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# Continuous-Time Detection Theory

Module by: Don Johnson. E-mail the author

## Introduction

In previous sections, we used a sampling approach to detect which of several signals was present in additive noise. While less general, an alternate approach can be used in situation where the additive noise is Gaussian. In such cases, the problem can be solved entirely in continuous-time without requiring sampling. This approach relies on the Karhunen-Loève expansion, which results in a representation of the received process rtrt . In general, this representation is an infinite dimensional vector; the critical result of continuous-time detection is that a finite-dimensional representation can always be found so that hypothesis testing results can be applied.

## Matched Filter Receiver for White Gaussian Noise Channels

The received signal rt r t is assumed to have one of MM forms

i,t,i0M1(0t<T):rt= s i t+nt i t i 0 M 1 0 t T r t s i t n t
(1)
where the s i t s i t comprise the signal set. nt n t is usually assumed to be statistically independent of the transmitted signal and a white, Gaussian process having spectral height N 0 2 N 0 2 . We represent the received signal with a Karhunen-Loève expansion. rt=j=1 r j φ j t=j=1( s i j + n j ) φ j t r t j 1 r j φ j t j 1 s i j n j φ j t where s i j s i j and n j n j are the representations of the signal s i t s i t and the noise nt n t , respectively. To have a Karhunen-Loève expansion, it suffices to choose φ j t φ j t so that the n j n j are pairwise uncorrelated. As nt n t is white, we may choose any φ j t φ j t we want! In particular, choose φ j t φ j t to be the set of functions which yield a finite-dimensional representation for the signals s i t s i t . A complete, but not necessarily orthonormal, set of functions that does this is s 0 t s M - 1 t ψ 0 t ψ 1 t s 0 t s M - 1 t ψ 0 t ψ 1 t where ψ j t ψ j t denotes any complete set of functions. We form the set φ j t φ j t by applying the Gram-Schmidt procedure to the set. With this basis, s i j =0 s i j 0 , jM j M . In this case, the representation of rt r t becomes rt={ s i j + n j   if  j0M1 n j   if  jM r t s i j n j j 0 M 1 n j j M so that we may write the model evaluation problem we are attempting to solve as 0 : rt=( s 0 0 + n 0 ) φ 0 t++( s 0 , M - 1 + n M - 1 ) φ M - 1 t+jM n j φ j t 0 : r t s 0 0 n 0 φ 0 t s 0 , M - 1 n M - 1 φ M - 1 t j j M n j φ j t 1 : rt=( s 1 0 + n 0 ) φ 0 t++( s 1 , M - 1 + n M - 1 ) φ M - 1 t+jM n j φ j t 1 : r t s 1 0 n 0 φ 0 t s 1 , M - 1 n M - 1 φ M - 1 t j j M n j φ j t We make two observations:
• We can consider the model evaluation problem that operates on the representation of the received signal rather than the signal itself. Recall that using the representation is equivalent to using the original process. We have thus created an equivalent model evaluation problem. For the binary signal set case, 0 : r= s 0 +n 0 : r s 0 n 1 : r= s 1 +n 1 : r s 1 n where nn contains statistically independent Gaussian components, each of which has variance N 0 2 N 0 2 .
• Note that the components are statistically independent of each other and that, for jM j M , the representation contains no signal-related information. Because these components are extraneous and will not contribute to improved performance, we can reduce the dimension of the problem to no more than MM by ignoring these components. By rejecting these noise-only components, we are effectively filtering out "out-of-band" noise, retaining those components related to the signals. Using eigenfunction related to the signals defines signal space, allowing us to ideally reject pure-noise components.

As a consequence of these observations, we have a model evaluation problem of the form r= r 0 r K - 1 = s i , 0 s i , K - 1 + n 0 n K - 1 r r 0 r K - 1 s i , 0 s i , K - 1 n 0 n K - 1 We know how to solve this problem; we compute i,i0K1: ϒ i r= N 0 2ln π i + s i ,r s i 22 i i 0 K 1 ϒ i r N 0 2 π i s i r s i 2 2 and choose the largest. The components of the signal and received vectors are given by s i j =0T s i t φ j td t s i j t 0 T s i t φ j t r j =0Trt φ j td t r j t 0 T r t φ j t Because of Parseval's Theorem, the inner product between representations equals the time-domain inner product between the represented signals. s i ,r=0T s i trtd t s i r t 0 T s i t r t Furthermore, s i 2=0T s i 2td t = E i s i 2 t 0 T s i t 2 E i , the energy in the i th i th signal. Thus, the sufficient statistic for the optimal detector has a closed form time-domain expression.

ϒ i r= N 0 2ln π i +0T s i trtd t E i 2 ϒ i r N 0 2 π i t 0 T s i t r t E i 2
(2)
This form of the minimum probability of error receiver is termed a correlation receiver (see Figure 1). Each transmitted signal and the received signal are correlated to obtain the sufficient statistic. These operations project the received signal onto signal space.

An alternate structure which computes the same quantities can be derived by noting that if ft f t and gt g t are nonzero only over 0 T 0 T , the inner product (correlation) operation can be written as a convolution followed by a sampler. 0Tftgtd t =ft*gTt|t=T t 0 T f t g t t T f t g T t Consequently, we can restructure the "correlation" operation as a filtering-and-sampling operation. The impulse responses of the linear filters are time-reversed, delayed versions of the signals in the signal set. This structure for the minimum probability of error receiver is known as the matched-filter receiver (see Figure 2). Each type of receiver has the same performance; however, the matched filter receiver is usually easier to construct because the correlation receiver requires an analog multiplier.

As we know, receiver performance is judged by the probability of error, which, for equally likely signals in a binary signal set, is given by

P e =Q s 0 s 1 2 N 0 2 P e Q s 0 s 1 2 N 0 2
(3)
The computation of the probability of error and the dimensionality of the problem can be assessed by considering signal space: The representation of the signals with respect to a basis. The number of basis elements required to represent the signal set defines dimensionality. The geometric configuration of the signals in this space is known as the signal constellation. Once this constellation is found, computing intersignal distances is easy.

## Glossary

signal space:
The representation of the signals with respect to a basis.

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