Digital Signal Processing. The Scientist and Engineers Guide. Steven W. Smith

Gauss distribution, see Gaussian Gaussian

See also Central limit theorem equation for, 26-31

Fourier transform of, 216-217, 423-425, 427-428

as filter kernel, 281-283, 400-401, 423-425, 427-428, 430-432

noise, 29-32 separability of, 404-406

Geophysics. See Seismology

Gibbs effect, 142, 203, 218-219, 286-287 GIF image, data compression, 488 Grayscale, image, 373, 387

Grayscale stretch, 390-391 Grayscale transforms, 390-394,433

Halftone image, 387, 433

Harmonics, 172-173, 220-222, 255, 355-358 Harvard Architecture, 84, 509

Hearing, 351-355

High fidelity audio, 358-362 High-pass filters, See Filters High speed convolution. See FFT

convolution Hilbert transformer, 621

Histogram, calculating, 19-26 Histogram equalization, 393-394 History of DSP, 1-3

Homogeneity, 89-90, 108, 185-186 Homomorphic processing, 370-372, 408-409 Hyperspace, 457, 463-464

Huffman encoding, 484-486, 500

IIR filters. See Filters, recursive Illumination flattening of images, 407-410 Immersion. See under Convolution Impedance, electrical, 563-566

Impulse, 100, 107-108

Impulse decomposition. See under decomposition

Impulse response

See also Convolution continuous systems, 244-245 defined, 108-109

examples of, 110-111, 128-132 two-dimensional (image), 397-398

Impulse train, 42-47

Independent variable (defined), 12

Infinite impulse response (IIR) filters. See Filters, recursive

Information

frequency domain encoded, 56, 265-266, 268-270

spatial domain encoded, 373-374, 424-425 time domain encoded, 56-57, 265-267

In-place computation, 233

Integral, discrete, See Running sum

Integral of continuous impulse, 244, 250-251 Integrated profile, 428-429

Interlaced decomposition. See under Decomposition

Interlaced video, 384-385 Interpolation. See Multirate

Iterative techniques, 444-445, 465-473 Iterative least squares technique (ILST), 444

JPEG image compression, 494-502

Karhunen-Loeve transform, 496 Kernel, filter. See Impulse response

Laplace transform, 334, 581-604 Layers, neural network, 459-460 Learning algorithm, neural network, 463 Least significant bit (LSB) (defined), 36 Lens, camera and eye, 376-379

Limiting resolution, images, 426 Line pair, 426

Line pair gauge, 425-426

Line scanning image acquisition, 386-387 Line spread function (LSF), 426-430 Linear phase, See under Phase

Linear predictive coding (LPC), 359, 366 Linear systems. See Linearity

Linearity

alternatives, 104-106 commutative property, 96 decomposition. See Decomposition

examples of linear and nonlinear systems, 94-95

Fourier transform, of the, 185-188 memoryless systems, 93 multiplication, 97

noise, adding, 97 requirements for, 89 sinusoidal fidelity, 92-94, 142 static linearity, 92-93 superposition, 98-100 synthesis, 98-99

Logarithmic scale. See Decibels Long integer, 72

Lossless data compression, 481-494

Lossy data compression, 481-482, 494-502 Low-pass filters. See under Filters Luminance signal, television, 386

LZW encoding, data compression, 488-494

Magnetic resonance imaging (MRI), 9-10, 450

Magnitude. See Polar notation

Matched filter, See under Filters, digital

Math coprocessor, 81

Mean, 13-17, 20-22, 434-436

Medical imaging, 2. See also X-ray imaging; Computed Tomography

Memory cache, 81-82 Memoryless system, 93 MFLOPS, 526 Microphonics, 172 MIPS, 526

Mix down, music, 5, 362

Modulation transfer function, 425-432 Morphing, 394

Morphological processing, 436-442 Moving average filter. See under Filters,

digital MPEG, 501-502

MTF. See Modulation transfer function Multiplexing, telephone, 4 Multiplication

amplitude modulation. See Amplitude modulation

frequency domain signals, 180-181, 312-316 image formation model, 378, 407-410

of time domain signals, 97 Multiprocessing, 529 Multirate techniques

compact disc DAC, 361 data conversion, 58-66 decimation, 60, 202-203

interpolation, 60, 202-203, 361 single bit ADC and DAC, 60-66

Mu law companding, 362-364 Music. See Audio processing

Natural frequency, 149, 164, 253 Nearest neighbor rounding, 396 Negative frequencies. See under Discrete

Fourier transform

Neural networks, 368, 451-480 Night vision systems, 392, 424, 436 Nodes, neural networks, 459-462 Nonlinear phase. See under Phase Noise

1/f noise, 172-173

ADC. See Quantization error

data compression, lossy, 481-482, 494-502 deconvolution, how noise limits, 304 digital generation, 29-32

image noise, 434-436

in math calculation. See Round-off error linearity of added, 97

Poisson noise, 434-436

speech, wideband noise reduction, 368-370 statistical noise, 18-19

step response sharpness vs. noise, 278-279 white noise, 172-173, 307

Normal distribution. See Gaussian NTSC television, 386

Nyquist rate (frequency), 41-42

Nuclear magnetic resonance imaging (NMR), 450

Octave, 357-358

Odd field in video, 384-385

Odd symmetry, 102, 196, 240-241 Off-line processing, 506

Offset binary, 68-69

Oil & mineral exploration. See Seismology Opening operation, 437

Optimal filters, 307-310, 465 Orthognal basis functions, 158-159 Output look-up table (image display),

387-389

Output transform (image display), 389-391 Overshoot. See under Filters, see Gibbs effect

Packbits, data compression, 483 PAL television, 386

Parallel processing, 529

Parallel stages with added outputs, 134, 616-619

Parameter space, 457

Parentheses, used to denote continuous signals, 87

Parseval's relation, 208 Passband. See under Filters

Passband ripple. See under Filters Periodic nature of the DFT. See under

Discrete Fourier transform

Phase

See also Polar notation

carries edge information, 191-192, 222-224 hearing, insensitivity to, 355-356

linear phase, 131-132, 188-191, 328-332 nonlinear phase, 131-132, 328-332 nuisances and ambiguities, 164-168 time domain shifting, effect on, 188-191 unwrapping, 167, 188-189

zero phase, 131-132, 188-191, 328-332 Phase lock loops, linearity of, 94

Phasor transform, 515

Piano keyboard frequencies, 353, 357-358 Pillbox, 400-401, 423-424, 427-428 Pipeline, 84

Pitch, 355-358

Pixel (picture element), 373

Polar notation, 161-168. See also Phase Polar-to-rectangular conversion, 162 Poles and zeros, 49-50, 334, 590-604 Point-by-point image acquisition, 387

Point spread function. See Impulse response Pointer, 507

Poisson statistics, 434-436

Positron emission tomography (PET), 449-450

PostScript image, data compression, 488 Precision, 32-34, 68, 72-76

Probability, 17-19

Probability distribution function (pdf), 19-24, 452-455

Probability mass function (pmf), 18, 19-24

Quantization error, 36-39 Quantization levels in images, 374 Quantization table, JPEG, 499-500 Quantum sink, 436

Radar, 1, 7, 88, 137, 222-224 Radians. See Natural frequency Range, 12

Random errors, 32-34

Random number generator, 29-32, 465 Real FFT, 239-242

Real time processing, 311, 506

Receiver operating characteristic (ROC), 454-455, 474-475

Reconstruction algorithms, CT, 444-450 Reconstruction filter. See under Filters,

analog

Rectangular-to-polar conversion, 162 Recursion equation

See also Filters, recursive first difference, 125-128 running sum, 125-128

moving average filter, 282-284 Reed-Solomon coding, 361 Region-of-convergence, 592 Resolution

frequency domain, 173-176 limiting resolution, images, 426 spatial domain, 423-432

Retina of the eye, 378-381

Ringing, step response. See under Filters, see Gibbs effect

RISC, 84

RLC circuits. See Electric circuit analysis ROC curve. See Receiver operating

characteristic

Rods and cones in the eye, 379-381 Roll-off. See under Filters, digital Root-mean-square (rms), 14 Round-off error, 73-76, 238, 284, 318,

332

Row major order, 384

Run-length encoding, data compression, 483-484, 500

Running sum, 126-128, 263 Reverberation in music, 5

s-domain, 581-587 s-plane, 581-587 Saccades of the eye, 381 Sampling, 35-44

Sampling aperture in images, 376, 423, 430-432

Sample spacing in images, 375-376, 423, 430-432

Sampling theorem, 39-45, 430-432 Scanning probe microscope, 387 SECAM television, 386

Separable image, 404-406 Seismology, 1, 7, 8, 451 Sharpening, image, 403-404 Shift and subtract, 402-403 Shift invariance, 89-92, 108 Sidebands in ADC, 44-45

Sidebands in AM. See Amplitude modulation Sigmoid function, 461-463, 471-472

Sign and magnitude, 68-69 Sign bit, 68-69

Signed fraction, 514 Signal (defined), 11, 87

Signal-to-noise ratio, 17, 432-436 Sinc function,

aliasing of, 212-214

DAC reconstruction, 46-47

Fourier transform of rectangular pulse, 212-215, 285-287

Two-dimensional (image), 400-401 Windowed-sinc filter, 283-285

Single precision, 70-76 Sinusoidal fidelity. See Linearity SIRT, 442

Skeletonization of binary images, 438-442 Smoothing filter. See under Filters

Space exploration. See Astrophotography Spectrogram of speech, 365-367

Speed

convolution vs. FFT convolution, 318 FIR vs. IIR filters, 346-350 hardware, 80-84

image convolution, 421-422 program language, 76-80 programming style, 84-86

Spatial domain (defined), 12, 373 Spatial resolution. See under Resolution Speak & spell, 6, 366

Spectral analysis, 169-176

Spectral inversion. See under Filters Spectral response of the eye, 380-381 Spectral reversal. See under Filters, digital Speech

digital recording, 6 generation, 6, 357, 364-368 recognition, 6-7, 364-368

speak & spell, 6

vocal tract simulation, 6

Sonar, 1, 7-8, 32-34, 88, 171-173, 222-224 Square PSF, 400-401

Stability, 339, 541, 600-601, 609 Standard deviation, 13-17, 20-22, 434-435 Stationary, 19

Static linearity, 92-93 Statistical variation. See Noise

Steepest decent, neural network learning, 470-473

Step decomposition. See under Decomposition

Step response. See under Filters Stereo audio, 362

Stopband (defined), 268 Stretch, grayscale, 390-391 Strong law of large numbers, 18 Subpixel interpolation, 395-396

Substitution, using complex numbers by, 557-559

Superposition. See under Linearity Symmetry, left-right. See zero phase under

Phase

Synthesis. See under Linearity; Fourier transform, Discrete Fourier transform

System (defined), 87 Systematic errors, 32-34

t-carrier system, 4

Target detection, 451-458 Tchebysheff. See Chebyshev Telecommunications, 4-5 Television, 206, 374, 384-386, 501

Text recognition, neural network, 465-476 TIFF image, data compression, 488 Timbre, 355-358

Time domain (defined), 12, 146-147 Time domain encoding. See under

Information Threshold, 453-458

Transfer function, 594, 611-612 Transform (defined), 146

Transform data compression, 494-500 Triangular pulse, 216-217, 281 Trigonometric functions, 85-86, 165-166 True-positive (true-negative), 453-454 Tschebyscheff. See Chebyshev

Two's complement, 69-70

µ255 law companding, 362-364 Unit circle, z-plane, 608-609 Unsigned integer, 68-69

Variance, 14

Von Neumann architecture, 509 Voiced sound in speech, 364 Voiceprint of speech, 365-367

Warping images, 394-396 Well, in CCDs, 382-384

Wiener filter. See under Filters, digital Windows

Bartlett window, 288

Blackman, 175-176, 282-283, 288 Hamming, 170-171, 174-176, 282-283,

288 Hanning, 288

raised cosine, 288

rectangular, 175-176, 212-215, 288

in spectral analysis, 169-176, 173-176

X-ray imaging

See also computed tomography airport baggage scanner, 402-403 detection by phosphor layer, 423-424 DSP improvements to, 9

measuring MTF, 425 image noise of, 434-436

z-domain, 605-610 z-plane, 605-610

z-transform, 334-335, 605-610 Zero phase. See under Phase Zeros. See Poles and zeros Zeroth-order-hold, DAC, 46-47

1/f noise: A type of random noise that increases in amplitude at lower frequencies. It is widely observable in physical systems, but not well understood. See white noise for comparison.

-3dB cutoff frequency: The division between a filter's passband and transition band. Defined as the frequency where the frequency response is reduced to -3dB (0.707 in amplitude).

"A" law: Companding standard used in Europe. Allows digital voice signals to be represented with only 8 bits instead of 12 bits by making the quantization levels unequal. See mu law for comparison.

AC: Alternating Current. Electrical term for the portion of a signal that fluctuates around the average (DC) value.

Accuracy: The error in a measurement (or a prediction) that is repeatable from trial to trial. Accuracy is limited by systematic (repeatable) errors. See precision for comparison.

Additivity: A mathematical property that is necessary for linear systems. If input a produces output p, and if input b produces output q, then an input of a+b produces an output of p+q.

Aliasing: The process where a sinusoid changes from one frequency to another as a result of sampling or other nonlinear action. Usually results in a loss of the signal's information.

Amplitude modulation: Method used in radio communication for combining an information carrying signal (such as audio) with a carrier wave. Usually carried out by multiplying the two signals.

Analysis: The forward Fourier transform; calculating the frequency domain from the time domain. See synthesis for comparison.

Antialias filter: Low-pass analog filter placed before an analog-to-digital converter. Removes

frequencies above one-half the sampling rate that would alias during conversion.

ASCII: A method of representing letters and numbers in binary form. Each character is assigned a number between 0 and 127. Very widely used in computers and communication.

Aspect ratio: The ratio of an image's width to its height. Standard television has an aspect ratio of 4:3, while motion pictures have an aspect ratio of 16:9.

Assembly: Low-level programming language that directly manipulates the registers and internal hardware of a microprocessor. See high-level language for comparison.

Associative property of convolution: Written as: ( a[n] tb[n] ) tc[n] ' a[n] t( b[n] tc[n]) . This is important in signal processing because it describes how cascaded stages behave.

Autocorrelation: A signal correlated with itself. Useful because the Fourier transform of the autocorrelation is the power spectrum of the original signal.

Backprojection: A technique used in computed tomography for reconstructing an image from its views. Results in poor image quality unless used with a more advanced method.

BASIC: A high-level programming language known for its simplicity, but also for its many weaknesses. Most of the programs in this book are in BASIC.

Basilar membrane: Small organ in the ear that acts as a spectrum analyzer. It allows different fibers in the cochlear nerve to be stimulated by different frequencies.

Basis functions: The set of waveforms that a decomposition uses. For instance, the basis functions for the Fourier decomposition are unity amplitude sine and cosine waves.

Bessel filter: Analog filter optimized for linear phase. It has almost no overshoot in the step response and similar rising and falling edges. Used to smooth time domain encoded signals.

Bidirectional filtering: Recursive method used to produce a zero phase filter. The signal is first filtered from left-to-right, then the intermediate signal is filtered from right-to-left.

Bilinear transform: Technique used to map the s-plane into the z-plane. Allows analog filters to be converted into equivalent digital filters.

Binning: Method of forming a histogram when the data (or signal) has numerous quantization levels, such as in floating point numbers.

Biquad: An analog or digital system with two poles and up to two zeros. Often cascaded to create a more sophisticated filter design.

Bit reversal sorting: Algorithm used in the FFT to achieve an interlaced decomposition of the signal. Carried out by counting in binary with the bits flipped left-for-right.

Blackman window: A smooth curve used in the design of filters and spectral analysis, calculated f r o m : 0.42 & 0.5 cos(2Bn /M ) % 0.08 cos(4Bn /M ) , where n runs from 0 to M.

Brightness: The overall lightness or darkness of an image. See contrast for comparison.

Butterfly: The basic computation used in the FFT. Changes two complex numbers into two other complex numbers.

Butterworth filter: Separates one band of frequencies from another; fastest roll-off while keeping the passband flat; can be analog or digital. Also called a maximally flat filter.

C: Common programming language used in science, engineering and DSP. Also comes in the more advanced C++.

C a r r i e r w a v e : T e r m u s e d i n a m p l i t u d e modulation of radio signals. Refers to the high frequency sine wave that is combined with a lower frequency information carrying signal.

Cascade: A combination of two or more stages where the output of one stage becomes the input for the next.

Causal signal: Any signal that has a value of zero for all negative numbered samples.

Causal system: A system that has a zero output until a nonzero value has appeared on its input (i.e., the input causes the output). The impulse response of a causal system is a causal signal.

Central Limit Theorem: Important theorem in statistics. In one form: a sum of many random numbers will have a Gaussian pdf, regardless of the pdf of the individual random numbers.

Cepstrum: A rearrangement of "spectrum." Used in homomorphic processing to describe the spectrum when the time and frequency domains are switched.

Charge coupled device (CCD): The light sensor in electronic cameras. Formed from a thin sheet of silicon containing a two-dimensional array of light sensitive regions called wells.

Chebyshev filter: Used for separating one band of frequencies from another. Achieves a faster roll-off than the Butterworth by allowing ripple in the passband. Can be analog or digital.

Chirp system: Used in radar and sonar. An impulse is converted into a longer duration signal before transmission, and compressed back into an impulse after reception.

Circular buffer: Method of data storage used in real time processing; each newly acquired sample replaces the oldest sample in memory.

Circular convolution: Aliasing that can occur in the time domain when frequency domain signals are multiplied. Each period in the time domain overflows into adjacent periods.

Circularity: The appearance that the end of a signal is connected to its beginning. This arises when considering only a single period of a periodic signal.

Classifiers: A parameter extracted from and representing a larger data set. For example: size of a region, amplitude of a peak, sharpness of an edge, etc. Used in pattern recognition.

Closing: A morphological operation defined as an erosion operation followed by a dilation operation.

Cochlea: Organ in the ear where sound in converted into a neural signal.

Cochlear nerve: Nerve that transmits audio information from the ear to the brain.

Coefficient-of-variation (CV): Common way of

stating the variation (noise) in data. Defined as: 100% × standard deviation / mean.

Commutative property of convolution: Written as: a[n] tb[n] ' b[n] ta[n] .

Companding: An "s" shaped nonlinearity allows voice signals to be digitized using only 8 bits instead of 12 bits. Europe uses "A" law, while the United States uses the mu law version.

Complex conjugation: Changing the sign of the imaginary part of a complex number. Often denoted by a star placed next to the variable. Example: if A ' 3 % 2 j , then A ( ' 3 & 2 j .

Complex DFT: The discrete Fourier transform using complex numbers. A more complicated and powerful technique than the real DFT.

Complex exponential: A complex number of the form: e a % bj . They are useful in engineering and science because Euler's relation allows them to represent sinusoids.

Complex Fourier transform: Any of the four members of the Fourier transform family written using complex numbers . See real Fourier transform for comparison.

Complex numbers: The real numbers (used in everyday math) plus the imaginary numbers (numbers containing the term j, where j ' & 1 ). Example: 3 % 2 j .

Complex plane: A graphical interpretation of complex numbers, with the real part on the x-axis and the imaginary part on the y-axis. This is analogous to the number line used with ordinary numbers.

Composite video: An analog television signal that contains synchronization pulses to separate the fields or frames.

Computed tomography (CT): A method used to reconstruct an image of the interior of an object from its x-ray projections. Widely used in medicine; one of the earliest applications of DSP. Old name: CAT scanner.

Continuous signal: A signal formed from continuous (as opposed to discrete) variables. Example: a voltage that varies with time. Often used interchangeably with analog signal.

Contrast: The difference between the bright-ness of an object and the brightness of the background. See brightness for comparison.

Converge: Term used in iterative methods to indicate that progress is being made toward a solution ("The algorithm is converging") or that a solution has been reached ("The algorithm has converged").

Convolution integral: Mathematical equation that defines convolution in continuous systems; analogous to the convolution sum for discrete systems.

Convolution kernel: The impulse response of a filter implemented by convolution. Also known as the filter kernel and the kernel.

Convolution sum: Mathematical equation defining convolution for discrete systems.

Cooley and Tukey: J.W. Cooley and J.W. Tukey, given credit for bringing the FFT to the world in a paper they published in 1965.

Correlation: Mathematical operation carried out the same as convolution, except a left-for-right flip of one signal. This is an optimal way to detect a known waveform in a signal.

Cross-correlation: The signal formed when one signal is correlated with another signal. Peaks in this signal indicate a similarity between the original signals. See also autocorrelation.

Cutoff frequency: In analog and digital filters, the frequency separating the passband from the transition band. Often measured where the amplitude is reduced to 0.707 (-3dB).

CVSD: Continuously Variable Slope Delta modulation, a technique used to convert a voice signal into a continuous binary stream.

DC: Direct Current. Electrical term for the portion of the signal that does not change with time; the average value or mean. See AC for comparison.

Decibel SPL: Sound Pressure Level. Log scale used to express the intensity of a sound wave: 0 dB SPL is barely detectable; 60 dB SPL is normal speech, and 140 dB SPL causes ear damage.

Decimation: Reducing the sampling rate of a digitized signal. Generally involves low-pass filtering followed by discarding samples. See interpolation for comparison.

Decomposition: The process of breaking a signal into two or more additive components. Often refers specifically to the forward Fourier transform,

breaking a signal into sinusoids.

Deconvolution: The inverse operation of convolution: if x[n] th[n] ' y[n] , find x[n] given only h[n] and y[n] . Deconvolution is usually carried out by dividing the frequency spectra.

Delta encoding: A broad term referring to techniques that store data as the difference between adjacent samples. Used in ADC, data compression and many other applications.

Delta function: A normalized impulse. The discrete delta function is a signal composed of all zeros, except the sample at zero that has a value of one. The continuous delta function is similar, but more abstract.

Delta-sigma: Analog-to-digital conversion method popular in voice and music processing. Uses a very high sampling rate with only a single bit per sample, followed by decimation.

Dependent variable: In a signal, the dependent variable depends on the value of the indepen-dent variable. Example: when a voltage changes over time, time is the independent variable and voltage is the dependent variable.

Difference equation: Equation relating the past and present samples of the output signal with past and present samples of the input signal. Also called a recursion equation.

Dilation: A morphological operation. When applied to binary images, dilation makes the objects larger and can combine disconnected objects into a single object.

Discrete cosine transform (DCT): A relative of the Fourier transform. Decomposes a signal into cosine waves. Used in data compression.

Discrete derivative: An operation for discrete signals that is analogous to the derivative for continuous signals. A better name is the first difference.

Discrete Fourier transform (DFT): Member of the Fourier transform family dealing with time domain signals that are discrete and periodic.

Discrete integral: Operation on discrete signals that is analogous to the integral for continuous signals. A better name is the running sum.

Discrete signal: A signal that uses quantized variables, such as a digitized signal residing in a computer.

Discrete time Fourier transform (DTFT):

Member of the Fourier transform family dealing with time domain signals that are discrete and aperiodic

Dithering: Adding noise to an analog signal before analog-to-digital conversion to prevent the digitized signal from becoming "stuck" on one value.

Domain: The independent variable of a signal. For example, a voltage that varies with time is in the time domain. Other common domains are the s p a t i a l d o m a i n ( s u c h a s i m a g e s ) a n d t h e frequency domain (the output of the Fourier transform).

Double precision: A standard for floating point notation that used 64 bits to represent each number. See single precision for comparison.

DSP microprocessor: A type of microprocessor designed for rapid math calculations. Often has a pipeline and/or Harvard architecture. Also called a RISC.

Dynamic range: The largest amplitude a system can deal with divided by the inherent noise of the system. Also used to indicate the number of bits u s e d i n a n A D C . C a n a l s o b e u s e d w i t h parameters other than amplitude; see frequency dynamic range.

Edge enhancement: Any image processing algorithm that makes the edges more obvious. Also called a sharpening operation.

Edge response: In image processing, the output of a system when the input is an edge. The sharpness of the edge response is often used as a measure of the resolution of the system.

Elliptic filter: Used to separate one band of frequencies from another. Achieves a fast roll-off by allowing ripple in the passband and the stopband. Can be used in both analog and digital designs.

End effects: The poorly behaved ends of a filtered signal resulting from the filter kernel not being completely immersed in the input signal.

Erosion: A morphological operation. When applied to binary images, erosion makes the objects smaller and can break objects into two or more pieces.

Euler's relation: The most important equation in complex math, relating sine and cosine waves with

complex exponentials.

Even/odd decomposition: A way of breaking a signal into two other signals, one having even symmetry, and the other having odd symmetry.

Even order filter: An analog or digital filter having an even number of poles.

False-negative: One of four possible outcomes of a target detection trial. The target is present, but incorrectly indicated to be not present.

False-positive: One of four possible outcomes of a target detection trial. The target is not present, but incorrectly indicated to be present.

Fast Fourier transform (FFT): An efficient algorithm for calculating the discrete Fourier transform (DFT). Reduces the execution time by hundreds in some cases.

FFT convolution: A method of convolving signals by multiplying their frequency spectra. So named because the FFT is used to efficiently move between the time and frequency domains.

Field: Interlaced television displays the even lines of each frame (image) followed by the odd lines. The even lines are called the even field, and the odd lines the odd field.

Filter kernel: The impulse response of a filter implemented by convolution. Also known as the convolution kernel and the kernel.

Filtered backprojection: A technique used in computed tomography for reconstructing an image from its views. The views are filtered and then backprojected.

Finite impulse response (FIR): An impulse response that has a finite number of nonzero values. Often used to indicate that a filter is carried out by using convolution, rather than recursion.

First difference: An operation for discrete signals that mimics the first derivative for continuous signals; also called the discrete derivative.

Fixed point: One of two common ways that computers store numbers; usually used to store integers. See floating point for comparison.

Flat-top window: A window used in spectral analysis; provides an accurate measurement of the amplitudes of the spectral components. The windowed-sinc filter kernel can be used.

Floating point: One of the two common ways that computers store numbers. Floating point uses a form of scientific notation, where a mantissa is raised to an exponent. See fixed point for comparison.

Forward transform: The analysis equation of the Fourier transform, calculating the frequency domain from the time domain. See inverse transform for comparison.

Fourier reconstruction: One of the methods used in computed tomography to calculate an image from its views.

Fourier series: The member of the Fourier transform family that deals with time domain signals that are continuous and periodic.

Fourier transform: A family of mathematical techniques based on decomposing signals into sinusoids. In the complex version, signals are decomposed into complex exponentials.

Fourier transform pair: Waveforms in the time and frequency domains that correspond to each other. For example, the rectangular pulse and the sinc function.

Fovea: A small region in the retina of the eye that is optimized for high-resolution vision.

Frame: An individual image in a television signal. The NTSC television standard uses 30 frames per second.

Frame grabber: A analog-to-digital converter used to digitize and store a frame (image) from a television signal.

Frequency domain: A signal having frequency as the independent variable. The output of the Fourier transform.

Frequency domain aliasing: Aliasing that occurs occurring in the frequency domain in response to an action taken in the time domain. Aliasing during sampling is an example.

Frequency domain convolution: Convolution carried out by multiplying the frequency spectra of the signals.

Frequency domain encoding: One of two main ways that information can be encoded in a signal. The information is contained in the amplitude, frequency, and phase of the signal's component sinusoids. Audio signals are the best example. See time domain encoding for comparison.

Frequency domain multiplexing: A method of combining signals for simultaneous transmis-sion by shifting them to different parts of the frequency spectrum.

Frequency dynamic range: The ratio of the largest to the lowest frequency a system can deal with. Analog systems usually have a much larger frequency dynamic range than digital systems.

Frequency resolution: The ability to distinguish or separate closely spaced frequencies.

Frequency response: The magnitude and phase changes that sinusoids experience when passing through a linear system. Usually expressed as a function of frequency. Often found by taking the Fourier transform of the impulse response.

Fricative: Human speech sound that originates as random noise from air turbulence, such as: s, f, sh, z, v and th. See voiced for comparison.

Full-width-at-half-maximum (FWHM): A common way of measuring the width of a peak in a signal. The width of the peak is measured at one-half of the peak's maximum amplitude.

Fundamental frequency: The frequency that a periodic waveform repeats itself. See harmonic for comparison.

Gamma curve: The mathematical function or look-up table relating a stored pixel value and the brightness it appears in a displayed image. Also called a grayscale transform.

Gaussian: A bell shaped curve of the general form: e x 2 . The Gaussian has many unique properties. Also called the normal distribution.

Gibbs effect: When a signal is truncated in one domain, ringing and overshoot appear at edges and corners in the other domain.

GIF: A common image file format using LZW (lossless) compression. Widely used on the world wide web for graphics. See TIFF and JPEG for comparison.

Grayscale: image A digital image where each pixel is displayed in shades of gray between black and white; also called a black and white image.

Grayscale stretch: Greatly increasing the contrast of a digital image to allow the detailed examination of a small range of quantization levels. Quantization levels outside of this range are displayed as saturated black or white.

Grayscale transform: The conversion function between a stored pixel value and the brightness that appears in a displayed image. Also called a gamma curve.

Halftone: A common method of printing images on paper. Shades of gray are created by various patterns of small black dots. Color halftones use dots of red, green and blue.

Hamming window: A smooth curve used in the design of filters and spectral analysis, calculated from: 0.54 & 0.46 cos(2Bn /M ) , where n runs from 0 to M.

Harmonics: The frequency components of a periodic signal, always consisting of integer multiples of the fundamental frequency. The fundamental is the first harmonic, twice this frequency is the second harmonic, etc.

Harvard Architecture: Internal computer layout where the program and data reside in separate memories accessed through separate busses; common in microprocessors used for DSP. See

Von Neumann Architecture for comparison.

High fidelity: High quality music reproduction, such as provided by CD players.

High-level language: Programming languages such as C, BASIC and FORTRAN.

High-speed convolution: Another name for FFT convolution.

Hilbert transformer: A system having the frequency response: Mag = 1, Phase = 90E, for all frequencies. Used in communications systems for modulation. Can be analog or digital.

Histogram equalization: Processing an image by using the integrated histogram of the image as the grayscale transform. Works by giving large areas of the image higher contrast than the small areas.

Histogram: Displays the distribution of values in a signal. The x-axis show the possible values the samples can take on; the y-axis indicates the number of samples having each value.

Homogeneity: A mathematical property of all linear systems. If an input x[n] produces an output of y[n] , then an input k x[n] produces an output of ky[n] , for any constant k.

Homomorphic: DSP technique for separating signals combined in a nonlinear way, such as by multiplication or convolution. The nonlinear