Phase and Group Delays The output yIn] 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(o)=argH(e/o) of the system For an input x{]=AcoS(00n+),-<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[n]=AH(e/oo )cos(oon+0(Oo)+d) Thus, the output lags in phase by 0(oo) radians Rewriting the above equation we get [n]=AH(e/oo )cosool n+ 0(o)+ 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 0(0o p(wo O 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 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 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 y a double-sideband suppressed carrier (DSB-SC) modulation with a carrier frequency o of a low-frequency sinusoidal signal of frequency o xn=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(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 componentsc () 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 ( ) ( )