Time-Frequency Analysis Time Frequency Analysis in Visual Signal Yetmen Wang AnCAD, Inc. AnCAD, Inc. 2007/7/12
Single frequency? Single frequency? • Is the squared periodical wave of single frequency? • With Fourier glasses, sine With Fourier glasses, sine wave is of single frequency while squared wave is of multiple frequencies multiple frequencies. • What is the definition of frequency? • What do we want to see?
Non-Periodical Signal of many frequencies? Non Periodical Signal of many frequencies? • Does a straight has a frequency? • Putting on Fourier glasses, we Putting on Fourier glasses, we see so many frequencies from a straight line. • A Again what do we want to see? i h t d t t ?
What do we want? Data perception. What do we want? Data perception. • Signal normally is composed of non-periodical part, periodical part, noise, and jump/discontinuity. For data perception we want to be able to separate all of them perception, we want to be able to separate all of them. And non-periodical signal is represented in time series plot, noise eliminated, jump/discontinuity taken cared plot, noise eliminated, jump/discontinuity taken cared without introducing undesired alias, and periodical signal shown in time-frequency plot.
TF Plot: Single frequency TF Plot: Single frequency
TF Plot: Change of frequency TF Plot: Change of frequency • Signal with abrupt change of frequency. × π ≤ < ⎧ ⎧ 0 . 30 cos( ( 2 10 t ) ) , , 0 t 1 = ⎨ ⎨ x ( ( t ) ) × π ≤ < ⎩ 0 . 30 cos( 2 20 t ) , 1 t 2
TF Plot: Change of frequency and amplitude TF Plot: Change of frequency and amplitude • Signal with abrupt change of frequency and amplitude × π ≤ < ⎧ 0 . 30 cos( 2 10 t ) , 0 t 1 = ⎨ x ( t ) × × π π ≤ ≤ < < ⎩ ⎩ 0 0 . 15 15 cos( cos( 2 2 20 20 ) ) , 1 1 2 2 t t t t
To be clarified To be clarified • What is non-periodical signal? • How to remove non-periodical signal? • What is frequency? What is instantaneous frequency? • What is noise? How to eliminate noise? • How to identified jump and discontinuity? • Signal trace is finite. How to eliminate end effects occurred very often in signal processing?
Removal of Non-Periodical Signal Frequency based filter q y Iterative Gaussian Filter EMD as filter
Containmanation of non-periodical Signal Containmanation of non periodical Signal 1 0 . 9 0.8 0 . 8 0.6 0 . 7 0.4 0 . 6 0.2 0 . 5 0 0 . 4 -0.2 0 . 3 -0.4 0 . 2 -0.6 0 . 1 -0.8 0 0 -1 0 1 2 3 4 5 6 7 8 9 10 -0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 12 10 1 0 8 8 6 6 4 4 2 2 0 0 0 1 2 3 4 5 6 7 8 9 10 -2 -1 0 1 2 3 4 5 6
Removal of Non-Periodical Signal Removal of Non Periodical Signal Frequency based filter
Monotonic Cubic Interpolation Monotonic Cubic Interpolation • Avoid Runge effect and end effect • Uniform or non-uniform over-sampling • Monotonic condition is slightly released
Spectrum Analyzer • For better spectrum analysis, the following steps are applied before doing FFT: • U i Using Gaussian filter to eliminate non-periodical part G i filt t li i t i di l t • Removal of signals before and after the first and the last zero crossings via interpolation • O Over sampling the signal via monotonic interpolation that the signal li th i l i t i i t l ti th t th i l length is a power of 2 • Odd function mapping to ensure the perfect periodic condition up to (N-1)/2-th order of discrete derivatives (N-1)/2-th order of discrete derivatives.
Iterative Gaussian Smoothing Filter • Removal of Non-Periodical Signal • Low Pass/High Pass/Band Pass Filter • Phase Perseverance through any number of iteration steps (theoretically proved) without ripple or artifact in time domain time domain • Derivative of signal (next version) • Broken Signal Reconstruction (next version) B k Si l R t ti ( t i )
EMD as filter EMD as filter
What is frequency? The need of instantaneous frequency. STFT Morlet Transform Enhanced Morlet Transform
Frequency definition (1) Frequency definition (1) • P Period is defined as number of i d i d fi d b f events in duration of time. And frequency is the inverse of period. i d • The concept of period is an average. So do the frequency. • Since there is no instantaneous period, there is no instantaneous frequency. • Putting on Fourier glasses, uncertainty principle comes into play for sinusoidal signal. y g That is, the lower the frequency to be identified, the longer the duration of signal.
Frequency definition (2) Frequency definition (2) • In classical wave theory, = ∇ ϕ frequency is defined as time k rate change of phase. g p ∂ ∂ ϕ ω = − • Frequency is an continuous ∂ concept. There is need to t clarify what we mean by clarify what we mean by ∂ k + ∇ ω = frequency at a certain moment. 0 ∂ t • Uncertainty principle does apply.
Short Term Fourier Transform Short-Term Fourier Transform • Frequency at a certain time is a distribution obtained from Fourier transform. The short period of signal applied to the Fourier transform contains the STFT: specific moment of interest. specific moment of interest. +∞ • In time-frequency analysis, ∫ ω = τ τ − − ωτ d τ i F ( t , ) f ( ) g ( t ) e such idea evolves as the − ∞ ∞ Sh Short-Term Fourier Transform t T F i T f (STFT). Note g is a windowing function. For Gaussian window, the transform For Gaussian window, the transform is also known as Gabor Transform.
Gabor transform Gabor transform • Same chirp signal processed by Gabor transform. (MATLAB) 500 450 400 350 300 Frequency 250 200 150 100 50 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time
Challenges in STFT Challenges in STFT • Catching low frequency component needs longer time. • Windowing is needed to avoid end effects. => STFT is suitable for band-limited signal like speech and sound.
Morlet Transform Morlet Transform • Scale property in signal is related to frequency property when mother wavelet is Morlet.. • Longer duration of wavelet is used to catch lower frequency component component. +∞ τ − = ∫ 1 t τ ψ τ * F ( t , s ) f ( ) ( ) d s s −∞ − ∞
Morlet transform Morlet transform • Morlet transform on a chirp signal. • In catching the high frequency In catching the high frequency spectrum, mother wavelet of short duration of time is used. The spectrum of such wavelet The spectrum of such wavelet suffers from wide span of frequency, resulting in low resolution, as shown in the l ti h i th right left plot.
Enhanced Morlet Transform Enhanced Morlet Transform • By applying Gaussian windowing in frequency domain and knowing that the crossed term of convolution between mother wavelet and signal is the cause of blur, the i l i th f bl th resolution of Morlet transform can be greatly improved by neglecting the crossed term. l ti th d t • The fine structure appears in high frequency region is caused by under sampling. d b d li The chirp signal is digitized with constant sampling rate.
Chirp signal with higher sampling rate Chirp signal with higher sampling rate • With higher sampling rate, the fine structure in TF plot disappeared. This is to assure pp the fine structure pattern is caused by under sampling.
Quadratic Chirp Signal Quadratic Chirp Signal 500 450 400 350 300 Frequency 250 200 150 100 50 0 0 0.5 1 1.5 2 2.5 3 3.5 Time Morlet-Jeng, MATFOR Spectrogram, MATLAB Spectrogram, MATLAB
Log unit Log unit
Voice “MATLAB” Voice MATLAB
Voice MATLAB Voice “MATLAB” S p e c tr o g r a m 3 5 0 0 3 3 0 0 0 0 0 0 2 5 0 0 Frequency 2 0 0 0 F 1 1 5 5 0 0 0 0 1 0 0 0 5 0 0 0 0 0 0 .0 5 0 .1 0 .1 5 0 .2 0 .2 5 0 .3 0 .3 5 0 .4 0 .4 5 T im e Morlet-Jeng Transform Spectrogram
Morlet-Jeng Transform Chi-Chi (921) Earthquake Chi Chi (921) Earthquake Morlet Transform by MATLAB
Sound of Fan Sound of Fan
Sound of Fan (IMF6) Sound of Fan (IMF6)
Sound of Machine Sound of Machine
Instantaneous Frequency and Instantaneous Frequency and Hilbert Transform
Instantaneous Frequency Instantaneous Frequency • A ti A time-series signal can be i i l b regarded as the x-axis projection of a planar slider motion. The slider moves ti Th lid along a rotating stick with axial & velocity , while the stick is a ( t ) spinning with an angular spinning with an angular ϕ & speed . ( t ) • The corresponding y-axis projection shares the same j ti h th frequency distribution as x-axis signal with 90 degree phase shift Such conjugate signal is shift. Such conjugate signal is evaluated from Hilbert Transform.
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