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

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Rohit Kumar 2023-6-28

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 ?

### 采纳的回答

Harsh Kumar 2023-7-5

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:
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Paul 2023-7-14
Last thing: the stem plot needs to be redone based on the updated code.

### 更多回答（1 个）

William Rose 2023-6-28

[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.
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William Rose 2023-6-29
@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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