1. Thomas Dohaney
COT 4810
Signal Processing

2. 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.

3. 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

4. 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

5. 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

6. Signal processing today
Today SP driven by
Commercial marketplace
Need to transfer information

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. Signal sampling Why we convert the signal to digital form. Trade offs.
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

13. 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. .

14. 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.

15. 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.

16. 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.

17. 1-d signals Seismic vibrations EEG and EKG Speech Sonar Audio Music
ph o - n - e t i c ia n

18. 2-d signals. Photographs Medical images Radar IED detection
Satellite data
Fax
Fingerprints

19. 3-d signals. Video Sequences Motion Sensing Volumetric data sets
Computed Tomography,
Synthetic Aperture Radar Reconstruction)

20. Why do we want to process a Signal?
Compare a transmitted and reflected signal
Find characteristics of a remote object
Recognize what’s 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

21. 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

22. 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

23. 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)

24. 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.

25. Discrete f0urier transform
Convert continuous DFT to discrete DFT.
Continuous version
Discrete version
Let  stand for (a primitive nth root of unity)
We get

26. 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…

27. FAST Fourier transform
found by recursively using N/2 factors of
For example N=23, then FFT of is *
Then FFT of quotient above is
O(nlogn)

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

29. 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.

30. 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.

31. 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

32. Image processing IN computer vision
Algorithms
Image segmentation
Scale Invariance
Object recognition and image understanding
Face detection

33. 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.

34. references
BORES. Introduction to DSP.
Dewdney, A. K. The New Turing Omnibus New York.
Irwin, David J. Industrial Electronics Handbook.
Smith, Steven W. The Scientist and Engineer’s Guide to Digital Signal Processing.
Wikipedia for some images.