Fourier Transform Reference Cheat Sheet

Fourier transforms express a function in terms of continuous frequencies, making them essential in calculus, differential equations, signal processing, and physics. This cheat sheet summarizes the angular frequency convention, where frequency is measured by $\omega$ in radians per unit. It helps students quickly compare definitions, properties, and common transform pairs without searching through a textbook.

The reference is especially useful when solving integrals, analyzing linear systems, or converting differential equations into algebraic equations.

The main formulas are the forward transform $F(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$ and the inverse transform $f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)e^{i\omega t}\,d\omega$ . Core ideas include linearity, scaling, shifting, differentiation, convolution, and energy preservation through Parseval's theorem. Common pairs such as Gaussians, exponentials, rectangular pulses, sines, cosines, and delta functions provide fast shortcuts for computation.

The most important skill is matching the convention and applying each property with the correct constants.

Key Facts

Using the angular frequency convention, the Fourier transform is $F(\omega)=\mathcal{F}\{f(t)\}=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$ .
The inverse Fourier transform is $f(t)=\mathcal{F}^{-1}\{F(\omega)\}=\frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)e^{i\omega t}\,d\omega$ .
Linearity means $\mathcal{F}\{af(t)+bg(t)\}=aF(\omega)+bG(\omega)$ for constants $a$ and $b$ .
Time shifting gives $\mathcal{F}\{f(t-t_0)\}=e^{-i\omega t_0}F(\omega)$ , while frequency shifting gives $\mathcal{F}\{e^{i\omega_0 t}f(t)\}=F(\omega-\omega_0)$ .
Scaling gives $\mathcal{F}\{f(at)\}=\frac{1}{|a|}F\left(\frac{\omega}{a}\right)$ for $a\ne 0$ .
Differentiation in time becomes multiplication in frequency: $\mathcal{F}\{f'(t)\}=i\omega F(\omega)$ when boundary terms vanish.
Convolution in time becomes multiplication in frequency: $\mathcal{F}\{(f*g)(t)\}=F(\omega)G(\omega)$ , where $(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau$ .
Parseval's theorem states $\int_{-\infty}^{\infty}|f(t)|^2\,dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}|F(\omega)|^2\,d\omega$ .

Vocabulary

Fourier transform: A transform that rewrites a function $f(t)$ as a frequency-domain function $F(\omega)$ using complex exponentials.
Inverse Fourier transform: The operation that reconstructs $f(t)$ from $F(\omega)$ using $f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega$ .
Angular frequency: The frequency variable $\omega$ measured in radians per unit, related to ordinary frequency by $\omega=2\pi f$ .
Convolution: An integral combination of two functions defined by $(f*g)(t)=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)\,d\tau$ .
Dirac delta function: A generalized function $\delta(t)$ with the sifting property $\int_{-\infty}^{\infty}f(t)\delta(t-a)\,dt=f(a)$ .
Transform pair: A matched pair $f(t)\leftrightarrow F(\omega)$ showing a function and its Fourier transform under a chosen convention.

Common Mistakes to Avoid

Mixing Fourier transform conventions is wrong because the factors of $2\pi$ change between definitions. If $F(\omega)=\int f(t)e^{-i\omega t}\,dt$ , then the inverse must include $\frac{1}{2\pi}$ .
Forgetting the absolute value in scaling gives the wrong amplitude. The correct rule is $\mathcal{F}\{f(at)\}=\frac{1}{|a|}F\left(\frac{\omega}{a}\right)$ , not $\frac{1}{a}F\left(\frac{\omega}{a}\right)$ in all cases.
Reversing the sign in the time-shift factor changes the phase in the wrong direction. The correct formula is $\mathcal{F}\{f(t-t_0)\}=e^{-i\omega t_0}F(\omega)$ .
Treating convolution as ordinary multiplication in the time domain loses the integral structure. The convolution $(f*g)(t)$ transforms to $F(\omega)G(\omega)$ , but $f(t)g(t)$ does not transform to $F(\omega)G(\omega)$ .
Ignoring distribution terms for constants and sinusoids leads to incomplete answers. For example, $\mathcal{F}\{1\}=2\pi\delta(\omega)$ and $\mathcal{F}\{e^{i\omega_0 t}\}=2\pi\delta(\omega-\omega_0)$ .

Practice Questions

1 Find $\mathcal{F}\{e^{-a|t|}\}$ for $a>0$ using $F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt$ .
2 If $\mathcal{F}\{f(t)\}=F(\omega)$ , find the Fourier transform of $3f(2t-4)$ in terms of $F(\omega)$ .
3 Use the differentiation property to transform the differential equation $y''(t)-4y(t)=g(t)$ into an algebraic equation involving $Y(\omega)$ and $G(\omega)$ , assuming boundary terms vanish.
4 Explain why a narrow pulse in the time domain tends to have a broad Fourier transform in the frequency domain.

Understanding Fourier Transform Reference

A Fourier transform measures how strongly each repeating wave is present in a signal. A single pure tone produces one concentrated frequency component. A short click contains a wide spread of frequencies.

This is not a flaw. It reflects a basic tradeoff. A signal tightly confined in time cannot be tightly confined in frequency.

This idea explains why a drum hit sounds broad and why a long, steady note has a clear pitch. The transform uses complex waves because they track size and timing together. The magnitude tells how much of a frequency is present.

The phase tells where that wave is aligned in time. Ignoring phase can seriously change the reconstructed signal, even when the magnitudes stay the same.

Several transform rules have a physical meaning that helps prevent mistakes. Delaying a sound recording does not change its frequency magnitudes, but it changes phase. Stretching a pulse in time makes its frequency content narrower.

Compressing the same pulse makes the frequency content wider. Taking a time derivative emphasizes rapid changes, so high frequencies receive greater weight. This is why differentiation can amplify noise in measured data.

In contrast, integration tends to reduce high frequency variation. When applying these rules, students should first identify the independent variable, then decide whether the change is a shift, a scale change, a modulation, or a derivative. Mixing these operations is a common source of incorrect signs and missing factors.

Convolution describes the output of many systems. Imagine a room responding to a short clap. The room has an impulse response that includes echoes and absorption.

Convolving the original sound with that response gives the sound heard in the room. The same model appears in camera blur, electrical circuits, heat flow, and image filters. Multiplication in frequency makes this process easier because each frequency can be treated separately.

A low pass filter keeps slow variations and reduces rapid ones. A high pass filter does the reverse. This viewpoint matters in real measurements, since a sensor often changes the signal before anyone analyzes it.

Delta functions need special care because they are not ordinary functions. They represent an ideal input concentrated at one instant or one location. Their main job is to select a value inside an integral or describe an ideal impulse.

Real pulses always have finite width and height, but a delta model can still be useful when that width is tiny compared with the timescale being studied. Parseval's theorem provides an important check on calculations. It says that total signal energy can be computed from either representation, with the required convention factor included.

Before trusting an answer, check units, signs in exponential terms, factors from scaling, and whether the function has enough decay for the usual integral rules to apply. Some important signals require generalized functions or limiting arguments rather than ordinary integrals.

Sign in to save

Sign in to save

Fourier Transform Reference Cheat Sheet

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Key Facts

Vocabulary

Common Mistakes to Avoid

Practice Questions

Understanding Fourier Transform Reference