Fourier Transforms and Spectral Analysis Chaiwoot Boonyasiriwat September 17, 2020
Discrete-Time Fourier Transform ▪ The discrete-time Fourier transform (DTFT) of a discrete-time signal x ( k ) with sampling interval T is the Z transform X ( z ) evaluated along the unit circle on complex plane: Analysis Equation ▪ “As a consequence, X ( f ) is well defined if and only if the region of convergence of X ( z ) includes the unit circle.” 2 Schilling and Harris (2012, p. 233)
Discrete-Time Fourier Transform ▪ “For a causal signal x ( k ), the DTFT X ( f ) exists if x ( k ) is absolutely summable or if the poles of X ( z ) all lie strictly inside the unit circle.” ▪ The DTFT X ( f ) is called the spectrum of x ( k ) and can be written in the polar form as where is the magnitude spectrum and is the phase spectrum of x ( k ). 3 Schilling and Harris (2012, p. 234)
Analysis and Synthesis Equations ▪ “The DTFT is called the analysis equation because it decomposes a signal into its spectral components.” ▪ “The signal x ( k ) can be recovered from its spectrum X ( f ) using the inverse transform” ▪ “The IDTFT is called the synthesis equation because it reconstructs or synthesizes the signal from its spectral components.” 4 See the derivation in Schilling and Harris (2012, p. 234, equation 4.2.3)
Properties of DTFT ▪ Since X ( f ) is periodic with period f s , the frequency is typically restricted to the interval [- f s /2, f s /2]. ▪ “For real signals, information about the spectrum is contained in the frequency range [0, f s /2] due to the symmetry of X ( f ). 5 Schilling and Harris (2012, p. 235)
Spectrum of Causal Exponential ▪ Consider the causal exponential ▪ The Z transform of this signal is ▪ Since the pole of X ( z ) is at z = c , DTFT converges only when | c | < 1. ▪ The spectrum of x ( k ) is ▪ The magnitude and phase spectra are 6 Schilling and Harris (2012, p. 235)
Spectrum of Causal Exponential Normalized frequency 7 Schilling and Harris (2012, p. 236)
Frequency Ranges of Some Signals 8 Schilling and Harris (2012, p. 236)
2 ( ) X f = L 9 Schilling and Harris (2012, p. 239)
IDTFT of Ideal Lowpass Filter “An ideal lowpass filter with a cutoff frequency of 0 < F c < f s /2 has a phase response of ( f ) = 0 and a magnitude response consisting of a rectangular window of radius F c .” The impulse response h low ( k ) can be computed by 10 Schilling and Harris (2012, p. 240)
IDTFT of Ideal Lowpass Filter 11 Schilling and Harris (2012, p. 240)
Basic DTFT Pairs 12 Schilling and Harris (2012, p. 241)
Drawbacks of DTFT The DTFT of causal signal x ( k ) is This leads to two drawbacks: ▪ An infinite number of floating point operations (FLOPs) is required to evaluate X ( f ). ▪ The transform must be evaluated at an infinite number of frequencies f . These drawbacks can be overcome by ▪ Focusing on finite signals ▪ Evaluating the transform at N equally spaced values of frequency f 13 Schilling and Harris (2012, p. 241)
Discrete Fourier Transform (DFT) If x ( k ) is absolutely summable, then For a sufficiently large value of N , X ( f ) can be approximated by the finite sum Then X ( f ) is evaluated at N discrete frequencies equally spaced over one period of X ( f ): Let z i be the point in the complex plane corresponding to frequency f i : . Since | z i | = 1 , the N evaluation points are equally around the unit circle. 14 Schilling and Harris (2012, p. 241)
Evaluation Points of DFT N = 8 15 Schilling and Harris (2012, p. 242)
DFT and IDFT DFT of causal N -point signal x ( k ) is then defined as IDFT: 16 Schilling and Harris (2012, p. 242-243)
Matrix Formulation of DFT DFT is a mapping from an input vector x to an output vector X : DFT is a linear transformation and can be represented by an N N matrix whose elements are . Example: N = 5, 17 Schilling and Harris (2012, p. 243)
Matrix Formulation of DFT and IDFT DFT can be represented in the matrix form as Multiplying both sides by W -1 yields Comparing the equations defining DFT and IDFT, we know that So the matrix form of IDFT is 18 Schilling and Harris (2012, p. 243-244)
Matrix Formulation of DFT: Example Suppose the input are Thus, N = 4 and DFT of x is 19 Schilling and Harris (2012, p. 244)
Matrix Formulation of IDFT: Example From the previous example, we have IDFT of X is 20 Schilling and Harris (2012, p. 245)
Fourier Series and Discrete Spectra ▪ “Periodic signals have a discrete spectrum.” ▪ “Suppose x a ( t ) is a periodic continuous-time signal with period T 0 and fundamental frequency F 0 = 1/ T 0 .” ▪ Then, x a ( t ) can be expanded into a complex Fourier series as ▪ The Fourier coefficient is ▪ Let x ( k ) = x a ( kT ) be the k th sample of using the sampling interval of T . ▪ We want to compute the spectrum of x ( k ). 21 Schilling and Harris (2012, p. 245)
Fourier Series and Discrete Spectra The IDTFT of X ( f ) = a ( f – F 0 ) is or So, the spectrum of the periodic signal x ( k ) is which is zero everywhere except at the harmonic frequencies iF 0 – discrete-frequency spectrum. 22 Schilling and Harris (2012, p. 245)
Fourier Coefficients Periodic continuous-time signal x a ( t ) can be approximated by truncating the Fourier series to M harmonics: Suppose that x a ( t ) is sampled at N = 2 M points using a sampling rate of f s = NF 0 . So, T 0 = 1/ F 0 = N / f s = NT . That is the N samples cover one period of x a ( t ). 23 Schilling and Harris (2012, p. 246)
Fourier Coefficients The Fourier coefficient can be approximated as and can be obtained from the DFT of the samples of x a ( t ). 24 Schilling and Harris (2012, p. 246)
Fourier Coefficients ▪ “For a real signal x a ( t ), c - i = c i * . Thus, the complete set of Fourier coefficients is ▪ The Fourier series can also be expressed as The coefficients d i and phase angle i can be obtained using the DFT by 25 Schilling and Harris (2012, p. 246)
DFT Properties 26 Schilling and Harris (2012, p. 251, 255)
Parseval’s Identity ▪ Let x and y be two N -point signals with DFT X and Y. ▪ It can be shown that ▪ When x = y , we then have ▪ The power density spectrum is ▪ The average power of N -point signal is 27 Schilling and Harris (2012, p. 254)
Fast Fourier Transform (FFT) ▪ The DFT of N -point signal x ( k ) is ▪ “The N 2 values of do not depend on x ( k ), so they can be precomputed and stored in an N N matrix.” ▪ “Each point X ( i ) requires N complex multiplications.” ▪ “So, the total number of complex FLOPs required to compute the entire DFT is .” ▪ The computational cost of DFT is of order O ( N 2 ). ▪ “A computational algorithm is of order O ( N p ) if and only if the number of FLOPs n satisfies 28 Schilling and Harris (2012, p. 256)
Decimation in Time FFT ▪ Suppose the number of data points is N = 2 r for some integer r . ▪ Decimate the N -point signal x ( k ) into two N /2-point signals x e ( k ) and x o ( k ) corresponding to the even and odd indices of x , respectively: ▪ “The DFT of x ( k ) then can be recast as two sums, one corresponding to the even values of k , and the other corresponding to the odd values of k .” 29 Schilling and Harris (2012, p. 256)
Decimation in Time FFT The DFT of x ( k ) then becomes where X e ( i ) = DFT{ x e ( k )} and X o ( i ) = DFT{ x o ( k )} are N /2- point transforms of the even and odd parts of x ( k ). 30 Schilling and Harris (2012, p. 257)
Decimation in Time FFT ▪ The total FLOPs for the even-odd decomposition is ▪ Breaking the merging formula into two cases where yields ▪ The periodic property of the N/2-point transforms gives ▪ W N also has the symmetry property 31 Schilling and Harris (2012, p. 257)
Decimation in Time FFT We then have, for , This computation is called the i th -order butterfly which reduces the number of multiplication from 2 to 1 but increases a storage by one scalar. Signal flow graph of the i th -order butterfly computation 32 Schilling and Harris (2012, p. 257)
FFT when N = 8 33 Schilling and Harris (2012, p. 258)
FFT ▪ The even-odd decomposition can be repeatedly applied. ▪ The even samples x e ( k ) and the odd samples x o ( k ) can each be further decimated so the N /2-point transform is turned into a pair of N /4-point transforms. ▪ Since N = 2 r , this process can be continued for r times. ▪ “In the end, we are left with a collection of elementary 2-point DFTs which are the zeroth- order butterfly.” ▪ “The algorithms is called the radix -two fast Fourier transform or FFT.” 34 Schilling and Harris (2012, p. 258)
Signal Flow Graph of FFT: N = 8 35 Schilling and Harris (2012, p. 259)
Computational Cost of FFT The number of FLOPs for the FFT of N -point signal is So, the FFT algorithm is of order O ( N log 2 N ). 36 Schilling and Harris (2012, p. 260)
IDFT using FFT The IDFT formula is and the complex conjugate of W N is We then have 37 Schilling and Harris (2012, p. 261-262)
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