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I am working my way through recorded lectures and a textbook related to DSP, and have come across a question that I am not sure how to answer. This is probably just due to how new I am to these concepts... if so, please forgive my beginner question.

Given a downsampling system such as:

$$ y[n] = \mathcal{T}\{x[n]\} = x[2n] $$

Clearly, the system is linear, but is not shift-invariant. So the system can not be represented as a convolution of the impulse response (in fact, the impulse response is just $h[n] = \mathcal{T}\{\delta[n]\} = \delta[n]$).

Now the question: is it possible to calculate a frequency response for this system? If this were an LSI, I would do this by taking the discrete Fourier transfer of $h[n]$, but that is not an option.

I thought I might be able to calculate it via:

$$ H(e^{i \omega}) = \frac{\mathcal{F}\{y[n]\}}{\mathcal{F}\{x[n]\}}\\ = \frac{e^{2 i \omega n}}{e^{i \omega n}} = e^{i \omega} $$

But that doesn't look right and I think that would only work if $\mathcal{T}$ were an LSI.

Is there another approach for calculating a frequency response? Or do frequency responses simply depend on a system being LSI?

Thanks!

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A linear time-varying (LTV) (or shift-variant) system does not have a frequency response in the conventional sense, which would mean that it is the eigenvalue of the system corresponding to the eigenfunction $e^{i\omega t}$.

You can, however, define an impulse response of an LTV system as the response to the shifted impulse $\delta[n-k]$, which is a two-dimensional function depending on both $n$ and $k$. The Fourier transforms of this two-dimensional impulse response with respect to $n$ or $k$ are called spreading function and TF (time-frequency) transfer function, respectively, which can be viewed as generalizations of the frequency response of a linear time- (or shift-)invariant system.

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