Unformatted text preview: )e jOk dn y[k] a s a s um o f responses t o all i nput c omponents 640 lO Fourier Analysis o f Discrete Time Signals which is precisely t he r elationship in Eq. (lO.57). Thus, F (n) is t he i nput s pectrum
a nd Y (n), t he o utput s pectrum (of t he e xponential components), is F (n)H(n) .
E xample 1 0.7
For a system with unit impulse response h[k] = (0.5)ku[k], determine the (zerostate)
response Y[k] for the input J[k] = (0.8)ku[k].
According t o Eq. (10.57) 10.6 Signal processing U sing D FT a nd F FT 641 similar to what we did in the continuoustime case by generalizing the frequency variable
j w to s = ( 7 + j w (from Fourier to Laplace transform) .
• • Y (n) = F (n)H(n) From the results in Eq. (10.37), we obtain F (n) _ 1 (10.60)  1 0.8ein Also, H (n) is the D TFT of (0.5)ku[k], which is obtained from Eq. (10.37) by substituting
= 0.5: y in
1
e
 1  0.5ein = e in _ 0.5 H (n) _ (10.61) Therefore
e i2n Y (n) = ( ein _ 0.5)(e in _ 0.8) 1 0.6 Signal processing by OFT and FFT
I n t his s ection, we use D FT (developed i n C hapter 5) a s a t ool, which allows
us t o utilize a digital c omputer for digital signal processing. T his s ignal processing
includes s pectral a nalysis o f d igital signals a nd LTID system analysis. B y s pectral
analysis, we m ean d etermining t he d iscrete t ime F ourier series ( DTFS) o f periodic
signals a nd d etermining F (n) from I[k) ( and vice versa) for aperiodic signals. As
a tool for LTID system analysis, D FT c an b e u sed as a software oriented solution
t o d igital filtering. D FT c an b e i mplemented on a digital c omputer b y a n efficient
algorithm, t he f ast Fourier transform ( FFT) also discussed i n C hapter 5. T he D FT
( using F FT) is t ruly t he workhorse o f m odern digital signal processing. 10.61 Computation o f DiscreteTime Fourier Series (DTFS) T he d iscretetime Fourier series (DTFS) equations (10.8) a nd (lO.9) are identical t o t he D FT e quations (5.18b) a nd (5.18a) within a scaling c onstant No. I fwe
l et I[k) = Nolk a nd V r = Fr i n Eqs. (lO.9) a nd (lO.8), we o btain We can express the righthand side as a sum of two firstorder terms (modified partial
fraction expansion as discussed in Sec. B.55) as follows:t
Y (n)
ein N ol Fr  dn = " I k e  jrflok
~
k =O (ein  0.5)(e in  0.8) 5
=~+ ejn  0.5 no 8 _ _3 _
_ ein  0.8 Consequently, 211"
No =; T his is precisely t he D FT p air i n Eqs. (5.18). For instance, t o c ompute t he D TFS
for t he p eriodic signal in Fig. lO.2a, we use t he values of l k = I[k)/No as
1 lk = { 032 0 ::; k ::; 4 5::; a nd 2 8::; k ::; 31 k ::; 27 We use these values in t he F FT a lgorithm discussed i n Sec. 5.22 t o o btain F r ,
which is t he s ame as V r •
According to Eq. (10.37), the inverse D TFT of this equation is This example demonstrates the procedure of determining an LTID system response
using DTFT. I t is similar to the method of Fourier transform in analysis of LTIC systems.
As in the case of Fourier transform, this method can be used only...
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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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