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Mathematical Methods

The Fourier transform: spectrum, convolution and sampling

From series to transform, the rectangular pulse and the sinc, properties, convolution and frequency response with proof, Parseval, duality and the sampling theorem.

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The Fourier series of Analysis 2 breaks a periodic signal into harmonics at discrete frequencies. But most real signals are not periodic: a pulse, a word, a data packet. The Fourier transform extends the series to any signal, replacing the sum over discrete frequencies with an integral over all frequencies. It is the tool used to think about a channel’s bandwidth, design filters and analyse modulation.

From series to transform

For a signal of period T, the complex series is f(t) = \sum c_k e^{ik\omega_0 t}, with \omega_0 = 2\pi/T and c_k = \frac1T\int_{-T/2}^{T/2}f(t)e^{-ik\omega_0 t}dt. Letting T\to\infty, the frequencies k\omega_0 get ever closer together, the sum becomes an integral and one obtains the pair

    \[ F(\omega) = \int_{-\infty}^{\infty}f(t)\,e^{-i\omega t}\,dt,\qquad f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)\,e^{i\omega t}\,d\omega \]

F(\omega) is the signal’s spectrum: the magnitude says how much “amplitude” there is at each frequency, the phase how the components line up. The transform certainly exists if f is absolutely integrable, \int|f|dt \lt \infty. Some texts use the frequency f in hertz instead of the angular frequency \omega: only the 2\pi factors change.

The fundamental example: the rectangular pulse

Let f(t) = 1 for |t| \lt T/2 and 0 elsewhere.

    \[ F(\omega) = \int_{-T/2}^{T/2}e^{-i\omega t}\,dt = \frac{e^{i\omega T/2} - e^{-i\omega T/2}}{i\omega} = T\,\frac{\sin(\omega T/2)}{\omega T/2} \]

the sinc function.

A rectangular pulse of duration T and its Fourier transform, the sinc function
Left, a pulse of duration T; right, its spectrum, a sinc of height T with its first zero at 2π/T. The shorter the pulse, the wider the spectrum.

The first zero of the spectrum is at \omega = 2\pi/T, that is at frequency 1/T. A 1 µs pulse occupies a bandwidth of the order of 1 MHz; a 1 ns pulse, 1 GHz. It is the uncertainty principle of signals: a signal cannot be both short in time and narrow in frequency. Precisely, the product of duration and bandwidth (measured as standard deviations) is at least \tfrac12. That is why sending more bits per second needs more bandwidth.

Properties

PropertyTimeFrequency
linearitya f + b ga F + b G
delayf(t - t_0)e^{-i\omega t_0}F(\omega)
modulatione^{i\omega_0 t}f(t)F(\omega - \omega_0)
scalingf(at)\frac{1}{|a|}F(\omega/a)
derivativef'(t)i\omega F(\omega)
convolution(f * g)(t)F(\omega)\,G(\omega)
productf(t)\,g(t)\frac{1}{2\pi}(F * G)(\omega)

Some readings:

  • a delay does not change the magnitude of the spectrum, only the phase, which becomes linear in \omega;
  • modulation shifts the spectrum to the carrier frequency: the principle of every radio, which “moves” the audio signal up around a high frequency to transmit it;
  • the derivative becomes multiplication by i\omega: linear differential equations become algebraic, as with phasors;
  • compressing a signal in time widens it in frequency: playing a tape at double speed raises every frequency by an octave.

Convolution

The convolution of two signals is

    \[ (f * g)(t) = \int_{-\infty}^{\infty}f(\tau)\,g(t - \tau)\,d\tau \]

For a linear time-invariant system (a filter, an amplifier, a transmission channel), the output is the convolution of the input with the impulse response h(t): y = x * h. In the frequency domain convolution becomes a product:

    \[ Y(\omega) = H(\omega)\,X(\omega) \]

Proof

    \[ \int\left(\int f(\tau)g(t - \tau)d\tau\right)e^{-i\omega t}dt = \int f(\tau)e^{-i\omega\tau}\left(\int g(t - \tau)e^{-i\omega(t - \tau)}dt\right)d\tau = F(\omega)\,G(\omega) \]

swapping the order of integration and setting s = t - \tau in the inner integral.

H(\omega) is the system’s frequency response: it says how much each frequency is amplified and phase-shifted. That is why filters are designed in frequency: an ideal low-pass has H = 1 up to the cut-off frequency and 0 beyond.

Parseval and energy

    \[ \int_{-\infty}^{\infty}|f(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|F(\omega)|^2\,d\omega \]

A signal’s energy can be computed in time or in frequency. |F(\omega)|^2 is the energy spectral density, and it is what a spectrum analyser shows.

Duality

The two formulas of the pair are almost identical: only the sign of the exponent and the 2\pi factor change. It follows that if f(t) has transform F(\omega), the signal F(t) has transform 2\pi f(-\omega). So, since the rectangular pulse has a sinc spectrum, a sinc signal in time has a rectangular spectrum: it is the perfectly band-limited signal, the basis of the sampling theorem.

The sampling theorem

If a signal contains no frequencies above f_{max}, it can be rebuilt exactly from samples taken at a frequency

    \[ f_s \gt 2f_{max} \]

(Nyquist-Shannon theorem). Sampling creates copies of the spectrum around every multiple of f_s; if f_s \gt 2f_{max} the copies do not overlap and a low-pass filter recovers the original. If they do overlap there is aliasing: high frequencies masquerade as low ones, and it cannot be undone. That is why CD audio is sampled at 44.1 kHz (hearing reaches about 20 kHz) and why every analogue-to-digital converter is preceded by an anti-aliasing filter.

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