Unable to get how to solve for mutltivariable function while calculating DTFT
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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 ?
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Accepted Answer
Harsh Kumar
on 5 Jul 2023
Edited: Harsh Kumar
on 14 Jul 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 Comments
Harsh Kumar
on 14 Jul 2023
Yes ,(n>=0) done already in the code and the second was a typing error. Thanks for pointing out and your time .
More Answers (1)
William Rose
on 28 Jun 2023
Edited: William Rose
on 28 Jun 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
.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421753/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421758/image.png)
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
.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421763/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421768/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421773/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421778/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421783/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1421788/image.png)
If you continue the algebra and deal with the complex numbers, you should find that Y is real.
3 Comments
Paul
on 28 Jun 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
on 29 Jun 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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