Non-overlapping Received Signals 1.4 2003/06/10 19:00:00 GMT-5 2003/09/08 13:34:23.392 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 Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu multipath channel If we restrict to observing r t over 0 T , the minimum P e receiver computes N 0 2 π i s i * r s i * 2 2 and chooses the largest. Here, s i * r t 0 T s i * t r t s i * 2 t 0 T s i * t 2 When we have a binary signal set, the probability of error is given by P e Q s 0 * s 1 * 2 N 0 One question of interest is what the "best" signal set at the transmitter to use to maximize the performance. We want to find the s i t that maximizes s 0 * s 1 * 2 for a given channel model. As usual, we constrain the transmitter signal energies by s i 2 E . Now s 0 * s 1 * 2 t 0 T s 0 * t 2 2 s 0 * t s 1 * t s 1 * t 2 As s i * s i h CH , we have t s i * t s j * t t α h CH t α s i α β h CH t β s j β Interchanging the order of integration, this integral can be written as β α t h CH t α h CH t β s i α s j β Defining the innermost integral as Q CH α β , we can write the squared distance between the received signals as s 0 * s 1 * 2 β α Q CH α β s 0 α s 0 β s 0 α s 1 β s 0 β s 1 α s 1 α s 1 β β α Q CH α β s 0 α s 1 α s 0 β s 1 β Q CH s 0 s 1 2 where Q CH x denotes the norm induced by Q CH α β . Q CH x 2 β α Q CH α β x α x β This quantity is a valid inner product because Q CH α β is a positive-definite, symmetric function of α and β. The relevant inner product can be written as Q CH s 0 s 1 2 Q CH s 0 2 2 Q CH s 0 s 1 Q CH s 1 2 Therefore, if we require Q CH s 0 2 Q CH s 1 2 , the maximum value of this norm occurs when Q CH s 0 s 1 Q CH s 1 2 . In other words, s 0 s 1 . Consequently, antipodal signals still have the best performance. Because of the weighting function Q CH · · in the inner product, certain signals are better than others. We seek to find those signals. According to Mercer's theorem, Q CH α β may be represented in an orthogonal expansion. Q CH α β j 1 λ j φ j α φ j β where φ j t and λ j are the eigenfunctions and eigenvalues of Q CH α β . As s 0 t j 1 s 0 j φ j t , the squared norm of a signal can be written Q CH s 0 2 β α Q CH α β s 0 α s 0 β j 1 λ j s 0 j 2 When we have an antipodal signal set, Q CH s 0 s 1 2 4 λ j s 0 j 2 . To constrain the energy of s 0 t means that s 0 j 2 E 0 . If h CH t τ δ t τ , Q CH α β δ α β λ j 1 for all j. Consequently, the distance with respect to Q CH α β is maximized for any signal having energy E 0 . On the other hand, if the λ j are not identical, the values of the representation s 0 j affect the distance. To maximize this distance, set s 0 j 0 except for that component corresponding to the largest eigenvalue λ max . Best results are obtained, therefore, when s 0 t E φ m t , where λ max j λ j . In this case, Q CH s 0 s 1 2 4 λ max E . With this optimum signal selection, P e Q 2 λ max E N 0 Consider a multipath channel that has the impulse response h CH t τ δ t τ a δ t τ Δ where Δ is the delay in the multipath and a is the gain (usually less than one). The kernal Q CH α β becomes in this case Q CH α β 1 a 2 δ α β a δ α β Δ a δ α β Δ The eigenfunctions of this kernal are determined by the equation λ φ t 1 a 2 φ t a φ t Δ a φ t Δ The solutions of this equation are sinusoids having frequencies harmonically related to Δ. φ k t A 2 k Δ t B 2 k Δ t λ k 1 a 2 In principal, any signal represented by a Fourier series having a period of Δ would yield the smallest probability of error over this multipath channel. When this solution is considered in detail, no time-limited solution to the eigenequation exists. Thus, this analysis does not yield a realistic signal set for transmitting digital information over a multipath channel. However, this result does suggest that a time-limited version of this signal set would result in a receiver insensitive to the multipath. Consider the antipodal signal set s 0 t E T Δ 0 t T Δ 0 T Δ t T s 1 t s 0 t The signal s 0 * t is given by s 0 * t E T Δ 0 t Δ 1 a E T Δ Δ t T Δ a E T Δ T Δ t T The probability of error obtained when a receiver uses this signal in its matched filter is P e Q 2 E N 0 1 a 2 2 a Δ T Δ The probability of error of the optimum (unobtainable) receiver is given by P e Q 2 1 a 2 E N 0 Thus, the suboptimum receiver can perform almost as well as the optimum one under certain conditions (i.e., Δ small compared to T Δ ).