An analog signal x(t) consists of the sum of two components The spectral characteristic...

An analog signal x(t) consists of the sum of two components The spectral characteristics of x(t) are shown in the sketch in Fig. P10.19. The signal x(t) is bandlimited to 40 kHz and it is sampled at a rate of 100 kHz to yield the sequence x(n).

It is desired to suppress the signal by passing the sequence x(n) through a digital lowpass filter. The allowable amplitude distortion on over the range Above 20 kHz, the filter must have an attenuation of at least 40 dB

(a) Use the Remez exchange algorithm to design the minimum-order linear-phase FIR filter that meets the specifications above. From the plot of the magnitude characteristic of the filter frequency response, give the actual specifications achieved by the filter.

(b) Compare the order Al obtained in part (a) with the approximate formulas given in equations (10.2.94) and (10.2.95).

(c) For the order M obtained in part (a), design an FIR digital lowpass filter using the window technique and the Hamming window. Compare the frequency response characteristics of this design with those obtained in pan (a).

(d) Design the minimum-order elliptic filter that meets the given amplitude specifications. Compare the frequency response of the elliptic filter with that of the FIR filter in part (a).

(e) Compare the complexity of implementing the FIR filter in part (a) versus the elliptic filter obtained in part (d). Assume that the FIR filter is implemented in the direct form and the elliptic filter is implemented as a cascade of two-pole filters. Use storage requirements and the number of multiplications per output point in the comparison of complexity.