DTFT Properties Example-Determine the dTFT Y(e/o)of yn]=(n+1)o"um]2a<1 et x[n]=a"'un]a<1 We can therefore write yn]=nx[n +xn From Table 3. 1, the dtFt of xn] is given X(e/0) c已 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 dtft of nx[n is given by dx(eu),d ae do do(1-ae Jo (1-e J)2 Next using the linearity property of the dtFT given in Table 3.2 we arrive at ae Jo Y(e/0)= (1-ae)21-aeo(1-aeo)2 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 DTFTV(e/o)of the sequence v[n] defined by dovn]+div[n-1=po8n+p,Sn-1] From Table 3. 1, the dtFTof Sn]is 1 Using the time-shifting property of the dtFT given in Table 3. 2 we observe that the dtft of 8[n-1]is e Jo and the dtFt of{n-1]ise/o(e° 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 dovln +divln-l=poo[n]+pon-1 as dov(e/o)+dye ov(e 20)=p0 t pye Solving the above equation we get V(e0) Po+pej do tdie 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 Spectrum The total energy of a finite-energy sequence In] is given by g=∑gln 1=-0 From Parseval's relation given in Table 3.2 we observe that f=∑|g 「G(eO)do 2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 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