In this problem, you will examine the use of the DFT to implement the filtering necessar...

In this problem, you will examine the use of the DFT to implement the filtering necessary for the discrete-time interpolation, or upsampling, of a signal. Assume that the discrete-time signal x[n] was obtained by sampling a continuous-time signal x_c(t) with a sampling period T . Moreover, the continuous-time signal is appropriately bandlimited; i.e., X_c(jΩ) = 0 for |Ω| ≥ 2π/T . For this problem, we will assume that x[n] has length N; i.e., x[n] = 0 for n < 0 or n > N − 1, where N is even. It is not strictly possible to have a signal that is both perfectly bandlimited and of finite duration, but this is often assumed in practical systems processing finite-length signals which have very little energy outside the band |Ω| ≤ 2π/T.

We wish to implement a 1:4 interpolation, i.e., increase the sampling rate by a factor of 4. As seen in Figure 4.23, we can perform this sampling rate conversion using a sampling rate expander followed by an appropriate lowpass filter. In this chapter, we have seen that the lowpass filter could be implemented using the DFT if the filter is an FIR impulse response. For this problem, assume that this filter has an impulse response h[n] that is N + 1 points long. Figure P8.70-1 depicts such a system, where H[k] is the 4N-point DFT of the impulse response of the lowpass filter. Note that both v[n] and y[n] are 4N-point sequences.

(a) Specify the DFT H[k] such that the system in Figure P8.70-1 implements the desired upsampling system. Think carefully about the phase of the values of H[k].

(b) It is also possible to upsample x[n] using the system in Figure P8.70-2. Specify System A in the middle box so that the 4N-point signal y2[n] in this figure is the same as y[n] in Figure P8.70-2. Note that System A may consist of more than one operation.

(c) Is there a reason that the implementation in Figure P8.70-2 might be preferable to Figure P8.70-1?