You have seen that N samples of the input signal result in N samples of the DFT. That is, the number of samples in both the time and frequency representations is the same. From the following equation, you see that regardless of whether the input signal x[i] is real or complex, X[k] is always complex, although the imaginary part may be zero.
for k = 0, 1, 2, , N 1
Thus, because the DFT is complex, it contains two pieces of information: the amplitude and the phase. It turns out that for real signals (x[i] real) such as those obtained from the output of one channel of a DAQ board, the DFT is symmetric with the properties
| X[k] | = | X[N k] |
and
phase ( X[k] ) = phase (X[N k] ).
The terms used to describe this symmetry are that the magnitude of X[k] is even symmetric, and phase(X[k]) is odd symmetric. An even symmetric signal is one that is symmetric about the y-axis, whereas an odd symmetric signal is symmetric about the origin, as shown in the following illustration.
The net effect of this symmetry is that there is repetition of information contained in the N samples of the DFT. Because of this repetition of information, only half of the samples of the DFT actually need to be computed or displayed, as the other half can be obtained from this repetition.
![]() | Note If the input signal is complex, the DFT will be nonsymmetrical and you cannot use this trick. |