A Tutorial on Fourier Analysis


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1 A Tutorial on Fourier Analysis Douglas Eck University of Montreal NYU March 26
2 1.5 A fundamental and three odd harmonics (3,5,7) fund (freq 1) 3rd harm 5th harm 7th harmm Sum of odd harmonics approximate square wave fund (freq 1) fund+3rd harm fund+3rd+5th fund+3rd+5th+7th
3 1 Sum of odd harmonics from 1 to
4 Linear Combination In the interval [u 1, u 2 ] a function Θ(u) can be written as a linear combination: Θ(u) = α i ψ i (u) i= where functions ψ i (u) make up a set of simple elementary functions. If functions are orthogonal (roughly, perpindicular; inner product is zero)then coefficients α i are independant from one another.
5 Continuous Fourier Transform The most commonly used set of orthogonal functions is the Fourier series. Here is the analog version of the Fourier and Inverse Fourier: X (w) = x(t) = + + x(t)e ( 2πjwt) dt X (w)e (2πjwt) dw
6 Discrete Fourier and Inverse Fourier Transform X (n) = x(k) = 1 N N 1 k= N 1 n= x(k)e 2πjnk/N X (n)e 2πjnk/N
7 Taylor Series Expansion f (x) = f (x )+ f (x ) (x x )+ f (x ) (x x ) 2 + f (x ) (x x ) or more compactly as f (x) = n= f (n) (x ) (x x ) n n!
8 Taylor series expansion of e jθ Since e x is its own derivative, the Taylor series expansion for f (x) = e x is very simple: We can define: e x = e jθ == n= (jθ) n n= x n n! = 1 + x + x x 3 n! 3! +... = 1 + jθ Θ2 2 j Θ3 3! +...
9 Splitting out real and imaginary parts All even order terms are real; all oddorder terms are imaginary: ree jθ = 1 Θ 2 /2 + Θ 4 /4!... ime jθ = Θ Θ 3 /3! + Θ 5 /5!...
10 Fourier Transform as sum of sines and cosines Observe that: cos(θ) = n>=;n is even ( 1) n/2 n! Θ n sin(θ) = n>=;n is odd ( 1) (n 1)/2 Θ n Thus yielding Euler s formula: e jθ = cos(θ) + j sin(θ) n!
11
12 Fourier transform as kernel matrix
13 Example Sum of cosines with frequencies 12 and 9, sampling rate = signal real part two cosines (freqs=9, 12) imag part
14 Example FFT coefficients mapped onto unit circle 1 FFT projected onto unit circle
15 Impulse response impulse response signal magnitude phase
16 Impulse response delayed impulse response signal magnitude phase
17 A look at phase shifting Sinusoid frequency=5 phase shifted multiple times. sinusoid freq=5 phase shifted repeatedly magnitude number of points shifted 3 2 angle freq= number of points shifted
18 Sin period 1 + period freq (mag) phase component sinusoids reconstructed signal using component sinusoids vs original reconstructed signal using ifft vs original
19 Aliasing The useful range is the Nyquist frequency (fs/2) 1 cos(21) cos(21) sampled at 24 Hz cos(45) cos(45) sampled at 24 Hz cos( 3) cos( 3) sampled at 24 Hz
20 Leakage Even below Nyquist, when frequencies in the signal do not align well with sampling rate of signal, there can be leakage. First consider a wellaligned exampl (freq =.25 sampling rate) a) b) c) Amplitude Magnitude (db) Magnitude (Linear) Sinusoid at 1/4 the Sampling Rate Time (samples) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample))
21 Leakage Now consider a poorlyaligned example (freq = ( /N) * sampling rate) a) b) c) Amplitude Magnitude (Linear) Sinusoid NEAR 1/4 the Sampling Rate Time (samples) Normalized Frequency (cycles per sample)) 3 Magnitude (db) Normalized Frequency (cycles per sample))
22 Leakage Comparison: a) 1 Sinusoid at 1/4 the Sampling Rate a) 1 Sinusoid NEAR 1/4 the Sampling Rate Amplitude.5.5 Amplitude Time (samples) b) Time (samples) b) Magnitude (Linear) Normalized Frequency (cycles per sample)) c) Magnitude (Linear) Normalized Frequency (cycles per sample)) c) Magnitude (db) Normalized Frequency (cycles per sample)) Magnitude (db) Normalized Frequency (cycles per sample))
23 Windowing can help Can minimize effects by multiplying time series by a window that diminishes magnitude of points near signal edge: a) 1 Blackman Window Amplitude.5 b) c) Magnitude (db) Magnitude (db) Time (samples) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample))
24 Leakage Reduced Comparison: a) 1 Sinusoid NEAR 1/4 the Sampling Rate b) a) 1 Sinusoid at 1/4 the Sampling Rate Amplitude.5.5 Amplitude.5.5 b) Time (samples) Time (samples) c) Magnitude (Linear) Magnitude (db) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample)) c) Magnitude (Linear) Magnitude (db) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample))
25 Convolution theorem This can be understood in terms of the Convolution Theorem. Convolution in the time domain is multiplication in the frequency domain via the Fourier transform (F). F(f g) = F(f ) F(g)
26 Computing impulse response The impulse response h[n] is the response of a system to the unit impulse function.
27 Using the impulse response Once computed, the impulse response can be used to filter any signal x[n] yielding y[n].
28 Examples
29 Filtering using DFT Goal is to choose good impulse response h[n]
30 Filtering using DFT Goal is to choose good impulse response h[n] Transform signal into frequency domain
31 Filtering using DFT Goal is to choose good impulse response h[n] Transform signal into frequency domain Modify frequency properties of signal via multiplication
32 Filtering using DFT Goal is to choose good impulse response h[n] Transform signal into frequency domain Modify frequency properties of signal via multiplication Transform back into time domain
33 Difficulties (Why not a perfect filter?) You can have a perfect filter(!)
34 Difficulties (Why not a perfect filter?) You can have a perfect filter(!) Need long impulse response function in both directions
35 Difficulties (Why not a perfect filter?) You can have a perfect filter(!) Need long impulse response function in both directions Very non causal
36 Difficulties (Why not a perfect filter?) You can have a perfect filter(!) Need long impulse response function in both directions Very non causal In generating causal version, challenges arise
37 Gibbs Phenomenon ideal lopass filter in frequency domain ideal filter coeffs in time domain truncated causal filter Gibbs phenomenon
38 Spectral Analysis Often we want to see spectral energy as a signal evolves over time
39 Spectral Analysis Often we want to see spectral energy as a signal evolves over time Compute Fourier Transform over evenlyspaced frames of data
40 Spectral Analysis Often we want to see spectral energy as a signal evolves over time Compute Fourier Transform over evenlyspaced frames of data Apply window to minimize edge effects
41 ShortTimescale Fourier Transform (STFT) X (m, n) = N 1 k= x(k)w(k m)e 2πjnk/N Where w is some windowing function such as Hanning or gaussian centered around zero. The spectrogram is simply the squared magnitude of these STFT values
42 Trumpet (G4) 5 4 Frequency Time [Listen]
43 Violin (G4) 5 4 Frequency Time [Listen]
44 Flute (G4) 5 4 Frequency Time [Listen]
45 Piano (G4) 5 4 Frequency Time [Listen]
46 Voice Frequency Time [Listen]
47 C Major Scale (Piano) Frequency Time [Listen]
48 C Major Scale (Piano) Log Spectrogram (ConstantQ Transform) reveals lowfrequency structure Frequency Time
49 TimeSpace Tradeoff spoken "Steven Usma" Amp
50 TimeSpace Tradeoff 4 Narrowband Spectrogram overlap=152 timepts= Frequency Time 4 Wideband Spectrogram overlap=3 timepts= Frequency Time
51 Autocorrelation and meter Autocorrelation long used to find meter in music (Brown 1993) Lag k autocorrelation a(k) is a special case of crosscorrelation where a signal x is correlated with itself: a(k) = 1 N N 1 n=k x(n) x(n k)
52 Autocorrelation and meter Autocorrelation long used to find meter in music (Brown 1993) Lag k autocorrelation a(k) is a special case of crosscorrelation where a signal x is correlated with itself: a(k) = 1 N N 1 n=k x(n) x(n k) Autocorrelation can also be defined in terms of Fourier analysis a = F 1 ( F(x) ) where F is the Fourier transform, F 1 is the inverse Fourier transform and indicates taking magnitude from a complex value.
53 time (seconds) time (seconds) lags (ms) Time series (top), envelope (middle) and autocorrelation (bottom) of excerpt from ISMIR 24 Tempo Induction contest (AlbumsCafe Paradiso8.wav). A vertical line marks the actual tempo (484 msec, 124bpm). Stars mark the tempo and its integer multiples. Triangles mark levels in the metrical hierarchy.
54 Fast Fourier Transform Fourier Transform O(N 2 ) Fast version possible O(NlogN) Size must be a power of two Strategy is decimation in time or frequency Divide and conquer Rearrange the inputs in bit reversed order Output transformation (Decimation in Time)
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