Unformatted text preview: ugh not
necessary) t hat No be a power of 2, t hat is No = 2m , where m is a n integer. o C omputer E xample C 3.1
C ompute a nd p lot t he t rigonometric a nd e xponential Fourier s pectra for t he p eriodic
signal in Fig. 3.7b (Example 3.3). 3 218 S ignal R epresentation b y O rthogonal S ets T he samples o f f (t) s tart a t t = 0 a nd t he l ast ( Noth) sample is a t t = To  T (the
last sample is n ot a t t = To because the sample a t t = 0 is identical t o t he sample a t t = To,
a nd t he n ext cycle begins a t t = To). At t he p oints of discontinuity, t he s ample value is
t aken as the average of t he values of t he function on two sides of t he discontinuity. Thus,
in t he present case, t he first sample (at t = 0) is n ot 1, b ut ( e"/2 + 1 )/2 = 0.604. To
determine No, we r equire t hat D n for n ~ N o/2 t o b e negligible. Because f (t) h as a j ump
discontinuity, D n d ecays r ather slowly as l /n. Hence, choice of No = 200 is acceptable
because the ( No/2)nd (100th) h armonic is a bout 0.01 ( about 1%) of t he fundamental.
However, we also require No t o b e power of 2. Hence, we shall take No = 256 = 28.
We write a nd save a MATLAB file (or program) c31.m to compute a nd p lot the
exponential Fourier coefficients.
% ( c31.m)
% M i s t he n umber o f c oefficients t o b e c omputed
T O=pi;NO=256;T=TO/NO;M=10;
t =O:T:T*(NOl); t =t';
f =exp(  t/2) ; f(l) = 0.604;
% f ft(f) i s t he F FT [ the s um o n t he r ighthand s ide o f E q. ( 3.86)]
D n=fft(f)/NO
[ Dnangle,Dnmag]=cart2pol(real(Dn),imag(Dn»;
k =O:length(Dn)l;k=k';
s ubplot(211) , stem(k,Dnmag)
s ubplot(212), s tem(k,Dnangle) 3.7 a ns= A mplitudes A ngles
0 .5043
0
0 .2446  75.9622
0 .1251  82.8719
0 .0837  85.2317
0 .0629  86.4175
0 .0503  87.1299
0 .0419  87.6048
0 .0359  87.9437
0 .0314  88.1977
0 .0279  88.3949 LT IC System Response to periodic Inputs A p eriodic s ignal c an b e e xpressed a s a s um o f e verlasting e xponentials (or
sinusoids). W e a lso k now h ow t o find t he r esponse o f a n L TIC s ystem t o a n e verlasting e xponential. F rom t his i nformation we c an r eadily d etermine t he r esponse
o f a n L TIC s ystem t o p eriodic i nputs. A p eriodic s ignal f (t) w ith p eriod To c an b e
e xpressed a s a n e xponential F ourier s eries
00 f (t) = 2 1l' D nejnwot w o=To n =oo e jwt " '.r'
i nput = =? H ( jw )e jwt 'v'
o utput T herefore, f rom l inearity p roperty
00 L
n =oo
'v"
input f (t) D nH(jnwo)ejnwot (3.87) n=oo
''    " . .    r esponse y (t) T he r esponse yet) is o btained i n t he f orm o f a n e xponential F ourier s eries, a nd is
t herefore a p eriodic s ignal o f t he s ame p eriod a s t hat o f t he i nput.
W e s hall d emonstrate t he u tility o f t hese r esults b y t he following e xample.
• E xample 3 .9
A fullwave rectifier (Fig. 3.20a) is used t o o btain a dc signal from a sinusoid sin t.
T he rectified signal f (t), d epicted in Fig. 3.18, is applied t o t he i nput of a a lowpass R C
filter, which...
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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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