Energy Density Spectrum The quantity s(o)=G(e/oy is called the energy density spectrum The area under this curve in the range ≤O≤π divided by2π is the energy of le 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 Spectrun Example-Compute the energy of the sequence hip[n]=sinc @n 0<n<0 ere ∑hP[n xLP(e0,2 H 2兀 where 0≤0≤0 H LP(eo C 0.0,<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 Spectrum Therefore ∑hpn=∫do=< n=-0 2兀 Hence, nLP [n]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 fregz can be used to compute the values of the dtft of a sequence described as a rational function in e J in the form of X(e)= p0+p1 ej*…+ PMe JoM 0 +a1e+ Ne voN +d at a prescribed set of discrete frequency points o=@ 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 (pi and di, respectively, at a prescribed set of frequencies between 0 and n(or 2 ri 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