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fourier analysis: signals and filters 253τ τ τΣFigure 10.6 A delay-line filter in which the signal at different time translations is scaled bydifferent amounts c i .And of course, if the signal’s input is a discrete sum, its output will remain a discretesum. (We restrict ourselves to nonrecursive filters [Pres 94].) In either case, we seethat knowledge of the filter coefficients c i provides us with all we need to knowabout a digital filter. If we look back at our work on the discrete Fourier transformin §10.4.1, we can view a digital filter (10.66) as a Fourier transform in which weuse an N-point approximation to the Fourier integral. The c n ’s then contain boththe integration weights and the values of the response function at the integrationpoints. The transform itself can be viewed as a filter of the signal into specificfrequencies.10.7.1 Digital Filters: Windowed Sinc Filters (Exploration) ⊙Problem: Construct digital versions of highpass and lowpass filters and determinewhich filter works better at removing noise from a signal.A popular way to separate the bands of frequencies in a signal is with a windowedsinc filter [Smi 99]. This filter is based on the observation that an ideal lowpass filterpasses all frequencies below a cutoff frequency ω c and blocks all frequencies abovethis frequency. And because there tends to be more noise at high frequencies thanat low frequencies, removing the high frequencies tends to remove more noise thansignal, although some signal is inevitably lost. One use for windowed sinc filters isin reducing aliasing by removing the high-frequency component of a signal beforedetermining its Fourier components. The graph on the lower right in Figure 10.7was obtained by passing our noisy signal through a sinc filter (using the programFilter.java given on the CD).If both positive and negative frequencies are included, an ideal low-frequencyfilter will look like the rectangular pulse in frequency space:( ) ωH(ω, ω c )=rect2ω c⎧⎨1, if |ω|≤ 1 2rect(ω)=,⎩0, otherwise.(10.67)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 253