
axis([0 4000 ‐1 1]); grid; subplot(2,1,2); plot(x2); title('Decimated signal'); xlabel('Sample Number'); ylabel('Signal Amplitude'); axis([0 2000 ‐1 1]); grid;
Use interp command in the above program to interpolate (up‐sample) the signal by a factor of 2.
5) Amplitude Clipping
clear x=[3 4 4 2 1 ‐4 4 ‐2]; len=length(x); y=x; hi=3; lo=‐3; for i=1:len if(y(i)>hi) y(i)=hi; elseif(y(i)
subplot(2,1,1); stem(x,'filled'); title('original signal'); xlabel('Sample number'); ylabel('Signal Amplitude'); subplot(2,1,2); stem(y,'filled'); title('Clipped Signal'); xlabel('Sample number'); ylabel('Signal Amplitude');
6) Signal Replication clear x=[1 2 3 2 1]; y=[x x x x]; subplot(2,1,1); stem(x,'filled'); title('Original Signal'); xlabel('Sample Number'); ylabel('Signal Amplitude'); axis([1 20 0 3]); grid; subplot(2,1,2); stem(y,'filled'); title('Replicated Signal');
xlabel('Sample Number'); ylabel('Signal Amplitude'); axis([1 20 0 3]); grid;
Lab # 8
OBJECTIVES OF THE LAB -----------------------------------------------------------------------------------------------------------This lab aims at the understanding of:
Making Signals Causal and Non‐Causal Convolution Properties of Convolution
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8.1
MAKING SIGNALS CAUSAL AND NON-CAUSAL
Causal Signals: A signal is said to be causal if it is zero for time t<0. A signal can be made causal by
multiplying it with unit step.
Example clc clear all close all t= ‐2:1/1000:2; x1 = sin(2*pi*2*t); subplot(3,1,1); plot(t,x1,'LineWidth',2); xlabel('time'); ylabel('signal amplitude'); title('sin(2*\pi*f*t)'); u = (t>=0); x2 = x1.*u; subplot(3,1,2); plot(t,u, 'r','LineWidth',2); xlabel('time'); ylabel('Signal Amplitude'); title('Unit Step'); subplot(3,1,3); plot(t,x2, 'k','LineWidth',2); xlabel('time'); ylabel('signal amplitude'); title('causal version of sin(2*\pi*f*t)'); figure; plot(t,x1,t,u,'‐.',t,x2,'LineWidth',2); text(0,1.2,'u(t)','FontSize',16); text(‐1.2,‐1.1,'x(t)','FontSize',16); text(0.8,‐1.1,'x(t)*u(t)','FontSize',16); axis([‐2 2 ‐1.5 1.5]);
Sample the signal given in above example to get its discrete‐time counterpart (take 10 samples/sec as sampling rate). Make the resultant signal causal. Display the lollipop plot of each signal.
A signal is said to be anti‐causal if it exists for values of n<0. Make the signal given in above example anti‐causal.
Create a function by name of sig_causal in matlab that has two input arguments: (i) a discrete ‐time signal, and (ii) a position vector. The function should make the given signal causal and return the resultant signal to the calling program. A non‐causal signal is shown in the Fig below. Write matlab code to make the signal causal by calling the above‐mentioned function. Plot the original non‐causal signal and the resultant causal signal.
8.2
CONVOLUTION
Use the matlab command conv(h, x) to find convolution where h – impulse response x – input signal
Example
clc clear all close all h = [1 2 3 4 5 4 3 2 1]; x = sin(0.2*pi*[0:20]);
y = conv(h, x); figure(1); stem(x); title('Discrete Filter Input x[n]'); xlabel('index, n'); ylabel('Value, x[n]'); figure(2); stem(y, 'r'); title('Discrete Filter Output y[n]'); xlabel('index, n'); ylabel('Value, y[n]');
Even though there are only 21 points in the x array, the conv function produces 8 more points because it uses the convolution summation and assumes that x[n] = 0 when n>20.
Convolve the following signals: x =[2 4 6 4 2]; h =[3 ‐1 2 1]; Plot the input signal as well as the output signal.
Convolve the signal x[n]=[1 2 3 4 5 6] with an impulse delayed by two samples. Plot the original signal and the result of convolution.
Convolution is associative. Given the three signal x1[n], x2[n], and x3[n] as: x1[n]= [3 1 1] x2[n]= [4 2 1] x3[n]=[3 2 1 2 3] Show that (x1[n] * x2[n]) * x3[n] = x1[n] * (x2[n] * x3[n]).
Convolution is commutative. Given x[n] and h[n] as: X[n]=[1 3 2 1] H[n]=[1 1 2] Show that x[n] * h[n] = h[n] * x[n].
Determine h[n] for the system:
yn kxn k
When x[n] = 2δ[n]. Plot the input signal, impulse response, and output signal.
Given the impulse response of the systems as: h[n]= 2δ[n] + δ[n‐1]+ 2δ[n‐2]+ 4δ[n‐3]+ 3δ[n‐4] If the input x[n] = δ[n]+ 4δ[n‐1] +3δ[n‐2] + 2δ[n‐3] is applied to the system, determine the output of the system.
Two systems are connected in cascade.
x[n]
h1[n]
h2[n]
y[n]
h1[n]=[1 3 2 1] h2[n]=[1 1 2] If the input x[n] = δ[n]+ 4δ[n‐1] +3δ[n‐2] +2δ[n‐3] is applied, determine the output.
Given the signals: x1[n]= 2δ[n] ‐3δ[n‐1]+ 3δ[n‐2] +4δ[n‐3] ‐2δ[n‐4] x2[n]= 4δ[n]+ 2δ[n‐1]+ 3δ[n‐2]
‐ δ[n‐3] ‐2δ[n‐4]
x3[n]= 3δ[n]+ 5δ[n‐1]‐3δ[n‐2] +4 δ[n‐3] Verify that x1[n] * (x2[n] * x3[n]) = (x1[n] * x2[n]) * x3[n] x1[n] * x2[n]= x2[n] * x1[n]
Lab # 9
OBJECTIVES OF THE LAB -----------------------------------------------------------------------------------------------------------This lab aims at the understanding of:
• • • •
Power of Continuous & Discrete time Signals Application of Fourier Series Synthesis of Square Wave Synthesis of Triangular Wave
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9.1 SIGNAL POWER Average power of continuous time signal can be calculated using the formula: T
P
=
1 T
2
∫ −T
| x(t ) |2
dt
2
To carry out the integral, Euler Approximation can be used. It simply tells that a definite integral can be approximated using a sum i.e. b
∫
x (t ) dt ≅
a
N −1
∑
x ( a
+ nΔt )Δt , Δt =
n =0
b−a N
In this method, the region over which integral is carried out is divided into N parts or intervals, each of duration Δt, such that function stays constant over those short intervals. Approximating function in this way is shown below. Note that as the number of intervals N is increased, the approximation gets better.
Approximating integrals using sums is a deep subject of numerical analysis by itself; therefore its further detail is out of scope. It is enough to know that Euler’s formula is easy to implement and produces good results for almost all the signals that will be studied here as long as N is selected large enough.
Example – Power of Continuous Time Cosine clc clear all close all
t = ‐1:0.005:0.995;
% time duration of given signal;
xt = cos(2*pi*t/2);
% generate signal
plot(t, xt);
% plot signal
xlabel('time, t'); ylabel('Amplitude, A'); title('Continuous Time Cosine');
abs_xt_2 = abs(xt).^2;
% take absolute square of signal
T = 2;
% length of interval
delta_t = 0.005;
% interval duration
pxt = sum(abs_xt_2)*delta_t/T
% power of given signal
pxt =
0.5000
Calculate the power of discrete‐time cosine signal with period 20, defined over interval 0:19 using the following formula:
P
=
1 N
N −1
∑ 0
| x[n] |2
9.2 FOURIER SERIES Fourier series theory states that a periodic wave can be represented as a summation of sinusoidal waves with different frequencies, amplitudes and phase values.
9.2.1
Synthesis of Square wave
The square wave for one cycle can be represented mathematically as: x(t) =
1
0 <= t < T/2
‐1
T/2 <= t < T
The Complex Amplitude is given by: Xk =
(4/j*pi*k)
for
k=±1, ±3, ±5…..
0
for
k=0,±2, ±4, ±6…..
For f = 1/T = 25Hz, only the frequencies ±25, ±50, ±75 etc. are in the spectrum.
i.
Effect of Adding Fundamental, third, fifth, and seventh Harmonics
Example clc clear all close all
t=0:0.0001:8; ff=0.5;
% WE ARE USING SINE FUNCTION BECAUSE FROM EXPONENTIAL FORM OF FOURIER % SERIES FINALLY WE ARE LEFT WITH SINE TERMS y = (4/pi)*sin(2*pi*ff*t); % COMPLEX AMPLITUDE = (4/(j*pi*k)) for k = 3:2:7 fh=k*ff;
x = (4/(k*pi))*sin(2*pi*fh*t); y=y+x; end
plot(t,y,'linewidth',1.5); title('A square wave with harmonics 1st, 3rd, 5th, and 7th'); xlabel('Time'); ylabel('Amplitude');
ii.
Effect of Adding 1st to 17th harmonics
Example clc clear all
t=0:0.0001:8; ff=0.5;
% WE ARE USING SINE FUNCTION BECAUSE FROM EXPONENTIAL FORM OF FOURIER
% SERIES FINALLY WE ARE LEFT WITH SINE TERMS y = (4/pi)*sin(2*pi*ff*t); % COMPLEX AMPLITUDE = (4/(j*pi*k)) for k = 3:2:17 fh=k*ff; x = (4/(k*pi))*sin(2*pi*fh*t); y=y+x; end
plot(t,y,'linewidth',1.5); title('A square wave with harmonics 1st‐17th'); xlabel('Time'); ylabel('Amplitude');
iii.
Effect of Adding 1st to 27th harmonics
Example
clc clear all
close all t=0:0.0001:8; ff=0.5;
% WE ARE USING SINE FUNCTION BECAUSE FROM EXPONENTIAL FORM OF FOURIER % SERIES FINALLY WE ARE LEFT WITH SINE TERMS y = (4/pi)*sin(2*pi*ff*t); % COMPLEX AMPLITUDE = (4/(j*pi*k)) for k = 3:2:55 fh=k*ff; x = (4/(k*pi))*sin(2*pi*fh*t); y=y+x; end
plot(t,y,'linewidth',1.5); title('A square wave with harmonics 1st to 27th'); xlabel('Time'); ylabel('Amplitude');
Write a program that plots the signal s(t). N
s (t )
=∑ n =1
s (t )
sin(2π nt ) n
= sin( 2π * t ) +
where n = 1, 3, 5, 7, 9
and N = 9
or
sin(6π * t ) sin(10π * t ) sin(14π * t ) sin(18π * t ) + + + 3 5 7 9
Write a program that plots the signal s(t) but with N = 100.
What do you conclude from TASKS 2 & 3?
9.2.2 Synthesis of Triangular wave The Complex Amplitude is given by: Xk = (‐8/*pi^2*k^2) for k is an odd integer 0
for k for k is an even integer
For f = 1/T = 25Hz
Example: Triangular wave with N=3 clc; clear all; close all t=0:0.001:5;
x=(‐8/(pi*pi))*exp(i*(2*pi*0.5*t)); y=(‐8/(9*pi*pi))*exp(i*(2*pi*0.5*3*t)); s=x+y;
plot(t,real(s),'linewidth',3); title('Triangular Wave with N=3'); ylabel('Amplitude'); xlabel('Time');
grid;
Example: Triangular wave with N=11 clc; clear all; close all
t=0:0.01:0.25; ff=25;
x1=(‐8/(pi^2))*exp(i*(2*pi*ff*t)); for k=3:2:21, fh=ff*k; x=(‐8/(pi^2*k^2))*exp(i*(2*pi*fh*t)); y=x1+x; end
plot(t,real(y),'linewidth',3); title('Triangular Wave with N=11'); ylabel('Amplitude'); xlabel('Time'); grid;
Lab # 10
OBJECTIVES OF THE LAB -----------------------------------------------------------------------------------------------------------This lab aims at the understanding of:
• Fourier Series Representation of Continuous Time Period Signals • Convergence of CT Fourier Series • Properties of CT Fourier Series o o o o o
Linearity Time Shifting Frequency Shifting Time Reversal Time Scaling
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10.1 FOURIER SERIES REPRESENTATION OF CONTINUOUS TIME PERIOD SIGNALS A signal expressed by the formula ∞
x (t ) =
∑a e k
k = −∞
∞
jkw0t
=
∑a e
jk ( 2 π ) t T
k
k = −∞
is periodic with period T, as it is linear combination of harmonically related complex exponentials that are all periodic with T. Any well‐behaving periodic function can be expressed as a linear combination of harmonically related complex exponentials. The representation of periodic signal in this way is known as Fourier series representation and the weight ak’s are referred to as Fourier series coefficients. Given a
periodic signal x(t), it is possible to determine its Fourier series coefficients through the following integral.
T
T
∫
ak = x(t )e
− jkw0t
−T
∫
dt = x (t )e
− jk ( 2 π T ) t
dt
−T
This integral can be done over any time interval of length T, the period of the signal x(t).
10.1.1
Synthesis of a Simple Periodic Signal
Following example demonstrates that the linear combination of harmonically related complex exponentials leads to a periodic function. The signal used in example is
x(t ) =
3
∑
ak e
jk ω 0t
k =−3
1 1 1 , where a0 = 1, a1 = a−1 = , a2 = a−2 = , a3 = a−3 = 4 2 3
Example – FS of CT Periodic Signal clc clear all close all
t = ‐3:0.01:3;
% duration of signal
% dc component for k=0 x0 = 1;
% first harmonic components for k=‐1 and k=1 x1 = (1/4)*exp(j*(‐1)*2*pi*t)+(1/4)*exp(j*(1)*2*pi*t);
y1 = x0 + x1;
% sum of dc component and first harmonic
% second harmonic components for k=‐2 and k=2 x2 = (1/2)*exp(j*(‐2)*2*pi*t)+(1/2)*exp(j*(2)*2*pi*t); y2 = y1 + x2;
% sum of all components until second harmonic
% third harmonic components for k=‐3 and k=3 x3 = (1/3)*exp(j*(‐3)*2*pi*t)+(1/3)*exp(j*(3)*2*pi*t); x = x0 + x1 + x2 + x3;
figure; subplot(3,2,1); plot(t,x1); axis([‐3 3 ‐2 2]); title('x1(t)');
subplot(3,2,2); plot(t,y1); axis([‐3 3 ‐0.2 2]); title('x0(t)+x1(t)');
subplot(3,2,3); plot(t,x2); axis([‐3 3 ‐2 2]); title('x2(t)');
subplot(3,2,4); plot(t,y2); axis([‐3 3 ‐1 3]); title('x0(t)+x1(t)+x2(t)');
subplot(3,2,5);
% sum of all components until third harmonic
plot(t,x3); xlabel('t'); axis([‐3 3 ‐1 1]); title('x3(t)');
subplot(3,2,6); plot(t,x); xlabel('t'); axis([‐3 3 ‐1 4]); title('x(t)=x0(t)+x1(t)+x2(t)+x3(t)')
In above example, ak’s are chosen to be symmetric about the index k=0, i.e. ak = a k. Select new ‐
ak’s on your own to alter this symmetry and form the new signal. What do you observe? Is x(t) a real signal when coefficients are not symmetric?
A continuous‐time periodic signal x(t) is real valued and has a fundamental period of T = 8. The non‐zero Fourier series coefficients for x(t) are
a1 = a−1 = 2,
a3 = a−*3 = 4 j
Express x(t) as linear combination of these coefficients.
A discrete‐time periodic signal x[n] is real valued and has a fundamental period of N = 5. The non‐zero Fourier series coefficients for x[n] are j
π
a0 = 1, a2 = a−2 = e , a4 = a− 4 = 2e *
*
4
j
π
3
Express x[n] as linear combination of given coefficients.
10.1.2
Synthesis of a Simple Periodic Signal
Once the Fourier series (FS) coefficient of a continuous time periodic signal is determined analytically using analysis equation, signal can be reconstructed using synthesis equation. Consider the periodic square wave signal defined as
| t |< T 1
⎧⎪1, x(t ) = ⎨ ⎪⎩0,
T 1 <| t |< T
2
where T is the time period and T1 is the duty cycle with FS coefficients
a0 =
2T 1 T
, a k =
sin( k 0T 1 ) k π
=
sin( k 2π (T 1 T )) k π
for k ≠ 0
In the following examples, first and ideal square wave is created and then square wave is approximated from its harmonics using synthesis equation by letting k in the partial sum go from –M to M instead of ‐∞ to +
∞ , where M is 10, 20, and 100. In all examples
T is taken as 1 sec. Example – Ideal Square Wave created by thresholding 1 Hz Cosine Wave % generate perfect square wave t = ‐1.5:0.005:1.5;
%duration of square wave
xcos = cos(2*pi*t);
%cosine wave of 1 Hz
xpsqw = xcos>0;
%thresholding cosine wave using relational operator
figure(1); set(gcf,'defaultaxesfontsize',9);
plot(t,xpsqw,'lineWidth',2); xlabel('t'); ylabel('x(t)'); title('Periodic Square Wave (T=1)'); axis([‐1.5 1.5 ‐0.1 1.1]); grid;
Example – FS coefficients for Square wave with period 1 sec & variable duty cycle k = ‐15:15;
%number of square wave coefficients
T = 1;
%time period of square wave
T1 = 1/4;
%duty cycle of square wave
ak1 = sin(k*2*pi*(T1/T))./(k*pi);
%square wave Fourier series coefficients
% Ignore the "divide by zero" warning that happens % because k in the denominator hits 0. We will now do % a manual correction for a0 ‐> ak1(16)
ak1(16) = 2*T1/T;
figure; set(gcf,'defaultaxesfontsize',8); subplot(3,1,1); stem(k,ak1,'filled'); ylabel('ak'); title('FS Coefficients for Periodic Square Wave (T=1, T1=1/4)');
T1 = 1/8; ak2 = sin(k*2*pi*(T1/T))./(k*pi); ak2(16) = 2*T1/T;
% Manual correction for a0 ‐> ak2(16)
subplot(3,1,2); stem(k,ak2,'filled'); ylabel('ak'); title('FS Coefficients for Periodic Square Wave... (T=1, T1=1/8)');
T1 = 1/16; ak3 = sin(k*2*pi*(T1/T))./(k*pi); ak3(16) = 2*T1/T;
% Manual correction for a0 ‐> ak3(16)
subplot(3,1,3); stem(k,ak3,'filled'); xlabel('k'); ylabel('ak'); title('FS Coefficients for Periodic Square Wave (T=1, T1=1/16)');
Considering the FS coefficients plot given below, what do you observe happens to the envelope of the coefficients when T1 is reduced from 1/4 to 1/16 with constant time period T?
Example – Reconstruction of Square Wave using 10 terms i.e. M=10 clc clear all close all
t = ‐1.5:0.005:1.5;
%square wave duration
T = 1;
%time period of square wave
T1 = 1/4;
%duty cycle of square wave
w0 = 2*pi/T;
%fundamental radian frequency of square wave
M = 10;
%number of coefficients
k = ‐M:M;
%2M+1 total coefficients to construct square wave
ak = sin(k*2*pi*(T1/T))./(k*pi); ak(M+1) = 2*T1/T;
x = zeros(1,length(t));
% Manual correction for a0 ‐> ak(M+1)
for k = ‐M:M x = x + ak(k+M+1)*exp(j*k*w0*t); end
figure; set(gcf,'defaultaxesfontsize',9); plot(t,real(x),'lineWidth',2); grid; xlabel('t'); ylabel('x(t)') ; title('Reconstruction from Fourier Series with 21 terms');
Example – Reconstruction of Square Wave using 20 terms i.e. M=20 clc clear all close all
t = ‐1.5:0.005:1.5; T = 1; T1 = 1/4; w0 = 2*pi/T; M = 20; k = ‐M:M; ak = sin(k*2*pi*(T1/T))./(k*pi);
% Manual correction for a0 ‐> ak(M+1) ak(M+1) = 2*T1/T; x = zeros(1,length(t)); for k = ‐M:M x = x + ak(k+M+1)*exp(j*k*w0*t); end
figure; set(gcf,'defaultaxesfontsize',9); plot(t,real(x),'lineWidth',2); grid; xlabel('t'); ylabel('x(t)'); title('Reconstruction from Fourier Series with 41 terms');
Example – Reconstruction of Square Wave using 100 terms i.e. M=100 clc clear all close all
t = ‐1.5:0.005:1.5; T = 1; T1 = 1/4;
w0 = 2*pi/T; M = 100; k = ‐M:M; ak = sin(k*2*pi*(T1/T))./(k*pi); ak(M+1) = 2*T1/T;
% Manual correction for a0 ‐> ak(M+1)
x = zeros(1,length(t)); for k = ‐M:M x = x + ak(k+M+1)*exp(j*k*w0*t); end
figure; set(gcf,'defaultaxesfontsize',9) plot(t,real(x),'lineWidth',2); grid;
xlabel('t'); ylabel('x(t)') ; title('Reconstruction from Fourier Series with 201 terms');
Considering the plots of square wave reconstructed using M = 10, 20, & 100 terms above, what do you observe about Gibb’s phenomena?
Given the following FS coefficients:
⎧1, ⎩2,
ak = ⎨
k even k odd
Plot the coefficients & reconstructed signal. Take the terms for reconstructed signal to be M = 10, 20, & 50. What effect do you see when M is varied?
Given the following FS coefficients:
⎧ jk , a =⎨ ⎩0, k
| k |< 3 otherwise
Plot the coefficients & reconstructed signal. Take 10 terms (M=10) for reconstructed signal.
10.2 PROPERTIES OF FOURIER SERIES 10.2.1
Linearity
Given two periodic signals x(t) and y(t) having same period, linearity property of FS representation can be expressed as FS
FS
FS
x(t ) ↔ a k , y (t ) ↔ bk ⇒ x(t ) + y (t ) ↔ a k + bk Where ak and bk are FS coefficients of x(t) and y(t) respectively. This property can be used in evaluating FS coefficients of a periodic signal that can be expressed as a linear combination of other periodic signals whose FS coefficients are known. Following example demonstrates the validity of linearity property using two periodic square wave signasl x(t) and y(t) with T = 1 and different T1 parameters.
⎧⎪1, x(t ) = ⎨ ⎪⎩0,
| t |< T 1
⎧⎪1, y (t ) = ⎨ ⎪⎩0,
| t |< T 1
T 1 <| t |< T
T 1 <| t |< T
T = 1 T 1 = 1
2
T = 1 T 1 = 1
2
Example – Demonstration of Linearity Property of FS clc clear all close all
% Generation of 1st signal x(t) t = ‐1.5:0.005:1.5; xcos = cos(2*pi*t);
4
8
xt = xcos>0;
figure(1); set(gcf,'defaultaxesfontsize',9) subplot(2,1,1); plot(t,xt,'lineWidth',2); xlabel('t'); ylabel('x(t)'); title('Peridoic Square Wave (T=1, T1=0.250)'); axis([‐1.5 1.5 ‐0.1 1.1]); grid;
% Generation of 2nd signal y(t) T = 1; T1 = 0.125; lenT = T/0.005; ytemp = zeros(1,lenT); lenT1 = T1/0.005; ytemp(round(lenT/2)‐lenT1:round(lenT/2)+lenT1‐1) = ones(1,2*lenT1);
yt = [ytemp ytemp ytemp 0]; % The last 0 added to make the size of yt equal to length(t)
subplot(2,1,2); plot(t,yt,'lineWidth',2); xlabel('t'); ylabel('y(t)') ; title('Periodic Square Wave (T=1, T1=0.125)'); axis([‐1.5 1.5 ‐0.1 1.1]); grid;
% FS coefficients of periodic square waves k = ‐50:50;
T1 = 0.25; ak = sin(k*2*pi*(T1/T))./(k*pi); ak(51)=2*T1/T;
% Manual correction for a0 ‐> ak(51)
T1 = 0.125; bk = sin(k*2*pi*(T1/T))./(k*pi); bk(51) = 2*T1/T;
% Manual correction for b0 ‐> bk(51)
% Application of linearity property of FS ck = ak+bk;
% Reconstruction with M=50 w0 = 2*pi/T; zt = zeros(1,length(t)); for k = ‐50:50 zt = zt + ck(k+51)*exp(j*k*w0*t); end
figure(2); set(gcf,'defaultaxesfontsize',9) plot(t,real(zt),'lineWidth',2); xlabel('t'); ylabel('z(t)=x(t)+y(t)'); title('Reconstruction from ak+bk''s with 101 terms'); grid;
Plot the FS coefficients and reconstruct the signal for the following sequence:
10.2.2
Time Shifting
The time shifting property of FS states that FS
FS
x(t ) ↔ ak ⇔ x(t − t 0 ) ↔ e
− jk ω 0t
ak ,
ω 0
= 2π T
Where x(t) is a periodic signal with FS coefficients ak and x(t‐t0)is the time shifted version of it. This property can be used to evaluate FS coefficients of a periodic signal that can be expressed as time shifted version of another periodic signal whose FS coefficients are known. Following example demonstrates the validity of time shifting property. Consider periodic square wave with period T = 1 and T1 = 0.25, its FS coefficients ak’s are
a0 =
2T 1 T
= 0.5, a =
sin( k 2π (T 1 T ))
k
k π
=
sin( k π / 2)) k π
for k ≠ 0
Let it be shifted by t0 = 0.25, then FS coefficients bk’s for x(t‐t0) = x(t‐0.25) can be found using time shifting property as
bk = e
− jk ω 0t 0
ak ,
t 0 = 0.25,
ω 0
= 2π T = 2π
Example – Demonstration of Time Shifting Property of FS clc clear all close all
% Generation of periodic square wave t = ‐1.5:0.005:1.5; xcos = cos(2*pi*t); xt = xcos>0;
% FS coefficients of periodic square wave k = ‐50:50; T = 1; T1 = 0.25; ak = sin(k*2*pi*(T1/T))./(k*pi);
ak(51)=2*T1/T;
% Manual correction for a0 ‐> ak(51)
% Amount of time shift t0 = 0.25;
% FS coefficients of the time shifted signal w0 = 2*pi/T; bk = ak.*exp(‐ j*k*w0*t0);
% Reconstruction from bk's with 101 terms (M=50) yt = zeros(1,length(t)); for k = ‐50:50 yt = yt + bk(k+51)*exp(j*k*w0*t); end
figure(1); set(gcf,'defaultaxesfontsize',9); subplot(2,1,1); plot(t,xt,'lineWidth',2); xlabel('t'); ylabel('x(t)'); title('Periodic Square Wave (T=1, T1=0.25)'); axis([‐1.5 1.5 ‐0.2 1.2]); grid;
subplot(2,1,2); plot(t,real(yt),'lineWidth',2); xlabel('t'); ylabel('y(t)=x(t‐0.5)'); title('Reconstruction from bk''s with 101 terms'); axis([‐1.5 1.5 ‐0.2 1.2]); grid;
Given the square wave, plot the FS coefficients and reconstruct the signal for the following: FS
x (t ) ↔ ak ⇔ e 10.2.3
FS
x(t ) ↔ a k − M ,
− jM ω 0t
ω
0
= 2π T
Time Reversal
The time reversal property of FS states that FS
FS
x(t ) ↔ ak ⇔ x( −t ) ↔ a−k , If FS representation for a periodic signal x(t) is known, then FS representation for the time reversed version of the signal x(‐t) can be determined through this property.
Example – Demonstration of Time Reversal Property of FS clc clear all
close all
% Generation of periodic square wave t = ‐1.5:0.005:1.5; xsin = sin(2*pi*t); xt = xsin>0;
% FS coefficients of periodic square wave k = ‐50:50; T = 1; T1 = 0.25; w0 = 2*pi/T; t0 = 0.25; ak = exp(‐ j*k*w0*t0).*sin(k*2*pi*(T1/T))./(k*pi); ak(51) = 2*T1/T;
% Manual correction for a0 ‐> ak(51)
% FS coefficients of the time reversed signal bk = fliplr(ak);
% Reconstruction from ak's with 101 terms (M=50) xtr = zeros(1,length(t)); for k = ‐50:50 xtr = xtr + ak(k+51)*exp(j*k*w0*t); end
% Reconstruction from bk's with 101 terms (M=50) yt = zeros(1,length(t)); for k = ‐50:50 yt = yt + bk(k+51)*exp(j*k*w0*t); end
figure(1);
set(gcf,'defaultaxesfontsize',8); subplot(3,1,1); plot(t,xt,'lineWidth',2); ylabel('x(t)'); title('Periodic Square Wave (T=1, T1=0.25)'); axis([‐1.5 1.5 ‐0.2 1.2]); grid;
subplot(3,1,2); plot(t,xtr,'lineWidth',2); ylabel('x(t)'); title('Periodic Square Wave Reconstructed from its FS'); axis([‐1.5 1.5 ‐0.2 1.2]); grid;
subplot(3,1,3); plot(t,real(yt),'lineWidth',2); xlabel('t'); ylabel('y(t)=x(‐t)'); title('Reconstruction from bk''s with 101 terms'); axis([‐1.5 1.5 ‐0.2 1.2]); grid;
Given the signal x(t) in task 7, do the following: a) Plot the time reverse version of the signal x(‐t) directly, b) Plot FS coefficients a k of time reversed signal, ‐
c) Plot the reconstructed time reversed signal using FS coefficients a k ‐
10.2.4
Time Scaling
The time scaling property of FS of FS states that ∞
x(t ) =
∑
∞
ak e
− jM ω t
⇒ x(α t ) ) =
k = −∞
∑
ak e
− jM ωα t
k = −∞
That is, if x(t) is periodic with period T and fundamental frequency w=2π/T, then time scaled version of x(t), x(αt) where
α
being positve real number has αw as its fundamental frequency
and the FS coefficients for x(αt) is same as those of x(t). Be careful about using the right i.e. scaled frequency (or period) in the reconstruction. Mathematically it can be expressed as: FS
x(t ) ↔ ak
x(t ) = x(t + T ), ω =
⇔
2π T
FS
x(α t ) ↔ ak
x(α t ) = x(α (t +
T α
)), ω = α
2π T
Example – Example – Demonstration of Time of Time Scaling Property of FS of FS having
α
clc clear all close all
% Generation of periodic of periodic square wave t = ‐1.5:0.005:1.5; xcos = cos(2*pi*t); xt = xcos>0;
% FS coefficients of periodic of periodic square wave k = ‐50:50; T = 1; T1 = 0.25; ak = sin(k*2*pi*(T1/T))./(k*pi); ak(51) = 2*T1/T;
% Manual correction for a0 ‐> ak(51)
% Time scaling parameters alp1 = 0.5;
% w's for the time scaled signals w0 = 2*pi/T; w1 = alp1*w0;
% Reconstruction from ak's with 101 terms (M=50) xat1 = zeros(1,length(t)); for k = ‐50:50 xat1 = xat1 + ak(k+51)*exp(j*k*w1*t); end
figure(1);
= 0.5
set(gcf,'defaultaxesfontsize',8); subplot(2,1,1); plot(t,xt,'lineWidth',2); ylabel('x(t)'); title('Periodic Square Wave (T=1, T1=0.25)'); axis([‐1.5 1.5 ‐0.2 1.2]); grid;
subplot(2,1,2); plot(t,real(xat1),'lineWidth',2); ylabel('x(t)'); title('Reconstruction from ak''s (alp1=0.5, w1=0.5*w0)'); axis([‐1.5 1.5 ‐0.2 1.2]); grid;
Given the periodic square wave x(t) with T = 1 & T1 = 0.25, rewrite the above code for time scaling when value of alpha of alpha is 2 i.e. x(αt) = x(2t).
Lab # 11
OBJECTIVES OF THE LAB -----------------------------------------------------------------------------------------------------------This lab aims at the understanding of:
•
• • •
Properties of CT Fourier Series o Multiplication o Conjugation & Conjugate Symmetry Characterization of LTI Systems Filtering Signals Delay Filter
------------------------------------------------------------------------------------------------------------
11.1 PROPERTIES OF FOURIER SERIES 11.1.1
Multiplication
Given two periodic signals x(t) and y(t) having same period T with FS coefficients ak and bk, there product signal x(t)y(t) is also periodic and its FS coefficients can be determined using multiplication property: FS
FS
∞
FS
x(t ) ↔ ak , y (t ) ↔ bk ⇒ x(t ) y (t ) ↔ ck =
∑a b l
k − l
l = −∞
Where ck is the discrete time convolution of two sequences ak’s and bk’s Following example demonstrates the validity of multiplication property using periodic square wave signal x(t) and cosing signal y(t). For the square wave signal with T = 1 and T1 = 0.25, FS coefficients ak’s are
a0 =
2T 1 T
= 0.5, a =
sin( k 2π (T 1 T ))
k
k π
=
sin( k π / 2)) k π
for k ≠ 0
For the cosine signal of 1 Hz, FS coefficients bk’s are
⎧0.5, ⎩0,
bk = ⎨
k = ±1 otherwise
= 0.5δ (k + 1) + 0.5δ ( k − 1)
To carry out convolution of ak and bk sequences, two approaches are used: first one is the direct implemention of sum formula for ck given in multiplication property using for loop; and second one is by using the following property of impulse function
x[k ] * δ (k ± k 0 ) = x(k ± k 0 ) Applying it on ak and bk:
ak * bk = ak * (0.5δ (k + 1) + 0.5δ (k − 1))
= 0.5a (k + 1) + 0.5a (k − 1) k
k
Example – Demonstration of Multiplication Property of FS clc clear all close all
% Generation of 1Hz cosine t = ‐1.5:0.005:1.5; yt = cos(2*pi*t); xt = yt>0;
% Generation of periodic square wave
% FS coefficients of periodic square wave k = ‐50:50; T = 1; T1 = 0.25; ak = sin(k*2*pi*(T1/T))./(k*pi); ak(51) = 2*T1/T;
% 1st way to create ck's % For loop implementation of convolution bk = zeros(1,201);
% FS coefficients of 1 Hz cosine (over k=‐100:100)
bk(100) = 0.5; bk(102) = 0.5; ck1 = zeros(1,101); for k = ‐50:50 u = 0; for L = ‐50:50 u = u + ak(L+51)*bk(k‐L+101); end ck1(k+51) = u; end
% 2nd way to create ck's % A simpler way of implementing convolution using properties of impulse bk = zeros(1,101);
% FS coefficients of 1Hz cosine (over k=‐50:50)
bk(50) = 0.5; bk(52) = 0.5; ck2 = 0.5*[ak(2:101) 0]+0.5*[0 ak(1:100)];
% Check the equality of ck1 and ck2 ck1‐ck2
% Reconstruction from ck1's with 101 terms (M=50) w0 = 2*pi/T; zt = zeros(1,length(t)); for k = ‐50:50 zt = zt + ck1(k+51)*exp(j*k*w0*t); end
figure(1); set(gcf,'defaultaxesfontsize',8); subplot(3,1,1); plot(t,xt,'lineWidth',2); ylabel('x(t)','lineWidth',2); title('Periodic Square Wave (T=1, T1=0.25)'); axis([‐1.5 1.5 ‐1.2 1.2]); grid;
subplot(3,1,2); plot(t,yt,'lineWidth',2); ylabel('y(t)'); title('1 Hz Cosine'); axis([‐1.5 1.5 ‐1.2 1.2]); grid;
subplot(3,1,3); plot(t,real(zt),'lineWidth',2); xlabel('t'); ylabel('z(t)=x(t)*y(t)'); title('Reconstruction from ck''s');
axis([‐1.5 1.5 ‐1.2 1.2]); grid;
Another way to compute FS coefficients of ck is by using matlab convolution function i.e. conv to convolve ak and bk sequences. Do the following steps to achieve this: a) Create cosine signal of 1 Hz and generate square wave from it b) Compute FS coefficients ak of square wave with T = 1 and T1 = 0.25 over k = ‐50:50. What is the length of sequence? c) Compute FS coefficients bk of cosine signal using equation
bk = 0.5δ (k + 1) + 0.5δ (k − 1) What is the length of bk? d) Convolve ak and bk using conv. What is the length of the sequence? e) Reconstruct the resultant signal from ck for M = 50. f) Plot x(t), y(t) and reconstructed signal.
Consider the following three continuous‐time signals with fundamental period of T = ½.
x(t ) = cos( 4π t ) y (t ) = sin( 4π t ) z (t ) = x(t ) y(t ) a) Determine the FS coefficients of x(t) and y(t) analytically. b) Use multiplication property to compute the coefficients of z(t) using matlab. c) Reconstruct the signal z(t) from its coefficients in matlab. d) Plot the signal x(t), y(t), and reconstructed signal z(t) with proper labeling.
11.1.2
Conjugation & Conjugate Symmetry
When complex conjugate of periodic signal x(t) is computed, corresponding FS coefficients are time reveresed i.e. flipped about k = 0 and complex conjugated. That is FS
FS
x(t ) ↔ ak ⇔ x (t ) ↔ a− k *
*
Following example demonstrates the validity of conjugation property. Cosider a comlex periodic signal z(t) that has a cosine x(t) and a square wave y(t) as its real and imaginary parts respoectively.
z = x(t ) + jy (t ) x(t ) = cos( 4π t )
| t |< 1 4 T 1 <| t |< 1 2
⎧1, y (t ) = ⎨ ⎩0, Where the coefficients are: FS
x(t ) ↔ ak , FS
y (t ) ↔ bk ,
2T 1
a0 =
T
,
⎧0.5, ⎩0,
bk = ⎨
ak =
sin( k π 2) k π
for k ≠ 0
k = ±1 otherwise
Given FS representations of x(t) and y(t), FS represenation of z(t) can be computed using the linearity property of FS. FS
z (t ) ↔ ck = ak + jbk
Example – Demonstration of Conjugation Property of FS clc clear all close all
% Generation of 1Hz cosine (Real part of our signal) t = ‐1.5:0.005:1.5; xt = cos(2*pi*t); yt = xt>0; zt = xt+j*yt;
% perodic complex valued signal
% FS coefficients of 1Hz cosine, xt (over k=‐50..50) ak = zeros(1,101); ak(50) = 0.5; ak(52) = 0.5;
% FS coefficients of periodic square wave, yt k = ‐50:50; T = 1; T1 = 0.25; bk = sin(k*2*pi*(T1/T))./(k*pi); bk(51) = 2*T1/T;
% FS coefficients of zt (using linearty property of FS) ck = ak+j*bk; % Flip ck's and conjugate dk = conj(fliplr(ck)); % Reconstruction from dk's with 101 terms (M=50) w0 = 2*pi/T; zrt = zeros(1,length(t)); for k = ‐50:50
zrt = zrt + dk(k+51)*exp(j*k*w0*t); end
figure(1); set(gcf,'defaultaxesfontsize',8); subplot(2,2,1); plot(t,xt,'lineWidth',2); ylabel('x(t)'); title('Real part of z(t), 1 Hz Cosine'); set(gca,'ylim',[‐1.2 1.2]); grid;
subplot(2,2,2); plot(t,yt,'lineWidth',2); ylabel('y(t)'); title('Imag. part of z(t), Periodic Square Wave'); axis([‐1.5 1.5 ‐1.2 1.2]); grid;
subplot(2,2,3); plot(t,real(zrt),'lineWidth',2); xlabel('t');ylabel('Re[zr(t)]'); title('Real part of Reconstruction from dk''s'); axis([‐1.5 1.5 ‐1.2 1.2]); grid;
subplot(2,2,4); plot(t,imag(zrt),'lineWidth',2); xlabel('t'); ylabel('Im[zr(t)]'); title('Imag. part of Reconstruction from dk''s'); axis([‐1.5 1.5 ‐1.2 1.2]);
grid;
11.2 CHARACTERIZATION OF LSI SYSTEMS There are different ways to characterize discrete‐time LSI systems including difference equation, impulse response, transfer function, and frequency response. We are considering here only difference equation.
Difference Equation A discrete‐time LSI system is characterized by the following general form difference equation
y(n) + a1y1(n-1) + a2y2(n-2) + … + aNyN(n-N) = b0x0(n) + b1x1(n-1) + … + bMxM(n-M) where x(n) and y(n) are input and output, respectively. Finite integers M and N represent the maximum delays in the input and output respectively.
The constants ai and bi are called the filter coefficients. Difference equation can be implemented in Matlab as follow:
y = filter(b, a, x) which takes the input signal in vector x and filter coefficient a and b. The Vector B constitutes the feed‐ forward filter coefficients and vector A constitutes the feed‐backward filter coefficients. In case of FIR filters, the feed‐backward filter coefficient consists of a single 1.
Example – Demonstration of Conjugation Property of FS
y[ n] + 2 y[n − 1] + 3 y[n − 2] = x[ n] + 3 x[ n − 1] + x[ n − 2] A = coefficients of y(n) = [1 2 3] B = coefficients of x(n) = [1 3 1] X = input sequence = [1 1 1 1 1]
Program
A = [1 2 3]; B = [1 3 1]; X = [1 1 1 1 1]; Y =filter (B,A,X)
Output: Y= 1
2
‐2
3
5
Given the following difference equation and input signal, calculate the output.
5 y[n] = −4 x[n] + x[n − 1] + 6 x[n − 3] x[ n] = [1 3 5 7 9]
11.3 FILTERING SIGNALS Matlab provides the command Filter for developing one‐dimensional digital filter. Y = filter(B,A,X)
It filters the data in vector X with the filter described by vectors A and B to create the filtered data Y. The filter is a "Direct Form II Transposed" implementation of the standard difference equation: a(1)*y(n) = b(1)*x(n) + b(2)*x(n‐1) + ... + b(nb+1)*x(n‐nb) ‐ a(2)*y(n‐1) ‐ ... ‐ a(na+1)*y(n‐na)
For FIR filters, keep a=1.
Example clc clear
b = [1 2 3 4 5 4 3 2 1]; a = 1; x = cos(0.2*pi*[0:20]); y = filter(b,a, x);
figure; subplot(2,1,1); stem (x); title('Discrete Filter Input x[n]'); xlabel('index, n'); ylabel('Value, x[n]');
subplot(2,1,2); stem(y, 'r'); title('Discrete Filter Output y[n]'); xlabel('index, n');
ylabel('Value, y[n]');
11.3.1
3-Point Averaging Filter
clc clear all close all
n=‐3:7;
% input signal x=[0 0 0 2 4 6 4 2 0 0 0];
% filter coefficients b=[1/3 1/3 1/3];
%feedforward filter coefficients of 3‐point averager
a=1;
%feedback filter coefficients
% output signal y=filter(b,a,x);
subplot(2,1,1); stem(n,x, 'filled'); xlabel('sample number'); ylabel('signal amplitude'); title('origianl signal');
subplot(2,1,2); stem(n,y, 'filled'); xlabel('sample number'); ylabel('signal amplitude'); title('Result of 3‐point averager: y[n]=(x[n]+x[n‐1]+x[n‐2])/3');
Modify above program (i.e. example of 3‐pt averaging filter) to implement
1. A 3‐point non‐causal averager 2. A centralized averager
A filter has coefficients bk = {1 3 ‐2 4 1}. An input signal x[n]= δ[n]
‐
3δ[n‐1] + 3δ[n‐2] + 4δ[n‐3] ‐ 2δ[n‐4]
is applied to the filter. Find the output y[n].
A non‐recursive discrete‐time system has filter coefficients {1, ‐2, 3, 2}. An input signal (shown in Fig) is applied to the system. Determine the discrete‐time output of the filtering. Write code to plot the original signal as well as the output on two subplots of the same figure. 2 2 1
‐2 ‐1
11.3.2
1
0 1 2 3
4
5
6
7
Application of 3-Point Averaging Filter to Smooth Signal
clc clear all close all
%duration of input signal n=0:40;
% input signal x=(1.02).^n + 0.5*cos(2*pi*n/8 + pi/4);
% filter coefficients b1=[1/3 1/3 1/3]; a1=1;
1
% plot of original signal subplot(2,1,1) stem(n,x, 'filled'); title('input signal'); xlabel('sample number'); ylabel('signal amplitude');
% output of 3‐ point filtering y1=filter(b1,a1,x);
subplot(2,1,2); stem(n,y1, 'filled'); title('output of 3‐point filtering'); xlabel('sample number'); ylabel('signal amplitude');
Develop a causal FIR filter that averages five samples of the input signal x[n]= 2δ[n] + 5δ[n‐1] + 8δ[n‐2] + 4δ[n‐3] + 3δ[n‐4]
Apply 7 point filtering filter on the input signal given in example of signal smoothing.
11.4 DELAY FILTER A delay system is a simple FIR filter. A delay by 2 is given by filter coefficients {0, 0, 1}.
Given the signal x[n]= δ[n] + 4δ[n‐1] +3δ[n‐2] + 2δ[n‐3]
Delay this signal by 2 using the above mentioned filter coefficients using filter command.
A delay filter delays an input signal. The amount of delay depends on filter coefficients. Design a delay filter that delays an input signal by 6 units.
Use the filter designed in task 10 to delay the input signal: x[n]= δ[n] ‐ 4δ[n‐1] +3δ[n‐2] + 2δ[n‐3] ‐ 6δ[n‐4] + 2δ[n‐6]
Write code to plot the input signal as well as the output signal.
Create a function that takes a signal and a delay count as arguments. The purpose of the function is to delay the signal by that amount (use filter command). Call this function from a main program to delay a required signal.
Lab # 12
OBJECTIVES OF THE LAB -----------------------------------------------------------------------------------------------------------This lab aims at the understanding of: • • • • •
Beat Notes Amplitude Modulation Determining Impulse Response of a System Determining Frequency Response of a System Designing Filters using matlab command FIR1
------------------------------------------------------------------------------------------------------------
12.1 BEAT NOTES When two sinusoidal signals with different frequencies are multiplied, a beat note is produced. To fulfill the spectrum representation, the resultant signal is expressed as an additive linear combination of complex exponential signals. For instance, consider a beat signal as the product of two sinusoids
x(t ) = sin(10π t ) cos(π t ) 1 cos 11π t − π 2 2
=
)
+
1 cos 9π t − π 2 2
)
Or in other words, beat note can be produced by adding two sinusoids with nearly identical frequencies, i.e.
x(t ) = cos(2π f 1t ) + cos(2π f 2 t )
1)
where f 1 f 2
= =
f c
=
f d
=
f c − f d f c + f d 1 ( f + f ) 2 2 1 1 ( f − f ) 2 2 1
first frequency → sec ond frequency →
→
center frequency
→
deviation
frequency
And, thus
x (t ) = sin(2π f d t ) cos(2π f c t ) =
1 cos (2π ( f c 2
=
1 cos(2π f t ) + 1 cos(2π f t ) 1 2 2 2
−
f d )t ) + 1 cos (2π ( f c 2
Example – Beat Note implementation for fc = 200 and fd = 20 Hz clc clear all close all
t = 0:0.00001:0.1;
fd = 20; Ad = 2; fc = 200; Ac = 5;
2) +
f d )t ) 3)
s1 = Ad * cos(2*pi*fd*t); s2 = Ac * cos(2*pi*fc*t);
X = s1.*s2;
figure; subplot(3,1,1); plot(t,s1,'linewidth',1.5); grid; ylabel('Amplitude'); xlabel('Time'); title('Difference Frequency Signal');
subplot(3,1,2); plot(t,s2,'linewidth',1.5); grid; ylabel('Amplitude'); xlabel('Time'); title('Center Frequency Signal');
subplot(3,1,3); plot(t,X,'linewidth',1.5); grid; ylabel('Amplitude'); xlabel('Time'); title('Beat Signal');
Modify the above beat signal code in such a way that the resultant beat signal & its envelope both are shown in following figure. (Hint: Envelope is made from deviation frequency signal).
12.2 AMPLITUDE MODULATION Amplitude modulation is the process of multiplying a low‐frequency signal by a high‐frequency sinusoid. It is a technique used to broadcast AM radio. The AM signal is the product of the form
x(t ) = v(t ) cos(2π f c t )
4)
where the frequency of the cosine (f c hertz) is much higher than any frequencies contained in the spectrum of v(t), which represent the data signal to be transmitted. The cosine wave is called the carrier signal, and its frequency is called the carrier frequency.
Example – AM Signal Implementation clc clear all close all
t = 0:0.00001:0.1;
fd =20; Ad = 2; fc = 200; Ac = 5;
s1 = Ad * cos(2*pi*fd*t); s2 = Ac * cos(2*pi*fc*t);
B = s1.*s2; AM = B + s2;
figure; subplot(3,1,1); plot(t,s1,'linewidth',1.5); grid; ylabel('Amplitude'); xlabel('Time'); title('Modulating Signal');
subplot(3,1,2); plot(t,s2,'linewidth',1.5); grid; ylabel('Amplitude'); xlabel('Time'); title('Carrier Signal');
subplot(3,1,3); plot(t,AM,'linewidth',1.5); grid; ylabel('Amplitude'); xlabel('Time'); title('Amplitude Modulated Signal');
Modify the above AM signal code in such a way that the resultant beat signal & its envelope both are shown in following figure. (Hint: Envelope is made from deviation frequency signal).
12.3 IMPULSE RESPONSE OF SYSTEM Given any system characteristics, its impulse response can be determined by following matlab function:
y = impz (b, a) Where ‘b’ are coefficients of inputs of inputs and ‘a’ that of outputs. of outputs. Following example illustrates its use:
Example – Example – Impulse Response of y[n] of y[n] = 0.2x[n] + 0.4x[n‐1] clc; clear all; close all
b = [0.2 0.4]; a = [1]; y = impz(b,a);
figure; stem(y,'filled'); title('Impulse Response of an of an FIR Filter'); xlabel('index, n'); ylabel('Value, x[n]'); axis([0 3 ‐0.5 1]);
Determine the impulse response of following of following difference equation:
y[n] − 1.56 y[n − 1] + 0.64 y[n − 2] = 0.2 x[n] + 0.4 x[n − 1] + 0.2 x[ n − 2] Considering the impulse response of system of system given in example & this task, what do you observe? State your findings.
12.4 FREQUENCY RESPONSE OF SYSTEM Given the coefficients of a of a filter, frequency response can be determined for a particular range of frequencies by using “Freqz”. The syntax is:
H = freqz (b, a, w) Where ‘b’ and ‘a’ are the filter coefficients and ‘w’ is the range of frequencies of frequencies for which the frequency response is desired to be determined. To plot the magnitude and phase of frequency of frequency response, use abs and angle respectively.
Example – Example – A Simple Lowpass Filter clc; clear all; close all
b = [1 2 1]; a = 1; w = ‐3*pi:1/100:3*pi;
H = freqz(b, a, w);
figure; subplot(3,1,1); plot(w, abs(H)); title('Magnitude of Frequency of Frequency Response of filter of filter with coefficients b_k={1, 2, 1}'); xlabel('\omega'); ylabel('H(\omega)'); grid;
subplot(3,1,2); plot(w, abs(H));
title('Zoomed view of the above graph from ‐\pi to \pi'); xlabel('\omega'); ylabel('H(\omega)'); axis([‐pi pi min(H) max(H)]); grid;
subplot(3,1,3); plot(w, angle(H)); title('Phase of Frequency Response of filter with coefficients b_k={1, 2, 1}'); xlabel('\omega'); ylabel('H(\omega)'); grid;
Plot the magnitude and phase of the frequency response of an FIR filter with coefficients b = {1 ‐2 4 ‐2 1}. Determine the type of filter.
A causal three point running average filter is given by
y[n] = ( x[n] + x[n − 1] + x[ n − 2]) 3 Find the frequency response of this filter from ‐3π to +3π.
Note: Freqz command can also be used as:
[ H W ] = freqz(b, a, n); Where H contains the frequency response, W contains the frequencies between 0‐π where the response is calculated, and n is the number of points at which to determine the frequency response. If n is missing, this value defaults to 512.
Example: A simple Lowpass filter clc clear all close all
b = [1 2 3 2 1]; a = [1];
[H W]=freqz(b, a);
figure; subplot(2,1,1); plot(W, abs(H), 'linewidth', 1.5); title('Magnitude of Frequency Response of filter with coefficients b_k={1, 2, 3, 2, 1}'); xlabel('\omega ‐‐‐‐>'); ylabel('H(\omega) ‐‐‐‐>'); grid;
subplot(2,1,2); plot(W, angle(H),'g', 'linewidth', 1.5); title('Phase of Frequency Response of filter with coefficients b_k={1, 2, 3, 2, 1}'); xlabel('\omega ‐‐‐‐>'); ylabel('H(\omega) ‐‐‐‐>'); grid;
Using the above‐mentioned syntax, determine the frequency response of a filter with filter coefficients {‐1, 3, ‐1}. Take 1024 points.
12.5 FILTER DESIGNING USING FIR1 Fir1 is used to design filter coefficients. The syntax of this command is:
b = fir 1( N , c); Where ‘b’ contains the filter coefficients returned by this command, ‘N’ is the order of filter, and ‘wc’ is the cutoff frequency normalized between 0
‐‐
1 where 1 refers to π. The number of coefficients created
is one more than the order of filter. By default, a lowpass filter is created. A highpass filter can be created using the argument ‘high’ at the end in the above command. A bandpass filter can be created using [Wa Wb] instead of Wc where frequencies between Wa and Wb are allowed to pass. If such a vector is specified instead of Wc, a bandpass filter is created by default. A bandstop filter can be created by using the argument ‘stop’ in the aforementioned command.
b = fir 1( N , ω c)
// creates coefficients of lowpass filter
b = fir 1( N , c, ' high ' )
// creates coefficien ts of highpass filter
b = fir 1( N , c,[Wa Wb])
// creates coefficien ts of bandpass filter
b = fir 1( N , ω c,[Wa Wb])
// creates coefficien ts of bandstop filter
After the coefficients are obtained, freqz command can be used to determine the frequency response. Note: For Highpass and bandstop filters, N must be even. Else Matlab will increase the order itself by 1.
Example: Designing High pass filter % High Pass filter clear, close all, clc
w = ‐pi:1/100:pi; b = fir1(4, 1/3, 'high') H = freqz(b, 1, w);
figure; subplot(2,1,1); plot(w/pi, abs(H)); title('High pass filter with cut‐off frequency of w_c=\pi/3'); xlabel('Normalized \omega = x\pi'); ylabel('H(\omega)'); grid;
b=fir1(1000, 1/3, 'high'); H=freqz(b, 1, w);;
subplot(2,1,2); plot(w/pi, abs(H)); title('High pass filter with cut‐off frequency of w_c=\pi/3'); xlabel('Normalized \omega = x\pi'); ylabel('H(\omega)'); grid;
Design a low‐pass filter with 7 coefficients and cutoff frequency of π/2.
Design a band‐pass filter which allows only the frequencies between π/6 and 2π/3 to pass and blocks the rest of frequencies. Design it for order 10 and 1000.
Design a band‐stop filter which stops the frequencies between π/3 and π/2 and allows the rest of frequencies to pass. Design it for order 10 and 1000.
Magnitude of an ideal high pass filter can be extracted from an ideal lowpass filter as Hhp=1‐Hlp and vice‐versa. Design a high‐pass filter with order 1000 and Wc=π/4. Using the high‐pass
