Consider a delta-modulation system with a first-order

predictor, defined as x n = w1 x n_1 where x[n] denotes the n-th sample

of a stationary signal of zero mean and the tap gain w1 is a constant. Assume

that the energy of an antipodal signal is described by _2 = E

a. Determine the optimal value of w1 which minimizes the

prediction error in the mmse sense _2 e = E xn _ x n 2 and find its

minimum value.

b. Find the optimal prediction gain which is defined as the

ratio of the average signal power to the minimum variance of the prediction

error.

c. If the variance of the prediction error

Consider a delta-modulation system with a first-order

predictor, defined as x n = w1 x n_1 where x[n] denotes the n-th sample

of a stationary signal of zero mean and the tap gain w1 is a constant. Assume

that the energy of an antipodal signal is described by _2 = E

a. Determine the optimal value of w1 which minimizes the

prediction error in the mmse sense _2 e = E xn _ x n 2 and find its

minimum value.

b. Find the optimal prediction gain which is defined as the

ratio of the average signal power to the minimum variance of the prediction

error.

c. If the variance of the prediction error is to be less

than or equal to _2 , determine the condition that the channel tap must

satisfy

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