The Frequency Response Note: Magnitude and phase functions are real functions of @, whereas the frequency response is a complex function of o If the impulse response hn is real then it follows from Table 3. 4 that the magnitude function is an even function of o H(e(°)=H(e and the phase function is an odd function of 6(0)=-((-0) 16 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 16 The Frequency Response • Note: Magnitude and phase functions are real functions of , whereas the frequency response is a complex function of • If the impulse response h[n] is real then it follows from Table 3.4 that the magnitude function is an even function of : and the phase function is an odd function of : ( ) ( ) − = j j H e H e () = −(−)
The Frequency Response Likewise, for a real impulse response h[n Hre(e/o)is even and him(e/o) is odd Example -Consider the M-point moving average filter with an impulse response given by mm=10,0≤n≤M-1 otherwise Its frequency response is then given by H(e0)=1∑e~0n Mn=0 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 17 The Frequency Response • Likewise, for a real impulse response h[n], is even and is odd • Example - Consider the M-point moving average filter with an impulse response given by • Its frequency response is then given by ( ) j re H e ( ) j Him e 0 otherwise 1 0 1 , / M, n M − h[n] = = − = − 1 0 1 ( ) M n j n M j H e e