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Properties of the Discrete Fourier Transform 169
real number. In addition, if N is even, then N/2 isalso an integer.
Then from (5.32)
X (N/2) = X ∗ ((−N/2)) N
= X ∗ (N/2)
which means that even the k = N/2 component is also real-valued.
This component is called the Nyquist component since k = N/2 means
that the frequency ω N/2 = (N/2)(2π/N) = π, which is the digital
Nyquist frequency.
The real-valued signals can also be decomposed into their even and odd
components, x e (n) and x o (n), respectively, as discussed in Chapter 2.
However, these components are not N-point sequences and therefore we
cannot take their N-point DFTs. Hence we define a new set of components
using the circular folding discussed above. These are called circular-even
and circular-odd components defined by
{
x ec (n) = △ x(0), n =0
1
2 [x(n)+x ((−n)) N ]= [x (n)+x (N − n)] , 1 ≤ n ≤ N − 1
1
2
{
x oc (n) = △ 0, n =0
1
2 [x(n) − x ((−n)) N ]= 1
2
[x (n) − x (N − n)] , 1 ≤ n ≤ N − 1
(5.33)
Then
DFT [x ec (n)] = Re [X(k)] = Re [X ((−k)) N
]
DFT [x oc (n)] = Im [X(k)] = Im [X ((−k)) N
]
(5.34)
Implication: If x(n) isreal and circular-even, then its DFT is also real
and circular-even. Hence only the first 0 ≤ n ≤ N/2 coefficients are
necessary for complete representation.
Using (5.33), it is easy to develop a function to decompose an N-point
sequence into its circular-even and circular-odd components. The following
circevod function uses the mod function given earlier to implement
the modulo-N operation.
function [xec, xoc] = circevod(x)
% signal decomposition into circular-even and circular-odd parts
% --------------------------------------------------------------
% [xec, xoc] = circevod(x)
%
if any(imag(x) ~= 0)
error(’x is not a real sequence’)
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