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Digital signal processing

  • Digitalsampled, discrete-time, quantized
  • 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

Functional magnetic resonance imaging

Challenges are measurement noise and intrinsic uncertainties in signal behavior.

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 0 2 .
  • 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:

  • x 1 N n 1 N x n (sample mean)
  • x 1 , , x N (the data itself)
  • x 1 x N (order statistic)
  • ( x 1 2 x 2 x 3 , x 1 x 3 )
A statistic cannot depend on unknown parameters .

Probability is used to model uncertainty.

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. Heads p 1 Tails Must decide p 1 4 vs. p 3 4 .
  • Step 2

    Collect data x 1 , , x N x i 1 Heads x i 0 Tails
  • Step 3

    Formulate a useful statistic k n 1 N x n If k N 2 , guess p 1 4 . If k N 2 , guess p 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 1 N 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.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
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Introduction about quantum dots in nanotechnology
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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