! Frequency Response of LTI Systems. " Magnitude Response. " Phase Response. " Example: Zero on Real Axis. ! We can define a magnitude response

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1 Lectue Outline ESE 53: Digital Signal Pocessing Lec 3: Febuay 23st, 207 Fequency Response of LTI Systems! Fequency Response of LTI Systems " agnitude Response " Simple Filtes " Phase Response " Goup Delay " 2 Fequency Response of LTI System Fequency Response of LTI System! LTI Systems ae uniquely detemined by thei impulse esponse y n = x k h n k = x k h k k=! We can wite the input-output elation also in the z-domain Y ( z) = H ( z) X ( z)! O we can define an LTI system with its fequency esponse Y ( e j ) = H ( e j ) X ( e j )! H(e j ) defines magnitude and phase change at each fequency! We can define a magnitude esponse! And a phase esponse Y ( e j ) = H ( e j ) X ( e j ) ( ) = H ( e j ) X ( e j ) Y e j Y ( e j j j ) = H ( e ) + X ( e ) 3 4 Phase Response Phase Response! Limit the ange of the phase esponse! Limit the ange of the phase esponse 5 6

2 Goup Delay Linea Diffeence Equations! Geneal phase esponse at a given fequency can be chaacteized with goup delay, which is elated to phase! oe late 7 8 agnitude Response agnitude Response 9 0 agnitude Response agnitude Response Example e j v d k 2 2

3 agnitude Response Example agnitude Response Example e j v 2 v 3 4 agnitude Response Example Simple Low Pass Filte v 2 v e j H(e j ) Simple Low Pass Filte Simple High Pass Filte H(e j ) 2 c 7 8 3

4 Simple High Pass Filte Simple High Pass Filte e j e j v 2 v H(e j ) v 2 v 2 c 9 20 Simple Band-Stop (otch) Filte Simple Band-Stop (otch) Filte Simple Band-Stop (otch) Filte Simple Band-Stop (otch) Filte H(e j ) H(e j )

5 Simple Band-Stop (otch) Filte Simple Band-Pass Filte H(e j ) Simple Band-Pass Filte Simple Band-Pass Filte H(e j ) Simple Band-Pass Filte Phase Response! Limit the ange of the phase esponse H(e j ) Lage α educes pass band

6 Phase Response Example Goup Delay! Geneal phase esponse at a given fequency can be chaacteized with goup delay, which is elated to phase ARG slope 32 Phase Response Example Goup Delay! Geneal phase esponse at a given fequency can be chaacteized with goup delay, which is elated to phase ARG 2 Fo linea phase system, goup delay is n d 33 - slope 34 Goup Delay Goup Delay slope - slope

7 Goup Delay ath Goup Delay ath H(z) = b 0 a 0 ( c k z ) ( d k z ) H(e j ) = b 0 a 0 ( c k e j ) ( d k e j ) H(z) = b 0 a 0 ( c k z ) ( d k z ) H(e j ) = b 0 a 0 ( c k e j ) ( d k e j ) ag of poducts is sum of ags ag[h(e j )] = ag[ c k e j ] ag[ d k e j ] gd[h(e j )] = gd[ c k e j ] gd[ d k e j ] Goup Delay ath gd[h(e j )] = gd[ c k e j ] gd[ d k e j ] ag[ e j ]! Look at each facto: ag[ e jθ e j ] = tan sin( θ) cos( θ) gd[ e jθ e j ] = 2 cos( θ) e jθ e j ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ]

8 ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] Penn ESE 53 Sping Khanna 43 Penn ESE 53 Sping Khanna 44 ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] ag = 0 ag Penn ESE 53 Sping Khanna ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] = ag ag

9 ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] ag[ e j ] = ag[(e j )e j ] = ag[e j ] ag[e j ] ag ag gd Goup Delay ath gd[h(e j )] = gd[ c k e j ] gd[ d k e j ]! Fo θ 0! Look at each facto: ag[ e jθ e j ] = tan sin( θ) cos( θ) gd[ e jθ e j ] = 2 cos( θ) e jθ e j ! agnitude Response! Fo θ=, how does zeo location effect magnitude, phase and goup delay?

10 ! Fo θ=, how does zeo location effect magnitude, phase and goup delay?! Fo θ=, how does zeo location effect magnitude, phase and goup delay? nd Ode IIR with Complex Poles 2 nd Ode IIR with Complex Poles phase magnitude magnitude goup delay Penn ESE 53 Sping Khanna 57 Penn ESE 53 Sping Khanna 58 Big Ideas Admin! Fequency Response of LTI Systems! HW 5 " agnitude Response " Due Fiday 3/3 " Simple Filtes " Phase Response " Goup Delay! Homewok solutions to be posted soon " 59 Penn ESE 53 Sping Khanna 60 0