Signal Processing and Linear Systems-B.P.Lathi copy

1027 in t he limit t he r ight hand side of eq 1029

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Unformatted text preview: a nd [see Sec. B.7-4] (10.42) 632 10 Fourier Analysis of Discrete- Time Signals F (n) =e j¥n .- e e J {l_ 1 - i¥-n •• • e - j {l/2 ( e-j'¥{l _ e j'¥{l) - D -21t e j {l/2(e- j {l/2 _ e j {l/2) sin (~n) sin (0.50.) (10.44) (P' "4 i lk) f orM=9 . . ."'. ' II .tII F igure 10.6b shows t he s pectrum F (n) for M = 9. N ote t hat t he s pectrum D r i n Fig. 10.2b [Eq. (10.19)] is a sampled version of F (n) in Fig. 1O.6b [Eq. (10.44)]: 0 .0= ~ 16 0 - It - It (10.43) sin ( 4.5n) sin (0.50.) 633 10.3 Properties of t he D TFT It o 21t It "4 r • •• n_ (a) lIt. 11'8"'. . . . 4 0 k(b) F ig. 1 0.7 Inverse Discrete-time Fourier transform of a periodic g ate s pectrum. T he signal flk] is d epicted in Fig. 1O.7b (for t he case n c = 1t/4). • E xercise E I0.3 Therefore, F (n) in Fig. 1O.6b is t he envelope o f D r ( within a multiplicative c onstant 32) • in Fig. 1O.2b. T he r eason for this behavior is discussed l ater in Sec. 10.6. £:. o Answer: F ind a nd s ketch t he a mplitude a nd p hase s pectra o f t he D TFT o f t he s ignal tlk] = '"tIki w ith 11'1 < l. C omputer E xample C I0.2 Do Example 10.5 using MATLAB. N O=512; f =[ones(I,5) z eros(I,NO-9) o nes(I,4»); F =fft(f); r =O:NO-l; W =r.*2*pi/512; p lot(W,F); x label('W');ylabel('F(W)');grid o n; 1 0.3 Properties o f t he DTFT T he l inearity p roperty [Eq. (10.35)] of D TFT h as been already discussed. O ther useful properties of the D TFT a re as follows: 0 T ime and Frequency Inversion • E xample 1 0.6 F ind t he inverse D TFT of t he r ectangular pulse s pectrum F (n) = r eet ( 2g ) w ith n c = ~ a nd r epeating a t t he intervals of 21t, a s shown in Fig. 1O.7a. c According to Eq. (10.30) f[k] = 2~ i: f [-k] From Eq. (10.31), the D TFT of f [-k] is 00 00 F (n)ejk{l dO. (10.46) F ( - 0.) { => D TFT{![-k]} = 2:= J [_k]e- irlk = 2:= J[m]e irlm = F (-o.) m =-oo k =-oo = J ...j{lc ejk{l dO. 21t -{lc M ultiplication by k: Frequency Differentiation 1 J.k{ll{lc = j 2ltk e -{lc kf[k] sin ( nck) { => j d:~o.) (10.47) = - -It-k- = n c s inc(nc k) It (10.45) T he result follows immediately by differentiating b oth sides of E q. (10.31) w ith respect t o o.. 634 10 Fourier Analysis of Discrete-Time Signals 10.4 Connection with t he C ontinuous-Time Fourier Transform ( FT) T ime-Shifting Property 635 00 h [k] If L * h [k] = h [m]h[k - m] m =-oo J[k] { :=> a nd F(O) t hen f[k - ko] F (O)e- jkoO { :=> ko a n integer Fl(O) (10.48) This property can be proved by direct substitution in t he e quation defining t he d irect transform. From Eq. (10.31) we o btain 00 f[k - ko] L { :=> = r Fl(U)F2(0 - u) du i2" T he t ime convolution p roperty is proved in C hapter 11 [Eq. (11.18)]. All we have t o do is replace z w ith e jo . T o prove t he frequency-convolution property (1O.50b), we have 00 L J[k - ko]e- jOk = f[m]e-jO(m+ko] m =-oo k =-oo 00 L = e - jOko * F2(0) f [m]e- jOm = e -jkooF(O) k =-oo Interchanging t he o rder o f s ummation a nd i ntegration, we o btain T his result shows t hat delaying a signal by ko u nits does n ot chang...
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