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Signal Processing

Thomas Dohaney COT 4810. Signal Processing. 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.

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Signal Processing

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  1. Thomas Dohaney COT 4810 Signal Processing

  2. 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. overview

  3. 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 Signal processing

  4. 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 Signal processing Goals

  5. 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 Signal processing needs

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

  7. 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. What is a signal?

  8. 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. Time Domain and Frequency Domain

  9. 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. The time domain

  10. 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. The frequency domain

  11. 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. Converting analog to digital signals

  12. 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 Signal sampling

  13. 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. . Signal sampling

  14. 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. Types of Signals.

  15. 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. Types of signals

  16. 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. Types of signals

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

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

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

  20. 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 Why do we want to process a Signal?

  21. 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 Why do we want to process A signal?

  22. 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 Techniques for processing a signal

  23. 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) Discrete Fourier transform

  24. DFT for continuous signals, not for digital signals. Discrete Fourier transform DFT Inverse DFT Plug in angular frequency f. DFT to get frequency. Inverse DFT to get time t.

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

  26. 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… Fast Fourier transform

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

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

  29. 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. 2-d Convolution

  30. 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. More 2-d convolution

  31. Algorithms • Edge Detection • Texture analysis • Object recognition and image understanding Image Processing IN Computer Vision Image can be classified as a Night Image Input Image Image can be classified as a Day Image Input Image

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

  33. 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. questions

  34. BORES. Introduction to DSP. http://www.bores.com/courses/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 Engineer’s Guide to Digital Signal Processing. http://www.dspguide.com/ • Wikipedia for some images. references

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