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I'll start off by saying, yes this is a homework problem, and no I do not want the answer. I simply need to know where to start.

The input to a system with a frequency response of $H(\omega)$ is $x(n)=A\sin(\omega_0 n+\theta)$

What is the expression for the systems output, $y(n)$.

As I said, I just am unsure of where to start.

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  • $\begingroup$ What's your math & systems background? Are you familiar with complex numbers and basic system theory (LTI) systems? This question is about one of the most fundamental properties of Linear Time Invariant systems so it's difficult to give a hint. There is no math required, you can just write down the answer directly. $\endgroup$ – Hilmar Feb 13 '15 at 21:53
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Hint:

I guess we may assume that the system is real-valued, i.e. its impulse response $h(n)$ is real-valued. This means that the response to the imaginary part of some complex input signal $\hat{x}(n)$ equals the imaginary part of the response to the complex signal $\hat{x}(n)$:

$$\mathcal{T}\{\Im\{\hat{x}(n)\}\}=\Im\{\mathcal{T}\{\hat{x}(n)\}\}$$

where $\mathcal{T}$ is the system operator.

Note that your input signal $x(n)$ is the imaginary part of

$$\hat{x}(n)=Ae^{j(\omega_0n+\theta)}\tag{1}$$

The frequency response is related to $h(n)$ by

$$H(\omega)=\sum_{n=-\infty}^{\infty}h(n)e^{-jn\omega}\tag{2}$$

The output $y(n)$ is given by the following convolution sum

$$y(n)=\sum_{k=-\infty}^{\infty}h(k)x(n-k)\tag{3}$$

Now use (3) with the complex input signal $\hat{x}(n)$ given by (1), make use of (2), and then take the imaginary part to obtain $y(n)$. The response of a system with frequency response $H(\omega)$ to a complex exponential $e^{j(\omega_0n+\theta)}$ is a fundamental result which you should remember.

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Let me make it easier for you to solve:

y(n) = x(n) (convolution operator) h(n)----> in Discrete domain[ as also wrote in equation 3 by Matt.]

But usually we avoid convolution. So in Frequency domain it is just multiplication.

Y(w) = X(w)*H(w) ---> in Freuquency domain.

To convert from discrete(n- domaain) to freq(w- domain) we use Fourier transform and to get back inverse fourier.

I hope now you will be easily able to do your homework now

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