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\noindent
Notes 9, Computer Graphics 2, 15-463

\vspace{.5in}
\begin{center}
{\LARGE \bf Glossary of Signal Processing Terms}
\end{center}

\term{signal}: a function of space or time, e.g. a picture is a 2-D 
signal.
\term{digital signal}: discrete in space and value.
\term{analog signal}: continuous in space and value.
\term{DSP}: digital signal processing
\\
\term{filter}: operator that takes a signal as input and produces a 
signal as output.
\term{linear filter}: $S$ is a linear filter if $S[\alpha f(x)+ \beta g(x)]= 
\alpha S[f](x)+ \beta S[g](x)$ for any $\alpha$, $\beta$.
\term{linear shift-invariant filter (LSI)}: $S$ is linear,
shift-invariant if it is linear and shifted input $f(x+t)$ yields output 
$S[f](x+t)$ for any shift $t$. Equivalently, if $S[f]$ can be written 
as the convolution of $f$ with some impulse response $h$.
In this case, we would write $S[f](x) = (f \conv h)(x)$ or
$S[f] = f \conv h$.
\term{convolution}: integral (if signals are continuous) or sum (if 
discrete) of product of signal and shifted filter. See notes 
for precise formulas.
\term{nonlinear filter}: a filter that is non linear, e.g. computes 
products, squares, or square roots of sample values.
Examples: Sobel edge filter, median filter.
Less common than LSI filters.
\\
\term{low pass filter}: filter that passes (allows through) low 
frequencies (those below the cutoff frequency) and stops 
(kills) high frequencies. Its frequency response will have a 
box-like shape. A sinc is the ``ideal'' low pass filter. 
Synonym: blur.
\term{high pass filter}: filter that passes high frequencies and 
stops low frequencies.
\term{support}: the domain over which a filter's impulse response is
nonzero.
Larger support typically makes a filter more expensive.
\term{FIR filter}: a Finite Impulse Response filter; one with finite
support.
Straightforward to compute.
Examples: box, triangle.
\term{IIR filter}: an Infinite Impulse Response filter; one with infinite
support.
Generally harder to compute and less common than FIR filters.
Computed using recursive filters, if they're computed at all.
Examples: sinc, Gaussian.
\term{recursive filter}: a filter whose output is written in terms of
previous samples of its output, e.g. $b[x] = \alpha b[x-1]+a[x]$.
\\
\term{box}: a function that is 0 except for some interval around 0, 
within which it is constant. In the spatial domain, a box is 
often used as a cheap blur filter. A box in the frequency domain makes an
ideal low pass filter.
\term{triangle}: a function that is 0 except for some interval 
around 0, within which its shape is an isosceles triangle. 
Standard definition: triangle = box$\conv$box.
\term{Gaussian}: a.k.a. the ``normal probability density function'', 
given by $A e^{-x^2/(2 \sigma^2)}$, where $\sigma$=standard deviation. 
To get unit integral, use $A = 1/(\sigma \sqrt{2 \pi})$.
Sometimes used as a low pass filter.
\term{sinc}: the ``ideal'' low pass filter.
$\mbox{sinc}(x)=\sin(\pi x)/(\pi x)$. Has 
value 0 for all integer $x$ except $x=0$, where it has value 1.
\\
\term{spatial domain}: in image processing, the domain of pixel coordinates
$x$ and $y$.
In audio processing, you speak of the ``time domain'' instead.
\term{frequency domain}: in image processing, the domain of horizontal and
vertical frequencies, $\omega_x$ and $\omega_y$.
\term{angular frequency}: frequency in radians per unit distance,
typically denoted $\omega$.
\term{rotational frequency}: frequency in cycles per unit distance,
typically denoted $f$, where $\omega=2 \pi f$.
\\
\term{Fourier Transform}: transformation of a signal into an 
integral of complex exponentials of all frequencies.
\term{convolution theorem}: states that convolution in the spatial 
domain is equivalent to multiplication in the frequency 
domain.
\term{DFT}: discrete Fourier transform. A variant of the Fourier 
transform for discrete, periodic signals. Involves finite 
sums, not infinite integrals.
\term{FFT}: fast Fourier transform. A fast algorithm for 
computing the DFT.
\term{complex exponential}: $e^{i\theta}=\cos\theta+i\sin\theta$
where $i=\sqrt{-1}$. 
A complex-valued sinusoid.
\term{roots of unity}: values of $z=e^{i\theta}$ for which $z^n=1$ for some
integer $n$.
These complex constants are coefficients in the DFT.
\term{butterfly}: a computation step of the FFT involving 
one complex multiply and two complex add/subtracts.
\\
\term{impulse, delta function}: a function $\delta(x)$ with value 0 
everywhere, except at $x=0$.  It has unit integral.
The continuous delta function has value infinity at $x=0$,
while the discrete delta function has value 1 at $x=0$.
\term{impulse response, point spread function}: output of a 
filter when given an impulse as input. Inverse Fourier transform of 
a filter's frequency response.
\term{gain}: ratio of output amplitude to input amplitude.
\term{frequency response}: function which gives the gain of a 
filter as a function of frequency. (A complex function.)
The Fourier transform of the filter's impulse response.
\term{magnitude response}: magnitude of the frequency 
response as a function of frequency.
\term{phase response}: phase of the frequency response
as a function of frequency:
atan(imaginary part/real part).
\term{dB (decibels)}: unitless measure of a number with a large 
dynamic range.
$\mbox{dB}(x) = -20\log_{10} (x)$.
Popular with electrical engineers.
Frequency responses
and signal to noise ratios are often measured in decibels.
A factor of 2 is about 6 dB.
\\
\term{sampling}: conversion of a continuous signal to a discrete 
signal.
\term{impulse train}: a sequence of uniformly-spaced, equal-height impulses.
\term{spectrum}: Fourier transform of a signal.
\term{bandwidth}: highest frequency in a bandlimited signal.
\term{bandlimited}: a signal that has a highest frequency.
\term{sampling frequency}: frequency of samples, $\omega_s = 2 \pi / T$
where $T$ is the sample spacing.
\term{Nyquist frequency}: half the sampling frequency. 
Frequencies above this will alias if the signal is sampled.
\term{aliasing}: masquerading of high frequencies as low frequencies
that occurs when a signal is improperly sampled.
\\
\term{reconstruction}: conversion of a discrete signal into a continuous signal.
\term{resampling}: sampling a discrete signal again at a new set of
sample points.
\term{decimation}: downsampling, scaling a signal down (samples farther apart
than previously).
\term{interpolation}: upsampling, scaling a signal up (samples closer together
than previously).
\term{point sampling}: sampling a signal at a single point,
as opposed to filtering before sampling.
Also known as ``pixel replication'', ``sample and hold'',
or ``box reconstruction''.
\term{rastering}: an artifact of poor reconstruction in which the
sampling grid is evident.  It's bad when point sampling is used,
better for bilinear interpolation.

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