Going one step further, we can use the bandpass FIR filter design technique to design a highpass FIR filter. To obtain the coefficients for a highpass filter, we need only modify the shifting sequence sshift(k) to make it represent a sampled sinusoid whose frequency is fs/2. This process is shown in Figure 529. Our final 31tap highpass FIR filter's hhp(k) coefficients are
Figure 529. Highpass filter with frequency response centered at fs/2: (a) generating 31tap filter coefficients hhp(k); (b) frequency magnitude response Hhp(m).
whose Hhp(m) frequency response is the solid curve in Figure 529(b). Because sshift(k) in Figure 529(a) has alternating plus and minus ones, we can see that hhp(k) is merely hlp(k) with the sign changed for every other coefficient. Unlike Hbp(m) in Figure 528(b), the Hhp(m) response in Figure 529(b) has the same amplitude as the original Hlp(m).
Again, notice that the hlp(k) lowpass coefficients in Figure 529(a) have not been modified by any window function. In practice, we'd use a windowed hlp(k) to reduce the passband ripple before implementing Eq. (521).
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