Notes on the Fourier Transform - Eli Bendersky's website

From Fourier Series to the Fourier Transform

Fourier series give us a powerful way to analyze periodic functions, but they fall short when a function never repeats. We can extend the idea of a Fourier series to non-periodic functions defined over the entire real line by considering what happens as the period grows without bound.

To see where this leads, take a non-periodic function defined on a finite interval and imagine its Fourier series with an ever-increasing assumed period. As the period grows, two things happen: the approximation requires more coefficients to stay accurate, and the spacing between adjacent harmonic frequencies shrinks. If we push this to the limit of an infinite period, the discrete set of Fourier coefficients becomes a continuous function of frequency. This continuous function is the Fourier transform.

Deriving the Transform from the Series

Starting from the complex exponential form of the Fourier series, we can reframe the equations in terms of the angular frequency ω_n = 2πn/L. The difference between consecutive frequencies is then Δω = π/L. Substituting these into the series and its coefficients, we take the limit as L → ∞ (which makes Δω → 0).

The sum over discrete frequencies becomes a Riemann sum, which in the limit turns into an integral. Carrying this through yields the core result—the Fourier transform pair. The forward transform maps a time-domain function f(t) to its frequency-domain representation F(ω):

The original function is recovered via the inverse transform, which is an integral over all frequencies.

A Concrete Example: The Triangular Pulse

Consider an odd, triangular pulse defined on [-L, L] and zero beyond that range. Because the function is odd, the cosine (real) part of its transform integral vanishes. The sine (imaginary) part remains and can be evaluated to give a purely imaginary result.

The transform F(ω) in this case is proportional to sin²(ωL/2) / ω². At ω = 0, this expression appears undefined due to division by zero, but applying L'Hôpital's rule shows the limit is finite. Since the transform is purely imaginary, its phase is either -π/2 or +π/2, depending on the sign of the numerator, and its magnitude follows the sinc-squared-like shape of the formula.

Plotting the magnitude and phase separately is the standard way to visualize a complex-valued transform. This frequency-domain view shows how strongly each frequency contributes to the original signal, and how each component is shifted in phase.

Understanding the Frequency Domain

When the independent variable represents time, the Fourier transform converts a signal into its frequency-domain representation. The two representations are dual: each fully describes the same function, just from a different perspective. A time-domain plot shows how a signal evolves, while a frequency-domain plot reveals its distribution across frequencies. Because the transform is generally complex-valued, every frequency component carries both a magnitude and a phase. This perspective is central to signal analysis, filter design, and systems theory.

Not every function has a Fourier transform. A sufficient condition is absolute integrability—that the integral of |f(t)| over the whole real line is finite. Under this condition, F(ω) exists, is continuous, and tends to zero as |ω| → ∞. This condition is not strictly necessary, but it covers the well-behaved functions typical in engineering. A practical additional assumption is that real-world signals—having finite energy—also vanish as |t| → ∞.

This also explains why Fourier transforms are not suitable for periodic functions: they are not absolutely integrable. For repeating signals, we return to Fourier series.

Key Properties of the Fourier Transform

Several properties make the transform invaluable for both analysis and computation. These follow directly from the defining integrals.

Linearity. The transform is linear because integration is linear. Scaling a function by a constant and adding signals transforms term by term, and the same holds for the inverse transform.

Time and frequency scaling. Stretching a function in time compresses its transform in frequency, and vice versa. If f(t) is scaled by a factor a, the transform of f(at) is (1/|a|)F(ω/a). This makes intuitive sense for signals: compressing a signal in time demands sharper transitions, shifting energy to higher frequencies.

Time shifting. Delaying a signal by a constant t₀ multiplies its transform by a phase factor e-jωt₀. The magnitude spectrum is unchanged; only the phase is affected.

Transforms of derivatives. Integration by parts shows that the transform of f'(t) is jωF(ω), provided f(t) vanishes at ±∞. This property is a cornerstone in solving differential equations with transforms.

The convolution theorem. Convolution in the time domain corresponds to simple multiplication in the frequency domain. The transform of (f * g)(t) is the product F(ω)G(ω). This result is pivotal in signal processing, where it turns complex convolution operations into straightforward pointwise multiplications.

Appendix: Riemann Sums as Integrals

The leap from the discrete Fourier series coefficients to the continuous transform hinges on recognizing a Riemann sum. For a function h(x) over an interval [a, b], we partition the interval into sub-intervals of width Δx and approximate the area under the curve with rectangles:

The sum of the rectangle areas is a Riemann sum. As the partition gets finer (Δx → 0) and the number of rectangles grows, the sum approaches the definite integral of h(x) over [a, b]—the Riemann integral. The exact point chosen within each sub-interval to evaluate the rectangle height becomes irrelevant in the limit, which is precisely why replacing the discrete frequency sum with an integral is valid.