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166 Chapter 5 THE DISCRETE FOURIER TRANSFORM
to compute its DFT. Therefore we use the modulo-N operation on the
argument (−n) and define folding by
{
x(0), n =0
x ((−n)) N
=
(5.28)
x(N − n), 1 ≤ n ≤ N − 1
This is called a circular folding. Tovisualize it, imagine that the sequence
x(n) iswrapped around a circle in the counterclockwise direction
so that indices n =0and n = N overlap. Then x((−n)) N can be
viewed as a clockwise wrapping of x(n) around the circle; hence the
name circular folding. In MATLAB the circular folding can be achieved
by x=x(mod(-n,N)+1). Note that the arguments in MATLAB begin
with 1. The DFT of a circular folding is given by
{
X(0), k =0
DFT [x ((−n)) N
]=X ((−k)) N
=
(5.29)
X(N − k), 1 ≤ k ≤ N − 1
□ EXAMPLE 5.9 Let x(n) =10(0.8) n , 0 ≤ n ≤ 10.
a. Determine and plot x ((−n)) 11
.
b. Verify the circular folding property.
Solution
a. MATLAB script:
>> n = 0:100; x = 10*(0.8) .^ n; y = x(mod(-n,11)+1);
>> subplot(2,1,1); stem(n,x); title(’Original sequence’)
>> xlabel(’n’); ylabel(’x(n)’);
>> subplot(2,1,2); stem(n,y); title(’Circularly folded sequence’)
>> xlabel(’n’); ylabel(’x(-n mod 10)’);
The plots in Figure 5.12 show the effect of circular folding.
b. MATLAB script:
>> X = dft(x,11); Y = dft(y,11);
>> subplot(2,2,1); stem(n,real(X));
>> title(’Real{DFT[x(n)]}’); xlabel(’k’);
>> subplot(2,2,2); stem(n,imag(X));
>> title(’Imag{DFT[x(n)]}’); xlabel(’k’);
>> subplot(2,2,3); stem(n,real(Y));
>> title(’Real{DFT[x((-n))11]}’); xlabel(’k’);
>> subplot(2,2,4); stem(n,imag(Y));
>> title(’Imag{DFT[x((-n))11]}’); xlabel(’k’);
The plots in Figure 5.13 verify the property.
□
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