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Unformatted text preview: ir s amples, they
m ust b e s ampled a t t he s ame r ate ( not below t he N yquist r ate o f e ither signal).
Let i k (0 ::; k :::; N l  1) a nd 9k (0 ::; k ::; N2  1) b e t he c orresponding discrete
sequences representing these samples. Now,
C(t) = f (t) a nd if we d efine three sequences as i k
t hent = T f(kT), Ck = tWe can show t hat 4
this equation. Ck = l imT_o!k * 9 (t) We generally t hink o f filtering in t erms of some hardwareoriented solution
(namely, building a circuit w ith R LC c omponents a nd o perational amplifiers). However, filtering also has a softwareoriented solution [a c omputer a lgorithm t hat yields
t he filtered o utput y (t) for a given i nput f (t) J. T his goal c an b e c onveniently accomplished by using t he D FT. I f f (t) is t he s ignal t o b e filtered, t hen Fro t he D FT o f ik,
is found. T he s pectrum Fr is t hen s haped (filtered) as desired by multiplying F r by
Hro where H r a re t he s amples of t he filter t ransfer f unction H (w) [Hr = H (rwo)J.
Finally, we t ake t he I DFT of F rHr t o o btain t he filtered o utput Yk [Yk = T y(kT)J.
T his p rocedure is d emonstrated i n t he following example.
• E xample 5 .8
T he signal f (t) in Fig. 5.18a is passed through an ideal lowpass filter of transfer
function H{w) depicted in Fig. 5.18b. Using DFT, find the filter output.
We have already found the 32point DFT of f (t) (see Fig. 5.16d). Next we multiply
Fr by Hr. To find H r , we recall using Fo = ~ in computing the 32point DFT of f (t).
Because Fr is 32periodic, Hr must also be 32periodic with samples separated by Hz.
This fact means t hat Hr must be repeated every 8 Hz or 1611' r ad/s (see Fig. 5.18c). The
resulting 32 samples of H r over (0 ~ w ~ 1611') are as follows: i Hr =
9k = T 9(kT), a nd Ck = T c(kT), i k * 9k * 9k. Because T # 0 in practice, there will be some error in {~
0.5 o ~ r ::; 7
9 and 25::; r ::; 31 23
r = 8 ,24
~ r ~ We multiply Fr with Hr. The desired output signal samples Yk are found by taking the
inverse DFT of F rH r . The resulting output signal is illustrated in Fig. 5.18d. Table 5.1
gives a printout of h , F r , H r , Yr , and Yk . • 352 5 Sampling
8 I (t) 1 H (ro) (a) 5.3 T he F ast Fourier Transform ( FFT) ( b) f, No.
0
I
2
3  0.5 0.5 2 t 5 7
8
9
10
\I  40  32  24  16  10 8 6 4 2 16 r _ o 2 24 32 40 4 :I'(Hz) 4  84 ..? 8 12
\3
I'
15
16
17
18
19
20
21 (c) 10 22 j\  16 ..•.
 8 ..•. o
F ig. 5.18 .8 16 31 Filtering f (t) through H(w). o C omputer E xample C 5.3
Solve Example 5.8 using MATLAB.
The MATLAB program of Example C5.2, where we obtained the 32point F r , should
be saved as an mfile, e.g., 'c52.m'. We can import Fr in the MATLAB environment by
the command 'c52'.
c52;
N O=32;k=0:NO1;
H =[Ones(1,8) 0 .5 z eros(1,15) 0.5 o nes(1,7)];
Y r=H.*Fr;
y k= ifft ( Yr);
s tem(k,yk)
0 5.3 T he F ast Fourier Transform (FFT)
T he n umber of computations required in performing the D FT was dramatically
reduced by a n a lgorithm developed by Tukey a nd Cooley in 1965. 5 T his al...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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