Unable to get how to solve for mutltivariable function while calculating DTFT

1 vue (au cours des 30 derniers jours)
Hi ,
I have the frequency response of a system given as:
H(e^(iθ))=1/(1-0.9*e^(-iθ))
and i am struggling to get the response of the above system when the input is
x(n)=0.5*cos((pi*n)/4)
.What i have tried and stucked at is how do i give the x(n) as a input to the function 'H' . . Can anybody help me out with this or other way ?

Réponse acceptée

Harsh Kumar
Harsh Kumar le 5 Juil 2023
Modifié(e) : Harsh Kumar le 14 Juil 2023
Hi Rohit,
I understand that you are trying to find the response of a system for a given input whose frequency response has been given already.
To do this, since the input is sinusoidal, you can use one of its property that the output will be just an amplified and phase differentiated version of itself when passed through the system.
Refer to the below code snippet for better understanding.
clc;
n = -25:1:25;
%system response at w=pi/4
w = pi/4;
H = 1/(1-0.9*exp(-1i*w));
mag = abs(H);
ang = angle(H);
%ouput=mag*sin(wt+ang) for sinosuidal input
output = 1/2*mag*cos((n*pi/4)+ang).*(n>=0);
stem(n,output);
You may refer to these documentation links for better understanding:
  7 commentaires
Harsh Kumar
Harsh Kumar le 14 Juil 2023
Yes ,(n>=0) done already in the code and the second was a typing error. Thanks for pointing out and your time .
Paul
Paul le 14 Juil 2023
Last thing: the stem plot needs to be redone based on the updated code.

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Plus de réponses (1)

William Rose
William Rose le 28 Juin 2023
Modifié(e) : William Rose le 28 Juin 2023
[Are you sure that is exactly how the frequency repsonse and the input funtion are defined?]
The frequency response is a function of θ. Let's assume .
The input x is a function of n. Let's assume n=t, so we have . Then we can write x(t) as , where . Now we recall that .
(assuming linearity)
If you continue the algebra and deal with the complex numbers, you should find that Y is real.
  3 commentaires
Paul
Paul le 28 Juin 2023
What is the advantage in changing the independent variable from n to t?
This solution is only the steady state solution, which might or might not meet the intent of the question.
William Rose
William Rose le 29 Juin 2023
@Paul,
Good point. I coud have, I should have kept n as n.
ALso a good point about steady state. The question has a transfer function and a sinusoidal input, so I assume that the "steady state" sinusoidal output is what is desired. If a transient solution were desired, then the initial condition of the output would need to be specified, and it is not.

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