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

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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' &quot;4 i lk) f orM=9 . . .&quot;'. ' 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 &quot;4 r • •• n_ (a) lIt. 11'8&quot;'. . . . 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] = '&quot;tIki w ith 11'1 &lt; 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.) { =&gt; 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) { =&gt; 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] { :=&gt; a nd F(O) t hen f[k - ko] F (O)e- jkoO { :=&gt; 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 { :=&gt; = r Fl(U)F2(0 - u) du i2&quot; 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...
View Full Document

Ask a homework question - tutors are online