Glossary 631
Glossary
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:
. This is (a[n]tb[n] )tc[n] ' a[n]t(b[n]tc[n])
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.
The Scientist and Engineer's Guide to Digital Signal Processing 632
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& 0.5cos(2Bn/M)% 0.08cos(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++.
Carrier wave: Term used in amplitude
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
Glossary 633
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 , then . A ' 3% 2j 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: . They are useful in engineering and e a % bj
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% 2j
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,
The Scientist and Engineer's Guide to Digital Signal Processing 634
breaking a signal into sinusoids.
Deconvolution: The inverse operation of
convolution: if , find given x[n]th[n] ' y[n] x[n]
only . Deconvolution is usually h[n] and y[n]
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
spatial domain (such as images) and the
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
used in an ADC. Can also be used with
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 635
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.
The Scientist and Engineer's Guide to Digital Signal Processing 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
form: . The Gaussian has many unique e x 2
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: , where n runs from 0.54 & 0.46cos(2Bn/M)
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 produces an x[n]
output of , then an input produces an y[n] kx[n]
output of , for any constant k. ky[n]
Homomorphic: DSP technique for separating
signals combined in a nonlinear way, such as by
multiplication or convolution. The nonlinear
Glossary 637
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 . In 3% 2j
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.
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 forward
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.
The Scientist and Engineer's Guide to Digital Signal Processing 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.
MIPS: 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
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 2B.
Negative frequencies: Sinusoids can be written
as a positive frequency: , or a negative cos(Tt )
frequency: . Negative frequencies are cos(&Tt )
included in the complex Fourier transform, making
it more powerful.
Normal distribution: A bell shaped curve of the
form: . Also called a Gaussian. e x 2
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.
Glossary 639
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 space, 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, , and , respectively. &j /TC jTL
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: , where M is Mcos(Tt% N)
the magnitude and N is the phase. See
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.
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 LSB. Also called quantization noise. 1/ 12
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 . In 3% 2j
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.
The Scientist and Engineer's Guide to Digital Signal Processing 640
Example: DSP algorithms for controlling echoes in
long distance telephone calls.
Reconstruction filter: A low-pass analog filter
placed after a digital-to-analog converter.
Smoothes the stepped waveform by removing
frequencies above one-half the sampling rate.
Rectangular form: Representing a sinusoid by
the form: , where A is called Acos(Tt ) % B sin (Tt )
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 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
2f , 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
changing with time (or other independent
variable).
Glossary 641
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:
. The B terms are often sinc (a) ' sin(Ba) /Ba
hidden in other variables, making it in the general
form: . Important because it is the sin(x) /x
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
of sound to detect submarines and other
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
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.
The Scientist and Engineer's Guide to Digital Signal Processing 642
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
forms, depending on how the signals are
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-domain: 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.
Index 643
Index
A-law companding, 362-364
Accuracy, 32-34
Additivity, 89-91, 185-187
Algebraic reconstruction technique (ART),
444-445
Aliasing
frequency domain, 196-200, 212-214,
220-222, 372
in sampling, 39-45
sinc function, equation for aliased, 212-214
time domain, 194-196, 300
Alternating current (AC), defined, 14
Amplitude modulation (AM), 204-206,
216-217, 370
Analysis equations. See under Fourier
transform
Antialias filter. See under filters- analog
Arithmetic encoding, 486
Artificial neural net, 458. See also Neural
network
Artificial reverberation in music, 5
ASCII codes, table of, 484
Aspect ratio of television, 386
Assembly program, 76-77, 520
Astrophotography, 1, 10, 373-375, 394-396
Audio processing, 5-7, 304-307, 311, 351-372
Audio signaling tones, detection of, 293
Automatic gain control (AGC), 370
Backprojection, 446-450
Basis functions
discrete Fourier transform, 150-152,
158-159
discrete cosine transform, 496-497
Bessel filter. See Filters, analog
Bias node in neural networks, 462-463
Bilinear interpolation, 396
Binary image processing. See Morphological
processing
Biquad, 600
Bit map to vector map conversion, 442
Bit reversal sorting in FFT, 229
Blacker than black video, 385
Blob analysis, 436. See also Morphological
processing
Brackets, indicating discrete signals, 87
Brightness in images, 387-391
Butterfly calculation in FFT, 231-232
Butterworth filter. See under Filters
C program, 67, 77, 520
Cascaded stages, 96, 133. See also under
Filters- recursive
Caruso, restoration of recordings by, 304-307
CAT scanner. See Computed tomography
Causal signals and systems, 130
CCD. See Charge coupled device
Central limit theorem, 30, 135-136, 407
Cepstrum, 371
Charge coupled device (CCD), 381-385,
430-432
Charge sensitive amplifier, in CCD, 382-384
Chebyshev filter. See under Filters
Chirp signals and systems, 222-224
Chrominance signal, in television, 386
Circular buffer, 507
Circularity. See under Discrete Fourier
transform
Classifiers, 458
Close neighbors in images, 439
Closing, morphological, 437
Coefficient of variation (CV), 17
Color, 376, 379-381, 386
Compact laser disc (CD), 359-362
Complex logarithm, 372
Complex numbers
addition, 553-554
associative property, 554
commutative property, 5054
complex number system, 551-554
complex plane, 552-553
conjugation, 193-194
distributive property, 554
division, 554, 557
Euler's relation, 556, 569-570
exponential form, 557, 569-570, 584
multiplication, 554, 557
polar notation, 555-557
rectangular notation, 552-556
sinusoids, representing, 559-561
The Scientist and Engineer's Guide to Digital Signal Processing 644
subtraction, 554
systems, representing, 561-563
Companding, 4, 358-359, 362-364
Compiler, 78, 546
Composite video, 384-386
Computed tomography (CT), 9, 411, 429,
442-450
Compression, data. See Data compression
Compression & expansion of signals,
200-203
Compression ratio in JPEG, 500
Connectivity analysis, 434. See also
Morphological processing
Continuous signal (defined), 11
Contrast, image 387-391, 432-434
Convolution
associative property, 133
circular, 182-184, 314
commutative property, 113, 132
continuous, 246-252
convolution machine, 116-119
discrete, 107-122
distributive property, 134
end effects, 118-121, 408
execution time, 140, 316, 318-422
frequency multiplication, by, 179-184.
See also FFT convolution
image, 397-410, 416-422
immersion of impulse response, 119, 410
input side view, 112-115, 246-247, 398,
418-423
left-for-right flip, 117, 138-140, 194,
416-421
neural networks, carried out by, 464-465
output side view, 116-121, 246-247,
399-400, 418-421
piecewise polynomial method, 250-252
separability, image convolution by,
404-407
sum of weighted inputs, viewed as, 122
Cooley, J.W. & Tukey, J.W., 225
Correlation, 136-140
autocorrelation, 137
convolution, carried out by, 138-140, 194,
416-422
correlation machine, 138-140
cross-correlation, 137
Fourier transform use of. See Discrete
Fourier transform
matched filter. See Filters
neural networks, carried out by, 464
radar and sonar use, 137
Counting statistics, 434-436
CRT display, point spread function of, 424
Cumulative distribution function, 28-29
CVSD modulation, 62-64
Data compression, 4, 10, 481-502
dB. See Decibels
dBm, 264
dBV, 264
DC offset
in Fourier transform, 152
linearity of, 97
DCT. See Discrete cosine transform
Decibels, 263-264
Decibels SPL, 352-356
Decimation. See Multirate
Decimation in frequency FFT, 234
Decimation in time FFT, 234
Decomposition
defined, 98
even/odd decomposition, 102-103, 240-242
Fourier decomposition, 104, 105, 147
impulse decomposition, 100-101
interlaced decomposition, 103-104, 228-229
step decomposition, 100-101
strategy for using, 98-100
Deconvolution, 179-180, 300-307
Delta encoding, data compression, 486-488
Delta function
continuous, 243-245
discrete, 107-109
Fourier transform of, 200, 209-212
identity for convolution, 123
two-dimensional (image), 397-398
Delta modulation, 60-66
Delta-sigma, 63-66
Dependent variable (defined), 12
Derivative, discrete. See First difference
Derivative of continuous step function,
250-252
DFT. See Discrete Fourier transform
Differential equations, RLC circuits, 563-564
Difference equation. See Recursion equation
Digital-to-analog conversion (DAC), 44-48
Dilatation. See Dilation
Dilation, 437. See also Morphological
processing
Direct current (DC) (defined), 14
Discrete signal (defined), 11
Discrete cosine transform (DCT), 496
Discrete Fourier transform (DFT)
See also Fourier transform; Fourier
transform pairs
analysis, 157-161, 567-580
basis functions, 150-152, 158-159
circular, 195. See also periodic
complex DFT, 225-227, 519-532
correlation method, 157-160
examples of, 142-143, 171, 181, 186, 193.
See also Fourier Transform pairs
forward DFT, 157-160, 567-580
Index 645
frequency resolution, 173-176
inverse DFT, 152-156, 573-580
Discrete Fourier transform (continued)
negative frequencies, 196-200, 209-211,
226-227, 569-575
orthognal basis functions, 158-159
periodic frequency domain, 196-200
periodic time domain, 194-196
real DFT, 141-160, 225-227, 519-521
spectral density, 156
spectral leakage, 175-176
synthesis, 152-156, 573-580
Discrete Time Fourier Transform (DTFT).
See Fourier transform
Distant neighbors in images, 439
Dithering, 38-39, 374
Division of frequency domain signals,
181-182
DN (digital number) in images, 374
Dolby stereo, 362
Domain (defined), 12
Double precision, 70-74, 284, 339
DSP microprocessor, 84
DTFT. See Fourier transform
Duality, 161, 210-212, 236
Dynamic range, 261-262, 378
Ear, 351-355
Echo control in telephones, 5
Echo location, 7-8. See also Radar; Sonar;
Seismology
Echoes in music, 5
Edge detection, 402-403, 416-417
Edge response, 426-430
EFM (8 to 14 modulation), 360
Electric circuit analysis
phasor transform method, 563-566
Laplace transform method, 592-599
Electroencephalogram (EEG), 292-293
End effect. See under Convolution
Erosion, 435. See also Morphological
processing
Euler's relation. See under Complex numbers
Even field in video, 384-385
Even/odd decomposition, 102, 240-242
Even symmetry, 102, 196, 240-241
Evolution, neural network learning, 470
Execution speed. See Speed
Exponential, two ways to generate, 606-607
Eye, 376-381, 404, 409, 432-434
False-positive (false-negative), 453-454
Fast Fourier transform (FFT), 180, 225-242
Feature extraction, 458
FFT convolution, 140, 179-184, 311-318, 411,
416-422
Field, television, 384-386
Filters, analog
See also Filters, digital; Filters, recursive
in ADC and DAC, 48-59
antialias, 48-49, 55-59, 172-173
Bessel, 49-59, 330, 361
Butterworth, 49-58, 600-604
Chebyshev, 49-58, 600-604
design methods, 49-59, 600-604
digital, compared to, 261-262, 343-345
elliptic, 54
frequency response, 52-54, 176-179, 598-599
high-pass, 51
low-pass, 49-54, 322, 343-345, 600-604
notch filter, 519-520, 592-597
overshoot, step response, 54-55
pulse response, 55
reconstruction filter for DAC, 44-49, 480
ringing, step response, 54-55
roll-off, 52-53
Sallen-key circuit, 49-50, 600-601
smoothing, 322
stability of analog filters, 541, 600-601
step response, 54-55, 322-323
switch-capacitor, 51-52
Filters, digital
See also Filters, analog; Filters, recursive
band-pass, 177-179, 268-274, 293
band-reject, 268-274, 291, 293
custom response, 297-310
cutoff frequency (defined), 268
edge enhancement, 400-401
even order filter, 603
finite impulse response (FIR), 263, 319
FIR vs. IIR, 346-350
frequency domain parameters, 268-270
frequency response, 176-179, 268-270, 562
high-pass, 110, 129, 268-274
low-pass, 110, 128-129, 268-270, 280,
285-288, 343-350, 400-401
matched filter, 138, 307-310, 416-419
moving average filter, 277-284, 307-310,
348-350, 406-407
odd order filter, 603
overshoot, step response, 266-267
passband (defined), 268
passband ripple (defined), 268
pulse response, 328-332
roll-off (defined), 268-269
smoothing, 110, 128-129, 280-282, 348-350
spectral inversion, 270-272, 293
spectral reversal, 273-274
step response, 262-263, 265-267, 338
stopband (defined), 268
stopband attenuation, 268-269, 293-296
time domain parameters, 266-268
The Scientist and Engineer's Guide to Digital Signal Processing 646
Filters, digital (continued)
transition band (defined), 268
Wiener filter, 308-310, 368-370
windowed-sinc filter, 216-217, 285-298,
346-348
Filters, recursive
See also Filters, analog; Filters, digital
band-pass filters, 326-327
band-reject filters, 326-327, 610-616
bidirectional recursive filtering, 330-332
Butterworth, 333-342, 601-603, 623-630
cascade stages, combining, 616-618
Chebyshev, 333-342, 346-348, 623-630
converting pole and zeros to recursion
coefficients, 610-615
custom response, 476-480
elliptic, 334, 602-603
FIR compared with IIR, 346-348
gain changes, 621-623
high-pass, 322-326, 334-338
infinite impulse response (IIR), 263, 284,
319-320
low-pass, 322-326, 334-338, 346-350
low-pass to low-pass transform, 628-629
low-pass to high-pass transform, 629
moving average filter, recursive, 282-284
narrow-band filters, 326-327
notch filters, 326-327, 612-614
overshoot, step response, 338
parallel stages, combining, 616-619
pulse response, 328-332
reconstruction filter for DAC, 44-55,
480
ringing, step response, 338
single pole recursive filters, 322-326,
348-350, 406-407
smoothing, 322-326, 348-350
spectral inversion, 619-621
stability of recursive filters, 339, 599,
598-599, 609
step response, 322-323, 338
transfer function of, 610-616
z-domain representation of, 610-616
Filter kernel (defined), 262
Filtered backprojection, 446-450
Fingerprint identification, 438-442
Finite impulse response. See Filters,
digital
FIR filters. See Filters, digital
First difference, 110-111, 125-128
Fixed point, 68-70, 514, 544
Flat-top window, 176
Floating point, 70-72, 514,544
Focusing, eyes and camera, 376-379
Formant frequencies in speech, 365
Fovea of the eye, 379-380
Fourier slice theorem, 411, 448-449
Fourier transform
See also Discrete Fourier transform;
Fourier transform pairs
analysis equations, 147, 157-161,
576-579
circular. See under Discrete Fourier
transform
data compression use of, 494-495
decomposition. See Decomposition
DFT. See Discrete Fourier Transform
discrete vs. periodic relationship, 222
discrete time Fourier transform (DTFT),
144-145, 178, 206-207, 213, 530-531
four types of, 143-145, 576-579
Fourier series, 144-145, 252, 255-260,
578-579
forward transform (defined), 147
Fourier transform, continuous, 144-145,
252-255, 579
images, Fourier transform of, 410-416
inverse transform (defined), 147
Jean Baptiste Joseph Fourier, 141
neural networks, can be carried out by,
464-465
periodic nature. See under Discrete
Fourier transform
real vs. complex, 146, 225-227, 576
real and imaginary parts, 148
scaling, 529-531
synthesis equations, 152-157, 577-579
why sinusoids are used, 142
Fourier Transform pairs, 209-224
chirp signals and systems, 222-224
delta function, 210-211
distorted sinusoid, 220-221
Gaussian, 216-217
Gaussian burst, 216-217
rectangular pulse, 212-213
shifted impulse, 210-211
sinc, 215-217
triangular pulse, 216-217
Fourier reconstruction (CT), 447-450
Frame grabber, 385
Frame, television, 384-385
Frequency domain (defined), 12, 147
Frequency domain encoding, See Information
Frequency domain multiplexing, 206
Frequency response, See Filters
Fricative sound in speech, 364-365
Full-width-half-maximum (FWHM), 424
Fully interconnected neural network, 460
Fundamental frequency. See Harmonics
Gamma curve, 389-394
Gamma ray detector, 301-305
Index 647
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
The Scientist and Engineer's Guide to Digital Signal Processing 648
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
Index 649
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
The Scientist and Engineer's Guide to Digital Signal Processing 650
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