Assume that observations of a sinusoidal signal s⁢l=A⁢sin2⁢π⁢f⁢l s l A 2 f l , l∈0…L−1 l 0 … L 1 , are contaminated by first-order colored noise as decribed in the example.

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, k∈1…K 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).

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.

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 ). ℳ 0 : r⁢l= s ( 0 ) ⁢l+n⁢l   l∈0…L−1 ℳ 0 : r l s ( 0 ) l n l   l 0 … L 1 ℳ 1 : r⁢l= s ( 1 ) ⁢l+n⁢l   l∈0…L−1 ℳ 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 .

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 ˜ 0 ⁢l+n⁢l   l∈0…L−1 ℳ 0 : r l s ˜ 0 l n l   l 0 … L 1 ℳ 1 : r⁢l= s ˜ 1 ⁢l+n⁢l   l∈0…L−1 ℳ 1 : r l s ˜ 1 l n l   l 0 … L 1 where s ˜ i ⁢l s ˜ i l equals s i ⁢l+w⁢l s i l w l , with w⁢l w l and n⁢l 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.

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 L−1 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. ℳ 0 : r⁢l∼𝒩0σ2   l∈0…L−1 ℳ 0 : r l 0 σ 2   l 0 … L 1 ℳ 1 : r⁢l∼{𝒩0σ2  if  l∈0…C−1𝒩mσ2  if  l∈C…L−1 ℳ 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.

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

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.

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 .

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 2⁢L 2 L with the observation interval remaining unchanged?

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 , i∈12 i 1 2 . Assume additive Gaussian noise is present.

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?

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.

The QAM (Quadrature Amplitude Modulation) signal set consists of signals of the form s i ⁢l= A i c ⁢cos2⁢π⁢ f c ⁢l+ A i s ⁢sin2⁢π⁢ 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.

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.

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.

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?

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.