ss_intro

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Introduction of Statistical Signal

Processing

• A signal is a set of observations taken sequentially in

time, space, or some other independent variable.

• A deterministic signal is one that may be reproduced

exactly with repeated measurements.

• A random signal is a signal that is not repeatable in a

predictable manner.

• Though random signals are evolving in time in an

unpredictable manner, their average statistical

properties exhibit considerable regularity and hence

they are described with statistical averages instead of

explicit equations.

• Main objectives of dealing with random signals:

1. the statistical description, modeling, and

exploitation of the dependence between the values

of one or more discrete-time signals and2. their application to theoretical and practical

problems.

• Random signals are described mathematically by

using the theory of probability, random variables and

stochastic processes.

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• In practice we deal with random signal by using

statistic techniques.

• Four major application areas:

1. Spectral estimation

2. Signal modeling

3. Adaptive filtering

4. Array processing

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Random signals

• A discrete-time signal or time series is a set of

observations taken sequentially in time, space, or

some other independent variable.

• In this class, we mainly concern discrete-time signal

and the terms signal, time series, or sequences will be

used to refer to a discrete-time signal.

• The key characteristics of a time series are that the

observations are ordered in time and that adjacentobservations are dependent.

• The dependency can be described with a scatter plot

or a quantity called correlation.

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• One can use past observations to predict future

values.

Prediction result Nature of the signal

Exact Deterministic

Not exact Random/stochastic

• The degree of their predictability is determined by the

dependence between consecutive observations.

• Special case: white noise is completely unpredictable.

• Fundamental characteristics of random signals: notable to precisely specify its values.

• The processing of discrete-time signals can be

classified as follows:

1. Signal analysis:

• Objective: To extract useful information that

can be used to understand the

signal generation process or

extract features that can be used

for signal classification purposes.

• Examples: spectral estimation & signal

modeling

• Applications: detection/classification of radar

and sonar targets, speech &

speaker recognition, etc.

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2. Signal filtering:

• Objective: To improve the quality of a

signal according to an acceptable

criterion of performance.

• Examples: Adaptive filtering and array

processing

• Applications: noise & interference cancellation,

echo cancellation, channelequalization, etc.

• Examples of random signals:

Speech signals, Electrophysiological signals and

geophysical signals, radar signals

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Random signal analysis

• Major areas of random signal analysis are

(1) statistical analysis of signal amplitude;

(2) analysis and modeling of the correlation among

the samples of an individual signal; and

(3) joint signal analysis of their interaction or

interrelationship.

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Amplitude distribution:

• Description of signal variability: mean, median,

variance and dynamic range

• The shape of the histogram tells the probability

density and is very important in applications such as

signal coding and event detection.

• The most popular distribution is Gaussian and, for

typical signals, we approximate their distribution as

Gaussian.

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• Significance of Gaussian distribution:

1. Many physical signals can be described by

Gaussian processes.

2. The central limit theorem states that any process

that is the result of the combination of many

elementary processes will tend, under quite

general conditions, to be Gaussian.

3. Linear systems preserve the Gaussianity of their

input signals.

Correlation and spectral analysis• Empirical normalized autocorrelation sequence (i.e. a

normalized estimate of the autocorrelation of x(n) by

using a set of its observations) of a zero-mean time

series x(n):

∑

∑ −

=ρ−

=

−

=1

0

2

1

0

*

)(

)()(

)( N

n

N

n

n x

ln xn x

l

• Implication of )(lρ

:

Property of a signal What does it means

Short-memory/short-

range dependence

)(lρ decay fast

Long-memory/long-

range dependence

)(lρ

decay slowly

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• The autocorrelation and the spectral density of a

signal form a Fourier transform pair.

• The spectral density function shows the distribution

of signal power or energy as a function of frequency.

• What is the connection between a signal and its

spectral, frequency plot and autocorrelation?

Joint signal analysis:

• In many applications, we are interested in the

relationship between 2 different random signals.

• Purposes:

1. To check if they are of the same or similar nature.

2. To ascertain and describe the similarity or

interaction between them so as to model or

identify the relevant system.

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Spectral estimation

• Spectral estimation is used to estimate the distribution

of energy or power of a signal from a set of

observations.

• The approaches for spectrum estimation may be

generally categorized into one of two classes.

• Nonparametric methods: Begin by first estimating

the autocorrelation sequence from a given set of data and then estimate the power spectrum by

Fourier transforming the estimated autocorrelation

sequence.

• Parametric methods: Use a model for the signal to

estimate the power spectrum.

• Applications: medical diagnosis, speech analysis,

radar & sonar, nondestructive fault detection, testing

of physical theories, and evaluating the predictability

of time series.

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Signal modeling

• In many theoretical and practical applications, we are

interested in (1) generating random signals with

certain properties or (2) obtaining an efficient

representation of real-world random signals that

captures a desired set of their characteristics in the

best possible way.

• The mathematical description that provides an

efficient representation of the 'essential' properties of

a signal is referred to as model.

• In general, a time series x(n) can be modeled as a

linear combination of values of w(n)

∑ −=∞

−∞=k

k nwk hn x )()()(

where w(n) is independent and identically distributed

(IID) observations.

• The list of desired features for a good model includes:

1. the number of model parameters should be as

small as possible

2. estimation of the model parameters from the data

should be easy, and

3. the model parameters should have a physically

meaningful interpretation.

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• Applications of a signal model:

• To achieve a better understanding of the physical

mechanism generating the signal

• To track changes in the source of the signal and

help identify their cause (e.g., EGG).

• To synthesize artificial signals similar to the natural

ones (e.g. speech).

• To extract parameters for pattern recognition

applications (e.g., speech recognition).

• To get an efficient representation of signals for data

compression (e.g. audio, video coding).• To forecast future signal behavior (e.g. predict

stock market indexes).

• Steps involved in signal modeling:

1. Select an appropriate model;

2. Select the 'right' number of parameters;

3. Fit the model to the actual data;

4. Verify the model.

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• Example: Speech model

• Classification of random signal models

• A more simple classification: AR, MA or ARMA

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Adaptive filtering

• Conventional digital filters with fixed coefficients are

designed to have a given frequency response chosen

to alter the spectrum of the input signal in a desired

manner.

• Many practical application problems can't be

successfully solved with fixed digital filters because

either (1) we do not have sufficient information todesign a digital filter with fixed coefficients or (2) the

design criteria change during the normal operation of

the filter.

• 3 basic modules of adaptive filters:

1. Filtering structure: forms the o/p of the filter

using measurements of the input signal or

signals.

2. Criterion of performance (COP): used to assess

the quality of the filter w.r.t. the requirements of

the oarticular application.

3. Adaptive algorithm: uses the value of COP andthe measurements of the input and desired

response (when available) to decide how to

modify the parameters of the filter to improve the

performance.

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• The input signals and the desirable response signal

(when available) are collectively referred to as the

signal operating environment (SOE) of the adaptive

filter.

• The design of any adaptive filter requires (1) a great

deal of a priori information about the SOE and (2) a

deep understanding of the particular application.

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• Applications of adaptive filters can be sorted for

convenience into 4 classes:

Application class Examples

Echo cancellation

Adaptive control

System identification

Channel modeling

Adaptive equalizationSystem inversion

Blind deconvolution

Adaptive predictive coding

Change detection

Signal prediction

Radio frequency

interference cancellation

Acoustic noise controlMultisensor

interference

cancellationAdaptive beamforming

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System identification:

• The goal is to estimate the parameters or the state of

the system.

• Assumption: The input of the filter is noise-free and

the desired response is corrupted by additive noise

that is uncorrelated with the input signal.

• Example: Acoustic echo cancellation

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• Problems:

1. The reverberations of the room: The microphone

picks up not only the speech coming from the

talker but also reflections from the walls and

furniture in the room.

2. Echos are created by the acoustic coupling

between microphone and the loudspeaker.

• Why an adaptive filter is used?

1. The echo path is usually unknown before actualtransmission begins.

2. The echo path is changing with time.

System inversion:

• The goal is to estimate and apply the inverse of the

system to recover the original signal.

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• Difficulties:

1. The input of the adaptive filter may be corrupted

by additive noise.

2. The desired response may not be available.

• Applications include channel equalization,

deconvolution and adaptive inverse control.

• Example: Channel equalizer:

• As the signal propagates through the channel, it's

delayed and attenuated in a frequency-dependentmanner.

• This distortion can be compensated for by using a

linear filter called an equalizer.

• A known training sequence is transmitted to train

the equalizer before a transmission session starts.

Signal prediction:

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• The goal is to estimate the value of a particular

sample )( 0n x of a random signal by using a set of

consecutive signal samples }|)({ 21 nnnn x ≤≤ .

Classification case

Forward prediction 20 nn >

Backward prediction 01 nn >

Smoothing/interpolation 0201 , nnnnn ≠<<

• Example: Linear predictive coding:

• Uses a linear predictor to estimate )(ˆ n x of the

present sample )(n x as a linear combination of the

M past samples.

∑ −==

M

k k k n xan x

1

)()(ˆ

• The coefficients k a are determined by minimizing

the objective function

}))(ˆ)({(})({22

n xn x E ne E J −==

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Multisensor interference cancellation:

• The key feature of this class of applications is the useof multiple sensors to remove undesired interference

and noise.

• A primary signal contains both the signal of interest

and the interference.

• Reference signals are picked up by other sensors to

cancel the undesired interference.

• The amount of correlation between the primary and

reference signals is measured and used to form an

estimate of the interference in the primary signal,

which is subsequently removed.

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• Example: Active noise control:

• The basic idea is to cancel acoustic noise using

destructive wave interference.

• In practice, the acoustic environment is unknown

and time-varying and hence adaptive filtering is

required.

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Array processing

• Array processing deals with techniques for the

analysis and processing of signals collected by a

group of sensors.

• Applications: radar, sonar, communications,

seismology, geophysical prospecting for oil and

natural gas, diagnostic ultrasound and multichannel

audio systems.

Spatial filtering or beamforming

• An array receives spatially propagating signals andprocesses them to emphasize signals arriving from a

certain direction.

• It acts as a spatially discriminating filter.

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• Note a frequency-selective FIR filter extracts signals

at a frequency of interest.

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Adaptive interference mitigation in radar systems.

• The desired information from these targets is (1) their

relative distance from the airborne platform (range),

(2) their angle w.r.t. the platform, and (3) their

relative speed.

• The processing of the radar:

1. Filter out undesired signals through adaptive

processing

2.

Determine the presence of targets, a processknown as detection

3. Estimate the parameters of all detected targets.

• The received signal is known as return.

• The angle of the target is determined through the use

of beamforming.

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• Two common types of interference:

1. Clutter: reflections of the radar signal from the

ground

2. Jamming: other transmitted energy at the same

operation frequency as the radar.

• We can adjust the beamformer weights s.t. signals

from the directions of the interference are rejected

while other direction are searched for targets.

• The resolution in angle of the beamformer is called a

beamwidth which can be improved by beamsplitting.

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Adaptive sidelobe canceler

• The high gain channel is known as the main channel

that contains both the signal and the jamming

interference.

• Auxiliary channels typically have much lower gain in

the direction in which the main channel is directed so

that they contain only the interference.

• The sidelobe canceler uses the auxiliary channels to

form an estimate of the interference in the main

channel.

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