Glossary

\int_{-\infty}^\infty|f(t)|\,dt is finite -- a stronger, whole-real-line condition than being L^1 on a bounded interval.

Two different frequencies producing the identical sample sequence at a given sampling rate.

The variable \omega a Fourier transform is a function of, in radians per unit time.

A filter applied before sampling to force a real signal's spectrum closer to band-limited.

The one frequency in [0,f_s/2] that a reconstruction from aliased samples will actually produce.

A signal whose spectrum is exactly zero past some finite frequency B.

The Fourier partial sum is the closest possible trigonometric polynomial of its degree, in the L^2 sense.

The fixed input reordering, a byproduct of splitting by successive bits, that the FFT's recursion needs.

A differential equation with conditions pinned at two points of the domain, not one starting point.

The two-input, one-multiply, two-output primitive that is the FFT's entire arithmetic content.

The running average of a Fourier series' first several partial sums, used to kill the Gibbs ripple.

A matrix built from one sequence, where every row is the row above it, shifted and wrapped around.

The wraparound version of convolution for two finite sequences, computed by a circulant matrix.

An orthogonal system with no nonzero vector orthogonal to every member of it.

A way to combine two functions, (f*g)(t)=\int f(\tau)g(t-\tau)\,d\tau, that is neither their sum nor their product.

The zero-frequency (\omega=0) spike in a signal's spectrum, corresponding to its constant part.

Writing a complicated periodic function as a sum of a few simple periodic pieces.

A subset of a normed space where every element of the space can be approximated arbitrarily well.

A generalized function, the limit of ever-narrower unit-area pulses, defined by what it does inside an integral.

The three conditions on a periodic function that guarantee its Fourier series converges.

The informal name signal processing gives to \|f\|_2^2.

The even-function mirror image of the Fourier sine series -- pure cosines plus a constant.

The pure-sine Fourier series an odd periodic function's expansion reduces to.

The frequency-domain function \hat f(\omega)=\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt built from a non-periodic f.

One output value X_k of a DFT, indexed by a slot number k, not a frequency in hertz on its own.

A function multiplied against a shifted 2D spectrum to keep or zero out frequency content by region.

The function H(\\omega) that specifies a filter, multiplying an input's transform to produce the output's.

The graph of |\hat f(\omega)| against \omega, for a non-periodic signal.

The pair of frequencies in a 2D transform, describing both how fast and which way a pattern oscillates.

A plot of how much of each frequency is mixed into a signal.

The universal overshoot fraction, about 8.95% of the jump size, in the Gibbs phenomenon.

A Fourier partial sum's persistent overshoot near a jump, which never shrinks away as more terms are added.

The sine or cosine series obtained by extending a function, defined only on half an interval, oddly or evenly.

How many full cycles a term completes while the fundamental completes one, in A\sin(nx+\varphi).

A complete inner product space -- every Cauchy sequence in it converges to a limit still inside it.

\cosh z=\tfrac12(e^z+e^{-z}) and \sinh z=\tfrac12(e^z-e^{-z}), a convenient exponential basis.

The time-domain function whose Fourier transform is a filter's frequency response.

An infinite row of evenly spaced unit Dirac impulses, used to model sampling as multiplication.

A vector space equipped with an inner product, generalizing the dot product.

The formula that recovers f back from its Fourier transform \hat f.

The size |\hat f(\omega)| and angle \arg\hat f(\omega) of a complex-valued spectrum, plotted separately.

A single separated wave-equation solution -- a standing wave with a fixed shape that only grows or shrinks in place.

A property of a chosen sampling rate: half of f_s, the highest frequency that rate can represent without aliasing.

A property of a signal: the minimum sampling rate, 2B, the theorem requires for its bandwidth B.

A harmonic above the fundamental, numbered starting at 1 -- so overtone number always equals harmonic number minus one.

The identity \|f+g\|^2+\|f-g\|^2=2\|f\|^2+2\|g\|^2, true only for norms that come from an inner product.

A function's shape copied every T to build a periodic signal out of a non-periodic slice.

The tiling effect sampling in time has on a signal's spectrum in frequency.

A function meeting all three of Dirichlet's conditions on one period.

A sequence of functions converges at every point separately, with no shared rate.

The version of the Cooley-Tukey FFT that splits by 2 at every level, requiring N a power of 2.

One of N equally spaced points on the unit circle, the exponential kernel a DFT sum is built from.

Recording a continuous signal's value only at evenly spaced instants, \Delta t apart.

A function of the form g(x)h(y), whose 2D transform is the product of the two functions' 1D transforms.

Integrating \delta(t-a) against f returns f(a) -- the defining rule for the Dirac delta.

The \sin-over-argument ripple shape, \sin(a\omega)/\omega, that is a rectangular pulse's Fourier transform.

How fast intensity changes per unit distance across an image -- independent of how bright the image is.

A wave whose spatial shape never travels -- it only grows, shrinks, and reverses in place.

The general pattern of eigenvalues and orthogonal eigenfunctions this chapter keeps rediscovering by hand.

The material constant governing how fast heat spreads through a rod.

What makes the same musical pitch sound different on different instruments -- the relative mix of harmonic amplitudes.

The ordinary graph of a signal's value against time.

A finite sum a_0+\sum_{n=1}^N(a_n\cos(n\omega_0x)+b_n\sin(n\omega_0x)).

The factor W_N^k=e^{-i2\pi k/N} that scales the odd half's DFT before combining it with the even half's.

A sequence of functions converges to its limit at the same rate everywhere on the domain.