Signalwaveform, sequnce of
measurements or observations
Processinganalyze, modify, filter,
synthesize
Examples of digital signals
sampled speech waveform
"pixelized" image
Dow-Jones Index
Dsp applications
Filtering (noise reduction)
Pattern recognition (speech, faces,
fingerprints)
Compression
A major difficulty
In many (perhaps most) DSP applications we
don't have complete or perfect knowledge of the signals we wishto process. We are faced with many
unknowns and
uncertainties .
Examples
noisy measurements
unknown signal parameters
noisy system or environmental conditions
natural variability in the signals encountered
How can we design signal processing algorithms
in the face of such uncertainty?
Can we model the uncertainty and incorporate
this model into the design process?
Statistical signal processing is the
study of these questions.
Modeling uncertainty
The most widely accepted and commonly used
approach to modeling uncertainty is
Probability
Theory (although other alternatives exist such as Fuzzy
Logic).
Probability Theory models uncertainty by
specifying the
chance of observing certain
signals.
Alternatively, one can view probability as
specifying the degree to which we
believe a
signal reflects the true
state of nature .
Examples of probabilistic models
errors in a measurement (due to an imprecise measuring
device) modeled as realizations of a Gaussian randomvariable.
uncertainty in the phase of a sinusoidal signal
modeled as a uniform random variable on
$\left[0 , 2\pi \right)$ .
uncertainty in the number of photons stiking a CCD per
unit time modeled as a Poisson random variable.
Statistical inference
A
statistic is a function of
observed data.
Suppose we observe
$N$ scalar values
${x}_{1},,{x}_{N}$ . The following are statistics:
Statistics are used to draw
conclusions about probability models.
Probability models our uncertainty about
signals we
may observe.
Statistics reasons from the measured signal to
the population of possible signals.
Statistical signal processing
Step 1
Postulate a probability model (or models) that reasonably
capture the uncertainties at hand
Step 2
Collect data
Step 3
Formulate statistics that allow us to interpret or
understand our probability model(s)
In this class
The two major kinds of problems that we will
study are
detection and
estimation . Most SSP problems fall under one of
these two headings.
Detection theory
Given two (or more) probability models, which
on best explains the signal?
Examples
Decode wireless comm signal into string of 0's and
1's
Pattern recognition
voice recognition
face recognition
handwritten character recognition
Anomaly detection
radar, sonar
irregular, heartbeat
gamma-ray burst in deep space
Estimation theory
If our probability model has free parameters,
what are the best parameter settings to describe the signalwe've observed?
Examples
Noise reduction
Determine parameters of a sinusoid (phase, amplitude,
frequency)
Adaptive filtering
track trajectories of space-craft
automatic control systems
channel equalization
Determine location of a submarine (sonar)
Seismology: estimate depth below ground of an oil
deposit
Detection example
Suppose we observe
$N$ tosses of an unfair coin. We
would like to decide which side the coin favors, heads or tails.
Step 1
Assume each toss is a realization of a Bernoulli random
variable.
$$(\mathrm{Heads})=p=1-(\mathrm{Tails})$$ Must decide
$p=\frac{1}{4}$ vs.
$p=\frac{3}{4}$ .
Step 2
Collect data
${x}_{1},,{x}_{N}$$${x}_{i}=1\equiv \text{Heads}$$$${x}_{i}=0\equiv \text{Tails}$$
Step 3
Formulate a useful statistic
$$k=\sum_{n=1}^{N} {x}_{n}$$ If
$k< \frac{N}{2}$ , guess
$p=\frac{1}{4}$ . If
$k> \frac{N}{2}$ , guess
$p=\frac{3}{4}$ .
Estimation example
Suppose we take
$N$ measurements of a DC voltage
$A$ with a noisy voltmeter. We would
like to estimate
$A$ .
Step 1
Assume a Gaussian noise model
$${x}_{n}=A+{w}_{n}$$ where
$({w}_{n}, (0, 1))$ .
Step 2
Gather data
${x}_{1},,{x}_{N}$
Step 3
Compute the sample mean,
$$(A)=\frac{1}{N}\sum_{n=1}^{N} {x}_{n}$$ and use this as an estimate.
In these examples (
and
), we solved detection and
estimation problems using intuition and heuristics (in Step 3).
This course will focus on developing principled
and mathematically rigorous approaches to detection and estimation,using the theoretical framework of probability and statistics.
Summary
DSPprocessing signals with computer
algorithms.
SSPstatistical DSPprocessing
in the presence of uncertainties and unknowns.
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