Energy Density Spectrun The quantity Sa(o)=G(eloy is called the energy density spectrum The area under this curve in the range π≤0≤π divided by2π is the energy of the sequence Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 6 Energy Density Spectrum • The quantity is called the energy density spectrum • The area under this curve in the range divided by 2 is the energy of the sequence − 2 ( ) ( ) = j gg S G e
Energy Density Spectrum Example-Compute the energy of the sequence O,1 spin SInc 00<1< Here T ∑hP[h ∫H Lp(evo 1=-0 2丌 Where 0≤0≤ H LP(eJo 0,0,<(≤兀 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 7 Energy Density Spectrum • Example - Compute the energy of the sequence • Here where [ ] sinc , c c LP n h n n = − = − =− h n H e d j LP n LP 2 2 ( ) 2 1 [ ] = c j c LP H e 0, 1, 0 ( )
Energy Density Spectrun Therefore ∑hp[n]2= c(= 2 C Hence, hpin] is a finite-energy sequence Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 8 Energy Density Spectrum • Therefore • Hence, is a finite-energy sequence = = − =− c n LP c c h n d 2 1 [ ] 2 h [n] LP
DTFT Computation Using MATLAB The function freqz can be used to compute the values of the DtfT of a sequence described as a rational function in e/ in the form of X(e/)p++…+e/oM 0+a1e-/o +.+d Ne JoN at a prescribed set of discrete frequency points o=@e Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 9 DTFT Computation Using MATLAB • The function freqz can be used to compute the values of the DTFT of a sequence, described as a rational function in in the form of at a prescribed set of discrete frequency points j N N j j M M j j d d e d e p p e p e X e − − − − + + + + + + = .... .... ( ) 0 1 0 1 = j e
DTFT Computation Using MATLAB For example the statement H= freqz(num, den, w) returns the frequency response values as a vector h of a dtft defined in terms of the vectors num and den containing the coefficients pil and idii, respectively, at a prescribed set of frequencies between 0 and n(or 2rd given by the vector w 10 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 10 DTFT Computation Using MATLAB • For example, the statement H = freqz(num,den,w) returns the frequency response values as a vector H of a DTFT defined in terms of the vectors num and den containing the coefficients and , respectively, at a prescribed set of frequencies between 0 and (or 2) given by the vector w { }i p { } di