% EE3220, Week 4, January 7, 2014, Dr. Durant



% Example to test numerically whether a function is (1) linear and (2)

% time-invariant in MATLAB.  Note: A numeric test is not a proof.  If a

% test of sufficient complexity indicates linearity or TI, it is probably,

% but not certainly, true.  If a numeric test shows a function is NOT

% linear or TI, then you know that it is definitely not.  So, we can prove

% falsity, but not truth.



x1 = [1 2 3]; % Two test input signals

x2 = [4 -1 -3]; % both assumed to start at n=0



x1p2 = x1+x2;



% In this example, the system is y(n) = x(n)^2



% Is the system linear (at least for the test signals)?

y1 = x1.^2; % apply system to x1

y2 = x2.^2;

y_x1p2 = x1p2.^2; % apply system AFTER summing inputs

y1p2 = y1 + y2; % apply summation AFTER system

error_seq_lin = y_x1p2 - y1p2 % no semicolon displays the error sequence

fprintf('The error relative to being a linear system is %g.\n',...

    sum(abs(error_seq_lin).^2))



% Is the system time-invariant?  Test using x1 and a delay of 2 samples

x1_d = [zeros(1,2) x1]; % delay by 2 samples

y_x1d = x1_d.^2; % apply the system AFTER the delay



y1d = [zeros(1,2) y1]; % apply the delay AFTER the system

error_seq_ti = y_x1d - y1d % display the time-invariance error sequence (0 if time invariant)

fprintf('The error relative to being a time-invariant system is %g.\n',...

    sum(abs(error_seq_ti).^2))



% In this case, the error relative to being a linear system is 404,

% therefore the system is not linear.



% In this case, the error relative to being a time-invariant system is 0,

% therefore we were unable to prove that the system is not time-invariant

% and we will assume it is time-invariant (and, it is indeed

% time-invariant).