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S 22.1 (P 4.3)
______________
Divide by 2*pi to express the three frequencies in cycles per
sample:
3/28, 9/35 and 17/48
Since these are rational (i.e., integer fractions), the
signal is periodic.
The period is the smallest integer
L such that each frequency can be expressed as k/L.
With irreducible fractions (as above), this is given by
the least common multiple (LCM) of the denominators:
LCM(28,35,48) = 1680
S 22.2 (P 4.7)
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The output y is also periodic with period L=7.
If
xL = x[0:6]
and
yL = y[0:6]
then the DFT's of xL and yL are related by
YL[k] = H(exp(j*2*pi*k/8))*XL[k]
for k=0:7.
The easiest way to obtain the values H(exp(j*2*pi*k/8))
numerically is to zeropad b to 2*L = 14 points, then take every
*other* coefficient in the DFT.
The MATLAB code is
b = [ 1 2 2 1 4 1 2 2 1 ].' ;
B = fft(b,14);
H = B(1:2:13);
xL = [ 1 1 0 3 1 2 0 ].' ;
XL = fft(xL);
YL = H.*XL;
yL = ifft(YL)
bar(0:6,yL)
and the resulting yL vector is
[20 3 20 10 21 14 12]'
Alternatively:
b = [ 1 2 2 1 4 1 2 2 1].' ;
B = fft(b,14);
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 Spring '08
 staff
 Frequency

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