Additive Colored Gaussian Noise Channels 1.6 2003/05/27 19:00:00 GMT-5 2003/09/08 14:31:29.971 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu Reijo Anttila reijo@cnx.rice.edu Colored Gaussian Noise Channels inverse kernel Fredholm integral Another important channel model occurs when the signal is uncorrupted save for the addition of Gaussian noise that is not necessarily white or stationary. We presume that correlation function of the noise is known. In particular we model the production of n t by passing white, Gaussian noise ω t through a linear filter. ()

We can convert this detection problem to one we have solved. Namely assume the existence of a whitening filter h ω t τ that changes n t into a white process. Consider passing r t through this whitening filter. ()

This procedure does not remove signal information as the whitening filter is reversible. We assume that R ω t u δ t u . We then have the detection of a known signal in white, Gaussian noise. We have solved this problem previously and know the minimum P e receiver computes the quantity π i Q n r s i Q n s i 2 2 where Q n r s i u t r t s i u Q n t u and Q n t u α h ω α t h ω α u . Consequently, all of the issues and solutions involved with the dispersive channel problem arise here. For example, antipodal signals are optimum. However, we shall see that there is a possibility of having communication over a noisy channel with no error! The inner product term Q n r s i can be written as a matched filter. u t r t s i u Q n t u t r t g i t where we define g i t as u s i u Q n t u . To find g i t more directly, we note that the correlation function of the output of the whitening filter is given by β α h ω t α h ω u β K n α β δ t u Convolving both sides of this equation with h ω t v , we have t h ω t v β α h ω t α h ω u β K n α β h ω u v Interchanging the order of integration results in β h ω u β α Q n α v K n α β h ω u v α Q n α v K n α β δ v β Because of this property, we can have that for any choice of φ i t β φ i β α Q n α v K n α β φ i v Let φ i t denote the eigenfunctions of K n α β . In this case, by interchanging integration order, we find that α Q n α v φ i α 1 λ i φ i v Consequently, eigenfunctions of K n are also eigenfunctions of Q n , and the eigenvalues are reciprocals of each other. Q n is thus termed the inverse kernel of K n . Using Mercer's Theorem, K n t u λ i φ i t φ i u Q n t u 1 λ i φ i t φ i u Recalling the defining equation for g i t , multiplying by K n t β and integrating β K n t β g i β β u s i u Q n β u K n t β u s i u β Q n β u K n t β As Q n · · is the inverse kernel of K n · · the last integral equals an impulse, and the impulse response g i t of the optimum receiver's matched filter is the solution of the integral equation β K n t β g i β s i t Despite the fact that this is an implicit equation for g i t , it is easier to solve than the explicit equation given above. The reason is that one is usually given the noise covariance function rather than its inverse kernel. Generally, one assumes the covariance function of the noise to be K n t u N 0 2 δ t u K c t u where K c t u denotes the colored noise component of K n t u [ K c t u has no impulses]. The equation for g i t becomes N 0 2 g i t u K c t u g i u s i t This expression is the Fredholm integral equation of the second kind. If N 0 0 (no white noise), we have the Fredholm integral of the first kind. Assume that n t is stationary; this situation implies that its covariance function has the form K n t u K n t u N 0 2 δ t u K c t u As we shall see, the absence or presence of the white noise term (i.e., does 𝒮 n f 0 as f or not) has a pronounced effect on the nature of g i t . In general, if N 0 0 , the solution will contain impulses, and, furthermore, the performance of the receiver will be very sensitive to assumptions one makes about the noise. If N 0 0 , the character of g i t changes completely: Answers make more sense, and the results are "nicer." To make these considerations more precise, we consider the important special case where 𝒮 n f is a rational function. 𝒮 n f N 2 f 2 D 2 f 2 where N · and D · are polynomials. Because 2 f is associated with the derivative in the time domain, we see that K n τ satisfies the differential equation: D p 2 K n τ N p 2 δ τ where ↔ p τ , ↔ p 2 τ 2 , ↔ p 3 τ 3 , etc. Now, we wish to solve s i t u 0 T K n t u g i u Multiplying by D p 2 , with p representing taking a derivative with respect to t, D p 2 s i t u 0 T D p 2 K n t u g i u Using the differential equation given previously that the noise covariance function must satisfy, we find that, t 0 t T D p 2 s i t u 0 T N p 2 δ t u g i u N p 2 g i t Consequently, the solution to our integral equation can be found by solving this differential equation. The solution consists of the particular solution g i P t and the homogeneous solution of N p 2 g i t 0 . When the observation interval is finite, the solution may also contain impulses at the boundaries.Letting n denote the order of the numerator polynomial and m the order of the denominator, the solution has the form g t g P t i 1 2 n a i g i h t k 0 m n 1 b k δ k t T i c k δ k t T f If the impulses are not included, the original integral equation may not be satisfied; this situation occurs when N 0 0 . When N 0 0 , the impulses are no longer necessary. Additional "funny things" happen when N 0 0 . If, for example, an antipodal signal set is used, Q s 0 s 1 2 4 Q s 0 2 j 1 s 0 j 2 λ j c N 0 2 where λ j c is an eigenvalue of K c t u . When N 0 0 , s 0 j 2 λ j c N 0 2 s 0 j 2 N 0 2 , which means that this distance will always be finite. On the other hand, if N 0 0 , s 0 j 2 λ j c may not converge. In such cases, the signals are infinitely far apart with respect to the distance measure induced by the inverse kernel, and the probability of error is zero! Generally speaking, such behavior is not obtained in actuality. White noise is usually present in the front end of a receiver. The colored noise component has power density spectrum and covariance function given by 𝒮 c f 2 α β β 2 2 f 2 K c τ α β τ The differential equation to solve when N 0 0 becomes t 2 s i t β 2 s i t 2 α β g t Obviously, we have an explicit equation for g t , meaning that g h t 0 . Adding in the impulses and substituting back into the integral equation, the total solution is g t 1 2 α β β s 0 s 0 δ t β s T s T δ t T β 2 s t 2 s t Note that if s t is discontinuous, we have doublets in the signal g t ! If, however, N 0 0 , the differential equation becomes N 0 2 t 2 g γ 2 g t t 2 s β 2 s t where γ 2 β 2 4 α β N 0 . Therefore, g h t a 1 γ t a 2 γ t and there are no impulses! Consequently, one usually assumes the presence of some white noise to refrain from obtaining rather bizarre answers.