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Non-Gaussian Detection Theory: Problems

Exercise 1

The additive noise in a detection problem consists of a sequence of statistically independent Laplacian random variables. The probability density of nl n l is therefore p n l n=12ⅇ-|n| p n l n 1 2 n The two possible signals are constant throughout the observation interval, equaling either +1 or -1.

1.a)

Find the optimum decision rule which could be used on a single value of the observation signal.

1.b)

Find an expression for the threshold in your rule as a function of the false-alarm probability.

1.c)

What is the threshold value for the "symmetric" error situation ( P F =1- P D P F 1 P D )?

1.d)

Now assume that two values of the observations are used in the detector (patience; to solve this problem, we need to approach it gradually). What is the decision rule for symmetric errors? How does this result generalize to LL observations?

Exercise 2

The threshold for non-parametric model evaluation is computed (somewhat glibly) in discussing Non-parametric Model Evaluation using the Central Limit Theorem. As warned in our previous discussion, such applications of the Central Limit Theorem need to be employed carefully.

2.a)

The critical parameters - Eux u x and σux2 u x - of the Central Limit Theorem approximation are claimed to equal 12 1 2 and 14 1 4 respectively under model ℳ 0 ℳ 0 . Verify this claim.

2.b)

How is the threshold in the non-parametric hypothesis test related to the number of observations from the point of view of the Central Limit Theorem?

2.c)

Contrast this answer with that determined by the non-parametric test. Which constraint dominates under what conditions?

2.d)

How many observations are needed in a non-parametric test to achieve a false-alarm rate of 10-2 10 -2 ? of 10-6 10 -6 ?

Exercise 3

What are the ranges of signal waveforms and probability density functions consistent with the appropriate model for uncertainty about a nominal?

3.a)

When considering waveform uncertainties, the uncertainty was modeled as an additive corrupting signal that has bounded energy. Assuming a sinusoidal nominal consisting of eight samples per period, sketch the worst-case and best-case signals for the detection problem described in the text.

3.b)

What other signals are consistent with this model? Do they fall within the envelopes of the best-case and worst-case signals? If so prove it; if not, find a counter example and provide some method of finding the others.

3.c)

The εε-contamination model for uncertain densities was somewhat different because we needed to maintain unit area. Characterize the range of densities consistent with this model.

Exercise 4

In the partially known signal waveform problem, we assumed that there was no uncertainty in the "no-signal" model ℳ 0 ℳ 0 . A residual signal corrupting the observations may well be present no matter what signal model is actually true. For example, hum (power-line noise) could be present no matter what signal is received.

4.a)

Re-derive the robust detector for the situation where a zero-energy signal and a non-zero energy signal can both be corrupted by a signal having energy Ec E c .

4.b)

Under what conditions will a solution exist?

4.c)

Contrast your detector with that found in the text. What are the worst-case signals? What signal is used in the matched-filter in the white noise case? Does the usual matched filter remain robust?

Exercise 5

The detector robust under signal waveform uncertainties was only briefly described in the text. This result has several interesting properties that warrant further exploration.

5.a)

Explicitly derive the detector using the Lagrange multiplier technique outlined in the text. Find the value of the Lagrange multiplier for the white-noise case.

5.b)

Don't take the authors' word that the worst-case signal is indeed that derived. By computing the probability of detection for a matched-filter detector that uses s o l s o l , find the worst-case signal s ω l s ω l - the one that minimizes P D P D - that satisfies the constraint s ω l= s o l+cl s ω l s o l c l , ∑ c 2 l≤Ec c 2 l E c . Do not assume the noise is white.

5.c)

The solution shown previously differs from the colored-noise detector that ignores signal uncertainties. By contrasting the signals to which the observations are matched, describe the differences. When are they similar? very different?

Exercise 6: Noisy Signal Templates

In many signal detection problems, the signal itself is not known accurately; for example, the signal could have been the result of a measurement, in which case the "signal" used to specify the observations is actually the actual signal plus noise. We want to determine how the measurement noise affects the detector.

The formal statement of the detection problem is ℳ 0 : r l = s ˜ l 0 + n l ℳ 0 : r l s ˜ l 0 n l ℳ 1 : r l = s ˜ l 1 + n l ℳ 1 : r l s ˜ l 1 n l for l∈0…L-1 l 0 … L 1 , where s ˜ l i s ˜ l i equals s l i + ω l s l i ω l , with ω l ω l and n l n l comprising white Gaussian noise having variances σ ω 2 σ ω 2 and σ n 2 σ n 2 , respectively. We know precisely what the signals s ˜ l i s ˜ l i are, but not the underlying "actual" signal.

6.a)

Find a detector for this problem.

6.b)

Analyze, as much as possible, how this detector is affected by the measurement noise, contrasting its performance with that obtained when the signals are known precisely.

Exercise 7

In robust detection problems in which we have signal uncertainties, the so-called "minimax" approach is to find the worst-case signals among received signal possibilities (the pair that would yield the worst performance if they were indeed transmitted), then use the likelihood ratio based on these signals instead of what is actually transmitted. Assume the signal transmissions are observed in the presence of additive white Gaussian noise. The received signal s ˜ i s ˜ i deviates from the nominal one s i s i by a finite mean-squared error: ∥ s ˜ i - s i ∥2≤ε2 s ˜ i s i 2 ε 2 . Prove or disprove: The receiver obtained when there is no uncertainty is robust.

Exercise 8

Consider a two-model problem where signals are observed in the presence of additive Laplacian noise having samples statistically independent of each other and the signal. ℳ 0 : rl= s 0 l+nl ℳ 0 : r l s 0 l n l ℳ 1 : rl= s 1 l+nl ℳ 1 : r l s 1 l n l for l∈0…L-1 l 0 … L 1 . To design an optimal signal set for this problem, we constrain each signal's energy to be less than EE.

8.a)

What signal choices provide optimal performance (the smallest error probabilities) when the signal-to-noise ratio is small?

8.b)

When the signal-to-noise ratio is large, do the choices change? If so, determine what choices, if any, would be optimal regardless of the signal-to-noise ratio.

Exercise 9

In Poisson channels, such as photon-limited optical communication systems, the received signal consists of the number of photons and when they occurred. The joint distribution of these quantities has the form Pr N 0 T =n|t= t 1 … t n =∏j=1nλ t j ⅇ-∫0Tλαdα t t 1 … t n N 0 T n j 1 n λ t j α 0 T λ α where λt λ t denotes the Poisson process's time-varying intensity, which must be non-negative. in optical digital communication, the signal sets consist of designed intensities, and these are typically received in the presence of a background light level. Phrased in terms of a model, ℳ i ℳ i : λt= λ i t+ λ d λ t λ i t λ d . In designing signal sets for the optical channel, the intensity's integral is typically bounded: ∫0T λ i αdα< Λ max α 0 T λ i α Λ max . What is the optimal binary signal set λ 0 t λ 1 t λ 0 t λ 1 t for the optical channel?

Exercise 10

Consider a two-model problem where signals are observed in the presence of additive Laplacian noise having samples statistically independent of each other and the signal. ℳ 0 : r l = s l 0 + n l ℳ 0 : r l s l 0 n l ℳ 1 : r l = s l 1 + n l ℳ 1 : r l s l 1 n l for l∈0…L-1 l 0 … L 1 To design an optimal signal set for this problem, we constrain each signal's energy to be less than EE. Assume that the number of observations LL is large.

10.a)