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Properties <strong>of</strong> the <strong>Frequency</strong> <strong>Response</strong>• The difference equation is?Periodicity <strong>of</strong>He jˆ • For any discrete-time LTI system, the frequency response isperiodic, that isHe j ˆ + 2 He jˆ• The pro<strong>of</strong> for <strong>FIR</strong> filters followsHe j ˆM + 2– b ke j ˆ + 2=k = 0=Mk = 0–b ke jˆ (6.19)(6.20)• This result is consistent with the fact that sinusoidal signalsare insensitive to 2 frequency shifts, i.e.,• In summary, the frequency response is unique on at most a2 interval, say in particular the interval – ˆ Conjugate Symmetry• The frequency response He jˆ is in general a complexquantity, in most cases obeys certain symmetry propertiese – j21=He jˆxn Xe j ˆ + 2nXe jˆ n j2n= = e =Xe jˆ nECE 2610 Signals and Systems 6–16
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