In Section 5.2 we introduced FIR filters with an averaging example, and that's where we first learned that the process of timedomain averaging performs lowpass filtering. In fact, successive timedomain outputs of an Npoint averager are identical to the output of an (N–1)tap FIR filter whose coefficients are all equal to 1/N, as shown in Figure 1111.
Figure 1111. An Npoint averager depicted as an FIR filter.
The question we'll answer here is "What is the frequency magnitude response of a generic Npoint averager?" We could evaluate Eq. (628), with all a(k) = 0, describing the frequency response of a generic Nstage FIR filter. In that expression, we'd have to set all the b(0) through b(N) coefficient values equal to 1/N and calculate HFIR(w)'s magnitude over the normalized radian frequency range of 0 w p. That range corresponds to an actual frequency range of 0 f fs/2 (where fs is the equivalent data sample rate in Hz). A simpler approach is to recall, from Section 5.2, that we can calculate the frequency response of an FIR filter by taking the DFT of the filter's coefficients. In doing so, we'd use an Mpoint FFT software routine to transform a sequence of N coefficients whose values are all equal to 1/N. Of course, M should be larger than N so that the sin(x)/x shape of the frequency response is noticeable. Following through on this by using a 128point FFT routine, our Npoint averager's frequency magnitude responses, for various values of N, are plotted in Figure 1112. To make these curves more meaningful, the frequency axis is defined in terms of the sample rate fs in samples/s.
Figure 1112. Npoint averager's frequency magnitude response as a function of N.
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