Frequency Response of FIR Filters

• We now use (6.10) to simplify (6.9)yn =A-- He jˆ 0 j ˆ e 0n + 2 ASuperposition and the Frequency Response-- H* e jˆ 0 j+ e2– ˆ 0n + = AH e jˆ 0e j ˆ 0n + + He jˆe – j ˆ 0n + +He jˆ+ ------------------------------------------------------------------------------------------------2(6.11)yn AH e jˆ 0 cosˆ 0n + + He jˆ 0=ˆ• We see that when a real sinusoid passes through an LTI system,such as an FIR filter (having real coefficients), the outputis also a real sinusoid which has picked up the magnitudeand phase of the system at = ˆ 0• The generalization (sum of sinusoids) of this result is whenNxn = X 0+ X kcosˆ kn + X k, (6.12)k = 1then the corresponding LTI system output isyn =X 0He j0 +Nk = 1X kHe jˆ k cos ˆ kn+ X k+He jˆ k(6.13)ECE 2610 Signals and Systems 6–7