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Properties <strong>of</strong> the <strong>Frequency</strong> <strong>Response</strong>• In particular if the coefficients <strong>of</strong> a digital filter are real, that*is b k= b kor hk = h * k, thenpro<strong>of</strong>–Conjugate Symmetry: He jˆ = H * e jˆH * e jˆ – b ke jˆ k * M* jˆ k= = b k e =• A consequence <strong>of</strong> conjugate symmetry is that(6.21)(6.22)(6.23)– We say that the magnitude response is an even function <strong>of</strong>ˆ and the phase is an odd function <strong>of</strong> ˆ• It also follows thatMk = 0Mk = 0b ke j– –ˆ kk = 0–He jˆ• When plotting the frequency response we can use the abovesymmetry to just plot over the interval 0 ˆ =He – jˆ = He jˆHe – jˆ = – He j ˆReHe – jˆ = ReHe jˆ(6.24)ImHe – jˆ = – ImHe jˆ– We say that the real part response is an even function <strong>of</strong> ˆand the imaginary part is an odd function <strong>of</strong> ˆECE 2610 Signals and Systems 6–17
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