代写Digital Signal Processing and Digital Filters Practice Sheet 5调试SPSS

Digital Signal Processing and Digital Filters

Practice Sheet 5

Fig. 1: Slope of Butterworth filter at the 3dB point. (Increasing filter orders, N = 1, 2, 3, 4).

1) Fig. 1 shows that as the order of an analog Butterworth filter is increased, the slope of |Ha (jω)|2 at the 3 dB cutoff frequency ωc also increases. Derive an expression for the slope of |Ha (jω)|2 at ωc as a function of the filter order N.

2) Fig. 2 shows the frequency response of an N-th order low-pass Butterworth filter at ω = 0. As we can see, the slope is zero at ω = 0. Furthermore, the figure suggests that as the filter order increases, the higher order derivatives are also zero. This is known as the maximally flat property.

Mathematically, show that the Buttherworth filter is indeed maximally flat at ω = 0, that is, the first 2N − 1 derivatives of |Ha (jω)| 2 are equal to zero at ω = 0.

3) Let Hc (s) be a continuous system defined by

(a) Using impulse invariance design a discrete filter from the continuous-time second-order filter

(b) For a = 0.1 and b = 4, convert Hc (s) into a digital IIR filter by means of the bilinear transformation. The digital filter should have a resonant frequency of ωr = 2/π.

Fig. 2: Slope of Butterworth filter at ω = 0. (Increasing filter orders, N = 1, 2, 3, 4).