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250 chapter 10Figure 10.4 Input signal f is filtered by h, and the output is g.Although there is just a single sine function in the denominator, there is aninfinite number of overtones as follows from the expansions(t) ≃ 1+0.9 sin t +(0.9 sin t) 2 +(0.9 sin t) 3 + ··· . (10.57)a. Compute the DFT S(ω). Make sure to sample just one period but to coverthe entire period. Make sure to sample at enough times (fine scale) toobtain good sensitivity to the high-frequency components.b. Make a semilog plot of the power spectrum |S(ω)| 2 .c. Take your input signal s(t) and compute its autocorrelation function A(τ)for a full range of τ values (an analytic solution is okay too).d. Compute the power spectrum indirectly by performing a DFT on the autocorrelationfunction. Compare your results to the spectrum obtained bycomputing |S(ω)| 2 directly.2. Add some random noise to the signal using a random number generator:y(t i )=s(t i )+α(2r i − 1), 0 ≤ r i ≤ 1, (10.58)where α is an adjustable parameter. Try several values of α, from small valuesthat just add some fuzz to the signal to large values that nearly hide the signal.a. Plot your noisy data, their Fourier transform, and their power spectrumobtained directly from the transform with noise.b. Compute the autocorrelation function A(t) and its Fourier transform.c. Compare the DFT of A(τ) to the power spectrum and comment on theeffectiveness of reducing noise by use of the autocorrelation function.d. For what value of α do you essentially lose all the information in theinput?The code Noise.java that performs similar steps is available on the CD.10.7 Filtering with Transforms (Theory)A filter (Figure 10.4) is a device that converts an input signal f(t) to an output signalg(t) with some specific property for the latter. More specifically, an analog filter isdefined [Hart 98] as integration over the input function:−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 250