We are interested in upsampling a sequence by a factor of 2, using a system of the form of Figure. However, the lowpass filter in that figure is to be approximated by a five-point filter with impulse response h[n] indicated in Figure. 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. 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 and in the system of Figure. You should find that the system of Figure is more efficient.

Transcribed Image Text:

Lowpass filter Gain = L Cutoff = #/L. Sampling period T Sampling period T = TL Sampling period T = T/L. h|n] ↑ 2 h|n] In] Part A † 2 2.