OE-8.2 Frequency Domain Analysis: DTFT & DFT
You can compute and apply the DTFT and DFT with their properties, relate the two, and perform circular convolution.
The DTFT is continuous in frequency and mostly a analytical tool; the DFT is what a computer can actually calculate, and the relationship between them is sampling in the frequency domain. Circular convolution is the DFT's native operation and it is not the linear convolution you want - zero-padding is how you get one from the other, and forgetting that produces wrong answers that look plausible. Practise the properties (shift, symmetry, Parseval) until they are reflexes; they turn hard problems into short ones.
Work through these
The discrete-time Fourier transform (DTFT)
The transform for a sequence, giving a continuous function of frequency. It is the theoretical tool, and the next transform is the computable version of it.
Properties of the DTFT
The properties that let you avoid recomputing from scratch. Convolution becoming multiplication is the one that matters most.
The discrete Fourier transform (DFT)
Sampling the transform above so a computer can hold it. Everything practical uses this one, so the relationship in two items' time deserves attention.
Properties of the DFT
The same kind of list for the computable version, with one important difference. Convolution here is circular rather than linear, which is the trap the last item of this topic addresses.
Relationship between the DFT and the DTFT
How the two transforms relate, which is where the differences above come from. Understanding this properly prevents most of the confusion in this subject.
Circular convolution
What the wraparound actually does to a convolution. Compute a small example by hand once and the next item will make immediate sense.
Linear convolution using the DFT (zero padding)
The fix: pad with zeros so the wraparound has nothing to corrupt. It is the practical recipe you will use every time you filter with a transform.
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