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Suppose I have the standard convolution of two functions $x$ and $y$, either continuous convolution or discrete convolution.

If both are periodic, the result of the convolution will also be periodic. But what if only one is periodic? Will the result be periodic under certain conditions?

Never saw any property about this.

Thanks.

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Assuming $y(t)$ is periodic, a shifted version of $y(t)$, $y(t+T_p) = y(t)$, where $T_p$ is the periodicity.

Convolution is integrating the product of $x(t)$ with a shifted (and flipped) version of $y(t)$. Since the shifted versions repeat after the periodicity $T_p$, the convolution integral result too keeps repeating with the same periodicity.

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Take a look at Periodic Convolution Progress can be made employing the Fourier analysis to perform Periodic convolution or circular convolution. The link I have given explains the situation well so I will not reiterate all the steps here. But in summary:

The Fourier Transform of the circular convolution has impulses at all (common) frequencies where the Fourier transforms of x(t) and h(t) have impulses. This result is the equivalent of the Convolution theorem in the context of periodic convolution.

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