华南理工大学：《数字信号处理》（双语版） Chapter 10 Phase and Group Delays

Phase and Group Delays The input can be rewritten as xn]=a cos(oen)+a cos(Oun) where oe=oc-Oo and Qu=oc+oo Let the above input be processed by an Lti discrete-time system with a frequency response H(e/o) satisfying the condition H(e/0)三1 for oe≤0≤Oln Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 6 Phase and Group Delays • The input can be rewritten as where and • Let the above input be processed by an LTI discrete-time system with a frequency response satisfying the condition [ ] cos( ) cos( ) 2 2 x n n un A A =  +   = c −o u = c +o ( ) j H e u j H e        ( ) 1 for

Phase and Group Delays The output yn] is then given by y[n]=A cos(on+0(o2))+acos(o, n+O(Ou) Acos ocn≠on)+0 6(0n)-6(0c) coS Oon+ 2 2 Note: The output is also in the form of a modulated carrier signal with the same carrier frequency Oc and the same modulation frequency @o as the input Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 7 Phase and Group Delays • The output y[n] is then given by • Note: The output is also in the form of a modulated carrier signal with the same carrier frequency and the same modulation frequency as the input [ ] cos( ( )) cos( ( )) 2 2 u u A A y n =  n +   +  n +           −     +        +   =  + 2 ( ) ( ) cos 2 ( ) ( ) cos  u  o u A cn n c o

Phase and Group Delays However. the two components have different phase lags relative to their corresponding components in the input Now consider the case when the modulated input is a narrowband signal with the frequencies o and o, very close to the carrier frequency Oc, 1. e. @o is very small 8 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 8 Phase and Group Delays • However, the two components have different phase lags relative to their corresponding components in the input • Now consider the case when the modulated input is a narrowband signal with the frequencies and very close to the carrier frequency , i.e. is very small  u c o

Phase and Group Delays In the neighborhood of o we can express the unwrapped phase response Ac(o)as 0c(0)≡6c(0)+ d0(o) do O=0 C by making a Taylors series expansion and keeping only the first two terms Using the above formula we now evaluate the time delays of the carrier and the modulating components 9 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 9 Phase and Group Delays • In the neighborhood of we can express the unwrapped phase response as by making a Taylor’s series expansion and keeping only the first two terms • Using the above formula we now evaluate the time delays of the carrier and the modulating componentsc  () c ( ) ( ) ( ) ( ) c c c c c c d d  −         + =

Phase and Group Delays In the case of the carrier signal we have ec(O1)+0e(0)0c(02 20c which is seen to be the same as the phase delay if only the carrier signal is passed through the system Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 10 Phase and Group Delays • In the case of the carrier signal we have which is seen to be the same as the phase delay if only the carrier signal is passed through the system c c c c c u c     −    +   − ( ) 2 ( ) ( ) 