Phase and Group Delays In the case of the modulating component we have ec(O4)-0c(00)c0e(0 O,-0 da The le parameter C g =0 is called the group delay or envelope delay caused by the system at =o Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 11 Phase and Group Delays • In the case of the modulating component we have • The parameter is called the group delay or envelope delay caused by the system at c d d c u c u c = − − − − ( ) ( ) ( ) c d d c g c = = − ( ) ( ) = c
Phase and Group Delays The group delay is a measure of the inearity of the phase function as a function of the frequency It is the time delay between the waveforms of underlying continuous-time signals whose sampled versions, sampled at t=nT are precisely the input and the output discrete-time signals 12 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 12 Phase and Group Delays • The group delay is a measure of the linearity of the phase function as a function of the frequency • It is the time delay between the waveforms of underlying continuous-time signals whose sampled versions, sampled at t = nT, are precisely the input and the output discrete-time signals
Phase and Group Delays If the phase function and the angular frequency o are in radians per second, then the group delay Is in seconds Figure below illustrates the evaluation of the phase delay and the group delay 6(U) A(o 0 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 13 Phase and Group Delays • If the phase function and the angular frequency are in radians per second, then the group delay is in seconds • Figure below illustrates the evaluation of the phase delay and the group delay
Phase and Group Delays Figure below shows the waveform of an amplitude-modulated input and the output generated by an lti system tftttftt t Inp t Output 14 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 14 Phase and Group Delays • Figure below shows the waveform of an amplitude-modulated input and the output generated by an LTI system
Phase and Group Delays Note The carrier component at the output is delayed by the phase delay and the envelope of the output is delayed by the group delay relative to the waveform of the underlying continuous-time input signal The waveform of the underlying continuous time output shows distortion when the group delay of the Lti system is not constant over the bandwidth of the modulated signal 15 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 15 Phase and Group Delays • Note: The carrier component at the output is delayed by the phase delay and the envelope of the output is delayed by the group delay relative to the waveform of the underlying continuous-time input signal • The waveform of the underlying continuoustime output shows distortion when the group delay of the LTI system is not constant over the bandwidth of the modulated signal