Question

1. (Simple differentiator) In Section 7.6.3, there is a thorough discussion on how to design a discrete-time differentiator. This problem set is a simplified version of it. Suppose that xc(t) is a continuous-time signal and we want to calculate dxc(t)/dt numerically. One obvious strategy is to approximate it by the definition of differentiation: lim and we can expect the approximation to be good if T>0 is small enough. Let x[n] - xc(nT). Then we have the following approximation dx 虱.nr ~ 〒(xfn]-xfn-1D t = (a) Consider y[n]-^(x[n]-x[n -1]) as a linear time-invariant system. Write down its impulse response. Calculate the frequency response and show that it can be expressed as a purely imaginary part multiplied with a linear phase term. (b) What is the group delay of the system? (c) Make a sketch of the magnitude response and show that, at low frequency range, the gain increases at approximately 6dB/octave

Answer 1

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