# Fourier transform The **Fourier transform** is the [[Integral_transform|integral transform]] that turns a function of time into a function of frequency by integrating it against a complex sinusoid at every frequency: `X(f) = ∫ x(t) exp(−j 2π f t) dt`. Where a [[Fourier_series]] describes a periodic [[Signal|signal]] by a discrete set of harmonics, the transform describes a signal of any shape by a continuous spectrum, a complex-valued function whose magnitude says how much of each frequency is present and whose phase says how the components are aligned. The inverse transform rebuilds the signal with the opposite sign in the exponent, so the [[Time_domain|time-domain]] and [[Frequency_domain|frequency-domain]] descriptions carry the same information. Its properties are what make it useful: [[Convolution|convolution]] in time becomes multiplication in frequency, differentiation becomes multiplication by j2πf, and a narrow pulse has a wide spectrum. Those rules let a filter be designed by its [[Frequency_response|frequency response]] and underlie [[Spectroscopy|spectroscopy]], [[Quantum_mechanics|quantum mechanics]] and [[Signal_processing|signal processing]]; in computation the integral becomes the [[Discrete_Fourier_transform|discrete Fourier transform]] of a sampled record. The primary microsim on this page, *Fourier transform*, lets the reader build a waveform from up to 12 harmonics, slide its fundamental across the frequency bins, switch the analysis window, and watch the magnitude spectrum of a 512-sample record respond: one clean line when a tone sits on a bin, a ridge of leakage when it sits between them. ## Definition For an integrable function x(t) the transform is `X(f) = ∫_{−∞}^{∞} x(t) exp(−j 2π f t) dt`, and the inverse is `x(t) = ∫_{−∞}^{∞} X(f) exp(j 2π f t) df`.[^j120] The two formulas differ only in the sign of the exponent: applying the forward formula where the inverse belongs returns x(−t), and applying the forward transform four times returns x(t).[^j146] Transform and inverse are written `x(t) ↔ X(f)`, a transform pair. ### Angular frequency (ω) With [[Angular_frequency|angular frequency]] ω = 2πf the pair becomes `X(ω) = ∫ x(t) exp(−jωt) dt` and `x(t) = (1/2π) ∫ X(ω) exp(jωt) dω`, and some authors split the 1/2π as 1/√(2π) on each side. The conventions describe the same spectrum with different scale factors, so a table of pairs is valid only in its own convention; the microsim prints the ω form but computes in bins. ### Lebesgue integrable functions The integral converges absolutely whenever ∫|x(t)| dt is finite, and then X(f) is bounded and continuous. A pure [[Sine_wave|sine wave]] or a constant is not integrable, and its transform exists only as a generalised function built on the [[Dirac_delta_function|Dirac delta function]]; square-integrable signals of finite [[Energy|energy]] are handled by Plancherel's theorem below. ## Background ### History Joseph Fourier introduced the sine and cosine integrals that make up the transform in his theory of heat, where an infinite solid replaces the finite bar; the mature account is his *Théorie analytique de la chaleur* of 1822.[^fourier1822] Michel Plancherel proved in 1910 that the transform preserves the energy of square-integrable functions.[^plancherel1910] It became a routine numerical tool when James Cooley and John Tukey published the fast algorithm in 1965.[^cooley1965] ### Complex sinusoids The kernel exp(−j2πft) is a unit phasor rotating clockwise at f turns per second, equal by Euler's formula to cos(2πft) − j sin(2πft).[^j19] Integrating x against it correlates the signal with a cosine and a sine of that frequency at once, and the two results are the real and imaginary parts of X(f); a real sinusoid is itself the sum of two phasors rotating in opposite directions.[^s31] ### Negative frequency Because a real cosine is the sum of a phasor at +f and one at −f, the transform of a real signal is two-sided: X(−f) is the conjugate of X(f), the magnitude is even and the phase odd.[^s31] The magnitude at +f alone accounts for half the amplitude; the microsim shows the positive half and divides by the record length, which is why a unit sine on an exact bin reads 0.5 in the sketch. ### Fourier transform for periodic functions A periodic signal has a transform made of impulses, the [[Fourier_series]] coefficients c_k weighting deltas at the harmonic frequencies k/T. The reverse view is more instructive: for a rectangular pulse train of height A, width Δ and period T, the coefficients are `c_k = A exp(−jπkΔ/T) sin(πkΔ/T)/(πk)`, and the scaled lines `T c_k`, plotted at f = k/T, all lie on the curve `P(f) = A exp(−jπfΔ) sin(πfΔ)/(πf)`, the transform of one pulse.[^j107][^j120] As T grows the lines crowd together on the fixed envelope, and in the limit the line spectrum becomes the continuous transform.[^j120] ### Sampling the Fourier transform Dually, sampling a signal makes its spectrum periodic: multiplying x(t) by a train of impulses spaced Δt = 1/fs apart gives the sum of shifted copies `X_s(f) = Σ_k X(f − k fs)`.[^tm108] An 11 Hz tone sampled at 30 Hz acquires copies at 19, 41, 49 and 71 Hz, all outside the band ±15 Hz, and is recovered exactly by a low-pass filter; sampled at 10 Hz it lands on a 1 Hz [[Aliasing|alias]], because the copies overlap unless the sampling rate exceeds twice the highest frequency present, the [[Nyquist–Shannon_sampling_theorem|sampling theorem]].[^tm108] Sampling the spectrum instead, as a [[Discrete_Fourier_transform|discrete Fourier transform]] does, makes the time record periodic. ## Units If t is in seconds and x in volts, X(f) is in volts per hertz: a density over frequency rather than an amplitude, which is why the transform of a finite-energy pulse is finite while that of a sinusoid is an impulse. The product of a time scale and a frequency scale is [[Dimensionless_quantity|dimensionless]]; the microsim's axes are sample and bin index, and its fundamental slider, labelled in hertz, is in cycles per 512-sample record. ## Properties ### Basic properties In the table x ↔ X and y ↔ Y are transform pairs and a, b, t0, f0 are constants. | Operation on x(t) | Effect on X(f) | Name | |---|---|---| | `a x(t) + b y(t)` | `a X(f) + b Y(f)` | Linearity | | `x(t − t0)` | `X(f) exp(−j 2π f t0)` | Time shifting | | `x(t) exp(j 2π f0 t)` | `X(f − f0)` | Frequency shifting | | `x(a t)` | `X(f/a)/|a|` | Time scaling | | `X(t)` | `x(−f)` | Symmetry (duality) | | `x*(t)` | `X*(−f)` | Conjugation | | x real | `X(−f) = X*(f)` | Real and imaginary parts | | `X(0)` | `∫ x(t) dt` | Zero frequency component | Time scaling has the most consequences: compressing a signal in time by a factor a stretches its spectrum by the same factor and lowers it by 1/|a|, so a short pulse is necessarily wideband.[^tm82] Duality says every pair reads in both directions, so rectangle and sinc are one pair whichever side is called time.[^tm82] Frequency shifting is [[Amplitude_modulation|amplitude modulation]], which moves a baseband spectrum of width W to a band 2W wide around the carrier.[^j124] ### Uniform continuity and the Riemann–Lebesgue lemma For an absolutely integrable signal the transform is uniformly continuous and tends to zero as |f| grows. Smoother signals decay faster: each integrable derivative adds a factor 1/f, so a jump gives a spectrum falling as 1/f, a corner as 1/f², and a Gaussian faster than any power. ### Plancherel theorem and Parseval's theorem For square-integrable signals the transform preserves energy, `∫ |x(t)|² dt = ∫ |X(f)|² df`, and inner products.[^tm82] The quantity |X(f)|² is an energy per unit frequency, the energy [[Spectral_density|spectral density]]; a cosine of amplitude A observed for T seconds has energy A²T/2 and power A²/2.[^tm96] The microsim's readout Σ|X|² is the discrete form of this sum over its positive bins. ### Convolution theorem Convolution in time is multiplication in frequency: if `y(t) = ∫ x(τ) h(t − τ) dτ` then `Y(f) = X(f) H(f)`, and dually a product of signals has the convolution of their spectra.[^tm82] A [[Linear_time-invariant_system|linear time-invariant]] system convolves its input with its [[Impulse_response|impulse response]], so its effect on any input is multiplication of the spectrum by the frequency response, which for a first-order RC [[Low-pass_filter|low-pass]] filter is `H(f) = 1/(1 + j 2π f R C)`.[^j124] ### Cross-correlation theorem The cross-correlation of x and y has transform X(f) Y*(f); with y = x it gives the [[Autocorrelation|autocorrelation]], whose transform is |X(f)|² with the phase discarded. A [[Matched_filter|matched filter]], which correlates a received waveform with a stored copy, therefore has the conjugate of the signal's spectrum as its response. ### Differentiation Differentiating x(t) multiplies X(f) by j2πf, and integrating divides by it. A [[Differential_equation|differential equation]] with constant coefficients becomes algebraic in f, and a system described by one has a rational frequency response, as the RC filter does. ### Eigenfunctions A Gaussian transforms to a Gaussian, `exp(−π t²) ↔ exp(−π f²)`. The Hermite functions, a Gaussian times each Hermite polynomial, form a complete set of eigenfunctions with eigenvalues (−j)ⁿ; they are also the stationary states of the [[Quantum_harmonic_oscillator|quantum harmonic oscillator]], which look the same in position and [[Momentum|momentum]] space. ### Inversion and periodicity Applying the transform twice gives x(−t) and four times the original, so as an operator it has period four and eigenvalues ±1 and ±j.[^j146] The inverse is the forward transform of the reflected function. ### Connection with the Heisenberg group Time shifts and frequency shifts do not commute: applied in the two orders they differ by a phase exp(±j2πf0t0). Shifts, modulations and phases together form the Heisenberg group, a [[Lie_group]] whose action on square-integrable functions unites the transform, the uncertainty principle and the [[Short-time_Fourier_transform|short-time Fourier transform]]. ## Complex domain ### Laplace transform Replacing the imaginary exponent by a complex one, s = σ + j2πf, gives the [[Laplace_transform]], `X(s) = ∫ x(t) exp(−st) dt`. The real part σ lets the integral converge for signals that grow or never decay, and the Fourier transform is the Laplace transform on the imaginary axis whenever that axis lies in the region of convergence. The kernel with σ ≠ 0 is a spiral, the complex-frequency signal `A exp(jφ) exp((−1/τ + j 2π f) t)` whose envelope falls to 1/e ≈ 0.368 at t = τ.[^j19] ### Inversion The Laplace inverse is a line integral up a vertical path inside the region of convergence, `x(t) = (1/2πj) ∫ X(s) exp(st) ds`, which on the path σ = 0 is the inverse Fourier integral; for rational transforms it is evaluated by residues at the poles. ## Fourier transform on Euclidean space In n dimensions the kernel is exp(−j2π ξ·x) with ξ a vector of spatial frequencies, and every one-dimensional rule has a counterpart. ### Uncertainty principle A signal cannot be narrow in both domains. With the standard deviations of |x|² and |X|² as widths, their product is at least 1/(4π), with equality only for the Gaussian, so a pulse of duration Δt has a spectrum of width at least of order 1/Δt. This is the mathematical content of Werner Heisenberg's principle of 1927, in which the position and momentum wavefunctions are a Fourier pair.[^heisenberg1927] ### Sine and cosine transforms For real signals the transform splits into a cosine transform of the even part and a sine transform of the odd part, the form in which Fourier first wrote it.[^fourier1822] The [[Discrete_cosine_transform|discrete cosine transform]] of [[Image_compression|image compression]] is the sampled cosine transform. ### Spherical harmonics In three dimensions the angular dependence of a function decomposes into spherical harmonics, which the transform maps to spherical harmonics of the same degree with a Bessel-function radial factor. ### Restriction problems Restricting a transform to a curved surface such as a sphere is meaningful for some function classes and not others, a question of [[Harmonic_analysis|harmonic analysis]] still open in higher dimensions. ## Fourier transform on function spaces ### On other Lp The transform is defined by the integral on absolutely integrable functions and by continuity on square-integrable ones, where Plancherel's theorem makes it an isometry. For Lp with 1 < p < 2 the Hausdorff–Young inequality bounds it in the dual space; beyond p = 2 the integral need not exist as a function. ### Tempered distributions Signals that are neither integrable nor square-integrable, such as a constant or a sinusoid, are transformed as tempered distributions, which act on rapidly decaying test functions. The transform of a constant is then a [[Dirac_delta_function|Dirac delta]], that of a cosine a pair of deltas at ±f0, and that of an impulse train another impulse train, the identity behind sampling.[^tm108] ## Generalizations ### Fourier–Stieltjes transform on measurable spaces A measure is transformed by integrating the kernel against it; the characteristic function of [[Probability_theory|probability theory]], the transform of a [[Probability_distribution|probability distribution]], is the leading example. ### Locally compact abelian groups The series, the transform on the line and the discrete transform are one construction on the circle, the line and the integers modulo N, the dual group of characters supplying the frequency variable; Pontryagin duality provides such a dual for every locally compact abelian group. ### Gelfand transform On a commutative Banach algebra the Gelfand transform maps each element to a function on the maximal ideals; for the convolution algebra of integrable functions on a group it is the Fourier transform. ### Compact non-abelian groups On a non-commutative compact group the characters give way to matrix-valued irreducible representations, and the Peter–Weyl theorem expands a square-integrable function in their entries. ## Alternatives The transform says which frequencies a signal contains but not when. The [[Short-time_Fourier_transform|short-time Fourier transform]] slides a window along the signal and gives a [[Spectrogram|spectrogram]] with a time–frequency trade-off set by the window length; the [[Wavelet_transform|wavelet transform]] uses short windows at high frequency and long ones at low; the [[Laplace_transform]] and [[Z-transform]] add a decay parameter; and the [[Hilbert_transform]] shifts every component by a quarter cycle to build the [[Analytic_signal|analytic signal]]. ## Example The rectangular pulse of height A and width Δ, starting at t = 0, has transform `P(f) = A exp(−jπfΔ) sin(πfΔ)/(πf)`: a sinc-shaped magnitude with a main lobe of width 2/Δ, zeros at every multiple of 1/Δ and the value AΔ at f = 0.[^j120] Halving Δ doubles the width of every lobe and halves the height, and a pulse train with Δ = 0.2 s puts lines on this one envelope at 1 Hz spacing for T = 1 s and 0.2 Hz for T = 5 s.[^j120][^tm96] The microsim shows the discrete counterpart. With the fundamental at 5.0 cycles per record, one flat harmonic and a rectangular window, the sketch draws a single bar at bin 5 with a peak magnitude of 0.5000 and a peak phase of −90.0°, a sine on an exact bin. At 4.5 the tone sits between bins and the bars spread to 0.337 at bin 4 and 0.302 at bin 5, with a tail still 0.074 high at bin 1: the record ends mid-cycle, and the transform of the truncated sine is a shifted sinc sampled off its zeros. The textbook version, a 4 Hz and a 4.5 Hz cosine observed for 1 s at 64 samples per second, gives a single bin of 0.5 against leakage across many bins.[^tm143] ## Applications ### Analysis of differential equations Because differentiation becomes multiplication by j2πf, a linear [[Partial_differential_equation|partial differential equation]] with constant coefficients transforms into an [[Ordinary_differential_equation|ordinary differential equation]] in the remaining variable. Fourier's own case is the model: transformed in space, the heat equation turns [[Heat_transfer|conduction]] into a decay of each spatial frequency at a rate proportional to its square, so high frequencies die first, the mechanism of [[Diffusion|diffusion]].[^fourier1822] Nonlinear equations have no such shortcut; the nonlinear Fourier transform of inverse scattering theory serves a few integrable equations only. ### Fourier-transform spectroscopy An interferometer that varies the path difference between two beams records an interferogram, the autocorrelation of the light, whose transform is the optical spectrum; such a [[Spectroscopy|spectrometer]] sees every wavelength at once. The same idea with a radio-frequency pulse and a free-induction decay is pulsed [[Nuclear_magnetic_resonance|nuclear magnetic resonance]], introduced by Richard Ernst and Weston Anderson in 1966, and [[Magnetic_resonance_imaging|magnetic resonance imaging]] reconstructs its images by inverting a sampled spatial spectrum.[^ernst1966] ### Quantum mechanics The position and momentum representations of a wavefunction are a Fourier pair with Planck's constant setting the scale, so the uncertainty principle of the transform is that of [[Quantum_mechanics|quantum mechanics]]. Transforming the [[Schrödinger_equation]] to momentum space turns its derivative term into multiplication. ### Signal processing In [[Signal_processing|signal processing]] filters are specified by their response, noise by its [[Spectral_density|power spectral density]] and modulation as a shift of spectrum, and a [[Radar]] pulse's range resolution by its bandwidth.[^j124] The measured transform is always a windowed one: observing a signal for T seconds multiplies it by a rectangular window whose transform is `T sinc(T f)`, so every line acquires a main lobe of width 2/T and side lobes, and two tones closer than about 1/T merge; a smoother [[Window_function|window]] such as the Hann doubles the main-lobe width but lowers the side lobes greatly.[^tm96] ## Other notations Engineering texts write the transform in f with j, mathematics texts in ω with i, and physics texts often with the opposite sign so that a plane wave exp(j(kx − ωt)) carries positive frequency; capital letters, a hat or the operator F denote the transform. The microsim's HUD writes F(ω) with i while the page writes X(f) with j; the spectra are the same function.[^s31] ## Computation methods ### Discrete Fourier transforms and fast Fourier transforms Numerically the integral becomes a sum over N samples, `X_k = Δt Σ_n x_n exp(−j 2π k n/N)`, the transform at the frequencies k/T for a record of length T = N Δt; most software omits the factor Δt.[^tm131] The direct sum costs N² operations and the [[Fast_Fourier_transform|fast Fourier transform]] about N log₂ N: at N = 4,096 that is 16,777,216 against 49,152, a saving of 341 times.[^tm131] The microsim uses the direct sum on 512 samples and shows 256 bins, dividing each magnitude by N so that a unit sine on an exact bin reads 0.5; the [[Discrete-time_Fourier_transform|discrete-time Fourier transform]] of the same samples is the continuous curve those bins sample.[^tm122] ### Symbolic integration of closed-form functions For a function given by a formula the transform is found by table, by the property rules or by contour integration; the textbook's odd two-sided exponential with decay rate α > 0, for instance, has the transform `−j 2π f/(α² + (2π f)²)`, purely imaginary because the signal is odd.[^tm82] ### Numerical integration of closed-form continuous functions When no closed form exists the integral is approximated by [[Numerical_integration|numerical integration]] of x(t) exp(−j2πft) on a grid, which is once more the discrete transform; truncating the infinite range is itself a window that leaks. ### Numerical integration of a series of ordered pairs Measured data arrive as pairs (t_n, x_n). Uniform samples go straight into the discrete transform; uneven ones can be weighted by their intervals or fitted with sinusoids by [[Least-squares_spectral_analysis|least-squares spectral analysis]], and either way the record length sets the frequency resolution. ## Tables of important Fourier transforms The pairs below use the f convention, with Π the unit rectangle and Λ the unit triangle; the functional rules are those of the basic-properties table. | x(t) | X(f) | Class | |---|---|---| | `Π(t/τ)` | `τ sinc(τ f)` | square-integrable | | `τ sinc(τ t)` | `Π(f/τ)` | square-integrable (duality) | | `τ sinc²(τ t)` | `Λ(f/τ)` | square-integrable | | `exp(−α t) u(t)`, α > 0 | `1/(α + j 2π f)` | square-integrable | | `exp(−π t²)` | `exp(−π f²)` | square-integrable (eigenfunction) | | `1` | `δ(f)` | distribution | | `δ(t − t0)` | `exp(−j 2π f t0)` | distribution | | `cos(2π f0 t)` | `[δ(f − f0) + δ(f + f0)]/2` | distribution | | `Σ_n δ(t − n Δt)` | `fs Σ_k δ(f − k fs)`, fs = 1/Δt | distribution | | `Π(x/a) Π(y/b)` | `a b sinc(a ξ) sinc(b η)` | two-dimensional | | `exp(−π |x|²)`, x in n dimensions | `exp(−π |ξ|²)` | n-dimensional | The sinc² pair, drawn in the textbook with τ = 2, is the rectangle convolved with itself, and a triangular spectrum of half-width W belongs to the time signal `W [sin(πWt)/(πWt)]²`.[^tm88][^j146] ## Microsims The *Fourier transform* sketch is a two-panel time-to-frequency visualiser: the upper panel shows 400 of the 512 samples of the waveform `Σ_{k=1}^{K} a_k sin(2π k F0 n/512)`, and the lower panel the magnitude of its discrete Fourier transform, one bar per bin for bins 1 to 63. The controls are the slider *F0, fundamental frequency* (0.5–5.0 in steps of 0.1, default 1.0; in cycles per record though labelled hertz), the slider *K, number of harmonics* (1–12, default 4), the *envelope* menu setting a_k to flat, 1/k, 1/k² or odd-only (square), and the *window* menu applying a rectangular, Hann, Hamming or Blackman window; the readouts give the number of bins (256), the peak bin, the peak magnitude, Σ|X|² and the phase at the peak. At the default setting the bars at bins 1 to 4 read 0.500, 0.250, 0.167 and 0.125 with a peak phase of −90.0°, because each component is a sine; at F0 = 1.0 the harmonics sit on adjacent bins, closer than a tapered window's main lobe, so the tapered windows behave as described only once F0 is 3 or more. A three.js companion on this page renders the passage from series to transform in depth, stretching a pulse train's period so that its lines crowd onto the sinc envelope. *Try:* Set K to 1, the envelope to flat and F0 to 5.0: one bar at bin 5 with peak magnitude 0.5000. Slide F0 to 4.5 and watch the bar split into 0.337 at bin 4 and 0.302 at bin 5 with leakage right down to bin 1; then switch the window to Hann and watch the pair become 0.212 and 0.212 with the tail gone by bin 7. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Fourier_transform) : [Wikitube](https://en.wikitube.io/wiki/Fourier_transform) Skeleton mirrored at revision 1373716538. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Fourier_series]] - [[Discrete-time_Fourier_transform]] - [[Discrete_Fourier_transform]] - [[Fast_Fourier_transform]] - [[Laplace_transform]] - [[Hilbert_transform]] - [[Spectral_density]] - [[Short-time_Fourier_transform]] - [[Wavelet_transform]] - [[PORTAL_Signal_Processing]] ## Notes The sketch is ILLUSTRATIVE: it computes a 512-point discrete Fourier transform of a windowed record, not the continuous integral, so every spectrum it shows is a windowed and sampled one, and its fundamental-frequency slider is in cycles per record rather than hertz. The continuous formulas in the article are the ones its bins sample. ## Citations Page numbers in the references are PDF pages of the open editions. ## References [^j19]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 19–22 (PDF pages): Euler's formula, the sinusoid as the real part of a rotating phasor and the decaying complex exponential with envelope 1/e at t = τ. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j107]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 107–109 (PDF pages): the periodic pulse train's Fourier coefficients. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j120]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 120–121 (PDF pages): the transform as the limit of the line spectrum, the transform of a single pulse, the pulse train drawn with Δ = 0.2 s at T = 1 s and 5 s, and the sinc(t) = sin(t)/t convention. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j124]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 119–125 (PDF pages): the RC low-pass response, the 2W bandwidth of amplitude modulation and the output spectrum Y = XH for any input. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j146]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 145–146 (PDF pages): the DC term of the pulse train, the reflection under a wrong exponent sign, the return to s(t) after four transforms and the time signal of a triangular spectrum. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^tm82]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 82–87 (PDF pages): the scale-change rule, the convolution theorem, Parseval's relation, the two-sided exponential example and the duality pair with τ = 2. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tm88]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 88–95 (PDF pages): the convolution integral, the Dirac shift property and the sinc² to triangle pair. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tm96]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 96–103 (PDF pages): the rectangular window transform T sinc(T f), the windowed cosine, two-tone resolution at separations of 1/T to 5/(2T), the energy and power of a windowed cosine and the Hann window. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tm108]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 108–115 (PDF pages): the sampling model, the periodic spectrum of a sampled signal, the sampling theorem and the 11 Hz tone sampled at 30 Hz and at 10 Hz. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tm122]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 122–123 (PDF pages): the discrete-time Fourier transform. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tm131]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 131–133 (PDF pages): the discrete Fourier transform pair with the Δt factor, its omission in software, and the N² against N log₂ N operation counts at N = 4,096. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tm143]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 143–144 (PDF pages): a 4 Hz and a 4.5 Hz cosine observed for 1 s at 64 samples per second. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^s31]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, pp. 29–32 (PDF pages): the j convention, the phasor and the conjugate pair that sums to a real cosine. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^fourier1822]: Fourier, J. *Théorie analytique de la chaleur*. Paris: Firmin Didot, 1822. [^plancherel1910]: Plancherel, M. "Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies". *Rendiconti del Circolo Matematico di Palermo* 30 (1910), pp. 289–335. [^cooley1965]: Cooley, J. W.; Tukey, J. W. "An algorithm for the machine calculation of complex Fourier series". *Mathematics of Computation* 19 (1965), pp. 297–301. [^heisenberg1927]: Heisenberg, W. "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". *Zeitschrift für Physik* 43 (1927), pp. 172–198. [^ernst1966]: Ernst, R. R.; Anderson, W. A. "Application of Fourier transform spectroscopy to magnetic resonance". *Review of Scientific Instruments* 37 (1966), pp. 93–102. ## External links - Fourier transform, live p5.js sketch: https://editor.p5js.org/sciencenibber/full/TPWK-3rjt - Fourier transform, fork the sketch in the p5.js editor: https://editor.p5js.org/sciencenibber/sketches/TPWK-3rjt <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->