*Study Guide*

631

**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.

**A n a l y s i s :**

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*

] t

*b*

[

*n*

] ) t

*c*

[

*n*

]

'

*a*

[

*n*

] t

(

*b*

[

*n*

] t

*c*

[

*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.

632

*The Scientist and Engineer's Guide to Digital Signal Processing*

**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(2

B

*n*

/

*M*

)

%

0.08 cos(4

B

*n*

/

*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*

] t

*b*

[

*n*

]

'

*b*

[

*n*

] t

*a*

[

*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.

*Glossary*

633

**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:**

M a t h e m a t i c a l e q u a t i o n 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*

,

634

*The Scientist and Engineer's Guide to Digital Signal Processing*

breaking a signal into sinusoids.

**Deconvolution:**

The inverse operation of convolution: if

*x*

[

*n*

] t

*h*

[

*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

*Glossary*

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.

635

**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.

636

**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

2 form:

*e x*

. 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.

*The Scientist and Engineer's Guide to Digital Signal Processing*

**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(2

B

*n*

/

*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 = 90

E

, 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

problem is converted to a linear one by an appropriate transform.

**Huffman encoding:**

Data compression method that assigns frequently encountered characters fewer bits than seldom used characters.

**Hyperspace:**

Term used in target detection and neural network analysis. One parameter can be graphically interpreted as a

*line*

, two parameters a

*plane*

, three parameters a

*space*

, and more than three parameters a

*hyperspace*

.

**Imaginary part:**

The portion of a complex number that has a

*j*

term, such as 2 in 3

%

2

*j*

. In the real Fourier transform, the

*imaginary part*

also refers to the portion of the frequency domain that holds the amplitudes of the sine waves, even though

*j*

terms are not used.

**Impulse:**

A signal composed of all zeros except for a very brief pulse. For discrete signals, the pulse consists of a single nonzero sample. For continuous signals, the width of the pulse must be much shorter than the inherent response of any system the signal is used with.

**Impulse decomposition:**

Breaking an

*N*

point signal into

*N*

signals, each containing a single sample from the original signal, with all the other samples being zero. This is the basis of convolution.

**Impulse response:**

The output of a system when the input is a normalized impulse (a delta function).

**Impulse train:**

A signal consisting of a series of equally spaced impulses.

**Independent variable:**

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

**Infinite impulse response (IIR):**

An impulse response that has an infinite number of nonzero values, such as a decaying exponential. Often used to indicate that a filter is carried out by using recursion, rather than convolution.

**Integers: **

Whole numbers:

þ

&

2,

&

1, 0, 1, 2,

þ

Also refers to numbers stored in fixed point

.

notation. See

*floating point*

for comparison.

**Interlaced decomposition:**

Breaking a signal into its even numbered and odd numbered samples.

Used in the FFT.

*Glossary*

637

**Interlaced video:**

A video signal that displays the even lines of each image followed by the odd lines. Used in television; developed to reduce flicker.

**Interpolation:**

Increasing the sampling rate of a digitized signal. Generally done by placing zeros between the original samples and using a low-pass filter. See

*decimation*

for comparison.

**Inverse transform:**

The synthesis equation of the

Fourier transform, calculating the time domain from the frequency domain. See

*f o r w a r d transform*

for comparison.

**Iterative:**

Method of finding a solution by gradually adjusting the variables in the right direction until convergence is achieved. Used in

CT reconstruction and neural networks.

**JPEG: **

A common image file format using transform (lossy) compression. Widely used on the world wide web for graphics. See

*GIF*

and

*TIFF*

for comparison.

**Kernel:**

The impulse response of a filter implemented by convolution. Also known as the

*convolution kernel*

and the

*filter kernel*

.

**Laplace transform:**

Mathematical method of analyzing systems controlled by differential equations. A main tool in the design of electric circuits, such as analog filters. Changes a signal in the time domain into the s-domain

**Learning algorithm:**

The procedure used to find a set of neural network weights based on examples of how the network should operate.

**Line pair:**

Imaging term for

*cycle*

. For example,

5 cycles per mm is the same as 5 line pairs per mm.

**Line pair gauge:**

A device used to measure the resolution of an imaging system. Contains a series of light and dark lines that move closer together at one end.

**Line spread function (LSF):**

The response of an imaging system to a thin line in the input image.

**Linear phase:**

A system with a phase that is a straight line. Usually important because it means the impulse response has left-to-right symmetry, making rising edges in the output signal look the same as falling edges. See also

*zero phase*

.

**Linear system:**

By definition, a system that has the properties of additivity and homogeneity.

638

**Lossless compression:**

Data compression technique that exactly reconstructs the original data, such as LZW compression.

**Lossy compression:**

Data compression methods that only reconstruct an approximation to the original data. This allows higher compression ratios to be achieved. JPEG is an example.

**Matched filtering:**

Method used to determine where, or if, a know pattern occurs in a signal.

Matched filtering is based on correlation, but implemented by convolution.

**Mathematical equivalence:**

A way of using complex numbers to represent real problems.

Based on Euler's relation equating sinusoids with complex exponentials. See

*substitution*

for comparison.

**Mean:**

The average value of a signal or other group of data.

**Memoryless:**

Systems where the current value of the output depends only on the current value of the input, and not past values.

**MFLOPS:**

Million-Floating-Point-Operations-

Per-Second; a common way of expressing computer speed. See

*MIPS*

for comparison.

**M I P S :**

Million-Instructions-Per-Second; a common way of expressing computer speed. See

*MFLOPS*

for comparison.

**Mixed signal:**

Integrated circuits that contain both analog and digital electronics, such as an

ADC placed on a Digital Signal Processor.

**Modulation transfer function (MTF):**

Imaging jargon for the

*frequency response*

.

**Morphing:**

Gradually warping an image from one form to another. Used for special effects, such as a man turning into a werewolf.

**Morphological:**

Usually refers to simple nonlinear operations performed on binary images, such as erosion and dilation.

**Moving average filter:**

Each sample in the output signal is the average of many adjacent samples in the input signal. Can be carried out by convolution or recursion.

**MPEG:**

Compression standard for video, such as digital television.

**Mu law:**

Companding standard used in the

*The Scientist and Engineer's Guide to Digital Signal Processing*

United States. Allows digital voice signals to be represented with only 8 bits instead of 12 bits by making the quantization levels unequal. See "

*A*

"

*law*

for comparison.

**Multiplexing:**

Combining two or move signals together for transmission. This can be carried out in many different ways.

**Multirate: **

Systems that use more than one sampling rate. Often used in ADC and DAC to obtain better performance, while using less electronics.

**Natural frequency:**

A frequency expressed in radians per second, as compared to cycles per second (hertz). To convert frequency (in hertz) to natural frequency, multiply by 2

B

.

**Negative frequencies: **

Sinusoids can be written as a positive frequency: frequency: cos(

&

T

*t*

) cos(

T

*t*

) , or a negative

. Negative frequencies are included in the complex Fourier transform, making it more powerful.

**Normal distribution:**

A bell shaped curve of the form:

*e x*

2

. Also called a

*Gaussian*

.

**NTSC:**

Television standard used in the United

States, Japan, and other countries. See

*PAL*

and

*SECAM*

for comparison.

**Nyquist frequency, Nyquist rate:**

These terms refer to the sampling theorem, but are used in different ways by different authors. They can be used to mean four different things: the highest frequency contained in a signal, twice this frequency, the sampling rate, or one-half the sampling rate.

**Octave:**

A factor of two in frequency.

**Odd order filter:**

An analog or digital filter having an odd number of poles.

**Opening: **

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

**Optimal filter:**

A filter that is "best" in some specific way. For example, Wiener filters produce an optimal signal-to-noise ratio and matched filters are optimal for target detection.

**Overlap add:**

Method used to break long signals into segments for processing.

**PAL:**

Television standard used in Europe. See

*NTSC*

for comparison.

**Parallel stages:**

A combination of two or more stages with the same input and added outputs.

**Parameter space:**

Target detection jargon. One parameter can be graphically interpreted as a

*line*

, two parameters a

*plane*

, three parameters a

*s p a c e*

, and more than three parameters a

*hyperspace*

.

**Parseval's relation:**

Equation relating the energy in the time domain to the energy in the frequency domain.

**Passband:**

The band of frequencies a filter is designed to pass unaltered.

**Passive sonar:**

Detection of submarines and other undersea objects by the sounds they produce.

Used for covert surveillance.

**Phasor transform:**

Method of using complex numbers to find the frequency response of

*RLC*

circuits. Resistors, capacitors and inductors become

*R*

,

&

*j*

/

T

*C*

, and

*j*

T

*L*

, respectively.

**Pillbox:**

Shape of a filter kernel used in image processing: circular region of a constant value surrounded by zeros.

**Pitch:**

Human perception of the fundamental frequency of an continuous tone. See

*timbre*

for comparison.

**Pixel:**

A contraction of "picture element." An individual sample in a digital image.

**Point spread function (PSF):**

Imaging jargon for the impulse response.

**Pointer:**

A variable whose value is the address of another variable.

**Poisson statistics:**

Variations in a signal's value resulting from it being represented by a finite number of particles, such as: x-rays, light photons or electrons. Also called

*Poisson noise*

and

*statistical noise*

.

**Polar form:**

Representing sinusoids by their magnitude and phase:

*M*

cos(

T

*t*

%

N

) , where

*M*

is t h e m a g n i t u d e a n d

N

i s t h e p h a s e . S e e

*rectangular form*

for comparison.

**Pole:**

Term used in the Laplace transform and ztransform. When the s-domain or z-domain transfer function is written as one polynomial divided by another polynomial, the roots of the denominator are the

*poles*

of the system, while the roots of the numerator are the

*zeros*

.

*Glossary*

639

**Pole-zero diagram:**

Term used in the Laplace and z-transforms. A graphical display of the location of the poles and zeros in the s-plane or zplane.

**Precision: **

The error in a measurement or prediction that is not repeatable from trial to trial.

Precision is determined by random errors. See

*accuracy*

for comparison.

**Probability distribution function (pdf):**

Gives the probability that a

*continuous*

variable will take on a certain value.

**Probability mass function (pmf):**

Gives the probability that a

*discrete*

variable will take on a certain value. See

*pdf*

for comparison.

**Pulse response:**

The output of a system when the input is a pulse.

**Quantization error:**

The error introduced when a signal is quantized. In most cases, this results in a maximum error of ±½ LSB, and an rms error of 1/ 12 LSB. Also called

*quantization noise*

.

**Random error:**

Errors in a measurement or prediction that are not repeatable from trial to trial. Determines

*precision*

. See

*systematic error*

for comparison.

**Radar:**

Radio Detection And Ranging. Echo location technique using radio waves to detect aircraft.

**Real DFT:**

The discrete Fourier transform using only real (ordinary) numbers. A less powerful technique than the complex DFT, but simpler. See

*complex DFT*

for comparison.

**Real FFT:**

A modified version of the FFT. About

30% faster than the standard FFT when the time domain is completely real (i.e., the imaginary part of the time domain is zero).

**Real Fourier transform:**

Any of the members of the Fourier transform family using only real (as opposed to imaginary or complex) numbers. See

*complex Fourier transform*

for comparison.

**Real part:**

The portion of a complex number that does not have the

*j*

term, such as 3 in 3

%

2

*j*

. In the real Fourier transform, the

*real*

part refers to the part of the frequency domain that holds the amplitudes of the cosine waves, even though no

*j*

terms are present.

**Real time processing:**

Processing data as it is acquired, rather than storing it for later use.

640

Example: DSP algorithms for controlling echoes in long distance telephone calls.

**Reconstruction filter:**

A low-pass analog filter p l a c e d a f t e r a d i g i t a l - t o - a n a l o g c o n v e r t e r .

Smoothes the stepped waveform by removing frequencies above one-half the sampling rate.

**Rectangular form:**

Representing a sinusoid by the form:

*A*

cos(

T

*t*

)

%

*B*

sin (

T

*t*

)

, where

*A*

is called the

*real part*

and

*B*

is called the

*imaginary part*

(even though these are not imaginary numbers).

**Rectangular window:**

A signal with a group of adjacent points having unity value, and zero elsewhere. Usually multiplied by another signal to select a section of the signal to be processed.

**Recursion coefficients:**

The weighing values used in a recursion equation. The recursion coefficients determine the characteristics of a recursive (IIR) filter.

**Recursion equation:**

Equation relating the past and present samples of the output signal with the past and present values of the input signal. Also called a

*difference equation*

.

**Region-of-convergence:**

The term used in the

Laplace and z-transforms. Those regions in the splane and z-planes that have a defined value.

**RGB encoding:**

Representing a color image by specifying the amount of red, green, and blue for each pixel.

**RISC:**

Reduced Instruction Set Computer, also called a DSP microprocessor. A fewer number of programming commands allows much higher speed math calculations. The opposite is the

Complex Instruction Set Computer, such as the

Pentium.

**ROC curve:**

A graphical display showing how threshold selection affects the performance of a target detection problem.

**Roll-off:**

Jargon used to describe the sharpness of the transition between a filter's passband and stopband. A

*fast*

roll-off means the transition is sharp; a

*slow*

roll-off means it is gradual.

**Root-mean-square (rms):**

Used to express the fluctuation of a signal around

*zero*

. Often used in electronics. Defined as the square-root of the mean of the squares. See

*standard deviation*

for comparison.

**Round-off noise:**

The error caused by rounding

*The Scientist and Engineer's Guide to Digital Signal Processing*

the result of a math calculation to the nearest quantization level.

**Row major order:**

A pattern for converting an image to serial form. Operates the same as

English writing: left-to-right on the first line, leftto-right on the second line, etc.

**Run-length encoding: **

Simple data compression technique with many variations. Characters that are repeated many times in succession are replaced by codes indicating the character and the length of the run.

**Running sum:**

An operation used with discrete signals that mimics integration of continuous signals. Also called the

*discrete integral*

.

**s-domain:**

The domain defined by the Laplace transform. Also called the

*s-plane*

.

**Sample spacing:**

The spacing between samples when a continuous image is digitized. Defined as the center-to-center distance between pixels.

**Sampling aperture:**

The region in a continuous image that contributes to an individual pixel during digitization. Generally about the same size as the sample spacing.

**Sampling theorem:**

If a continuous signal composed of frequencies less than

*f*

is sampled at

2

*f*

, all of the information contained in the continuous signal will be present in the sampled signal. Frequently called the

*Shannon*

sampling theorem or the

*Nyquist*

sampling theorem.

**SECAM:**

Television standard used in Europe.

See NTSC for comparison.

**Seismology:**

Branch of geophysics dealing with the mechanical properties of the earth.

**Separable:**

An image that can be represented as the product of its vertical and horizontal profiles.

Used to improve the speed of image convolution.

**Sharpening:**

Image processing operation that makes edges more abrupt.

**Shift and subtract:**

Image processing operation that creates a 3D or embossed effect.

**Shift invariance:**

A property of many systems.

A shift in the input signal produces nothing more than a shift in the output signal. Means that the characteristics of the system do not c h a n g i n g w i t h t i m e ( o r o t h e r i n d e p e n d e n t variable).

*Glossary*

**Sigmoid:**

An "s" shaped curve used in neural networks.

**Signal:**

A description of how one parameter varies with another parameter. Example: a

*voltage*

that varies with

*time*

.

**Signal restoration:**

Returning a signal to its original form after it has been changed or degraded in some way. One of the main uses of

*filtering*

.

**Sinc function:**

Formally defined by the relation:

*sinc*

(

*a*

)

'

*sin*

(

B

*a*

) /

B

*a*

. The

B

terms are often hidden in other variables, making it in the general form:

*sin*

(

*x*

) /

*x*

. Important because it is the

Fourier transform of the rectangular pulse.

**Single precision:**

A floating point notation that used 32 bits to represent each number. See

*double precision*

for comparison.

**Single-pole digital filters:**

Simple recursive filters that mimic

*RC*

high-pass and low-pass filters in electronics.

**Sinusoidal fidelity:**

An important property of linear systems. A sinusoidal input can only produce a sinusoidal output; the amplitude and phase may change, but the frequency will remain the same.

**Sonar:**

Sound Navigation And Ranging. The use o f s o u n d t o d e t e c t s u b m a r i n e s a n d o t h e r underwater objects.

*Active sonar*

uses echo location, while

*passive sonar*

only listens.

**Source code:**

A computer program in the form written by the programmer; distinguished from

*executable code*

, a form that can be directly run on a computer.

**Spatial domain:**

A signal having distance (space) as the independent variable. Images are signals in the spatial domain.

**Spectral analysis:**

Understanding a signal by examining the amplitude, frequency, and phase of its component sinusoids. The primary tool of spectral analysis is the Fourier transform.

**Spectral inversion:**

Method of changing a filter kernel such that the corresponding frequency response is flipped top-for-bottom. This can change low-pass filters to high-pass, band-pass to band-reject, etc.

**Spectral leakage:**

Term used in spectral analysis.

Since the DFT can only be taken of a finite length

641 signal, the frequency spectrum of a sinusoid is a peak with tails. These tails are referred to as

*leakage*

from the main peak.

**Spectral reversal:**

Technique for changing a filter kernel such that the corresponding frequency response is flipped left-for-right. This changes low-pass filters into high-pass filters.

**Spectrogram: **

Measurement of how an audio frequency spectrum changes over time. Usually displayed as an image. Also called a

*voiceprint*

.

**Standard deviation:**

A way of expressing the fluctuation of a signal around its average value.

Defined as the square-root of the average of the deviations squared, where the deviation is the difference between a sample and the mean. See

*root-mean-square*

for comparison.

**Static linearity:**

Refers to how a linear system acts when the signals are not changing (i.e., they are

*DC*

or

*static*

). In this case, the output is equal to the input multiplied by a constant.

**Statistical noise:**

Variations in a signal's value resulting from it being represented by a finite number of particles, such as: x-rays, electrons, or light photons. Also called

*Poisson statistics*

and

*Poisson noise*

.

**Steepest descent:**

Strategy used in designing iterative algorithms. Analogous to finding the bottom of a valley by always moving in the downhill direction.

**Step response:**

The output of a system when the input is a step function.

**Stopband:**

The band of frequencies that a filter is designed to block.

**Stopband attenuation:**

The amount by which frequencies in the stopband are reduced in amplitude, usually expressed in decibels. Used to describe a filter's performance.

**Substitution:**

A way of using complex numbers to represent a physical problem, such as electric circuit design. In this method,

*j*

terms are added to change the physical problem to a complex form, and then removed to move back again. See

*mathematical equivalence*

for comparison.

**Switched capacitor filter:**

Analog filter that uses rapid switching to replace resistors. Made as easy-to-use integrated circuits. Often used as antialias filters for ADC and reconstruction filters for DAC.

642

*The Scientist and Engineer's Guide to Digital Signal Processing*

**Synthesis:**

The inverse Fourier transform, calculating the time domain from the frequency domain. See

*analysis*

for comparison.

**System:**

Any process that produces an output signal in response to an input signal.

**Systematic error:**

Errors in a measurement or prediction that are repeatable from trial to trial.

Systematic errors determines

*accuracy*

. See

*random error*

for comparison.

**Target detection:**

Deciding if an object or condition is present based on measured values.

**TIFF: **

A common image file format used in word processing and similar programs. Usually not compressed, although LZW compression is an option. See

*GIF*

and

*JPEG*

for comparison.

**Timbre:**

The human perception of harmonics in sound. See

*pitch*

for comparison.

**Time domain:**

A signal having time as the independent variable. Also used as a general reference to any domain the data is acquired in.

**Time domain aliasing:**

Aliasing occurring in the time domain when an action is taken in the frequency domain. Circular convolution is an example.

**Time domain encoding:**

Signal information contained in the shape of the waveform. See

*frequency domain encoding*

for comparison.

**Transfer function:**

The output signal divided by the input signal. This comes in several different f o r m s , d e p e n d i n g o n h o w t h e s i g n a l s a r e represented. For instance, in the s-domain and zdomain, this will be one polynomial divided by another polynomial, and can be expressed as

*poles*

and

*zeros*

.

**Transform:**

A procedure, equation or algorithm that changes one group of data into another group of data.

**Transform compression:**

Data compression technique based on assigning fewer bits to the high frequencies.

*JPEG*

is the best example.

**Transition band:**

Filter jargon; the band of frequencies between the passband and stopband where the roll-off occurs.

**True-negative:**

One of four possible outcomes of a target detection trial. The target is not present, and is correctly indicated to be not present.

**True-positive: **

One of four possible outcomes of a target detection trial. The target is present, and correctly indicated to be present.

**Unit circle:**

The circle in the z-plane at

*r*

'

1 .

The values along this circle are the frequency response of the system.

**Unit impulse:**

Another name for

*delta function*

.

**Von Neumann Architecture:**

Internal computer layout where both the program and data reside in a single memory; very common. See

*Harvard*

*Architecture*

for comparison.

**Voiced:**

Human speech sound that originates as pulses of air passing the vocal cords. Vowels are an example of voiced sounds. See

*fricative*

for comparison.

**Well:**

Short for

*potential well*

; the region in a

CCD that is sensitive to light.

**White noise:**

Random noise that has a flat frequency spectrum. Occurs when each sample in the time domain contains no information about the other samples. See

*1/f noise*

for comparison.

**Wiener filter:**

Optimal filter for increasing the signal-to-noise ratio based on the frequency spectra of the signal and noise.

**Windowed-sinc:**

Digital filter used to separate one band of frequencies from another.

**z - d o m a i n : **

The domain defined by the ztransform. Also called the

*z-plane*

.

**z-transform:**

Mathematical method used to analyze discrete systems that are controlled by difference equations, such as recursive (IIR) filters. Changes a signal in the time domain into a signal in the z-domain.

**Zero: **

A term used in the Laplace & z-transforms.

When the s-domain or z-domain transfer function is written as one polynomial divided by another polynomial, the roots of the numerator are the

*zeros*

of the system. See also

*pole*

.

**Zero phase:**

A system with a phase that is entirely zero. Occurs only when the impulse response has left-to-right symmetry around the origin. See also

*linear phase*

.

**Zeroth-order hold:**

A term used in DAC to describe that the analog signal is maintained at a constant value between conversions, resulting in a staircase appearance.

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