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 componentsc () 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 ( ) ( )