Introduction of Fourier Analysis and Time-frequency Analysis

Transcription

1 Introduction of Fourier Analysis and Time-frequency Analysis March 1, 2016

2 Fourier Series Fourier transform Fourier analysis Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them. Joseph Fourier ( )

3 Fourier Series Fourier transform Frequency and spectrum Signal model: a signal f (t) R contains components of different frequencies For physical meaning, the term frequency is defined in terms of sinusoidal function cos(ωt) and sin(ωt), where ω is the angular frequency (rad/s) The spectrum shows the intensity and phase of the sinusoidal functions composing f (t) Fourier analysis facilitates this model Fourier analysis is a classic method of retrieving the spectrum of a signal

4 Fourier Series Fourier transform Fourier series A periodic signal f (t) R can be represented in terms of an infinite sum of sines and cosines. f (t) = a 0 + a n cos(ω n t) + b n sin(ω n t) (1) a 0 = π π f (t)dt, a n = n=1 π π n=1 f (t) cos(ω n t)dt, b n = π π f (t) sin(ω n t)dt The frequencies of these sines and cosines are constructed by a harmonic series, containing a fundamental frequency (i.e., ω 0 ) and its integer multiples (harmonics); ω n = nω 0. These sines and cosines form an orthogonal basis, and the Fourier coefficients are the projection of f (t) on the basis.

5 Fourier Series Fourier transform Illustration A synthetic viewpoint:

6 Fourier Series Fourier transform Example: a note C4 played on a piano Fundamental frequency f 0 = Hz

7 Fourier Series Fourier transform Physical interpretation For a bounded vibrating string with length L: Fundamental mode with wavelength λ 0 = L 2 Fundamental frequency f 0 = 2v L 1st harmonic frequency 2f 0 2nd harmonic frequency 3f 0 Pitched musical signal f 0 + higher order harmonics periodic

8 Fourier Series Fourier transform More examples (a) Piano (b) Trumpet (c) Violin (d) Flute

9 Fourier Series Fourier transform Phase We can write Equation (1) as f (t) = a 0 + n=1 a 2 n + b 2 n cos(ω n t + φ n ) (2) where φ n = tan 1 b n a n (3)

10 Fourier Series Fourier transform Phasor, complex number, and negative frequency We can also write Equation (1) as f (t) = a 0 + n=1 c n 2 ejωnt + c n 2 e jωnt (4) where e jωnt = cos(ω n t) + j sin(ω n t), c n = a n + jb n When written in phasor forms, a real-valued signal is decomposed into components with positive and negative frequencies For real-valued signals (e.g., most of the audio signals), negative frequencies are redundant For complex-valued signals, negative frequencies are informative

11 Fourier Series Fourier transform Fourier transform The Fourier transform is a generalization of the Fourier series, by changing the sum to an integral. A signal f (t) R can be represented as F (ω) = The inverse Fourier transform f (ω) = Denote F (ω) = Ff (t) and f (t) = F 1 F (ω) f (t)e jωt dt, (5) F (ω)e jωt dω. (6)

12 Fourier Series Fourier transform Example (1): impulse The Dirac delta function f (t) = δ(t) = { 1, t = 0, 0, t 0. (7) F (ω) = δ(t)e jωt dt = e jω 0 = 1. (8)

13 Fourier Series Fourier transform Example (2): cosine f (t) = cos(ηt) (9) F (ω) = = j cos(ηt)e jωt dt cos(ηt) cos(ωt)dt ( = 1 2 cos(ηt) sin(ωt)dt (= 0) ) for ω = ±η = 1 2 δ(ω η) + 1 δ(ω + η) (10) 2 It is also evident that Fe jηt = δ(ω η)

14 Fourier Series Fourier transform Example (3): rectangular pulse f (t) = { 1, T 2 < t < T 2, 0, others (11) T 2 F (ω) = e jωt dt = 1 T jω e jωt T 2 T 2 2 = sin T ω ω (12) In a weak sense, lim T F (ω) = δ(ω) lim T 0 f (t) = δ(t)

15 Fourier Series Fourier transform Example (4): Gaussian f (t) =e at2 F (ω) = = e at2 e jωt dt e at2 cos(ωt)dt j π e at2 sin(ωt)dt = (odd function!) 2a e ω 2 4a. (13)

16 Fourier Series Fourier transform Example (4): Gaussian (cont d) The Fourier transform of a Gaussian is still a Gaussian f (t) = e t2 2 is an eigenfunction of the Fourier transform We also have lim T F (ω) = δ(ω) and lim T 0 f (t) = δ(t)

17 Fourier Series Fourier transform Translation and modulation Shifted by t 0 in the time domain multiplied by a phase shift in the frequency domain Modulated by e jω 0t in the time domain shifted by ω 0 in the frequency domaiin f (t t 0 )e jωt dt = f (t)e jω 0t e jωt dt = f (u)e jω(t 0+u) dt = e jωt 0 F (ω)(14) f (t)e j(ω ω 0)t dt = F (ω ω 0 )(15)

18 Fourier Series Fourier transform Convolution Convolution of f (t) and g(t): f (t) g(t) = f (τ)g(t τ)dτ (16) The convolution theorem: for F (ω) = Ff (t) and G(ω) = Fg(t) then F (f (t) g(t)) = F (ω)g(ω) (17)

19 Fourier Series Fourier transform Magnitude spectrum, additivity and interference Magnitude spectrum: F (ω) If f (t) = f A (t) + f B (t), then F (ω) = F A (ω) + F B (ω) But F (ω) F A (ω) + F B (ω)! Constructive interference and destructive interference

20 Sampling Quantization Spectral leakage Sampling Sampling: make a continuous-time signal be discrete-time A discrete-time signal x(n), n Z, sampled from f (t), t R, by a sampling period T x(n) = f (nt ) (18) Sampling frequency f s = 1 T, we have n = tf s Audio signals: f s = 8 khz, 16 khz, khz, 44.1 khz, etc. T =? f s =?

21 Sampling Quantization Spectral leakage Aliasing Limited sampling rate implies limited information of the spectrum Nyquist-Shannon sampling theorem: if the highest frequency of a signal f (t) is f h, then the signal can be perfectly reconstructed by a sampled signal with sampling rate of at least 2f h If sampling rate smaller than 2f h : aliasing

22 Sampling Quantization Spectral leakage Aliasing effects in image, video and sound Video example and Audio example

23 Sampling Quantization Spectral leakage Quantization Quantization make a analog-value signal digital Bit depth: number of bits of information in each sample Example: bit depth = 4 Audio signals: 16-bit, 24-bit or more For an audio signal with 2-channel, 16-bit and sampling rate 44.1 khz, what is its bit rate?

24 Sampling Quantization Spectral leakage Types of signal Can you tell the difference among the following terms? Continuous ( 連續 ) Analog ( 類比 ) Discrete ( 離散 ) Digital ( 數位 )

25 Sampling Quantization Spectral leakage Spectral leakage We are never able to measure a signal of infinite length Fourier transform on a finite support [T A, T B ]: F (ω) = TB T A f (t)e jωt dt Spectral leakage: the spectrum of a sinusoidal function is never a impulse Heisenberg s uncertainty: better localization in time implies worse localization in frequency

26 Interpretation of DFT Fast Fourier transform (FFT) DFT For a discrete-time signal (usually sampled from a continuous-time signal) x R N, x = [x(0), x(1),..., x(n 1)] X (k) = N 1 0 x(n)e j2πkn N (19) Let X R N, X = [X (0), X (1),..., X (N 1)] and ω = j2π N ω ω 2 ω (N 1) X = Fx, F = (20) 1 ω N 1 ω 2(N 1) ω (N 1)(N 1)

27 Interpretation of DFT Fast Fourier transform (FFT) Interpretation of DFT (1) The discrete Fourier transform approximates the continuous Fourier transform (Riemann sum approximation) ˆx(ω) = n Z x(n)e jωn = f (nt )e jωn n Z f (nt )e jωt dt t R = 1 f (nt )e jωt T dt T t R = 1 ( ω ) T F (21) T

28 Interpretation of DFT Fast Fourier transform (FFT) Interpretation of DFT (2) Frequency resolution = fs N When N is even: (the distance between two bins) X N 1 X (0) = x(n) f =0 n=0 N 1 N 1 X (1) = x(n)e j2πn N = x(n)e j2π( fs t) N n=0 n=0 f = f s N. ( ) N 1 N 2 1 = x(n)e j2πn ( N 1) N 2 f = f s 2 f s N n=0

29 Interpretation of DFT Fast Fourier transform (FFT) Interpretation of DFT (3) X ( ) N 1 N X = x(n)e j2πn ( N N 2 ) f = f s 2 2 n=0 ( ) N 1 N = x(n)e j2πn ( N 1) N 2 f = f s 2 + f s N. n=0 N 1 X (N 1) = x(n)e j2πn ( N 1) N 2 f = f s N n=0

30 Interpretation of DFT Fast Fourier transform (FFT) Example (1): normal case A chirp signal f (t) := sin(0.003πt 2 ) for t 0

31 Interpretation of DFT Fast Fourier transform (FFT) Example (2): aliasing A chirp signal f (t) := sin(0.004πt 2 ) for t 0

32 Interpretation of DFT Fast Fourier transform (FFT) Example (3): approximation property

33 Interpretation of DFT Fast Fourier transform (FFT) Fast Fourier transform (FFT) Complexity of N-point DFT: O(N 2 ) Cooley-Tukey FFT algorithm: O(N log N) Utilizing the redundancy of the DFT matrix F

34 Interpretation of DFT Fast Fourier transform (FFT) Algorithm

35 Non-stationary signals Implementation Non-stationarity of signals Fourier analysis assumes that the amplitude/frequency/phase of a signal do not change over time But this never happens in real world Music is non-stationary: onset, offset, pitch change, etc. f (t) = m A m (t) cos(ω m (t)t + φ m (t)) (22) How to model the temporal dynamics of a non-stationary signal? How to model time and frequency information at the same time?

36 Non-stationary signals Implementation Example

37 Non-stationary signals Implementation Dennis Gabor ( ) Nobel Prize in Physics (1971) Known for Holography, Time-frequency analysis, etc. Google Doodle on June 5, 2010

38 Non-stationary signals Implementation Short-time Fourier transform (STFT): capture local information through a sliding window function h(t) S x (h) (t, ω) = x(τ)h(τ t)e jωτ dτ, (23) Also have magnitude and phase Spectrogram: S (h) x 2

39 Non-stationary signals Implementation Sliding window and short-time spectrum

40 Non-stationary signals Implementation Window function For better temporal continuity, better control of spectral leakage and less ripple artifacts (than using rectangular window) ), n = N 2,..., N 2 Hann window: h(t) = cos ( 2πn N Comparison of rectangular, triangular and Hann windows:

41 Non-stationary signals Implementation Illustration (1): window type Hann window (up) and rectangular window (bottom)

42 Non-stationary signals Implementation Illustration (2): window length Short Hann window (32 ms, up) and long Hann window (128 ms, bottom)

43 Non-stationary signals Implementation Illustration (3): scaling Spectrogram and db-scaled spectrogram of a scale played with the piano

44 Non-stationary signals Implementation Implementation of STFT (1) Slice the signal into frames of segments Multiply the short segments by a window function Do FFT for each segment!

45 Non-stationary signals Implementation Implementation of STFT (2) Discrete STFT: X (n, k) = H: time step (hop size) N 1 m=0 x(m + nh)h(m)e j2πkm N (24) The index k corresponds to the frequency f (k) := kfs N The index n corresponds to the time t(n) := nh f s Example: for an audio signal sampled at f s = 44.1 khz, use window size N = 4096 samples and hop size H = 1024 samples. Could you specify the time-frequency grid in the STFT representation?

46 Fourier Analysis Non-stationary signals Implementation Example (1): piano solo fs = 44.1 khz, H = 441, N = 4096 (left) and N = 1024 (right) Play sound

47 Fourier Analysis Non-stationary signals Implementation Example (2): voice fs = 44.1 khz, H = 441, N = 4096 (left) and N = 1024 (right) Play sound

48 Fourier Analysis Non-stationary signals Implementation Example (3): rock fs = 44.1 khz, H = 441, N = 4096 (left) and N = 1024 (right) Play sound

49 Fourier Analysis Non-stationary signals Implementation Example (4): symphony fs = 44.1 khz, H = 441, N = 4096 (left) and N = 1024 (right) Play sound