Some textbooks like (Numerical recipes the art of scientific computing) derive the DFT as a Riemann sum of the CTFT. With this in mind it would be natural then to approximate the identity(adsbygoogle = window.adsbygoogle || []).push({});

##y(t)=x*h=\mathcal{F}^{-1}\big\{XH\big\}##

with the mathlab code y=ifft(fft(x).*fft(h)) which roughly means that my response is the inverse DFT of the product of the DFTs.

I have been reading recently that this approach isn't really valid in the case of the DFT. The relevant identity in the discrete realm is

##\mathcal{F}^{-1}(XH)_{n}=\sum_{l=0}^{N-1} x_l (y_N)_{n-l}##

which is called a circular convolution.

But so far, my ifft(fft...) approach yields results that are completely compatible with the analytical results.

I also want to mention that I have checked for the well-known result that convolving with a shifted impulse, shifts your response along the domain; and the result that scaling an impulse, scales the response.

I've implemented impulses by inputting the coefficient in the dirac-delta function in some position on a vector, without really justifying why this works.

So therefore I have two questions:

Why is the y=ifft(fft(x).*fft(h)) approach valid and compatible with the theory? Is this the way its supposed to be? How can I justify such a thing? Why aren't the step factors ##dt## involved in this approach?

Why are impulses correctly modeled by only inputting the coefficient that would correspond to the dirac-delta function? (This seems intuitive to me, but I wouldn't like to say my opinion as I would prefer to listen to you mostly).

Finally, if I want to analyze another signal like cos(t), I do have to include the step ##dt## in order to get results congruous with the analytical results. Why is this occurring ?

Thanks!

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# Analyzing RC response with convolution theorem and fft.

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