4.2 The Fourier Transform

Transcription

1 4.2. THE FOURIER TRANSFORM The Fourier Transform Inroducion One way o look a Fourier series is ha i is a ransformaion from he ime domain o he frequency domain. Given a signal f (), finding is Fourier series on an inerval [ L, L], amouns o finding how much each frequency of he form 2nπ conribues o he signal. The Fourier ransform is similar bu i is defined L over he enire real line ha is on (, ). I also covers all he possible frequencies, no jus frequencies of he form 2nπ. The Fourier ransform has L proven o be an incredible ool in mahemaics as well as many applied fields Definiions The Fourier ransform is an inegral ransform, defined over he enire real line, ha is for x [, ]. I decomposes a funcion of ime (some signal) ino he frequencies ha make up ha signal. The Fourier ransform and is inverse always go in pair. They can be viewed as ransformaions beween he ime domain and he frequency domain. Definiion Le f (x) be some 1-dimensional signal ; x (, ). 1. The Fourier ransform of f is defined by 2. Is inverse is defined by Le us make some remarks: F {f (x)} (ω) f (ω) 1 f (x) F 1 { f (ω) } (x) 1 f (x) e iωx dx f (ω) e iωx dω Remark The Fourier ransform is a coninuous funcion of ω. If we hink of ω as he frequency, hen f (ω) gives us he conribuion he frequency ω makes o he signal. Remark The analogy wih Fourier series is ha he Fourier ransform corresponds o he coeffi ciens of he Fourier series. Remark Using he inverse Fourier ransform, we can recover he signal from is Fourier ransform. Remark The Fourier ransform allows us o idenify he frequencies which play an imporan role in a signal. Remark The Fourier ransform allows us o filer a signal by removing cerain frequencies from i.

2 58CHAPTER 4. TIME-FREQUENCY ANALYSIS: FOURIER TRANSFORMS AND WAVELETS An Example Though we will no focus on compuing Fourier ransforms, we give an example of how i is compued. Example Find he Fourier ransform of f (x) e α x where x (, ) and α >. By definiion, f (w) 1 1 f (x) e iωx dx e iωx e α x dx Recalling he heory of improper inegrals, we wrie e iωx e α x dx e iωx e α x dx + e iωx e αx dx + e (α iω)x dx + e iωx e α x dx e iωx e αx dx e (α+iω)x dx We do each inegral separaely. Similarly e (α iω)x dx lim lim 1 α iω α + iω e (α+iω)x dx lim e (α iω)x dx e (α iω)x α iω e (α+iω)x dx lim α + iω 1 α + iω α iω e (α+iω)x

3 4.2. THE FOURIER TRANSFORM 59 Figure 4.8: Blue: graph of f (x) e x Green: graph of f Therefore f (w) + α iω ) ( 1 α + iω 2 π α The graphs of f and f are shown in figure The Fas Fourier Transform (FFT) In heory a signal is a coninuous funcion of x (or ) and is ransform a coninuous funcion of ω. In pracice, o compue he FFT of a signal, we sample he signal, in oher words we measure he signal a a finie number of ime values. Hence, insead of having an inegral ransform (he Fourier ransform), we have a discree ransform, also known as he discree Fourier ransform or DFT.One problem wih he DFT is ha he number of operaions needed o evaluae i are

4 6CHAPTER 4. TIME-FREQUENCY ANALYSIS: FOURIER TRANSFORMS AND WAVELETS of he order of n 2 ha is he algorihm id O ( n 2). For large n, his is significan. In he 196,s, Cooley and Tukey developed a much faser algorihm o compue he DFT (see he Cooley and Tukey algorihm).i is now commonly known as he Fas Fourier Transform (FFT) algorihm. This algorihm is so good ha i is considered one of he op en algorihms of he wenieh cenury. I reduced he number of operaions needed o compue he DFT drasically. The FFT algorihm is O (n log n). The key feaures of his algorihm are: 1. I has a low operaion coun: O (n log n). 2. I finds he ransform on an inerval [ L, L]. I is imporan o discreize his inerval so ha is number of poins is a power of The FFT has excellen accuracy properies Exercises 1. Find some applicaions which can be derived from he saemen made in remark Be specific enough so ha i should be clear from wha you wrie wha he applicaions are. 2. Explain how he saemen made in remark can be implemened and find some applicaions. Be specific enough so ha i should be clear from wha you wrie wha he applicaions are and how hey could be implemened.

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