We are interested in upsampling a sequence by a factor of 2, using a system of the form...

We are interested in upsampling a sequence by a factor of 2, using a system of the form of Figure 4.23. However, the lowpass filter in that figure is to be approximated by a five-point filter with impulse response h[n] indicated in Figure P4.58-1. In this system, the output y1[n] is obtained by direct convolution of h[n] with w[n].

(a) A proposed implementation of the system with the preceding choice of h[n] is shown in Figure P4.58-2. The three impulse responses h₁[n], h₂[n], and h₃[n] are all restricted to be zero outside the range 0 ≤ n ≤ 2. Determine and clearly justify a choice for h₁[n], h₂[n], and h₃[n] so that y₁[n] = y₂[n] for any x[n], i.e., so that the two systems are identical.

(b) Determine the number of multiplications per output point required in the system of Figure P4.58-1 and in the system of Figure P4.58-2. You should find that the system of Figure P4.58-2 is more efficient.