50  Fourier transforms

From periodic decomposition to continuous frequency analysis

The Fourier series produces a discrete spectrum: a list of amplitudes at frequencies \pi/L, 2\pi/L, 3\pi/L, \ldots spaced \pi/L apart. As L \to \infty, the spacing between frequencies shrinks to zero, the list of amplitudes becomes a continuous function of frequency, and the sum becomes an integral. This is the Fourier transform.

The transition from series to transform is not just a mathematical technicality. It extends the analysis to any signal that decays to zero as |t| \to \infty — a one-shot pulse, a finite-duration transient, or any square-integrable function. The series was restricted to functions that repeat forever. The transform has no such restriction.

50.1 What this chapter helps you do

Symbols to keep handy

These are the bits of notation you'll see a lot. If a line of symbols feels like a fence, read it out loud once, then keep going.

  • {f}(): Fourier transform of f at frequency omega

  • f * g: f convolved with g

  • (): f hat of omega

  • F(): F of omega

  • (t): Dirac delta at t

  • e^{-it}: complex exponential kernel

Definitions to keep handy

These are the words we keep coming back to. If one feels slippery, come back here and steady it before you push on.

  • Fourier transform: A way to describe a non-periodic signal by how much of each frequency it contains.

  • spectrum: The frequency-domain picture of a signal: which frequencies are present and how strongly.

  • convolution: How linear systems combine an input with a response kernel; in frequency it becomes multiplication.

  • Dirac delta: An idealised impulse used to model an instantaneous event.

  • sinc: A bell-shaped oscillating function that appears as the transform of a box/window.

This chapter teaches you to move between time and frequency for signals that do not repeat forever. You will learn to:

  • understand the Fourier transform as the L \to \infty limit of Fourier series
  • compute transforms from the definition in simple cases and from tables in practical cases
  • use properties (shifts, scaling, differentiation) to avoid re-integrating from scratch
  • use convolution as the “system response” operation, and multiplication as its frequency-domain shortcut

Watch for this

  • A short event in time spreads out in frequency; a narrow band of frequencies spreads out in time. This is the time-bandwidth tradeoff.
  • Keep conventions straight: where the 2\pi sits and the sign in the exponential.
  • When a function is not absolutely integrable (step, constant), the transform still exists in a distribution sense, but you should treat it as a limit/idealisation.

50.2 From Fourier series to the Fourier integral

Consider a function f of period 2L with Fourier series:

This starts from the complex Fourier-series form introduced at the end of the previous chapter, where the real coefficients a_n and b_n were bundled into the single sequence c_n.

f(x) = \sum_{n=-\infty}^{\infty} c_n\,e^{in\pi x/L}, \qquad c_n = \frac{1}{2L}\int_{-L}^{L} f(x)\,e^{-in\pi x/L}\,dx

Define \omega_n = n\pi/L and \Delta\omega = \pi/L (the spacing between successive frequencies). Then:

f(x) = \sum_{n=-\infty}^{\infty} \frac{\Delta\omega}{2\pi}\left[\int_{-L}^{L} f(t)\,e^{-i\omega_n t}\,dt\right]e^{i\omega_n x}

The rewrite uses a single substitution: since \Delta\omega = \pi/L, the prefactor 1/(2L) in c_n equals \Delta\omega/(2\pi). Replacing it turns each coefficient into (\Delta\omega/2\pi)\times[\text{integral}], and the whole sum becomes a Riemann sum with step size \Delta\omega — ready to become an integral.

Let L \to \infty: \Delta\omega \to d\omega, the sum becomes an integral, and we obtain the Fourier integral representation:

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

The inner bracket is the Fourier transform of f evaluated at \omega.

50.3 The Fourier transform

How to read the transform pair

  • Symbol: F(\omega)=\mathcal{F}\{f\}(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt

  • Reads as: “F of omega is the Fourier transform of f”

  • Means: F(\omega) measures how much of frequency \omega is present in f(t)

  • Use when: frequency analysis, filtering, and linear time-invariant (LTI) systems

  • Common misread: \omega is angular frequency (rad/s), not cycles per second

  • Symbol: f(t)=\mathcal{F}^{-1}\{F\}(t)

  • Reads as: “f is the inverse Fourier transform of F”

  • Means: you can reconstruct the original signal from its spectrum

  • Use when: you have modified F(\omega) (filtering) and need the time-domain result

  • Common misread: inversion is not a separate guess; it is part of the same transform pair

The Fourier transform of f(t) is:

F(\omega) = \mathcal{F}\{f\}(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt

The inverse Fourier transform recovers f from F:

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

Together these are a transform pair: f(t) \leftrightarrow F(\omega). Knowing either one determines the other completely (for well-behaved functions).

Conventions. There are three common conventions for where the 2\pi factor sits and whether the sign in the exponent is -i\omega t or +i\omega t. The one above is the engineering convention (also used by Kreyszig and Stroud). The physics convention places 1/\sqrt{2\pi} on both the forward and inverse transform. When using a reference or software tool, check which convention it follows.

Existence. The Fourier transform is guaranteed to exist (as an ordinary integral) when f is absolutely integrable: \int_{-\infty}^{\infty}|f(t)|\,dt < \infty. For the larger class of square-integrable functions (\int|f|^2 < \infty), it exists in the L^2 sense. Important special cases — the unit step, the delta function, the constant function — require the theory of distributions but yield well-defined transforms. For engineering purposes, if the signal is physically realisable and has finite energy (or decays reasonably fast), you can safely treat its transform as existing.

50.4 Key transform pairs

50.4.1 Rectangular pulse

The rectangular pulse (or gate function) of width 2a:

f(t) = \begin{cases} 1 & |t| < a \\ 0 & |t| > a \end{cases}

Its transform:

F(\omega) = \int_{-a}^{a} e^{-i\omega t}\,dt = \left[\frac{e^{-i\omega t}}{-i\omega}\right]_{-a}^{a} = \frac{e^{-i\omega a} - e^{i\omega a}}{-i\omega} = \frac{2\sin(\omega a)}{\omega} = 2a\,\operatorname{sinc}(\omega a)

where \operatorname{sinc}(x) = \sin(x)/x.

The time–bandwidth tradeoff. The width of the main lobe of |F(\omega)| is approximately 2\pi/a. A narrow pulse (a small) has a wide spectrum. A wide pulse (a large) has a narrow spectrum. This is the mathematical statement of the tradeoff the bat exploits.

50.4.2 Gaussian

The Gaussian f(t) = e^{-\alpha t^2} (\alpha > 0) transforms to:

F(\omega) = \sqrt{\frac{\pi}{\alpha}}\,e^{-\omega^2/(4\alpha)}

also a Gaussian. The Gaussian is its own Fourier transform (up to scaling). It minimises the time–bandwidth product, which is why it is used in pulse shaping and wavelet theory.

50.4.3 Dirac delta

The Dirac delta is not an ordinary function — it has no well-defined value at any point, and you cannot draw its graph. It is defined entirely by what it does inside an integral: \int_{-\infty}^{\infty}\delta(t)\phi(t)\,dt = \phi(0) for any smooth \phi. Its transform:

\mathcal{F}\{\delta(t)\} = \int_{-\infty}^{\infty}\delta(t)\,e^{-i\omega t}\,dt = e^{0} = 1

The delta function has a flat (white) spectrum — it contains all frequencies equally. This is why a sharp impulse (the tap of a hammer on a structure) excites all resonant frequencies simultaneously.

The inverse: \mathcal{F}\{1\} = 2\pi\delta(\omega) — a constant signal contains only zero frequency. This means the delta is best understood as an idealised sampling object inside integrals, not as an ordinary spike-shaped function.

50.5 Properties of the Fourier transform

These properties make the Fourier transform useful in practice.

Linearity. \mathcal{F}\{\alpha f + \beta g\} = \alpha F + \beta G.

Time shifting. \mathcal{F}\{f(t - t_0)\} = e^{-i\omega t_0}F(\omega). Shifting a signal in time multiplies its spectrum by a complex exponential — it changes the phase but not the magnitude.

Frequency shifting. \mathcal{F}\{e^{i\omega_0 t}f(t)\} = F(\omega - \omega_0). Multiplying by a complex exponential shifts the spectrum. This is the mathematical basis of modulation: shifting a baseband signal to a carrier frequency. In engineering practice, modulation uses a real cosine \cos(\omega_0 t) = \frac{1}{2}(e^{i\omega_0 t} + e^{-i\omega_0 t}), which shifts the spectrum to both +\omega_0 and -\omega_0 (creating upper and lower sidebands); the complex exponential form isolates the single-sideband case.

Scaling. \mathcal{F}\{f(at)\} = \frac{1}{|a|}F\!\left(\frac{\omega}{a}\right). Compressing time by factor a expands the spectrum by the same factor — another form of the time–bandwidth tradeoff.

Differentiation. \mathcal{F}\{f'(t)\} = i\omega\,F(\omega). Differentiation in time corresponds to multiplication by i\omega in frequency. This turns differential equations into algebraic equations.

Convolution theorem.

\mathcal{F}\{f * g\}(\omega) = F(\omega)\cdot G(\omega)

where (f*g)(t) = \int_{-\infty}^{\infty}f(\tau)g(t-\tau)\,d\tau is the convolution of f and g. Equivalently, \mathcal{F}\{fg\} = \frac{1}{2\pi}F * G.

Parseval’s theorem. \int_{-\infty}^{\infty}|f(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|F(\omega)|^2\,d\omega.

This means total signal energy can be computed either in time or in frequency. In practice, Parseval lets you answer energy questions after transforming to the domain where the calculation is easier.

50.6 The convolution theorem in engineering

How to read convolution

  • Symbol: (f*h)(t)=\int_{-\infty}^{\infty}f(\tau)h(t-\tau)\,d\tau
  • Reads as: “f convolved with h”
  • Means: output at time t is a weighted sum of shifted copies of h, weighted by the input history
  • Use when: LTI systems (filters), where h is the impulse response
  • Common misread: convolution is not ordinary multiplication; it mixes across time

The convolution theorem is the most practically important property. A linear time-invariant (LTI) system is completely characterised by its impulse response h(t): if you drive the system with input f(t), the output is the convolution y(t) = (f * h)(t).

Computing convolution directly requires integrating over all past input values. In the frequency domain, this becomes multiplication:

Y(\omega) = F(\omega)\cdot H(\omega)

where H(\omega) = \mathcal{F}\{h\} is the transfer function of the system.

This is why filters are designed in the frequency domain. A low-pass filter that passes frequencies |\omega| < \omega_c and blocks higher frequencies has H(\omega) = 1 for |\omega| < \omega_c and H(\omega) = 0 otherwise. Computing H(\omega) \cdot F(\omega) is multiplication of two functions. Computing the equivalent convolution would mean implementing an integral.

In discrete time, the same structure holds: the discrete Fourier transform converts discrete convolution to pointwise multiplication. The FFT computes this in O(N\log N) time, which is why all modern digital signal processing systems work this way.

50.7 Solving differential equations with Fourier transforms

The differentiation property converts a linear ODE with constant coefficients into an algebraic equation in the frequency domain.

Example. Solve y' + 2y = e^{-|t|} for y(t).

Take the Fourier transform of both sides:

i\omega\,Y(\omega) + 2Y(\omega) = F(\omega)

(i\omega + 2)Y(\omega) = F(\omega)

Y(\omega) = \frac{F(\omega)}{i\omega + 2}

Now we need F(\omega) = \mathcal{F}\{e^{-|t|}\} = \int_{-\infty}^{\infty}e^{-|t|}e^{-i\omega t}\,dt = \frac{2}{1+\omega^2}.

To see this quickly, split at t=0: [ {-}^{} e{-|t|}e{-it},dt ={0}^{} e{-t}e{-it},dt+_{0}^{} e{-t}e{it},dt =+ =. ] So Y(\omega) = \frac{2}{(1+\omega^2)(i\omega+2)}.

Write this in partial fractions:

\frac{2}{(1+\omega^2)(i\omega+2)} = \frac{A}{i\omega+2} + \frac{B\omega + C}{1+\omega^2}

Matching numerators gives A = 2/5, B = -2/5, C = 4/5, so:

Y(\omega) = \frac{2/5}{i\omega+2} + \frac{(-2\omega + 4)/5}{1+\omega^2}

Each term matches a known transform pair: \mathcal{F}^{-1}\{1/(i\omega+2)\} = e^{-2t} for t > 0 (causal exponential); \mathcal{F}^{-1}\{1/(1+\omega^2)\} = \frac{1}{2}e^{-|t|}; and \mathcal{F}^{-1}\{\omega/(1+\omega^2)\} gives a derivative term. Inverting term by term:

y(t) = \frac{1}{5}\left(2e^{-2t} - e^{-|t|} + \operatorname{sgn}(t)e^{-|t|}\right)

The Fourier transform has reduced the differential equation to algebra.

50.8 Connection to the Laplace transform

If you have met the Laplace transform in an ODE course, this section shows how it connects to the Fourier transform. If not, you can skip it — nothing later in this chapter depends on it.

The Laplace transform \mathcal{L}\{f\}(s) = \int_0^{\infty}f(t)e^{-st}\,dt is a one-sided transform: it only sees t \geq 0 (causal signals). Replacing s = i\omega recovers the Fourier transform of the causal part of f.

The Laplace transform is preferred for initial-value problems (ODEs with t \geq 0 and given initial conditions) because it handles them directly via \mathcal{L}\{f'\} = sF(s) - f(0). The Fourier transform is preferred for steady-state and bilateral analysis (signals for all t).

In circuit analysis: the transfer function H(s) in Laplace domain becomes H(i\omega), the Fourier domain transfer function, when you restrict to purely sinusoidal steady-state inputs.

50.9 Where this goes

This chapter is the bridge from periodic mode decomposition (Fourier series) to continuous spectral analysis. The next chapter in this block uses the transform as a solver tool: many PDEs become algebraic in \omega after transforming in space or time, then you invert back.

More broadly, the transform is the foundation for frequency-response thinking: LTI systems become multiplication by a transfer function, and filtering becomes “shape the spectrum, then invert.”

50.10 What you can do now

You can now compute Fourier transforms in basic cases, use the property table to avoid redoing integrals, and use convolution/multiplication to reason about LTI system behaviour. You should also be able to keep track of conventions and interpret \omega as angular frequency.

50.11 Exercises

50.11.1 Exercise 1: Transform of a rectangular pulse

Compute the Fourier transform of f(t) = \begin{cases}1 & |t| \leq 1 \\ 0 & |t| > 1\end{cases}. Identify the first zero of |F(\omega)|.

50.11.2 Exercise 2: Time-shifting theorem

Given that \mathcal{F}\{e^{-|t|}\} = \frac{2}{1+\omega^2}, find \mathcal{F}\{e^{-|t-3|}\} and \mathcal{F}\{e^{-|t|}e^{2it}\}.

50.11.3 Exercise 3: Differentiation property — solving an ODE

Use the Fourier transform to solve y' + 3y = e^{-2|t|}.

(The inversion uses only partial fractions and the known pairs from this chapter — no complex analysis required.)

50.11.4 Exercise 4: Convolution theorem — LTI system response

A low-pass filter has impulse response h(t) = e^{-2t}u(t) (where u(t) is the unit step: u(t) = 1 for t \geq 0, 0 otherwise). The input is f(t) = e^{-t}u(t). Find the output y(t) = (f * h)(t) using the Fourier transform.

50.11.5 Exercise 5: Parseval’s theorem — energy in the frequency domain

The voltage pulse across a 1 Ω resistor is v(t) = e^{-|t|}. Use Parseval’s theorem to find the total energy dissipated.