D Fourier transforms in MATLAB

we can extract all of the independent information of the Fourier transform of a sampled real signal /(t) from only one half of the discrete Fourier transform values, say the first half 

Even for a real signal /(t), the Fourier transform F co) is complex therefore, it is common practice to compute the real-valued, noimegative power spectrum F ( ) vs. (o. The quantity F co) 0 is a measure of the contribution to /(f) from dynamics with a frequency co. The symmetry F = F leads to the property F(2o) - ty )p = F co ), and thus all independent information in the power spectrum is contained in the lower-frequency half 0 co tumax- 

The discrete Fourier transform and its inverse (9.54) are computed in MATLAB using fft and ifft respectively. The following code demonstrates their use for the signal /(f) = sin(f) - -2 cos(2f), which has two peaks in the power spectrum at = 1 and co = 2. The signal is sampled at uniform times over the period [0, /2 r], such that P = In and At = 2P/N, where N = 2 for some integer e. 

The realO operation is done to remove any near-zero imaginary contributions that are introduced due to numerical error. 

In the example above, the sampling interval At was sufficiently small to observe all contributing frequencies co to F(cti). Let us now consider what happens to the discrete FT when f(t) is not bandwidth-limited. Consider sampling the signal with N = 2 points during an interval 2P, P = in. The maximum resolvable frequency is 

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