8. Write computer programs to obtain the Fourier transform, normalized power spectrum, autocorrelation and probability density function of the following functions:

(1)

(2) is white noise with a zero mean.

(3) is a superposition of twenty harmonics, whose amplitudes decrease quadratically with increasing frequency.

Develop the programs by interpreting the integrals as an arithmetic sum (i.e., an ensemble average). Study the effect of number of points and the total signal length on the accuracy of the answers obtained.

Compute the Fourier transform by the FFT algorithm. Compare the CPU times for the FFT algorithm with the ensemble average approach referred above.

Comment on developing the signal , when one of its statistics such as PDF or spectrum is specified.