Consider a real, discrete-time, LTI system whose impulse response is h[n]. The original input x(t) is considered to be an ergodic signal that has been sampled at 44.1 kHz to produce x[n]. The output y[n] = h[n] ⊗ x[n] is, therefore, an ergodic signal as well. The filter represented by h[n] has a Fourier transform H(Ω). The relation between the power spectral density S_xx(Ω) of the input x[n] and the power spectral density S_yy(Ω) of the output y[n] is given by Equation 6.21.

In this first section, x[n] = g[n] a white noise signal with a Gaussian amplitude distribution and g_F [n] is the filtered version of g[n], that is, g_F [n] = h[n] ⊗ g[n]. Below you can view and listen to

1.5

second samples of both g[n] and g_F [n].

1. Is the stochastic signal g_F [n] a “white” noise signal? Explain your reasoning.


2. Based upon the data presented, why is it reasonable to conclude that h[n] is a lowpass filter as opposed to a highpass filter, bandpass filter, or band-reject filter? Be sure to use the “Zoom” slider.


3. When you listen to each of the signals, does this support your answer to the previous question?


4. Using the “Zoom” slider, examine the estimates of the autocorrelation functions of the input and output signals. Estimate the FWHM (full-width, half-maximum) of φ_gg(τ=kT) and φ_FF(τ=kT) where T is the sampling interval.


5. How do you explain the change in the estimates of the probability density function between g[n] and g_F [n]?


6. Filters—particularly low pass filters—are frequently described as one-pole filters or two-pole filters or three-pole filters, and so forth. How many poles does the filter h[n] have?

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