The cross-correlation function between two continuous-time real signals x(t) and y(t) is...

The cross-correlation function between two continuous-time real signals x(t) and y(t) is

The autocorrelation function of a signal x(t) is obtained by setting y(t) = x(t) in eq. (P2.67-1.):

(a) Compute the autocorrelation function for each of the two signals x₁(t) and x₂(t) depicted in Figure P2.67(a).

(b) Let x(t) be a given signal, and assume that x(t) is of finite duration—i.e., that x(t) = 0 for t < 0 and t > T. Find the impulse response of an LTI system so that is the output if x(t) is the input.

(c) The system determined in part (b) is a matched filter for the signal x(t). That this definition of a matched filter is identical to the one introduced in Problem 2.66 can be seen from the following:

Let x(t) be as in part (b), and let y(t) denote the response to x(t) of an LTI system with real impulse response h(t). Assume that h(t) = 0 for t < 0 and for t > T. Show that the choice for h(t) that maximizes y(T), subject to the constraint that

a fixed positive number, (P2.67-2)

is a scalar multiple of the impulse response determined in part (b). [Hint: Schwartz's inequality states that

for any two signals u(t) and v(t). Use this to obtain a bound on y(T).]

d) The constraint given by eq. (P2.67-2) simply provides a scaling to the impulse response, as increasing M merely changes the scalar multiplier mentioned in part (c). Thus, we see that the particular choice for h(t) in parts (b) and (c) is matched to the signal x(t) to produce maximum output. This is an extremely important property in a number of applications, as we will now indicate. In communication problems, one often wishes to transmit one of a small number of possible pieces of information. For example, if a complex message is encoded into a sequence of binary digits, we can imagine a system that transmits the information bit by bit. Each bit can then be transmitted by sending one signal, say, x₀(t), if the bit is a 0, or a different signal x₁(t) if a 1 is to be communicated. In this case, the receiving system for these signals must be capable of recognizing whether x₀(t) or x₁ (t) has been received. Intuitively, what makes sense is to have two systems in the receiver, one tuned to x₀(t) and one tuned to x₁ (t), where, by "tuned," we mean that the system gives a large output after the signal to which it is tuned is received. The property of producing a large output when a particular signal is received is exactly what the matched filter possesses.

In practice, there is always distortion and interference in the transmission and reception processes. Consequently, we want to maximize the difference be-tween the response of a matched filter to the input to which it is matched and the response of the filter to one of the other signals that can be transmitted. To illustrate this point, consider the two signals x₀(t) and x₁ (t) depicted in Figure P2.67(b). Let L₀ denote the matched filter for x₀(t), and let L₁ denote the matched filter for x₁ (t).

(i) Sketch the responses of L₀ to x₀(t) and x₁ (t). Do the same for L₁ .

(ii) Compare the values of these responses at t = 4. How might you modify x₀(t) so that the receiver would have an even easier job of distinguishing between x₀(t) and x₁ (t) in that the response of L₀ to x₁ (t) and L₁ to x₀(t) would both be zero at t = 4?