Suppose we wish to test the noise alone hypothesis H0 : xt = nt against the signal-plus-noise hypothesis H1 : xt = st+nt, where st and nt are uncorrelated zero-mean stationary processes with spectra fs(ω) and fn(ω). Suppose that we want the test over a band of L = 2m + 1 frequencies of the form

ωj:n + k/n, for k = 0, ±1, ±2, . . . , ±m near some fixed frequency ω. Assume that both the signal and noise spectra are approximately constant over the interval.

(a) Prove the approximate likelihood-based test statistic for testing H0 against H1 is proportional to T = X k

|dx(ωj:n + k/n)|

2

 1 fn(ω) − 1 fs(ω) + fn(ω)



.

(b) Find the approximate distributions of T under H0 and H1.

(c) Define the false alarm and signal detection probabilities as PF = P{T >

K|H0} and Pd = P{T > k|H1}, respectively. Express these probabilities in terms of the signal-to-noise ratio fs(ω)/fn(ω) and appropriate chi-squared integrals.

Section 4.9