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Frequency Domain Filtering

Module by: Don Johnson

Summary: Explains the complication of dealing with both continuous and discrete frequency.

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Before detailing this procedure, let's clarify why so many new issues arose in trying to develop a frequency-domain implementation of linear filtering. The frequency-domain relationship between a filter's input and output is always true: Yⅇⅈ2πf=Hⅇⅈ2πfXⅇⅈ2πf Y 2 f H 2 f X 2 f . This Fourier transforms in this result are discrete-time Fourier transforms; for example, Xⅇⅈ2πf=∑nxnⅇ-ⅈ2πfn X 2 f n n x n 2 f n . Unfortunately, using this relationship to perform filtering is restricted to the situation when we have analytic formulas for the frequency response and the input signal. The reason why we had to "invent" the discrete Fourier transform (DFT) has the same origin: The spectrum resulting from the discrete-time Fourier transform depends on the continuous frequency variable ff. That's fine for analytic calculation, but computationally we would have to make an uncountably infinite number of computations.

note:

Did you know that two kinds of infinities can be meaningfully defined? A countably infinite quantity means that it can be associated with a limiting process associated with integers. An uncountably infinite quantity cannot be so associated. The number of rational numbers is countably infinite (the numerator and denominator correspond to locating the rational by row and column; the total number so-located can be counted, voila!); the number of irrational numbers is uncountably infinite. Guess which is "bigger?"
The DFT computes the Fourier transform at a finite set of frequencies — samples the true spectrum — which can lead to aliasing in the time-domain unless we sample sufficiently fast. The sampling interval here is 1K 1 K for a length-KK DFT: faster sampling to avoid aliasing thus requires a longer transform calculation. Since the longest signal among the input, unit-sample response and output is the output, it is that signal's duration that determines the transform length. We simply extend the other two signals with zeros (zero-pad) to compute their DFTs.

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