The Frequency Response Then we can write v[n]=H(e/oe! Thus for a complex exponential input signal e on, the output of an lti discrete-time system is also a complex exponential signal of the same frequency multiplied by a complex constant H(e/) Thus eJon is an eigen function of the system Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 11 The Frequency Response • Then we can write • Thus for a complex exponential input signal , the output of an LTI discrete-time system is also a complex exponential signal of the same frequency multiplied by a complex constant • Thus is an eigen function of the system j j n y n H e e [ ] = ( ) ( ) j H e j n e j n e
The Frequency Response The quantity H(e/)is called the frequency response of the lti discrete-time system H(eo) provides a frequency-domain description of the system H(eu) is precisely the dtft of the impulse response hn of the system 12 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 12 The Frequency Response • The quantity is called the frequency response of the LTI discrete-time system • provides a frequency-domain description of the system • is precisely the DTFT of the impulse response {h[n]} of the system ( ) j H e ( ) j H e ( ) j H e
The Frequency Response H(e ), in general, is a complex function of o with a period 2 It can be expressed in terms of its real and Imaginary parts H(e10)=H12(e10)+jHmn(e0) or, in terms of its magnitude and phase, H(e)=h(elo)- f(0) where 0(o)=argH(eo) Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 13 The Frequency Response • , in general, is a complex function of with a period 2p • It can be expressed in terms of its real and imaginary parts or, in terms of its magnitude and phase, where ( ) j H e ( ) ( ) ( ) = + j im j re j H e H e j H e ( ) ( ) ( ) = j j j H e H e e ( ) arg ( ) = j H e
The Frequency Response The function H(ejo)is called the magnitude response and the function 0(o) is called the phase response of the lti discrete-time system Design specifications for the lti discrete time system, in many applications, are given in terms of the magnitude response or the phase response or both 14 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 14 The Frequency Response • The function is called the magnitude response and the function is called the phase response of the LTI discrete-time system • Design specifications for the LTI discretetime system, in many applications, are given in terms of the magnitude response or the phase response or both ( ) j H e ()
The Frequency Response In some cases. the magnitude function is specified in decibels as G(o)=20logoH(e)dB where G(o)is called the gain function The negative of the gain function A(0)=-G(0) is called the attenuation or loss function Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 15 The Frequency Response • In some cases, the magnitude function is specified in decibels as where G() is called the gain function • The negative of the gain function is called the attenuation or loss function H e dB j ( ) 20log ( ) 10 G = A () = −G ()