March 13, 2007

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System Identification & Parameter Estimation

Lecture 6: Perturbation signal design

WB2301

Alfred C. Schouten (a.c.schouten@tudelft.nl)

Faculty of 3MEDepartment of Biomechanical Engineering

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Contents

• Sources for error in estimators: • Aliasing• Leakage• Signal-to-noise ratio

• Improve the estimate• Optimal perturbation signals• Filtered white noise• Multisine signals

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Recording of signals

• In practical situations a limited time is recorded and digitized with a given sample frequency

• What errors can/do occur as a result• The sampling theorem (also Nyquist or Shannon)

• Signal > Fs/2 => aliasing• Resolution in frequency domain

• Δf=T-1

• What if frequency does not ‘fit’ on Δf

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Example aliasing• Matlab: AliasingDemo.m

• 9 Hz sine• Fs = 10 Hz

• 9 Hz sine > Fs/2, and a 1 Hz sine is ‘seen’.

• Frequencies above the Nyquist frequency are mirrored

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How to prevent aliasing

• Apply anti-aliasing filter, before digitizing!• Analog filter! • Rule of thumb: sample frequency should be at least

2.5 times the cut-off frequency (preferably more)

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Example leakage• Matlab: LeakageDemo.m

• 1 second observation=> resolution: 1Hz

• 5 Hz and 5.5 Hz sine

• Only frequencies with integer number of period are correctly observed!• Other frequencies will ‘leak’ to neighboring frequencies

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Effect of observation time

• Signal has infinite and a finite observation is made• Theoretical concept: signal is multiplied with a

window• window(t) = 1 within observation time• window(t) = 0 elsewhere

• What is effect of windowing?

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Window and it’s Fourier transform• Note: multiplication in time-domain is convolution in frequency-domain (and vice versa)

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Theory leakage

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Theory leakage

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How to prevent leakage

• White noise contains all frequencies, so leakage will always occur (at least in theory)

• Can we make ‘leakage’ free signals?

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Signal-to-Noise ratio

• SNR: ratio between signal power and noise power• Definition:

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noise

signal

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AA

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dBSNR

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SNR

log20log10)( 1010

2

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Signal-to-Noise Ratio (SNR)

• Improve SNR by increasing signal (u), or decreasing noise (n)

• Increase signal: turn up the volume!• Decrease noise: average (in time or frequency)

? y(t)u(t)

n(t)

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Ideal perturbation

Properties• Persistently exciting• Introducing no bias and varianceAdditional• Long enough to gain sufficient frequency resolution• Short enough to limit measurement time

(Humans: fatigue, attention, etc)

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White noise vs. colored noise

• White noise • Leakage and aliasing, bad SNR

• Colored noise (=filtered white noise)• Improved SNR, no aliasing, leakage not solved. • Note that discrete noise is ‘colored’ by sample

frequency

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Example colored noise: boost the signal

• Matlab: ColoredNoise.m

• If white and colored noise have the same variance• The power in colored noise is concentrated in a limited

number of frequencies • => the power per frequency will be higher (parseval!)• => better SNR (within the bandwidth)

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Improving the estimate 1

Bias (structural errors):• Main causes

• finite observation of stochastic input (leakage!)• frequency averaging

• Cure• application of 'leakage free' deterministic

perturbation signals• moderate frequency averaging• apple method that do not need frequency averaging

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Improving the estimate 2

Variance (random errors):• Main causes

• noise• Cure

• improve SNR• averaging (time or frequency domain)

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Multisine signals• With a limited observation time only the frequencies

with an integer number of period can be ‘seen’ by spectral estimators.

• Idea: construct signal with all frequencies with integer number op period

• => Multisine signals

• Advantages:• no leakage, no aliasing• better SNR compared to white noise

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Example multisine signal

• Matlab: xxx.m

• If white noise and multisine have same variance• The power in multisine is concentrated in a limited

number of frequencies • => the power per frequency is higher in multisine

(parseval!)• => better SNR• => no leakage, no aliasing

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Cresting

• A multisine signal (as white noise) has many outliers• Probability density function is gaussian (matlab)• Reduces the variance of the multisine signal, by

removing outliers: cresting• Crest factor = max|x(t)| / σx

• Even more improved SNR

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More advanced multisine tricksBasically two possibilities to improve multisine• Reduce number of frequencies

• Power per frequency can be increased, better SNR• Example: linear frequency spacing• Example: (quasi-)logarithmic frequency spacing

• Shape of the gain per frequency• shape input signal such that output signal becomes

flat (ideal case for white noise disturbance on the output)

• shape input signal such that output signal has same shape as output noise

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Reducing variance

• Average in time• Losing frequency resolution• Improves SNR

• Average adjacent frequencies• Can introduce bias

• Average in frequency domain• Multiple realizations or chop in multiple segments. Calculate

spectral densities for each segment and average spectral densities over the segments (‘Welch’ method)

• Losing frequency resolution• Improves SNR

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Assignment 6: compare the signals

Effect of perturbation signal on estimate of the FRF:• White noise• Filtered white noise• Multisine• Crested multisine• Crested multisine with gain optimized for system• Logarithmically-distributed crested multisine• Log-distributed crested multisine with optimized gain

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Next week

• So far identification of SISO systems in open and closed loop• SISO: single-input single-output

Next week:• Multivariable closed-loop system identification

• MIMO: multi-input multi-output