SC · Fourier

The Fourier Transform

Stretch the period of a periodic signal’s discrete spectrum to infinity and the lines melt into a continuous spectrum. That is the Fourier transform. See the reciprocal link between time and frequency: the narrower a rectangular pulse, the wider its spectrum.

Narrow the pulse, widen the spectrum

The top is a rectangular pulse; below is its Fourier transform, a sinc spectrum. The dashed curve is the target spectrum. Move the pulse width T to lay the blue sinc onto the target. Notice: narrowing in time widens it in frequency.

rectangular pulse x(t)

spectrum X(ω) = sinctarget spectrum

pulse width T2.60

Pulse width and first zero

T = 2.60   ω₀ = 2π/T = 2.42

The pulse width differs from the target, so the sinc’s width is off too. Time width and frequency width are reciprocals: narrow the pulse and the spectrum widens.

Width off

Stretch the period to infinity

The harmonics of a Fourier series sit a spacing ω₀ = 2π/T apart. Grow the period T and that spacing narrows; stretch T to infinity (the signal happens only once, aperiodic) and the spacing vanishes to 0. Then the separate lines merge into a gap-free continuous spectrum X(ω). This limit, where the discrete sum Σ becomes an 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 a sinc function. If the pulse width is T, the sinc’s first zero falls at ω = 2π/T. So the narrower the pulse (smaller T), the farther out the zero and the wider the main lobe — because a shorter signal needs more frequencies gathered to make its rapid change. The width in time and the width in frequency are reciprocals whose product is roughly constant: this is the uncertainty principle for signals.

Transform and inverse

Analysis measures how much of each frequency is in a time signal: X(ω) = ∫ x(t) e^(−jωt) dt. Synthesis gathers those frequency components back into a time signal: x(t) = (1/2π) ∫ X(ω) e^(+jωt) dω. Differing only in the sign of the exponent, the two are 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 working calculus.

Back to the first screen

Widening the pulse narrowed the sinc; narrowing it widened the sinc. The product of the two widths stayed roughly the same, because time and frequency are tied as reciprocals. The first zero always stood at ω = 2π/T. A signal that happens only once, aperiodic, has not a few harmonics but a continuous spectrum across all frequencies. Reading and writing that continuous spectrum by integral is the Fourier transform, the period-to-infinity limit of the Fourier series.

The Fourier transform X(ω) = ∫ x(t) e^−jωt dt maps an aperiodic signal to a continuous spectrum; the inverse is x(t) = (1/2π) ∫ X(ω) e^+jωt dω. It is the period-to-infinity limit of the Fourier series, with the harmonic lines melting into a continuum. A rectangular pulse transforms to a sinc, and a pulse of width T has its first zero at ω = 2π/T. Narrowing in time widens it in frequency (reciprocity / uncertainty, Δt·Δω ≳ const).

On to the next unit

Computing the transform by integral every time is laborious. The next unit, Fourier transform properties, collects the rules the transform obeys: linearity, how a time shift acts on phase, the reciprocal effect of time scaling on frequency scaling, and above all that convolution in time becomes multiplication in frequency. With this one property, computing an LTI system’s output simplifies to a multiplication. The convolution of chapter B and the frequency of chapter C meet here.