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README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain the slide with highlighted notes.
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STFT as Filter Bank Introduction to Wavelet Transform Yen-Ming Lai Doo-hyun Sung November 15, 2010 ENEE630, Project 1
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Wavelet Tutorial Overview DFT as filter bank STFT as filter bank Wavelet transform as filter bank
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Discrete Fourier Transform
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DFT for fixed w_0
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fix specific frequency w_0
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DFT for fixed w_0 pass in input signal x(n)
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DFT for fixed w_0 modulate by complex exponential of frequency w_0
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DFT for fixed w_0 summation = convolve result with “1”
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Why is summation convolution?
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start with definition
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Why is summation convolution? Let
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Why is summation convolution?
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convolution with 1 equivalent to summation
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DFT for fixed w_0 summation = convolve result with “1”
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DFT for fixed w_0 output X(e^jw_0) is constant
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DFT for fixed w_0
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input signal x(n)
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DFT for fixed w_0 fix specific frequency w_0
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DFT for fixed w_0 modulate by complex exponential of frequency w_0
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DFT for fixed w_0 summation = convolve with “1”
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DFT for fixed w_0 Transfer function H(e^jw)
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DFT for fixed w_0 summation = convolution with 1
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DFT for fixed w_0 i.e. impulse response h(n) = 1 for all n
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DFT for fixed w_0
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output X(e^jw_0) is constant
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Frequency Example
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Arbitrary example
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Frequency Example modulation = shift
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Frequency Example convolution by 1 = multiplication by delta
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DFT as filter bank
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fix specific frequency w_0
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DFT as filter bank one filter bank
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DFT as filter bank w continuous between [0,2pi)
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DFT as filter bank … uncountably many filter banks
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DFT as filter bank … Uncountable cannot enumerate all (even with infinite number of terms)
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DFT as filter bank … bank of modulators of all frequencies between [0, 2pi)
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DFT as filter bank … bank of identical filters with impulse response of h(n) = 1
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Short-Time Fourier Transform
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two variables
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Short-Time Fourier Transform frequency
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Short-Time Fourier Transform shift
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Short-Time Fourier Transform shifted window function v(k)
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Short-Time Fourier Transform let dummy variable be n instead of k
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Short-Time Fourier Transform fix frequency w_0 and shift m
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Short-Time Fourier Transform pass in input x(n)
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Short-Time Fourier Transform multiply by shifted window and complex exponential
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Short-Time Fourier Transform summation = convolve with 1
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Short-Time Fourier Transform output constant determined by frequency w_0 and shift m
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Short-Time Fourier Transform
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fix frequency w_0 and shift m
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Short-Time Fourier Transform pass in input x(n)
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Short-Time Fourier Transform multiply by shifted window and complex exponential
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Short-Time Fourier Transform summation = convolve with 1
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Short-Time Fourier Transform output constant determined by frequency w_0 and shift m
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Short-Time Fourier Transform
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dummy variable k instead of n
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Short-Time Fourier Transform shift is n (previously m)
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Short-Time Fourier Transform rewrite
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Short-Time Fourier Transform multiply by e^-jwn and e^jwn
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Short-Time Fourier Transform n is shift variable
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Short-Time Fourier Transform LTI system
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Short-Time Fourier Transform Impulse response
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Short-Time Fourier Transform flipped window modulated by +w
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Short-Time Fourier Transform modulation by -w
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Short-Time Fourier Transform
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fixed shift n
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Short-Time Fourier Transform fixed frequency w_0
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Short-Time Fourier Transform convolution with modulated window Multiplication by shifted window transform freq domain
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Short-Time Fourier Transform modulation by – w_0 shift by –w_0 freq domain
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input X(e^jw)
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transfer function (window transform shifted by +w_0)
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LTI system output
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final output after shift by –w_0
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STFT as filter bank
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fixed shift n
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STFT as filter bank fixed frequency w_0
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STFT as filter bank fixed shift n, fixed shift w_0 = one filter bank
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STFT as filter bank
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fixed shift n
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STFT as filter bank let w vary between [0, 2pi)
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STFT as filter bank uncountably many filters since w in [0, 2 pi)
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STFT as filter bank bandpass filters separated by infinitely small shifts
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transfer function (window transform shifted by +w_0)
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STFT as filter bank demodulators
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STFT as filter bank segments of X(e^jw)
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final output after shift by –w_0
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Fix number of frequencies
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STFT as filter bank bank of M filters
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STFT as filter bank M band pass filters
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Uniformly spaced frequencies
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STFT as filter bank bank of M filters becomes…
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STFT as filter bank uniform DFT bank
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M band pass filters …
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STFT as filter bank Let E_k(z)=1 for all k
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STFT as filter bank Let E_k(z)=1 for all k
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STFT as filter bank window becomes rectangle M samples long ……
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Uncertainty principle
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wide window
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Uncertainty principle narrow bandpass
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Uncertainty principle wide window poor time resolution
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Uncertainty principle narrow bandpass good frequency resolution
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Uncertainty principle narrow window
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Uncertainty principle wide bandpass
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Uncertainty principle narrow window good time resolution
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Uncertainty principle wide bandpass poor frequency resolution
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Special case narrowest window
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Special case widest bandpass
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Special case narrowest window perfect time resolution
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Special case widest bandpass no frequency resolution
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v(n)=delta(n)
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v(n)=delta(n)=1 if n=0, 0 otherwise
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v(n)=delta(n)
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STFT DFT
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v(m)=delta(m)
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v(0)=delta(0)=1 STFT DFT
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LTI system output band limited
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can decimate in time domain
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Decimation in time domain
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LTI system output band limited
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Decimation in time domain maximal decimation (total of M samples across M channels)
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Copies in frequency domain
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LTI system output band limited
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Copies in frequency domain copies after maximal decimation
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Decimated STFT M M M
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M M M uniformly spaced versions of same window filter
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Decimated STFT M M M constant maximal decimation
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Decimated STFT M M M decimation by M samples window shift of M
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Decimated STFT (sliding window)
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time axis
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Decimated STFT (sliding window) frequency axis
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Decimated STFT (sliding window) shift window by integer multiples of M
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Decimated STFT M M M decimation by M samples window shift of M
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Decimated STFT (sliding window) calculate M uniformly spaced samples of DFT
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Decimated STFT M M M uniformly spaced versions of same window filter
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Decimated STFT (sliding window)
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time axis
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Decimated STFT (sliding window) frequency axis
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Decimated STFT (sliding window) shift window by integer multiples of M
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Decimated STFT M M M decimation by M samples window shift of M
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Decimated STFT (sliding window) calculate M uniformly spaces samples of DFT
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Decimated STFT M M M uniformly spaced versions of same window filter
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Decimated STFT (sliding window) uniform sampling of time/frequency
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Decimated STFT (sliding window) M fixes sampling
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Decimated STFT M M M decimation by M samples M versions of same window filter
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Decimated STFT let decimation vary
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Decimated STFT (sliding window) let window shifts vary
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Decimated STFT let window transforms vary
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Decimated STFT (sliding window) let window transforms vary
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Wavelet Transform
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let decimation vary
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Wavelet Transform let window shifts vary
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Wavelet Transform let window transforms vary
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Wavelet Transform let window transforms vary
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Wavelet Transform let window transforms vary
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Summary
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Decimated STFT M M M decimation by M samples M versions of same window filter
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Decimated STFT (sliding window) M fixes sampling
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Decimated STFT (sliding window) uniform sampling of time/frequency
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Wavelet Transform let decimation vary let window transforms vary
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Wavelet Transform let window shifts vary let window transforms vary
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Wavelet Transform non-uniform sampling of time/frequency grid
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Reference Multirate Systems and Filter Banks by P.P. Vaidyanthan, pp.457-486
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