SC · Fourier

The Fourier Transform

Stretch the period of a periodic signal's discrete spectrum without end and the lines melt into a continuous spectrum: that is the Fourier transform. See for yourself the reciprocal relation of time and frequency, in which narrowing a rectangular pulse widens its spectrum.

Narrow the pulse, widen the spectrum

The upper figure is a rectangular pulse of width T, and the lower figure is its Fourier transform, a sinc spectrum. The gold dashed curve is the target spectrum to match: move the pulse width T to lay the blue sinc over it. Narrowing the pulse in time widens the sinc in frequency, and the target is the shape for T = 1.4.

rectangular pulse x(t)

spectrum X(ω) = sinctarget spectrum

pulse width T2.60

Pulse width and first zero

T = 2.60   ω = 2π/T = 2.42

The pulse width differs from the target, so the sinc's width is off as well. Time width and frequency width are reciprocals of each other, so narrowing the pulse widens the spectrum.

Width off

Stretching the period without end

The harmonics of a Fourier series are spaced ω₀ = 2π/T apart. Enlarging the period T narrows this spacing, and stretching T without end, until the signal happens only once and becomes aperiodic, makes the spacing vanish to 0. The separate lines then merge into a gap-free continuous spectrum X(ω), and this limit, in which the discrete sum Σ turns into the integral ∫, is the Fourier transform.

ObserveX(ω) = ∫ x(t) e^−jωt dt

Analysis multiplies by e^−jωt and integrates.

Choosex(t) = (1/2π) ∫ X(ω) ? dω

Synthesis is the inverse with +jωt.

Fill innarrow pulse → ?

A narrow pulse gives a wide spectrum.

On your ownΔt · Δω ?

Time width times frequency width is roughly constant.

Narrower means wider

The transform of a rectangular pulse is the sinc function. For a pulse of width T the sinc's first zero is at ω = 2π/T, so the narrower the pulse (the smaller T), the farther the zero and the wider the main lobe. That is because a shorter signal needs more frequencies working together to produce its rapid change. The width in time and the width in frequency are reciprocals whose product stays roughly constant, and this is the uncertainty principle of signals. The figure scales the peak to 1 so that only the widths are compared, but the true peak of the transform equals the pulse's area T.

Transform and inverse

Analysis measures how much of each frequency a time signal contains, X(ω) = ∫ x(t) e^(−jωt) dt, and synthesis gathers those frequency components back into a time signal, x(t) = (1/2π) ∫ X(ω) e^(+jωt) dω. Apart from the sign of the exponent the two form a complete pair, so you can start in either domain and cross to the other. The next unit collects the rules this transform obeys into a toolkit for calculation.

Back to the first screen

Widening the pulse narrowed the sinc and narrowing it widened the sinc, while the product of the two widths stayed roughly the same. That is because time and frequency are tied together as reciprocals, and the first zero always stood at ω = 2π/T. An aperiodic signal that happens only once has not a few harmonics but a continuous spectrum spread over every frequency. Reading and writing that continuous spectrum with integrals is the Fourier transform, the limit of a Fourier series whose period is stretched without end.

The Fourier transform X(ω) = ∫ x(t) e^−jωt dt carries an aperiodic signal into a continuous spectrum, and the inverse transform is x(t) = (1/2π) ∫ X(ω) e^+jωt dω. It is the infinite-period limit of the Fourier series, in which the harmonic lines melt into a continuum. A rectangular pulse transforms into a sinc, and for a pulse of width T the first zero is at ω = 2π/T. Narrow in time means wide in frequency (reciprocity and uncertainty, Δt·Δω ≳ const).

On to the next unit

Working out the transform by integration every time is tedious. The next unit, Fourier transform properties, collects the rules the transform obeys: linearity, the effect of a time shift on phase, the reciprocal effect of time scaling on frequency scaling and, above all, the property that convolution in the time domain becomes multiplication in the frequency domain. That one property simplifies computing an LTI system's output to a multiplication, and it is where the convolution of chapter B meets the frequency of chapter C.