Find the output of the digital integrator in Prob XXXXXXXXXXwhen the input x[n] = 0 and the initial condition is y[−1] = 1. Repeat the problem for the integrator in Prob XXXXXXXXXXComment on the two...

Find the output of the digital integrator in Prob. 5.1-1 when the input x[n] = 0 and the initial condition is y[−1] = 1. Repeat the problem for the integrator in Prob. 5.1-2. Comment on the two results.

Prob. 5.1-1

A realization of a digital integrator that uses a backward difference approximation is represented as

y[n] − y[n − 1] = T x[n],

where T is the sampling interval. Assuming zero initial condition y[−1] = 0, solve this equation iteratively when the input x[n] is

(a) δ[n]

(b) u[n]

Prob. 5.1-2

A digital integrator that uses a trapezoidal approximation is characterized by the difference equation

where T is the sampling interval. Assuming zero initial condition y[−1] = 0, solve this equation iteratively when the input x[n] is

(a) δ[n]

(b) u[n]

Comment on the performance of this system as an integrator.