In this and other laboratory exercises we will sometimes use the “digital” frequency Ω and sometimes use the “analog” frequency ω = 2πf. Although the free use of both concepts in one exercise may be a bit unorthodox, the sampling frequency is 44.1 kHz so there should not be any confusion as to their relation.

In Chapters 4 and 5 we defined white noise and pink noise and presented two mathematical examples: Example: White noise and Example: Pink noise . In this and the subsequent laboratory exercise we will look at various “physical” examples.

In the top-left panel below, the Gaussian noise signal g[n] is displayed and in the top-right, the amplitude histogram h[g] as a measure of p[g], the Gaussian probability density function based upon the data in g[n] ¹. In the lower-right panel the normalized power spectral density S_gg(Ω) is displayed as computed from Equation 5.31. In the associated left panel, the normalized autocorrelation function φ_gg[k] is displayed—not from application of Equation 5.13 —but computed, instead, from the inverse Fourier transform of S_gg(Ω) as in Equation 5.23.

This seemingly indirect route allows us to use the computationally efficient FFT routine. This is an example of where we have used ω = 2πf (in the lower-right panel) together with Ω (in the formula).

1. Based upon a stochastic signal of length N, how much faster is the computation of φ_gg[k] using the FFT algorithm compared to a direct computation of the autocorrelation?


2. What does this imply about a speed improvement for
   1.5
   seconds of signal sampled at 44.1 kHz?

The “Zoom” slider allows you to “zoom in” on the center of each display. On the top row a smaller sample of g[n] is displayed and the amplitude histogram from that segment is displayed. On the bottom row the centers of φ_gg[k] and S_gg(Ω) are displayed but based upon the complete

1.5

second long signal g[n].

3. Based upon the data shown in the bottom row, is it reasonable to conclude that g[n] is a sample of a white noise process? Explain your reasoning.

Proceeed to another part of this exercise by choosing the variant below.

¹ The random signals presented here and in other laboratory exercises are based upon the random number generators contained in Mathematica and Javascript. Professor Donald Knuth in his excellent book “Seminumerical Algorithms” presents a number of such algorithms together with John von Neumann’s warning that “Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin”.   ↵