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fourier analysis: signals and filters 249x10 3 0 5 10 15 20 25 30 35 40 453.53.02.52.0P1.51.00.50.0Power Spectrum (with Noise)Frequency1086y420Function y(t) + Noise After Low pass Filter0 2 4 6 8 10 12t (s)1086y420Initial Function y(t) + Noise0 2 4 6 8 10 12t (s)x10 2 Autocorrelation Function A(tau)1.41.21.0A0.80.60.40 2 4 6 8 10 12tau (s)Figure 10.3 From bottom left to right: The function plus noise s(t) + n(t), the autocorrelationfunction versus time, the power spectrum obtained from autocorrelation function, and thenoisy signal after passage through a lowpass filter.The function |S(ω)| 2 is the power spectrum we discussed in §10.4. For practical purposes,knowing the power spectrum is often all that is needed and is easier tounderstand than a complex S(ω); in any case it is all that we can calculate.As a procedure for analyzing data, we (1) start with the noisy measured signaland (2) compute its autocorrelation function A(t) via the integral (10.50). Becausethis is just folding the signal onto itself, no additional functions or input is needed.We then (3) perform a DFT on the autocorrelation function A(t) to obtain the powerspectrum. For example, in Figure 10.3 we see a noisy signal (lower left), the autocorrelationfunction (lower right), which clearly is smoother than the signal, andfinally, the deduced power spectrum (upper left). Notice that the broadband highfrequencycomponents characteristic of noise are absent from the power spectrum.You can easily modify the sample program DFT.java in Listing 10.1 to computethe autocorrelation function and then the power spectrum from A(τ). We presenta program NoiseSincFilter/Filter.java that does this on the CD.10.6.1 Autocorrelation Function Exercises1. Imagine that you have sampled the pure signals(t)=11 − 0.9 sin t . (10.56)−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 249