Fourier Transform Glossary

Aliasing

Aliasing occurs when a continuous signal is sampled at a rate too low to capture its highest frequency components, causing those high-frequency components to appear as lower-frequency artifacts in the sampled data. This violates the Nyquist–Shannon sampling theorem. For example, a 900 Hz tone sampled at 1000 Hz will appear as a 100 Hz tone in the resulting discrete signal. Anti-aliasing filters are applied before sampling to remove frequencies above the Nyquist limit.

Amplitude

Amplitude is the maximum displacement of a signal from its equilibrium (zero) value. In a sinusoidal waveform, it represents the peak value of the oscillation. The amplitude of a frequency component can be extracted from the Fourier transform by computing the magnitude of the corresponding complex coefficient. Amplitude is distinct from power, which is proportional to the square of amplitude.

Formula:  A\sin(2\pi f t)

Bandwidth

Bandwidth is the range of frequencies over which a signal or system operates, typically measured as the difference between the highest and lowest significant frequency components. In communications, it determines how much information a channel can carry per unit time. A signal with components from 200 Hz to 3400 Hz has a bandwidth of 3200 Hz. Filters are often characterized by their passband bandwidth.

Complex Number

A complex number has the form $z = a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i = \sqrt{-1}$. Complex numbers are essential in Fourier analysis because each frequency bin in a DFT produces a complex value encoding both magnitude and phase. The polar form $z = r e^{i\theta}$ directly separates these two quantities.

Convolution

Convolution is a mathematical operation that combines two signals to produce a third, describing how the shape of one is modified by the other. In continuous time it is defined as $(f * g)(t) = \int_{-\infty}^{\infty} f(\tau)\,g(t - \tau)\,d\tau$. A key property — the convolution theorem — states that convolution in the time domain equals pointwise multiplication in the frequency domain, making FFT-based convolution extremely efficient for long signals.

Cutoff Frequency

The cutoff frequency is the boundary frequency at which a filter begins to significantly attenuate the input signal. For a low-pass filter, frequencies below the cutoff pass through with little change while those above are progressively reduced. The cutoff is commonly defined as the −3 dB point, where the output power drops to half the passband value. Choosing the right cutoff frequency is critical in anti-aliasing, noise removal, and audio equalization.

DFT (Discrete Fourier Transform)

The Discrete Fourier Transform converts a finite sequence of equally-spaced samples into a same-length sequence of complex numbers representing sinusoidal frequency components. It is defined as $X[k] = \sum_{n=0}^{N-1} x[n]\,e^{-i 2\pi k n / N}$. The DFT is the mathematical foundation for spectral analysis of sampled data; the FFT is simply a fast algorithm for computing it.

Discrete Signal

A discrete signal is one that is defined only at specific, usually equally-spaced, points in time (or space). It is the result of sampling a continuous signal and is represented as a sequence of numbers $x[n]$. All digital audio, images, and sensor data are discrete signals. The DFT and FFT operate exclusively on discrete signals.

Euler's Formula

Euler's formula states that $e^{i\theta} = \cos\theta + i\sin\theta$. It provides a compact notation for sinusoidal signals using complex exponentials and is the backbone of Fourier analysis. The special case $e^{i\pi} + 1 = 0$ (Euler's identity) elegantly links five fundamental constants. In the DFT, each basis function is a complex exponential derived directly from this formula.

FFT (Fast Fourier Transform)

The Fast Fourier Transform is a family of algorithms that compute the DFT in $O(N \log N)$ time instead of the naïve $O(N^2)$. The most famous variant is the Cooley–Tukey radix-2 algorithm, which recursively splits the DFT into smaller sub-problems. The FFT has been called one of the most important algorithms of the 20th century and underpins real-time audio processing, telecommunications, medical imaging, and more.

Filter

A filter is a system that selectively modifies the frequency content of a signal. Common types include low-pass (passes low frequencies), high-pass (passes high frequencies), band-pass (passes a specific range), and band-stop (rejects a specific range). Filters can be implemented in hardware (analog) or software (digital, using difference equations or FFT-based multiplication). Digital filters are categorized as FIR (finite impulse response) or IIR (infinite impulse response).

Fourier Series

A Fourier series decomposes a periodic function into an infinite sum of sines and cosines (or complex exponentials) at integer multiples of the fundamental frequency. For a function with period $T$, the coefficients capture how much energy resides in each harmonic. The Fourier series is a precursor to the Fourier transform, which extends the idea to non-periodic signals by letting the period tend to infinity.

Fourier Transform

The Fourier transform maps a time-domain signal to the frequency domain, revealing the amplitude and phase of each constituent frequency. The continuous Fourier transform is defined as $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(t)\,e^{-i 2\pi \xi t}\,dt$. It is one of the most powerful tools in mathematics, physics, and engineering, enabling spectral analysis, filtering, compression, and solving differential equations.

Frequency

Frequency is the number of complete cycles a periodic signal completes per unit of time, measured in hertz (Hz). A 440 Hz tone completes 440 oscillations every second. Angular frequency $\omega = 2\pi f$ is often used in mathematical formulations. The Fourier transform decomposes arbitrary signals into their constituent frequencies.

Frequency Domain

The frequency domain is a representation of a signal in terms of its frequency components rather than its variation over time. While a time-domain plot shows amplitude vs. time, a frequency-domain plot (spectrum) shows amplitude (or power) vs. frequency. Many operations — such as filtering, compression, and feature extraction — are far simpler and more intuitive when performed in the frequency domain.

Frequency Resolution

Frequency resolution is the smallest frequency difference that can be distinguished in a DFT, calculated as $\Delta f = f_s / N$, where $f_s$ is the sampling rate and $N$ is the number of samples. A longer signal (larger $N$) yields finer frequency resolution. Zero-padding can interpolate the spectrum but does not truly improve resolution — only a longer observation window achieves that.

Harmonic

A harmonic is a frequency component that is an integer multiple of a fundamental frequency. If the fundamental is $f_0$, the second harmonic is $2f_0$, the third is $3f_0$, and so on. Harmonics determine the timbre (tone color) of musical instruments and speech. The Fourier series naturally expresses periodic signals as sums of harmonics.

Hertz (Hz)

Hertz is the SI unit of frequency, defined as one cycle per second. Named after physicist Heinrich Hertz, it is used to quantify oscillation rates in signals, sampling rates of digital systems, clock speeds, and electromagnetic wave frequencies. Common multiples include kilohertz (kHz = 1,000 Hz), megahertz (MHz = 10⁶ Hz), and gigahertz (GHz = 10⁹ Hz).

Impulse Response

The impulse response of a linear time-invariant (LTI) system is the output produced when the input is a unit impulse (Dirac delta). It completely characterizes the system: the output for any input signal can be found by convolving that input with the impulse response. In the frequency domain, the Fourier transform of the impulse response yields the transfer function (frequency response) of the system.

Inverse FFT (IFFT)

The Inverse Fast Fourier Transform converts frequency-domain data back into the time domain, reconstructing the original signal from its spectral representation. Mathematically, $x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k]\,e^{i 2\pi k n / N}$. The IFFT is essential in spectral processing workflows: transform to the frequency domain, modify the spectrum (e.g., filter), then inverse-transform back to time.

Leakage

Spectral leakage occurs when a finite-length signal segment does not contain an exact integer number of cycles of a frequency component. The energy of that component "leaks" into adjacent frequency bins, smearing the spectrum. Window functions (Hann, Hamming, Blackman, etc.) taper the signal edges toward zero, significantly reducing leakage at the cost of slightly wider main lobes.

Linear Time-Invariant (LTI)

A system is linear time-invariant if it obeys both superposition (linearity) and time-shift invariance. LTI systems are fully described by their impulse response, and their output for any input is the convolution of that input with the impulse response. Most classical filter design, control theory, and signal processing theory is built on the LTI framework, because the Fourier transform diagonalizes LTI systems into independent frequency channels.

Magnitude

In the context of the Fourier transform, magnitude is the absolute value of a complex spectral coefficient: $|X[k]| = \sqrt{\text{Re}(X[k])^2 + \text{Im}(X[k])^2}$. It indicates the strength (amplitude) of a particular frequency component, regardless of phase. A magnitude spectrum plot shows how signal energy is distributed across frequencies.

Nyquist Frequency

The Nyquist frequency is half the sampling rate: $f_{Nyq} = f_s / 2$. It represents the highest frequency that can be faithfully represented in a sampled signal. Any frequency content above the Nyquist frequency will alias to a lower frequency unless removed by an anti-aliasing filter before sampling. For CD-quality audio sampled at 44,100 Hz, the Nyquist frequency is 22,050 Hz.

Nyquist Rate

The Nyquist rate is the minimum sampling rate required to avoid aliasing for a given signal, equal to twice the signal's maximum frequency: $f_s \geq 2 f_{max}$. If the highest frequency in a signal is 20 kHz (the upper limit of human hearing), the Nyquist rate is 40 kHz. Practical systems sample slightly above the Nyquist rate to allow for imperfect anti-aliasing filters.

Phase

Phase describes the offset of a sinusoidal component relative to a reference point, typically expressed in radians or degrees. In the Fourier transform, each complex coefficient encodes phase as $\phi = \text{atan2}(\text{Im}(X[k]),\,\text{Re}(X[k]))$. While often less visually intuitive than magnitude, phase carries critical information — altering the phase spectrum while keeping magnitudes fixed can drastically change a signal's shape.

Power Spectrum

The power spectrum (or power spectral density) shows how the power of a signal is distributed across frequencies. It is computed as the squared magnitude of the Fourier transform: $P[k] = |X[k]|^2$. Unlike the magnitude spectrum, the power spectrum discards phase information entirely. It is widely used in vibration analysis, audio processing, and telecommunications to quantify signal energy at each frequency.

Sampling Rate

The sampling rate (or sample rate) is the number of samples taken per second when digitizing a continuous signal, measured in hertz. Common rates include 8 kHz (telephony), 44.1 kHz (CD audio), 48 kHz (professional audio/video), and 96 kHz (high-resolution audio). The sampling rate determines the Nyquist frequency and thus the maximum frequency the digital system can represent.

Sidelobe

Sidelobes are the secondary peaks that appear in the frequency response of a window function or filter, flanking the main lobe. They are a consequence of applying a finite-length window to a signal and contribute to spectral leakage. Different window functions trade off main-lobe width against sidelobe level — for instance, a Hamming window has lower sidelobes than a rectangular window but a wider main lobe.

Signal

A signal is any quantity that varies over time, space, or another independent variable, carrying information. Examples include audio waveforms, radio transmissions, ECG recordings, seismic data, and pixel intensity in images. Signals can be continuous (analog) or discrete (digital). Signal processing — including Fourier analysis — extracts, modifies, and interprets the information encoded in signals.

Sinusoid

A sinusoid is a smooth, periodic waveform described by the sine or cosine function: $x(t) = A\sin(2\pi f t + \phi)$, where $A$ is amplitude, $f$ is frequency, and $\phi$ is phase. Sinusoids are the fundamental building blocks of Fourier analysis — the Fourier transform expresses arbitrary signals as weighted sums (or integrals) of sinusoids at different frequencies.

Spectral Density

Spectral density (or power spectral density, PSD) quantifies how the power of a signal is spread across frequencies on a per-hertz basis. Unlike the power spectrum of a finite-length signal, PSD is normalized by frequency resolution, producing a density function with units such as V²/Hz. PSD is the standard tool for characterizing random and stochastic signals, noise floors, and broadband processes.

Spectrogram

A spectrogram is a visual representation of the frequency content of a signal as it changes over time. It is generated by computing the Short-Time Fourier Transform (STFT) — dividing the signal into overlapping segments, applying a window function, and computing the FFT of each segment. The result is a heat map with time on the x-axis, frequency on the y-axis, and color (or brightness) indicating magnitude or power.

Spectrum

A spectrum is the frequency-domain representation of a signal, showing the distribution of its frequency components. The magnitude spectrum displays how much of each frequency is present; the phase spectrum shows the phase offset of each component. Spectra can be one-sided (showing only positive frequencies) or two-sided. The term also refers broadly to the range of frequencies in electromagnetic radiation, light, or sound.

STFT (Short-Time Fourier Transform)

The Short-Time Fourier Transform analyzes how a signal's frequency content evolves over time by applying the Fourier transform to successive windowed segments of the signal. It is defined as $\text{STFT}\{x(t)\}(\tau, f) = \int x(t)\,w(t-\tau)\,e^{-i2\pi ft}\,dt$, where $w$ is the window function. The STFT faces a time-frequency trade-off: a longer window improves frequency resolution but reduces time resolution, and vice versa.

Time Domain

The time domain is the default representation of a signal, showing how its value changes over time. An oscilloscope display or raw audio waveform is a time-domain view. The Fourier transform converts a time-domain signal into the frequency domain and back (via the inverse transform), allowing engineers to choose the most convenient representation for a given task.

Transfer Function

A transfer function describes the input–output relationship of an LTI system in the frequency domain. It is the Fourier (or Laplace / Z) transform of the system's impulse response: $H(f) = \mathcal{F}\{h(t)\}$. Multiplying the input spectrum by the transfer function yields the output spectrum. Transfer functions are central to filter design, control systems, and audio equalization.

Waveform

A waveform is a graphical representation of a signal's amplitude over time. Common waveform shapes include sine, square, triangle, and sawtooth waves. Each shape has a unique harmonic structure — for instance, a square wave contains only odd harmonics. In digital audio, the waveform is a sequence of sample values; in analog electronics, it is a continuous voltage or current variation.

Window Function

A window function is a weighting function applied to a finite segment of a signal before computing its DFT, designed to reduce spectral leakage. Common windows include Hann, Hamming, Blackman, and Kaiser. Each window trades off main-lobe width (frequency resolution) against sidelobe level (leakage suppression). The rectangular window (no weighting) has the narrowest main lobe but the highest sidelobes.

Zero-Padding

Zero-padding is the practice of appending zeros to a signal before computing its DFT, increasing the transform length. This interpolates additional points in the frequency domain, producing a smoother-looking spectrum, but does not improve true frequency resolution (which depends on the original signal duration). Zero-padding is also used to extend a signal's length to a power of two for efficient radix-2 FFT computation.