Final answer:
The solution to the LCCDE for the input signal x[n] = δ[n−1] is an initial impulse response followed by an exponential decay, derived by substituting the input into the given equation and applying the recursive relationship.
Step-by-step explanation:
The student is asking for help in solving a linear constant coefficient difference equation (LCCDE) associated with a discrete-time linear time-invariant (DT-LTI) system. The LCCDE is given as y[n] − 0.7y[n−1] = x[n], and the student is tasked with finding the response of the system to an input signal x[n] = δ[n−1], which is a unit shift of the discrete-time delta function.
To find the response of the system, we need to substitute the input signal into the equation and solve for y[n]. In this case, when n = 1, x[1] = δ[0] which equals 1, and for all other n, x[n] will be 0. This allows us to find the initial condition for y[n] and then apply the recursive relationship given by the LCCDE to find the complete response of the system.
Assuming zero initial conditions, that is y[n] = 0 for n < 0, at n = 1 we get y[1] = 1. Now, for n > 1, we can use the recursive relation to find y[n] by plugging in the value of y[n−1] into the equation y[n] − 0.7y[n−1] = 0 (since x[n] = 0 for n > 1). The result is a sequence of values for y[n] that decay exponentially, due to the factor 0.7y[n−1] in the recursive relation.
The response of the system can be summarized as an initial impulse response at n = 1 followed by an exponential decay for subsequent values of n.