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    Lab6 Discrete-Time Signals and Systems
    Activity 1:
    A discrete-time system can be described in one of two ways. One is by the unit-pulse response of the system and the other is by the difference equation. In the first case if we are given an input and want to find the output we have to convolve the system response with the input signal. In the second case we have to solve the difference equation for the output when the input signal is substituted into the difference equation. The result should be the same no matter what approach is used.
    A discrete-time system has the following unit-pulse response:
    for
    Correspondingly the following difference equation describes the behavior of the system:
    Matlab Tip: Remember that you can easily get help on a command in Matlab by typing help with the command at the prompt:
    EDU>>help conv
    Matlab Tip: Also don’t forget that in Matlab when you do certain operations on an array element-by-element you must precede the operator with a period. For instance in Matlab if you define an array n as follows:
    EDU>>n=[0:20];
    then the first term of the unit-pulse response is written as 0.5.^n
    Matlab Tip: Sometimes it is useful to know the size of arrays you have created. This can be done in several ways. For example you can use the command size:
    EDU>> size(n)
    ans =
    1 21
    The size of the array is returned. In this case the array n has a size of 1 row by 21 columns.
    Alternatively you can switch the display from the Current Directory to the Workspace. Then all current variables and arrays and their sizes are displayed. Just click on Workspace in the upper left corner as indicated below.
    The Matlab function recur is not a function which comes with Matlab or any of the toolboxes you have installed. Instead it is a that must be added to your Matlab directory.
    The Matlab function recur can be found in a zip file at the Mathworks site at the following location:
    http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=2148
    Extract the file recur.m to the current directory shown at the top of the Matlab window.
    On the other hand you can copy and paste the script below into a text file like Notepad and save it as recur.m in the current directory shown at the top of the Matlab window.
    function y = recur(abnxx0y0);
    %
    % y = recur(abnxx0y0)
    % solves for y[n] from:
    % y[n] + a1*y[n-1] + a2*y[n-2]… + an*y[n-N]
    % = b0*x[n] + b1*x[n-1] + … + bm*x[n-M]
    %
    % a b n x x0 and y0 are vectors
    % a = [a1 a2 … aN]
    % b = [b0 b1 … bM]
    % n contains the time values for which the solution will be computed
    % y0 contains the initial conditions for y in order
    % i.e. y0 = [y[n0-N] y[n0-N+1] …y[n0-1]]
    % where n0 represents the first element of n
    % x0 contains the initial conditions on x in order
    % i.e. x0 = [x[n0-M]…x[n0-1]]
    % the output y has length(n)
    %
    N = length(a);
    M = length(b)-1;
    if length(y0) ~= N
    error(‘Lengths of a and y0 must match’)
    end
    if length(x0) ~= M
    error(‘Length of x0 must match length of b-1′)
    end
    y = [y0 zeros(1length(n))];
    x = [x0 x];
    a1 = a(length(a):-1:1); % reverses the elements in a
    b1 = b(length(b):-1:1);
    for i=N+1:N+length(n)
    y(i) = -a1*y(i-N:i-1)’ + b1*x(i-N:i-N+M)’;
    end
    y = y(N+1:N+length(n));
    To get information on how to use the function at the Matlab prompt type:
    EDU>>help recur
    It might be helpful to change the index of the difference equation such that the highest index is n rather than n+2.
    HW6 Discrete-Time Signals and Systems
    x[0] = 4 x[1] = 1 x[3] = -1 x[n] = 0 for all other integers n.
    v[0] = 1 v[1] = -2 v[2] = 3 v[3] = -4 v[n] = 0 for all other integers n.
    Compute the convolution x[n]*v[n] for n 0.

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