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Course by: Don Johnson. E-mail the author

# Discrete-Time Detection: Problems

Module by: Don Johnson. E-mail the author

## Exercise 1

Assume that observations of a sinusoidal signal sl=Asin2πfl s l A 2 f l , l0L1 l 0 L 1 , are contaminated by first-order colored noise as decribed in the example.

### 1.a)

Find the unit-sample response of the whitening filter.

### 1.b)

Assuming that the alternative model is the sole presence of the colored Gaussian noise, what is the probaiblity of detection?

### 1.c)

How does this probability vary with signal frequency ff when the first-order coefficient is positive? Does your result make sense? Why?

## Exercise 2

In space-time decoding systems, a common bit stream is transmitted over several channels simultaneously but using different signals. r ( k ) r ( k ) denotes the signal received from the k th k th channel, k1K k 1 K , and the received signal equals s ( k , i ) + n ( k ) s ( k , i ) n ( k ) . Here ii equals 0 or 1, corresponding to the bit being transmitted. Each signal has length LL. n ( k ) n ( k ) denotes a Gaussian random vector with statistically independent components having mean zero and variance σ k 2 σ k 2 (the variance depends on the channel).

### 2.a)

Assuming equally likely bit transmissions, find the minimum probability of error decision rule.

### 2.b)

What is the probability that your decision rule makes an error?

### 2.c)

Suppose each channel has its own decision rule, which is designed to yield the same miss probability as the others. Now what is the minimum probability of error decision rule of the system that combines the individual decisions into one?

## Exercise 3

The performance for the optimal detector in white Gaussian noise problems depends only on the distance between the signals. Let's confirm this result experimentally. Define the signal under one hypothesis to be a unit-amplitude sinusoid having one cycle within the 50-sample observation interval. Observations of this signal are contaminated by additive white Gaussian noise having variance equal to 1.5. The hypotheses are equally likely.

### 3.a)

Let the second hypothesis be a cosine of the same frequency. Calculate and estimate the detector's false-alarm probability.

### 3.b)

Now let the signals correspond to square-waves constructed from the sinusoids used in the previous part. Normalize them so that they have the same energy as the sinusoids. Calculate and estimate the detector's false-alarm probability.

### 3.c)

Now let the noise be Laplacian with variance 1.5. Although no analytic expression for the detector performance can be found, do the simulated performances for the sinusoid and the square-wave signals change significantly?

### 3.d)

Finally, let the second signal be the negative of the sinusoid. Repeat the calculations and the simulation for Gaussian noise.

## Exercise 4

Physical constraints imposed on signals can change what signal set choices result in the best detection performance. Let one of two equally likely discrete-time signals be observed in the presence of white Gaussian noise (variance/sample equals σ2 σ 2 ). 0 : rl= s ( 0 ) l+nl   l0L1 0 : r l s ( 0 ) l n l   l 0 L 1 1 : rl= s ( 1 ) l+nl   l0L1 1 : r l s ( 1 ) l n l   l 0 L 1 We are free to choose any signals we like, but there are constraints. Average signals power equals l ls2lL l l s l 2 L , and the peak power equals max l l s2l l s l 2 .

### 4.a)

Assuming the average signal power must be less than P ave P ave , what are the optimal signal choices? Is your answer unique?

### 4.b)

When the peak power P peak P peak is constrained, what are the optimal signal choices?

### 4.c)

If P ave = P peak P ave P peak , which constraint yields the best detection performance?

## Exercise 5

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 0 : rl= s ˜ 0 l+nl   l0L1 0 : r l s ˜ 0 l n l   l 0 L 1 1 : rl= s ˜ 1 l+nl   l0L1 1 : r l s ˜ 1 l n l   l 0 L 1 where s ˜ i l s ˜ i l equals s i l+wl s i l w l , with wl w l and nl n l comprising white Gaussian noise having variances σ w 2 σ w 2 and σ n 2 σ n 2 , respectively. We know precisely what the signals s ˜ i l s ˜ i l are, but not the underlying "actual" signal.

### 5.a)

Find a detector for this problem.

### 5.b)

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

## Exercise 6

One of the more interesting problems in detection theory is determining when the probability distribution of the observations differs from that in other portions of the observation interval. The most common form of the problem is that over the interval 0 C 0 C , the observations have one form, and that in the remainder of the observation interval C L1 C L 1 have a different probability distribution. The change detection problem is to determine whether in fact a change has occurred and, if so, estimate when that change occurs.

To explore the change detection problem, let's explore the simple situation where the mean of white Gaussian noise changes at the C th C th sample. 0 : rl𝒩0σ2   l0L1 0 : r l 0 σ 2   l 0 L 1 1 : rl{𝒩0σ2  if  l0C1𝒩mσ2  if  lCL1 1 : r l 0 σ 2 l 0 C 1 m σ 2 l C L 1 The observations in each case are statistically independent of the others.

### 6.a)

Find a detector for this change problem when mm is a known positive number.

### 6.b)

Find an expression for the threshold in the detector using the Neyman-Pearson criterion.

### 6.c)

How does the detector change when the value of mm is not known?

## Exercise 7

Noise generated by a system having zeros complicates the calculations for the colored noise detection problem. To illustrate these difficulties, assume the observation noise is produced as the output of a filter governed by the difference equation

a , b ,ab:nl=anl1+wl+bwl1 a b a b n l a n l 1 w l b w l 1
(1)
where wl w l is white, Gaussian noise. Assume an observation duration sufficiently long to capture the colored effects.

### 7.a)

Find the covariance matrix of this noise process

### 7.b)

Calculate the Cholesky factorization of the covariance matrix

### 7.c)

Find the unit-sample response of the optimal detector's whitening filter. If it weren't for the finite observation interval, would it indeed have an infinite-duration unit-sample reponse as claimed here? Describe the edge-effects of your filter, constrasting them with the case when b=0 b 0 .

## Exercise 8

The calculation of the sufficient statistic in spectrally based detectors (see this equation) can be simplified using signal processing notions. When the observations are real-valued, their spectra are conjugate symmetric.

### 8.a)

Exploiting this symmetry, how can the calculations of this equation be simplified? Be careful to note "spectral edge-effects."

### 8.b)

Can your formula be manipulated to depend on the power spectra of the signals and the observations? Why or why not?

## Exercise 9

Here, we claimed that the relation between noise bandwidth and reciprocal duration of the observation interval played a key role in determining whether DFT values were approximately uncorrelated. While the statements sound plausible, their veracity should be checked. Let the covariance function of the observation noise be K n l=a|l| K n l a l .

### 9.a)

How is the bandwidth (defined by the half-power point) of this noise's power spectrum related to the parameter aa? How is the duration (defined to be two time constants) of the covariance function related to aa?

### 9.b)

Find the variance of the length-LL DFT of this noise process as a function of frequency index kk. This result should be compared with the power spectrum calculated in the previous part; they should resemble each other when the "memory" of the noise - the duration of the covariance function - is much less than LL while demonstrating differences as the memory becomes comparable to or exceeds LL.

### 9.c)

Calculate the covariance between adjacent frequency indices. Under what conditions will they be approximately uncorrelated? Relate your answer to the relations of aa to LL found in the previous part.

## Exercise 10

The results derived in Exercise 9 assumed that a length-LL Fourier Transform was computed from a length-LL segment of the noise process. What will happen if the transform has length 2L 2 L with the observation interval remaining unchanged?

### 10.a)

Find the variance of DFT values at index kk.

### 10.b)

Assuming the conditions in Exercise 9 for uncorrelated adjacent samples, now what is the correlation between adjacent DFT values?

## Exercise 11

A common application of spectral detection is the detection of sinusoids in noise. Not only can one sinusoidal signal be present, but several. The problem is to determine which combination of sinusoids is present. Let's first assume that no more than two sinusoids can be present, the frequencies of which are integer fractions of the number of observations: f i = k i L f i k i L , i12 i 1 2 . Assume additive Gaussian noise is present.

### 11.a)

What is the detection procedure for determining if one of the sinusoids is present while ignoring the second? The amplitude of the sinusoid is known.

### 11.b)

How does the probability of detection vary with the ratio of the signal's squared amplitude to the noise variance at the frequency of the signal? Sketch this result for several false-alarm probabilities.

### 11.c)

Now assume that none, either, or both of the signals can be present. Find the detector that best determines the combination present in the observations. Does your answer have a simple interpretation?

### 11.d)

Derive a detector that determines the number of sinusoids present in the input.

## Exercise 12

In a discrete-time detection problem, one of two, equally likely, length-LL sinusoids (each having a frequency equal to a known multiple of 1L 1 L ) is observed in additive colored Gaussian noise. Signal amplitudes are also unknown by the receiver. In addition, the power spectrum of the noise is uncertain: what is known is that the noise power spectrum is broadband, and varies gently across frequency. Find a detector that performs well for this problem. What notable properties (if any) does your receiver have?

## Exercise 13

A sampled signal is suspected of consisting of a periodic component and additive Gaussian noise. The signal, if present, has a known period LL. The number of samples equals NN, a multiple of LL. The noise is white and has known variance of σ2 σ 2 . A consultant (you!) has been asked to determine the signal's presence.

### 13.a)

Assuming the signal is a sinusoid with unknown phase and amplitude, what should be done to determine the presence of the sinusoid so that a false-alarm probability criterion of 0.1 is met?

### 13.b)

Other than its periodic nature, now assume that the signal's waveform is unknown. What computations must the optimum detector perform?

## Exercise 14

The QAM (Quadrature Amplitude Modulation) signal set consists of signals of the form s i l= A i c cos2π f c l+ A i s sin2π f c l s i l A i c 2 f c l A i s 2 f c l where A i c A i c and A i s A i s are amplitudes that define each element of the signal set. These are chosen according to design constraints. Assume the signals are observed in additive Gaussian noise.

### 14.a)

What is the optimal amplitude choice for the binary and quaternary (four-signal) signal sets when the noise is white and the signal energy is constrained ( l l s i 2l<E l l s i l 2 E )? Comment on the uniqueness of your answers.

### 14.b)

Describe the optimal binary QAM signal set when the noise is colored.

### 14.c)

Now suppose the peak amplitude ( max l l | s i l|< A max l s i l A max ) is constrained. What are the optimal signal sets (both binary and quaternary) for the white noise case? Again, comment on uniqueness.

## Exercise 15

Because sources rarely remain still, a common problem in radar applications is uncertainty of the signal frequency because of Doppler shifts. This effect is usually small, but conceivably could be large enough for a sinusoidal signal to "wander" from its presumed frequency index.

### 15.a)

Design a spectrally based detection strategy which examines a range of frequencies, testing for the presence of a sinusoid in any one of the bins or no sinusoid in any bin.

### 15.b)

Predict the detection loss (or gain) relative to the known frequency case (no Doppler shift) by constrasting the signal-to-noise ratio terms in the detection probability expression for each detector.

### 15.c)

Contrast the performance of this detector with a square-law detector that only considers the frequency band in question.

## Exercise 16

I confess that unknown-delay problem given previously is not terribly relevant to active sonar and radar ranging problems. In these cases, the signal's delay, measured with respect to its emission by a source, presents the round-trip propagation time from the source to the energy-reflecting object and back. Hence, delay is proportional to twice the range. What makes the example overly simplistic is the independence of signal amplitude on delay.

### 16.a)

Because of attenuation due to spherical propagation, show that the received signal energy is inversely related to the fourth power of the range. This result is known as the radar equation.

### 16.b)

Derive the detector that takes the dependence of delay and amplitude into account, thereby optimally determining the presence of a signal in active radar/sonar applications, and produces a delay estimate, thereby simultaneously providing the object's range. Not only determine the detector's structure, but also how the threshold and false-alarm probability are related.

### 16.c)

Does the object's reflectivity need to be known to implement your detector?

## Exercise 17

Derive the frequency domain counterpart of the CFAR detector, explicitly indicating its independence on noise variance. Constrast this detector to that derived in Exercise 15: how do they differ? How much penalty in performance is paid by uncertainty in the noise variance?

## Exercise 18

CFAR detectors are extremely important in applications because they automatically adapt to the value of noise variance during the observations, allowing them to be used in varying noise situations. However, as described in Exercise 16, unknown delays must be also considered in realistic problems.

### 18.a)

Derive a CFAR detector that also takes unknown signal delay into account.

### 18.b)

Show that your detector automatically incorporates amplitude uncertainties.

### 18.c)

Under the no-signal model, what is the distribution of the sufficient statistic?

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