华南理工大学电子与信息学院：《数字信号与处理》（英文版）Lecture 10 Phase and Group Delays

Phase and Group Delays The output yn]of a frequency-selective LTI discrete-time system with a frequency response H(e/o)exhibits some delay relative to the input x[n caused by the nonzero phase response 0(a)=argh(eJo) of the system · For an input xn]=Acos(00n+中),-∞<n<0 Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 1 Phase and Group Delays • The output y[n] of a frequency-selective LTI discrete-time system with a frequency response exhibits some delay relative to the input x[n] caused by the nonzero phase response of the system • For an input ( ) j H e ( ) arg{ ( )}    = j H e x[n] = Acos(on + ), −  n  

Phase and Group Delays the output is y=AH(e0)co0n+00,)+) Thus, the output lags in phase by 8(Oo) radians Rewriting the above equation we get 0() y[n]=AH(eJ0o )cosmo n+0+d O Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 2 Phase and Group Delays the output is • Thus, the output lags in phase by radians • Rewriting the above equation we get [ ] = ( ) cos( + ( ) + )  o o j y n AH e o n ( )  o          +          =  +  o o o j y n AH e o n ( ) [ ] ( ) cos

Phase and Group Delays This expression indicates a time delay. known as phase delay, at o=@o given by τ(0l)= 0(Oo) Now consider the case when the input Signal contains many sinusoidal components with different frequencies that are not harmonically related Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 3 Phase and Group Delays • This expression indicates a time delay, known as phase delay, at given by • Now consider the case when the input signal contains many sinusoidal components with different frequencies that are not harmonically related = o o o p o      = − ( ) ( )

Phase and Group Delays n this case, each component of the input will go through different phase delays when processed by a frequency-selective LTI discrete-time system Then, the output signal, in general, will not look like the Inp ut signal The signal delay now is defined using a different parameter Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 4 Phase and Group Delays • In this case, each component of the input will go through different phase delays when processed by a frequency-selective LTI discrete-time system • Then, the output signal, in general, will not look like the input signal • The signal delay now is defined using a different parameter

Phase and Group Delays To develop the necessary expression consider a discrete-time signal xn obtained by a double-sideband suppressed carrier DSB-SC) modulation with a carrier frequency o of a low-frequency sinusoidal signal of frequency @o x[n=acos(oon cos(ocn) Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 5 Phase and Group Delays • To develop the necessary expression, consider a discrete-time signal x[n] obtained by a double-sideband suppressed carrier (DSB-SC) modulation with a carrier frequency of a low-frequency sinusoidal signal of frequency : o c x[n] Acos( n)cos( n) = o c

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(e0)=1 for oe≤0≤02 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 V[n]=A cos(oen+0(oD))+ cos(o,n+O(Ou)) Acos on+ e(o4)+6(0 0(04)-((0c coS Oon+ 2 Note: The output is also in the form of a modulated carrier signal with the same carrier frequency @c 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 ditterent 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 o.i. e o, is very small 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 ec(o)as d0(0) 0(0)=0(0c)+ 0=0 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 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 0(01)+0(0)0(0) 2c 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 ( ) ( ) 