Complex-conjugate property
If the sequence fxn, 0 n N À 1g is real, then
XÀk X
Ã
k XN À k, 0 k N À 1,
7:1:14
where X
Ã
k is the complex conjugate of X(k). Or equivalently,
XM k X
Ã
M À k, 0 k M,
7:1:15
where M N=2 if N is even, and M N À 1=2 if N is odd. This property shows that
only the first (M 1) DFT coefficients are independent. Only the frequency compon-
ents from k 0 to k M are needed in order to completely define the output. The rest
can be obtained from the complex conjugate of corresponding coefficients, as illustrated
in Figure 7.2.
The complex-conjugate (or symmetry) property shows that
ReXk ReXN À k, k 1 , 2, . . . , M À 1
7:1:16
and
ImXk ÀImXN À k, k 1 , 2, . . . , M À 1:
7:1:17
Thus the DFT of a real sequence produces symmetric real frequency components and
anti-symmetric imaginary frequency components about X(M). The real part of the DFT
output is an even function, and the imaginary part of the DFT output is an odd
function. From (7.1.16) and (7.1.17), we obtain
jXkj jXN À kj, k 1 , 2, . . . , M À 1
7:1:18
and
fk ÀfN À k, k 1 , 2, . . . , M À 1:
7:1:19
Because of the symmetry of the magnitude spectrum and the anti-symmetry of the phase
spectrum, only the first M 1outputs represent unique information from the input
signal. If the input to the DFT is a complex signal, however, all N complex outputs
could carry information.
X(0) X(1) ... X(M
-2) X(M-1) X(M) X(M+1) X(M+2) ... X(N-1)
Complex conjugate
real
real
Figure 7.2 Complex-conjugate property
DISCRETE FOURIER TRANSFORM
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