An analog signal x(t) consists of the sum of two components x₁(t) and x₂(t). The spectral characteristics of x(t) are shown in figure. 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 desire to suppress the signal x₂(t) by passing the sequence x(n) through a digital lowpass filter. The allowable amplitude distortion on |X₁(f)| is ?2% (?₁ = 0.02) over the range 0 ? |F| ? 15 kHz. Above 20 kHz, the filter must have an attenuation of at least 40 dB (?₂ = 0.01).

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

(b) Compare the order M obtained in part (a) with the approximate formulas given in equations (8.2.94) and (8.2.95).

(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 part (a).

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

(e) Compare the complex 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.

Transcribed Image Text:

iX(F) Spectrum of Spectrum of 20 40 Frequency in kilohertz (b) -20 log10 (5,82)– 13 м (8.2.94) 14.6 Af Doo(81, 82) – f(8, 62) (Af)? (8.2.95) Af