华南理工大学：《数字信号处理》（双语版） Chapter 7 DTFT Properties

DTFT Properties Example- Determine the DTFT y(e/)of yr]=(m+1)pm],a<1 Let x[n]=a"u[n],a<1 We can therefore write n=nxn+xn From Table 3. 1, the dtFt of x[n] is given Y(e0) 1-ae y Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 1 DTFT Properties • Example - Determine the DTFT of • Let • We can therefore write • From Table 3.1, the DTFT of x[n] is given by y[n] = (n +1) [n],  1 n x[n] =  [n],  1 n y[n] = n x[n]+ x[n] −   −  = j j e X e 1 1 ( ) ( ) j Y e

DTFT Properties Using the differentiation property of the dtFT given in Table 3. 2. we observe that the dtftof nxin is given by dX(e/o). d de o J o1-0e Ja ae Jo Next using the linearity property of the dtFT given in Table 3. 2 we arrive at 0 ae Y(e/o) (1-ae)21-ae0(1-ae-0) Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 2 DTFT Properties • Using the differentiation property of the DTFT given in Table 3.2, we observe that the DTFT of is given by • Next using the linearity property of the DTFT given in Table 3.2 we arrive at nx[n] 2 1 (1 ) ( ) 1 −  −  −   −  =          − =  j j j j e e d e d j d dX e j 2 2 (1 ) 1 1 1 (1 ) ( ) −  −  −  −   − = − + −  = j j j j j e e e e Y e

DTFT Properties Example-Determine the dtfT V(e/o)of the sequence vIn] defined by dovIn]+div[n-1]=poo[n]+p 8[n-1 From Table 3. 1, the dtFt of Sn]is 1 Using the time-shifting property of the dtFT given in Table 3.2 we observe that the dtft of Sn-1]is e /o and the dtFT of v[n-1]is e v(el Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 3 DTFT Properties • Example - Determine the DTFT of the sequence v[n] defined by • From Table 3.1, the DTFT of is 1 • Using the time-shifting property of the DTFT given in Table 3.2 we observe that the DTFT of is and the DTFT of is v[n −1] [ ] [ 1] [ ] [ 1] d0 v n + d1 v n − = p0 n + p1 n −  [n] [n −1] − j e ( ) j V e ( ) − j j e V e

DTFT Properties Using the linearity property of Table 3. 2 we then obtain the frequency-domain representation of dovIn]+div[n-1]=poS[n]+p, 8[n-1 as J +dle e P0+p1 Solving the above equation we get V(e/o)=Po t p1e /o JQ 0 +1e Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 4 DTFT Properties • Using the linearity property of Table 3.2 we then obtain the frequency-domain representation of as • Solving the above equation we get [ ] [ 1] [ ] [ 1] d0 v n + d1 v n − = p0 n + p1 n −  −   −  + = + j j j j d V e d e V e p p e 0 1 0 1 ( ) ( ) −  −   + + = j j j d d e p p e V e 0 1 0 1 ( )

Energy Density Spectrun The total energy of a finite-energy sequence gIn is given by E g=△gln 1=-0 From Parseval's relation given in Table 3.2 we observe that T=∑ IG(eJo) do 2丌 丌 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 5 Energy Density Spectrum • The total energy of a finite-energy sequence g[n] is given by • From Parseval’s relation given in Table 3.2 we observe that 2 [ ] g n g n  =− E =  2 2 1 [ ] ( ) 2 j g n g n G e d       =− − E = =  

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