# Consider a delta-modulation system with a first-order predictor, defined as x n = w1 x n-1 where…

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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