Figure depicts a system for interpolating a signal by a factor of L, where

 and the lowpass filter interpolates between the nonzero values of x_e[n] to generate the upsampled or interpolated signal x_i[n]. When the lowpass filter is ideal, the interpolation is referred to as baldlimited interpolation. As indicated in Section 4.6.2, simple interpolation procedures are adequate in many applications. Two simple procedures often used are zero-order-hold and linear interpolation. For zero-order-hold interpolation, each value of x[n] is simply repeated L times; I.e.,

 Linear interpolation is described in Section4.6.2  

(a) Determine an appropriate choice for the impulse response of the lowpass filter in Figure 4.24 to implement zero-hold interpolation. Also, determine the corresponding frequency response.

(b) Equation (4.29) specifies the impulse response for linear interpolation. Determine the corresponding frequency response. (You may find it helpful to use the fact that h_׀in[n] us triangular and consequently corresponds to the convolution of two rectangular sequences.)

(c) Sketch the magnitude of the filter frequency response for zero-order-hold and linear interpolation. Which is a better approximation to ideal band limited interpolation?

Lowpass filter Gain = L x{n] *, [n} x, [1] Cutoff = #/L Sampling period T' = T/L Sampling period T Sampling period T' = T/L S x|1/L]. n = 0, L. 2L.etc.... xe[1] = 0. otherwise. xe[0]. x[L]. x.[21]. 1. .... L -- 1, 1. n = L. L n = 2 L. 21+ 1. n = ), 2L - 1. x, [n] = [1 - |n|/ L. | < L. otherwise. (4.92) 0.