Fourier Transform Technique

The complex function is defined as the Fourier transform of and is calculated as

The normalized power spectrum can then be calculated as

where is the RMS value of . It is possible to show that and from a Fourier transform pair, i.e.

Methods of calculating Fourier transforms are well-established. In particular, the fast Fourier transform (FFT) algorithm has found wide usage both in software and in hardware applications in signal processing. Hence it is to be understood that integrals appearing in the Fourier transforms defined above can be readily determined.

Though the integrals given above are complex-valued, the property guarantees that is purely real. On the other hand, the Fourier integral for is to be interpreted as the real part of the complex function. Typical autocorrelation functions and power spectrum are sketched in Figure 1.10.