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Detection of Signals in Noise

Module by: Don Johnson Based on: White Gaussian Noise by Don Johnson

Summary: Detection theory is specialized to the most common decision problem that occurs in signal processing: determining which signal was received in the presence of additive noise.

Far and away the most common decision problem in signal processing is determining which of several signals occurs in data contaminated by additive noise. Specializing to the case when one of two possible of signals is present, the data models are

  • ℳ 0 : Rl= s 0 l+Nl , 0≤l<L ℳ 0 : R l s 0 l N l , 0 l L
  • ℳ 1 : Rl= s 1 l+Nl , 0≤l<L ℳ 1 : R l s 1 l N l , 0 l L
where s i l s i l denotes the known signals and Nl N l denotes additive noise modeled as a stationary stochastic process. This situation is known as the binary detection problem: distinguish between two possible signals present in a noisy waveform.

We form the discrete-time observations into a vector: R=R0…RL-1T R R 0 … R L 1 . Now the models become

  • ℳ 0 : R= s 0 +N ℳ 0 : R s 0 N
  • ℳ 1 : R= s 1 +N ℳ 1 : R s 1 N
To apply our detection theory results, we need the probability density of RR under each model. As the only probabilistic component of the observations is the noise, the required density for the detection problem is given by pR| ℳ i r=pNr- s i p R ℳ i r p N r s i and the corresponding likelihood ratio by Λr=pNr- s 1 pNr- s 0 Λ r p N r s 1 p N r s 0 Much of detection theory revolves about interpreting this likelihood ratio and deriving the detection threshold.

Additive White Gaussian Noise

By far the easiest detection problem to solve occurs when the noise vector consists of statistically independent, identically distributed, Gaussian random variables, what is commonly termed white Gaussian noise. The mean of white noise is usually taken to be zero1 and each component's variance is σ2 σ 2 . The equal-variance assumption implies the noise characteristics are unchanging throughout the entire set of observations. The probability density of the noise vector evaluated at r- s i r s i equals that of a Gaussian random vector having independent components with mean s i s i . pNr- s i =12πσ2L2ⅇ-12σ2r- s i Tr- s i p N r s i 1 2 σ 2 L 2 1 2 σ 2 r s i r s i The resulting detection problem is similar to the Gaussian example we previously examined, with the difference here being a non-zero mean---the signal---under both models. The logarithm of the likelihood ratio becomes r- s 0 Tr- s 0 -r- s 1 Tr- s 1 ≷ ℳ 0 ℳ 1 2σ2lnη r s 0 r s 0 r s 1 r s 1 ≷ ℳ 0 ℳ 1 2 σ 2 η The usual simplifications yield in rT s 1 - s 1 T s 1 2-rT s 0 - s 0 T s 0 2 ≷ ℳ 0 ℳ 1 σ2lnη r s 1 s 1 s 1 2 r s 0 s 0 s 0 2 ≷ ℳ 0 ℳ 1 σ 2 η The model-specific components on the left side express the signal processing operations for each model.2

Each term in the computations for the optimum detector has a signal processing interpretation. When expanded, the term s i T s i s i s i equals ∑l=0L-1 s i 2l l 0 L 1 s i l 2 , the signal energy E i E i . The remaining term, rT s i r s i , is the only one involving the observations and hence constitutes the sufficient statistic ϒ i r ϒ i r for the additive white Gaussian noise detection problem. ϒ i r=rT s i ϒ i r r s i An abstract, but physically relevant, interpretation of this important quantity comes from the theory of linear vector spaces. In that context, the quantity rT s i r s i would be termed the projection of rr onto s i s i . From the Schwarz inequality, we know that the largest value of this projection occurs when these vectors are proportional to each other. Thus, a projection measures how much alike two vectors are: they are completely alike when they are parallel (proportional to each other) and completely dissimilar when orthogonal (the projection is zero). In effect, the projection operation removes those components from the observations which are orthogonal to the signal, thereby generalizing the familiar notion of filtering a signal contaminated by broadband noise. In filtering, the signal-to-noise ratio of a bandlimited signal can be drastically improved by lowpass filtering; the output would consist only of the signal and "in-band" noise. The projection serves a similar role, ideally removing those "out-of-band" components (the orthogonal ones) and retaining the "in-band" ones (those parallel to the signal).

Matched Filtering

The projection operation can be expanded as rT s i =∑l=0L-1rl s i l r s i l 0 L 1 r l s i l another signal processing interpretation emerges. The projection now describes a finite impulse response (FIR) filtering operation evaluated at a specific index. To demonstrate this interpretation, let hl h l be the unit-sample response of a linear, shift-invariant filter where hl=0 h l 0 for l<0 l 0 and l≥L l L . Letting rl r l be the filter's input sequence, the convolution sum expresses the output. rk*hk=∑l=k-L-1krlhk-l r k h k l k L 1 k r l h k l Letting k=L-1 k L 1 , the index at which the unit-sample response's last value overlaps the input's value at the origin, we have rk*hk|k=L-1=∑l=0L-1rlhL-1-l k L 1 r k h k l 0 L 1 r l h L 1 l Suppose we set the unit-sample response equal to the index-reversed, then delayed signal. hl= s i L-1-l h l s i L 1 l In this case, the filtering operation becomes a projection operation. rk* s i L-1-k|k=L-1=∑l=0L-1rl s i l k L 1 r k s i L 1 k l 0 L 1 r l s i l Figure 1 depicts these computations graphically.

Figure 1: The detector for signals contained in additive, white Gaussian noise consists of a matched filter, whose output is sampled at the duration of the signal and half of the signal energy is subtracted from it. The optimum detector incorporates a matched filter for each signal compares their outputs to determine the largest.
Figure 1 (mf.png)

The sufficient statistic for the i th i th signal is thus expressed in signal processing notation as rk* s i L-1-k|k=L-1- E i 2 k L 1 r k s i L 1 k E i 2 . The filtering term is called a matched filter because the observations are passed through a filter whose unit-sample response "matches" that of the signal being sought. We sample the matched filter's output at the precise moment when all of the observations fall within the filter's memory and then adjust this value by half the signal energy. The adjusted values for the two assumed signals are subtracted and compared to a threshold.

Detection Performance

To compute the performance probabilities, the expressions should be simplified in the ways discussed in previous sections. As the energy terms are known a priori they can be incorporated into the threshold with the result ∑l=0L-1rl s 1 l- s 0 l ≷ ℳ 1 ℳ 0 σ2lnη+ E 1 - E 0 2 l 0 L 1 r l s 1 l s 0 l ≷ ℳ 1 ℳ 0 σ 2 η E 1 E 0 2 The left term constitutes the sufficient statistic for the binary detection problem. Because the additive noise is presumed Gaussian, the sufficient statistic is a Gaussian random variable no matter which model is assumed. Under ℳ i ℳ i , the specifics of this probability distribution are ∑l=0L-1rl s 1 l- s 0 l∼Nmivari l 0 L 1 r l s 1 l s 0 l N mi vari where the mean and variance of the Gaussian distribution are given respectively by mi=∑ s i l s 1 l- s 0 l mi s i l s 1 l s 0 l vari=σ2∑ s 1 l- s 0 l2 vari σ 2 s 1 l s 0 l 2 Note that the variance does not depend on model. The false-alarm probability is given by P F =Qσ2lnη+ E 1 - E 0 2-m0var12 P F Q σ 2 η E 1 E 0 2 m0 var 1 2 The signal-related terms in the numerator of this expression can be manipulated so that the false-alarm probability of the optimal white Gaussian noise detector is succinctly expressed by P F =Qlnη+12σ2∑ s 1 l- s 0 l21σ∑ s 1 l- s 0 l212 P F Q η 1 2 σ 2 s 1 l s 0 l 2 1 σ s 1 l s 0 l 2 1 2

Note that the only signal-related quantity affecting this performance probability (and all of the others as well) is the ratio of the energy in the difference signal to the noise variance. The larger this ratio, the better (i.e., smaller) the performance probabilities become. Note that the details of the signal waveforms do not greatly affect the energy of the difference signal. For example, consider the case where the two signal energies are equal ( E 0 = E 1 =E E 0 E 1 E ); the energy of the difference signal is given by 2E-2∑ s 0 l s 1 l 2 E 2 s 0 l s 1 l . The largest value of this energy occurs when the signals are negatives of each other, with the difference-signal energy equaling 4E 4 E . Thus, equal-energy but opposite-signed signals such as sine waves, square-waves, Bessel functions, etc. all yield exactly the same performance levels. The essential signal properties that do yield good performance values are elucidated by an alternate interpretation. The term ∑ s 1 l- s 0 l2 s 1 l s 0 l 2 equals ∥s1-s0∥2 s 1 s 0 2 , the L2 L 2 norm of the difference signal. Geometrically, the difference-signal energy is the same quantity as the square of the Euclidean distance between the two signals. In these terms, a larger distance between the two signals means better performance.

Example 1: Detection, Gaussian example

A common detection problem is to determine whether a signal is present ( ℳ 1 ℳ 1 ) or not ( ℳ 0 ℳ 0 ). To model the latter case, the signal equals zero: s 0 l=0 s 0 l 0 . The optimal detector relies on filtering the data with a matched filter having a unit-sample response based on the signal that might be present. Letting the signal under ℳ 1 ℳ 1 be denoted simply by sl s l , the optimal detector consists of rl*sL-1-l|l=L-1-E2 ≷ ℳ 1 ℳ 0 σ2lnη l L 1 r l s L 1 l E 2 ≷ ℳ 1 ℳ 0 σ 2 η or rl*sL-1-l|l=L-1 ≷ ℳ 1 ℳ 0 γ l L 1 r l s L 1 l ≷ ℳ 1 ℳ 0 γ The false-alarm and detection probabilities are given by P F =QγE12σ P F Q γ E 1 2 σ P D =QQ-1 P F -Eσ P D Q Q P F E σ Figure 2 displays the probability of detection as a function of the signal-to-noise ratio Eσ2 E σ 2 for several values of false-alarm probability. Given an estimate of the expected signal-to-noise ratio, these curves can be used to assess the trade-off between the false-alarm and detection probabilities.

Figure 2: The probability of detection is plotted versus signal-to-noise ratio for various values of the false-alarm probability P F P F . False-alarm probabilities range from 10-1 10 -1 down to 10-6 10 -6 by decades. The matched filter receiver was used since the noise is white and Gaussian. Note how the range of signal-to-noise ratios over which the detection probability changes shrinks as the false-alarm probability decreases. This effect is a consequence of the non-linear nature of the function Q· Q · .
Figure 2 (pdformanypf.png)

The important parameter determining detector performance derived in this example is the signal-to-noise ratio Eσ2 E σ 2 : the larger it is, the smaller the false-alarm probability is (generally speaking). Signal-to-noise ratios can be measured in many different ways. For example, one measure might be the ratio of the rms signal amplitude to the rms noise amplitude. Note that the important one for the detection problem is much different. The signal portion is the sum of the squared signal values over the entire set of observed values - the signal energy; the noise portion is the variance of each noise component - the noise power. Thus, energy can be increased in two ways that increase the signal-to-noise ratio: the signal can be made larger or the observations can be extended to encompass a larger number of values.

To illustrate this point, how a matched filter operates is shown in Figure 3. The signal is very difficult to discern in the presence of noise. However, the signal-to-noise ratio that determines detection performance belies the eye. The matched filter output demonstrates an amazingly clean signal.

Figure 3: The signal consists of ten cycles of sin ω 0 l ω 0 l with ω 0 =2π0.1 ω 0 2 0.1 . The middle panel shows the signal with noise added. The lower portion depicts the matched-filter output. The detection threshold was set for a false-alarm probability of 10-2 10 -2 . Even though the matched filter output crosses the threshold several times, only the output at l=L-1 l L 1 matters. For this example, it coincides with the peak output of the matched filter.
Figure 3 (mfout1.png)

Footnotes

  1. The zero-mean assumption is realistic for the detection problem. If the mean were non-zero, simply subtracting it from the observed sequence results in a zero-mean noise component.
  2. If more than two signals were assumed possible, quantities such as these would need to be computed for each signal and the largest selected.

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