1Signal Processing9-21-20052Opening Discussion■Do you have any questions about the quiz?■What did we talk about last class?3Signal Processing■The general topic for today is signal processing. This is very important to a number of different scientific fields.■You already saw some basic signal processing in Chapter 18 on basic data analysis. There you were introduced to the filter function as well as the ltitr function.■For both of these, the idea is that you want to make new elements that are combinations of elements that had been near them. Filter is sometimes called a moving box filter.4Fourier Series■The most common forms of signal processing are probably those based on Fourier analysis. The idea behind Fourier analysis is that we can express any function as a sum or integral of different sine and cosine waves.■For a Fourier series we want to express a function f(x) in the following form.■f x=a0∑n=1∞[ancos nxbnsin nx]5More Fourier Series■The terms in this are given by the following.a0=12L∫−LLf xdxan=1L∫−LLf xcosn xLdxan=1L∫−LLf xcosn xLdx6Complex Exponents of e■It actually simplified things many times if we don't split the sin and cosine apart and instead us a nice property of complex exponentiation.■■Matlab uses this form and it is a nice thing to know in general. For Fourier analysis, it turns places where we would have to do two integrals into places where we only do one.eix=cos xi sin x7Fourier Transforms■Matlab doesn't actually solve a Fourier series, instead it solves a Fourier transform. That is a continuous form that uses integrals.8Fast Fourier Transform■Technically what Matlab uses is a function called a fast Fourier transform. To keep it fast, the number of elements passed into it needs to be a power of 2.■The fft function will go from a signal series and return a series of the frequencies.■The ifft function does the opposite.■Notice that we don't pass in the time values so the indexes are assumed to be the times. We have to scale that back to whatever range we actually want.9Reminders■Assignment #4 is due Friday at
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