In this problem you are to estimate the parameters of two, real, ergodic noise processes based upon information from their estimated autocorrelation function φ[k] and/or estimated power spectral density S(Ω). In the following:

• μ is the mean value of the random process;
• σ is the standard deviation of the random process;
• N is the number of samples of the random process;
• φ₀ = E{ φ[k = 0] } is the expected value of the autocorrelation function at k = 0 as computed from the signal sample, and;
• S₀ = E{ S[Ω = 0] } is the expected value of the power spectral density at Ω = 0 as computed from the signal sample.

The key is to develop two expressions f and g that will enable us to estimate the relevant parameters.

1. Find an expression based upon E{ φ[k] } that relates μ, σ, N, and φ₀ in the form φ₀ = f(μ, σ, N).


2. Find an expression based upon E{ S[Ω] } that relates μ, σ, N, S₀, and φ₀ in the form S₀ = g(μ, σ, N, φ₀).

Note that these two expressions f and g are independent of the distribution being studied so long as μ and σ exist and are finite.

The two processes we will analyze are a) an exponential distribution for the amplitude at each time sample leading to a random signal e[n] and b) a Poisson distribution at each time sample leading to a random signal P[n].

We begin with a

1.5

second sample of an exponentially-distributed random process whose probability density function p(x|α) = p(x) = αe^–αx where α > 0 and x ≥ 0. The single parameter α is randomly chosen from the set {0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, 25.6}. Each time you perform this exercise by pressing the “retry” button below () another sample of the process will be generated but with a new, probably different value for the parameter α of the random process.

1. Using the expression f, N, and the measured value φ₀ estimate the value of α that was used to generate this sample. Call this estimate α_f.


2. Using the expression g, N, and the measured value S₀ as well as the measured φ₀ estimate the value of α that was used to generate this sample. Call this estimate α_g.


3. By what percentage do α_f and α_g differ from each other? By what percentage do they differ from the true value α?


4. Can you think of a reason why we might have intentionally suppressed the display of the labels on the vertical axis of the signal e(t)?