Understanding the Discrete Fourier Transform: Properties and Applications, Study notes of Environmental Science

Download Understanding the Discrete Fourier Transform: Properties and Applications and more Study notes Environmental Science in PDF only on Docsity! ESS 522 Spring 2007 6-1 6. Properties and Application of the Discrete Fourier Transform In this lecture we are going briefly review the different kinds of Fourier transforms and then discuss some practical aspects of using the discrete Fourier transform (DFT). Different Flavors of Transforms. We have come across four types of transform, which we will review here: 1. Continuous time ⇔ Continuous frequency. Non-periodic continuous functions in time and frequency are related by the integral transform G f( ) = g(t)exp !i2" ft( )dt !# # $ g t( ) == G( f )exp i2" ft( )df !# # $ (6-1) We can write the Fourier transform in terms of its amplitude and phase G( f ) = G( f ) exp i! f( )"# $% (6-2) where G(f) is the amplitude spectrum θ(f) is the phase spectrum and |G(f)|2 is the energy or power spectrum. For many applications we are more interested in the amplitude/power spectrum than the phase 2. Continuous time ⇔ Discrete frequency. Periodic continuous functions in time transform to form Fourier Series that are discrete (but not periodic) in frequency. Gj = 1 T g(t)exp ! i2" jt T # $% & '( dt 0 T ) g t( ) == Gj exp i2" jt T # $% & '( j=!* * + (6-3) 3. Discrete time ⇔ Continuous frequency. From the symmetry of the Fourier transform pair we can infer functions that are periodic and continuous in frequency yield discrete (but not periodic) functions in time G f( ) == G k!t( )exp "i2# fk!t( ) k="$ $ % gk = !t G( f )exp i2# fk!t( )df 0 1 !t & (6-4) (note that the integral is taken over one period in the periodic frequency function). ESS 522 Spring 2007 6-2 4. Discrete time ⇔ Discrete frequency. When both the time and frequency functions are periodic, then they are both discrete. This yields the discrete Fourier transform (DFT) Gk = gj exp ! 2"ikj N # $% & '( j=0 N !1 ) gj = 1 N Gk exp 2"ikj N # $% & '( k=0 N !1 ) (6-5) The Fast Fourier Transform (FFT). From the expression for the discrete Fourier transform shown in equation (6-5), it is clear that calculating each term for a real time series requires N multiplications of a real number and a complex exponential or 2N multiplications. Now all, N terms require 2N2 operations which is reduced to N2 when we remember the symmetry properties of the Fourier transform. However, it turns that the Fourier transform can be computed in only N logN operations when the number of samples is a power of 2 (i.e., N = 2M where M is an integer). We will not cover the details in class, but we can explain it briefly as follows. The forward transform of equation (6-5) can be written in as two series over the odd and even terms to yield Gk = g2 j exp ! 2"ik2 j N # $% & '( j=0 N /2!1 ) + g2 j+1 exp ! 2"ik 2 j +1( ) N # $% & '(j=0 N /2!1 ) = g2 j exp ! 2"ikj N / 2 # $% & '( j=0 N /2!1 ) + exp ! 2"ik N # $% & '( g2 j+1 exp ! 2"ikj N / 2 # $% & '( j=0 N /2!1 ) = Pk + exp ! 2"ik N # $% & '( Qk (6-6) We have expressed Gk in terms of a sum of two Fourier transforms Pk and Qk, each for a time series of N/2 samples. The reason why this reduces the number of calculations is that because the periodicity of the Fourier transform we can write P k = P k+ N 2 (6-7) so the calculation of all the terms in P and Q requires only half the number of calculations required for all the terms in G. Now if N is a power of 2 we can keep on subdividing the Fourier series until we are left with series of two-point sequences. The book keeping becomes fairly complex to express, but the net result is that the number of operations is reduced to N logN, an enormous computational saving for long time series. Without the FFT, Fourier transforms would be much less prevalent in computational data analysis Even and Odd Number of Samples. An N sample time series with a sample interval Δt will have a length or period of NΔt. The spacing of frequency samples is 1/T = 1/NΔt and the Nyquist frequency (or “folding” frequency) is 1/2Δt.