Signal Processing

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Signal Processing

• Thomas Dohaney
• COT 4810

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overview

• Signal Processing Goals, Needs, Applications.
• What is a signal?
• Types of signals.
• Reasons to process signals.
• Analog to Digital conversion.
• Digital Filters.
• Time domain and frequency domain.
• Discrete Fourier Transforms and Fast Fourier
  Transforms and their properties.
• Image processing and Computer Vision.

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Signal processing

• An area in Computer Science that is unique by the
  type of data it uses, signals.
• Signals are sensory data from physical systems .
• Vibrations
• Visual images
• Voltage
• Sound

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Signal processing Goals

• Signal Processing is
• Mathematics
• Algorithms
• Techniques
• To manipulate signals
• Lots of goals
• Enhancement of visual images
• Recognition and generation of speech
• Compression of data for storage and transmission
• Object detection
• Image enhancement

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Signal processing needs

• 1960s and 1970s
• Digital computers first became available
• Computers were expensive
• SP was limited to only a few critical
  applications.
• Pioneering efforts were made in four key areas.
• RADAR and SONAR
• Oil Exploration
• Space Exploration
• Medical Imaging

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Signal processing today

• Today SP driven by
• Commercial marketplace
• Need to transfer information

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What is a signal?

• A signal is a function that conveys information,
  generally about the state or behavior of a
  physical system.
• Analog signals are continuous time, continuous
  amplitude.
• Digital signals are discrete time, discrete
  amplitude.

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Time Domain and Frequency Domain

• Many ways that information can be contained in a
  signal.
• Manmade signals.
• AM
• FM
• Single-sideband
• Pulse-code modulation
• Pulse-width modulation
• Only two ways that are common for information to
  be represented.
• Information represented in the time domain,
• Information represented in the frequency domain.

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The time domain

• Domain describes when something occurs
• What the amplitude of the occurrence was
• Each sample in the signal indicates
• What is happening at that instant, and the
• Level of the event
• If something occurs at time t, the signal
  directly provides information on the time it
  occurred, the duration, and the development over
  time.
• Contains information that is interpreted without
  reference to any other part of the sample.

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The frequency domain

• Frequency domain is considered indirect.
• Information is contained in the overall
  relationship between many points in the signal.
• By measuring the frequency, phase, and amplitude,
  information can be obtained about the system
  producing the motion.

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Converting analog to digital signals

• Converting continuous time, continuous amplitude
• To discrete time, discrete amplitude
• To convert to a digital signal we must sample it
  at a rate, so there is enough information to
  reconstruct it, and not leave any information
  out.

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Signal sampling

• Why we convert the signal to digital form.
• Software implementations
• Accuracy can be controlled
• Repeatable
• Noise is minimal
• Operations are easier to implement
• Digital storage is cheap
• Security
• Price and performance
• Trade offs.
• Loss of information
• AD and DA conversion requires additional hardware
• Speed of processors is limited
• Round off errors

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Signal sampling

• Nyquist sampling theorem.
• The lower bound of the rate at which we should
  sample a signal, in order to be guaranteed there
  is enough information to reconstruct the original
  signal is 2 times the maximum frequency.
• Now in its digital form,
• we can process the signal
• in some way.
• .

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Types of Signals.

• 1-D signals.
• Sound and Vibrations.
• Signals used to extract statistical
  characteristics, and construct a mathematical
  model of the signal.
• Output signal is entered into the mathematical
  model, if only white noise is observed it is
  normal, it is abnormal if there is a lack of
  white noise.
• Typically used to diagnose a system in that they
  are used to detect abnormality and deterioration.

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Types of signals

• 2-D signals.
• Considered to be an image signal.
• Signal is distorted in the digitizing process
  based on signal to noise ratio. (blur, movement,
  arithmetic, or color distortion).
• Typically to determine measurement of an object
  in an image, image restoration, visualization to
  extract physical information, pattern
  recognition, image inspection and fault
  detection.

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Types of signals

• 3-D signals.
• Computer vision.
• Signal is obtained by visual sensor composed of
  many two dimensional images, or by measuring
  distance of an object (using electromagnetic
  wave, or laser) and adding this information to an
  object in a 2-d signal.
• Typically used in automation, remote sensing.

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1-d signals

• Seismic vibrations
• EEG and EKG
• Speech
• Sonar
• Audio
• Music

ph - o - n - e - t - i -
c - ia - n
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2-d signals.

• Photographs
• Medical images
• Radar
• IED detection
• Satellite data
• Fax
• Fingerprints

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3-d signals.

• Video Sequences
• Motion Sensing
• Volumetric data sets
• Computed Tomography,
• Synthetic Aperture Radar Reconstruction)

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Why do we want to process a Signal?

• Compare a transmitted and reflected signal
• Find characteristics of a remote object
• Recognize whats in a signal
• Target detection
• Speech recognition
• Image analysis
• Predict a future value of the signal
• Stock market prediction
• Interpolate missing values of a signal
• Conceal lost video packets
• Restore a signal that has been degraded
• Noise removal
• Echo cancellation

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Why do we want to process A signal?

• Obtain a visual representation of a signal
• Extract information
• Enhance a signal
• Image contrast enhancement
• Compress a signal
• Faster transmission
• Less storage space
• Synthesize a realistic example of a signal
• Speech generation and synthesis
• Image texture generation
• Choose specific input signals to control a
  process
• Face detection
• Motion detection

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Techniques for processing a signal

• A system is a function that produces an output
  signal in response to an input signal.
• An input signal can be broken down into a set of
  components, called an impulses.
• Impulses are passed through a system resulting in
  output components, which are synthesized into an
  output signal.
• Convolution is a way of combining two signals to
  form a third.
• Discrete Fourier Transforms
• Properties of Fourier Transforms

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Discrete Fourier transform

• Given the time domain, the process of calculating
  the frequency domain is called DFT.
• Given frequency domain the process of calculating
  the time domain is inverse DFT.
• O(n2)

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Discrete Fourier transform

• DFT for continuous signals, not for digital
  signals.

DFT
Inverse DFT
Plug in angular frequency f.
DFT to get frequency.
Inverse DFT to get time t.
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Discrete f0urier transform

• Convert continuous DFT to discrete DFT.
• Continuous version
• Discrete version
• Let ? stand for (a primitive nth root of
  unity)
• We get

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Fast Fourier transform

• The algorithm views the problem as computing a
  polynomial for ?
  instead of k.
• The theory of polynomials says P(?k) is found by
  the remainder of
• In FFT, For N 23, finding the remainder for
  P(?k) is done by

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FAST Fourier transform

• found by recursively using N/2
  factors
• of
• For example N23, then FFT of is
• Then FFT of quotient above is
• Then FFT of quotient above is
• O(nlogn)

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Properties of FFT

• FFT can apply to 1-d, 2-d, multidimensional
  signals.
• Linearity
• Scaling
• Shifting
• Conjugation
• Convolution
• Differentiation

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2-d Convolution

• Convolution is combining two signals to form a
  third.
• A delta function is a normalized response
  (signal).
• Example of an image convolved with a 3x3 delta
  function.
• Example of an image convolved with a 3x3 impulse
  response.

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More 2-d convolution

• A is the impulse response padded with zeros.
• Output image C is the sum of the components of B
  convolved with A.
• Represents overlap between the two signals.

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Image Processing IN Computer Vision
Image can be classified as a Night Image
Input Image

• Algorithms
• Edge Detection
• Texture analysis
• Object recognition and image understanding

Image can be classified as a Day Image
Input Image
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Image processing IN computer vision

• Algorithms
• Image segmentation
• Scale Invariance
• Object recognition and image understanding
• Face detection

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questions

• 1) True or False, Discrete Fourier Transforms
  will transform one function in terms of another.
• 2) List one instance of signal processing used in
  any field today.

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references

• BORES. Introduction to DSP. http//www.bores.com/c
  ourses/intro/index.htm
• Dewdney, A. K. The New Turing Omnibus. 2001. New
  York.
• Irwin, David J. Industrial Electronics Handbook.
• Smith, Steven W. The Scientist and Engineers
  Guide to Digital Signal Processing.
  http//www.dspguide.com/
• Wikipedia for some images.