Fourier Transform of $y(t)=x(t-c)\sum_{n=-\infty}^{\infty}\delta(t-nT)$

Question

I struggle to understand the solution of an exercise and would be grateful for your help.

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We have the following signal :
$$y(t)=x(t-c)\sum_{n=-\infty}^{\infty}\delta(t-nT)$$

The Fourier Transform of $x(t-c)$ is $$e^{-2\pi i f c} \hat{x}(f)$$ The Fourier Transform of $\sum_{n=-\infty}^{\infty}\delta(t-nT)$ is $$\frac{1}{T}\sum_{k=-\infty}^{\infty}\delta(t-\frac{k}{T})$$

Remembering that the FT of a product is a convolution, and also remembering the Sifting Property of the Delta Function, we have $$\hat{y}(f)=\frac{e^{-2\pi i f c}}{T}\sum_{k=-\infty}^{\infty}\hat{x}(f-\frac{k}{T})e^{2\pi i c k/T}$$

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My question is: where does this $e^{2\pi i c k/T}$ come from ?

Thanks for your help !

Answer 1

You need to apply the Sifting Property of the Delta Function to $e^{-2\pi i f c} \hat{x}(f)$, not just $\hat{x}(f)$. That results in the $e^{2 \pi i c k/T}$.