Thomas Dohaney COT 4810 Signal Processing

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.

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

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

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

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

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.

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.

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.

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.

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.

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

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. .

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.

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.

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.

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

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

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

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

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

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

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)

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.

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

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…

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)

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

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.

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.

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

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

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.

references 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.