Problem

In this problem, we consider the discrete-time counterparts of the zero-order hold and f...

In this problem, we consider the discrete-time counterparts of the zero-order hold and first-order hold, which were discussed for continuous time in Sections 7.1.2 and7.2

Let x[n] be a sequence to which discrete-time sampling, as illustrated in Figure 7.31, has been applied. Suppose the conditions of the discrete-time sampling theorem are satisfied; that is, , where is the sampling frequency am . The original signal x[n] is then exactly recoverable from by ideal lowpass filtering, which, as discussed in Section 7.5, corresponds to band-limited interpolation.

The zero-order hold represents an approximate interpolation whereby every sample value is repeated (or held) N — 1 Successive times, as illustrated in Figure P7.50(a) for the case of N = 3. The first-order hold represents a linear interpola4 between samples, as illustrated in the same figure.

(a) The zero-order hold can be represented as an interpolation in the form or eq. (7.47) or, equivalently, the system in Figure P7.50(b). Determine and sketch ho[n] for the general case of a sampling period N.

(b) x[n] can be exactly recovered from the zero-order-hold sequence using an appropriate LTI filter as indicated in Figure P7.50(c). Determine and sketch

(c) The first-order-hold (linear interpolation) can be represented as an interpolation in the form of eq. (7.47) or, equivalently, the system in Figure P7.50(d). Deter-mine and sketch for the general case of a sampling period N.