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