hadas benisty, yekutiel avargel and israel cohen · hadas benisty, yekutiel avargel and israel...
TRANSCRIPT
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ADAPTIVE SYSTEM IDENTIFICATION USING
TIME-VARYING FOURIER TRANSFORM
Hadas Benisty, Yekutiel Avargeland Israel Cohen
Presented by: Idan Igra
Technion, July 2013
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Motivation
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Long vs. short window
� In some processing schemes:
� Long window analysis is more accurate in
terms of steady-state error
� Short window analysis yields faster
convergence
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Long vs. short window
� varying window length could solve this tradeoff
� Assuming we can identify interesting points
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Mathematics
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Time Varying STFT
� Time Varying-STFT deals with the following issues, like STFT is:
� Transformation
� Decimation factor
� Overlap
� Inverse-Transformation
� Completeness condition
� Analysis windows
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Time Varying STFT
� Time-varying Fourier Transform:
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Time Varying STFT vs. STFT
� Remember the original STFT:
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N(t)
� N(t) is a piece-wise constant function:
N(t) = Nv, tv-1 < t < tv, v = 1, 2,…, V
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Decimation Factor
� Decimation factor (Lv) also time-varying
� Piece-wise constant, like Nv.
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Overlap
� And what about overlap?
� Still constant:
Nv / Lv = const
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Inverse TV-STFT
� The inverse transform:
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Inverse TV-STFT
� Completeness condition:
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Analysis window
� Analysis window:
� Should preserve continuity (why?)
� And constant overlap…
� Solution: interlacing windows
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Analysis window
� Interlaced hamming windows:
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Applications
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System Identification
� y(n) = Measured signal
� x(n) = Input signal (to estimate)
� ξ(n) = Additive noise (unknown)
� h(n) = Unknown LTI system
� Based on NLMS approximation.
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System Identification
� For the simulations:
� ξ(n) ~ N(0, σξ2), where SNR = 30dB
� h(n) = w(n)β(n)e-0.03n
� β(n) ~ N(0, σβ2)
� w(n) is rectangular window by length of Nh
� STFT overlap was 50%.
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System Identification
� Updating estimated system coefficients at transient frame: cubic interpulation.
� Zero-padding DFT produced similar results.
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System Identification: Noise
� System identification with White Gaussian noise (as input signal):
� x(n) ~ N(0, 1)
� 0 < n < 9,200 [samples]: Nh = 16 [samples]
� n > 9,200 [samples]: Nh ≠ 16 [samples]
� Pre-knowledge: the change time.
� Window length was changed on the beginning
and after 9,200 samples for fast convergence
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System Identification: Noise
� White Gaussian noise: time varying window length
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System Identification: Noise
� White Gaussian noise: smoothed error
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System Identification: AEC
� System identification for Acoustic Echo Cancellation (AEC).
� x(n) is a speech signal
� Sample rate = 16kHz.
� Room echo path: h(n)
� t < 2[sec]: Nh = 16[samples].
� Changed after 2 seconds.
� Again, pre-knowledge about the change.
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System Identification: AEC
� AEC: Time varying window length
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System Identification: AEC
� AEC: Results.
� (a) Far-end signal
� (b) Near-end signal
� (c)-(f): Error signals: 128,512,1024 & Time varying
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Similar work
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Reducing computational cost
� Adapting the time-frequency resolution over time: AR-STFT [Qaiser et al, 2008].
� For reducing computational cost.
� Controlling the A/D sampling rate.
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Optimize processing quality
� Define the window length to maximize a measure of short-time time-frequency concentration.
� Investigating also other transformations except STFT: Wavelet and cone-kernel.
� By Jones et al, 1994.
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Optimize processing quality
� (a) short, (b) medium, (c) long STFT.
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Overcome impulse noise
� Varying window length can be used for reducing impulse noise [Wei, Bi, 2003].
� By optimizing window length to some signal-characteristics bombastic words…
� Rotation direction.
� Chirp rate.
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Overcome impulse noise
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Overcome impulse noise
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Similar work
Much more varying window-length uses and manipulation on the net
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My twist…
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Phoneme adaptation
� Best window length for varying speech signal
� System Identification applications adapted
window length only to the changing system
� Why not adapting also to the changing input
signal?
� For example: Adapting to different
phonemes
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Phoneme adaptation
� The experiment:
� Gaussian noise with a given variance.
� SNR = 10dB.
� Time-varying Wiener filtering.
� Offline processing.
� Known phoneme division over time (for
example by preprocessing).
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Phoneme adaptation
... אדום
... כחול
! נפל
Time divisor
Phoneme
recognition (given)
Time varying
Wiener filtering
+
Gaussian noise
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Phoneme recognition
� Phoneme recognizer returns one out of four phoneme types (changed on time):
� Silent,
� White Noise (ssss, fff, etc.),
� Vowel (aaa, eee…),
� Or impulse (d, t, …).
� Pre-recognized manually for the experiment purpose.
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Phoneme recognition
0 0.5 1 1.5 2 2.5 3 3.5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
time (s)
am
plit
ude
Original
Silent
Impulse
Vowel
White
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Time divisor
� Time divisor decides the window length
� May change over time.
� Depends on phoneme type.
� Constant length: Simple wiener filtering.
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Time divisor
� Time divisors tested:
� Per phoneme window length.
� Short convergence time divisor:
� Short window length right after phoneme type
change.
� Long window length later until next phoneme type change.
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� Motivation:
� Following empirical experiment, the error is
changed depends on:
� Phoneme type
� Window length
� � Optimize window length for a phoneme
type may results in better performance.
Per phoneme window length
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� Motivation:
2 3 4 5 6 7 8 9 10
x 10-3
10-4
10-3
10-2
Average error for each phoneme type
window length (ms)
avera
ge s
quare
d e
rror
Overall
Silent
Impulse
Vowel
White
Per phoneme window length
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� Motivation:
� Similar to Adaptive System Identification
idea.
� Really? Wiener Filtering vs. NLMS.
� But adaptation according to signal instead
of system.
Short convergence time divisor
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Phoneme adaptation
Silent Impulse Vowel White Overall0
0.01
0.02
0.03
0.04
0.05
0.06Error per phoneme (with musical noise reduction)
Optimal length per phoneme
Uniform optimal length
Short stft for convergence
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Phoneme adaptation
0 0.5 1 1.5 2 2.5 3 3.5-1
0
1
am
plit
ude
Original
Silent
Impulse
Vowel
White
0 0.5 1 1.5 2 2.5 3 3.5-0.5
0
0.5Optimal length per phoneme - error
am
plit
ude
0 0.5 1 1.5 2 2.5 3 3.5-0.5
0
0.5Uniform optimal length - error
am
plit
ude
0 0.5 1 1.5 2 2.5 3 3.5-0.5
0
0.5Short stft for convergence - error
am
plit
ude
time (s)
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Phoneme adaptation disadvantages
� Large error on length replacement
� Tried to improve by:� Very small alpha (0.5) on length replacement
� Using old filtering for a while after replacement
� Old filter is not optimal for the new size (need further investigation why).
� Except large mathematical error, inconvenient listening phenomena on length replacement:
� We didn’t discuss the computational cost…
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Thanks!
Thanks for
listening!
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References
� Hadas Benisty, Yekutiel Avargel and Israel Cohen. Adaptive System Identification using Time-Varying Fourier Transform. Department of Electrical Engineering, Technion - Israel Institute of Technology.
� Saeed Mian Qaisar, Laurent Fesquet, and Marc Renaudin. An Adaptive Resolution Computationally Efficient Short-Time Fourier Transform. Proceeding of World academy of science, engineering and technology volume 31, July 2008 ISSN 1307-6884.
� Douglas L. Jones, Richard G. Baraniuk. A Simple Scheme for Time-Frequency Representations. IEEE Transactions on Signal Processing, Vol. 42, No. 12, Dec. 1994.
� Wei, Y. M.; Bi, G. A. Robust STFT with Adaptive Window Length and Rotation Direction. International conference on information, communications and signal processing; ICIS-PCM 2003. 4th, International conference on information, communications and signal processing; ICIS-PCM 2003; 827-829.
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Musical signal
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
am
plit
ude
Original
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
0
0.5Optimal length per phoneme - error
am
plit
ude
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
0
0.5Uniform optimal length - error
am
plit
ude
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
0
0.5Short stft for convergence - error
am
plit
ude
time (s)
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Applications
� Application #3 – Per phoneme length.
2 3 4 5 6 7 8 9 10
x 10-3
10-3
10-2
10-1
Average error for each phoneme type
window length (ms)
avera
ge s
quare
d e
rror
Overall
Silent
Impulse
Vowel
White
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Applications
� Application #3 – Per phoneme length.
Silent Impulse Vowel White Overall0
0.02
0.04
0.06
0.08
0.1
0.12Error per phoneme (with musical noise reduction)
Optimal length per phoneme
Uniform optimal length
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Applications
� Application #3 – Per phoneme length.
Silent Impulse Vowel White Overall0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09Error per phoneme (no musical noise reduction)
Optimal length per phoneme
Uniform optimal length
![Page 54: Hadas Benisty, Yekutiel Avargel and Israel Cohen · Hadas Benisty, Yekutiel Avargel and Israel Cohen. Adaptive System Identification using Time-Varying Fourier Transform. Department](https://reader033.vdocuments.net/reader033/viewer/2022050422/5f913542fc66b4405e75c3b4/html5/thumbnails/54.jpg)
Applications
� Application #3 – Per phoneme length.
0 0.5 1 1.5 2 2.5 3 3.5-1
0
1
am
plit
ude
Original
Silent
Impulse
Vowel
White
0 0.5 1 1.5 2 2.5 3 3.5-0.5
0
0.5Optimal length per phoneme - error
am
plit
ude
0 0.5 1 1.5 2 2.5 3 3.5-0.5
0
0.5Uniform optimal length - error
am
plit
ude
time (s)