Why does the sine preset give a tall bar at one bin?

When the sine completes an integer number of full cycles across exactly N samples, its energy aligns with a single DFT basis vector (up to the conjugate symmetry mirror at N−k). That produces a concentrated peak. If you painted a sine-like wave with a non-integer number of cycles, energy would spread across neighboring bins—an example of leakage tied to implicit rectangular windowing.

Why is the impulse preset’s spectrum spread out instead of a single spike?

A single nonzero sample is the discrete analogue of a very short pulse. In the frequency domain, short time localization forces a broad spectral footprint. Conversely, a narrowband tone in frequency requires many oscillations across the window—illustrating the reciprocal relationship between duration and bandwidth.

Are the green bars power, amplitude, or something else?

They show the magnitude |X[k]| = √(Re(X[k])² + Im(X[k])²) of each complex coefficient. The display scales bars by the largest magnitude so peaks are easy to compare; absolute numbers appear in the sidebar for DC, the strongest interior bin, and Nyquist.

Does this replace a full FFT library for real engineering work?

No. It is a teaching FFT with a fixed length, no window functions beyond the implicit rectangle, no zero-padding controls, and no phase plot. Production spectrum analyzers add windowing, overlap, calibration, and often use hardware-accelerated transforms.

FFT Magnitude Spectrum

Paint or preset a 256-point signal; radix-2 FFT shows |X[k]| vs bin (DC to Nyquist).

About this model

The discrete Fourier transform (DFT) reveals how much of each sinusoidal frequency is present in a finite sequence of samples. This simulator fixes N = 256 real-valued samples x[n], computes a radix-2 fast Fourier transform (FFT) in software, and plots the magnitudes |X[k]| for frequency bins k = 0 through N/2, covering the DC component, positive-frequency bins, and the Nyquist bin. The time-domain view shows the sample sequence as a polyline; users can load presets (pure sine with an integer number of cycles in the window, two-tone mixture, band-limited square wave, centered impulse, or uniform noise) or paint the waveform directly, which switches to a custom buffer. Peak normalization in the spectrum panel emphasizes relative energy distribution. The companion Fourier Series page builds periodic waveforms by adding harmonics in the time domain; this page performs the complementary frequency-domain analysis on a fixed-length window, foreshadowing the continuous Fourier transform, windowing effects, and spectral leakage when a non-integer number of cycles is captured.

Who it's for: Undergraduate students in signals and systems, numerical analysis, or physics labs introducing spectral analysis, and anyone who already saw Fourier series and wants the discrete transform picture.

Key terms

Discrete Fourier Transform (DFT)
Fast Fourier Transform (FFT)
Frequency bin
DC component
Nyquist frequency
Magnitude spectrum
Time–frequency intuition
Spectral leakage

How it works

Sample a real signal on N = 256 points, then take a radix-2 FFT. The lower panel shows magnitude |X[k]| for bins k = 0 … N/2 (DC, positive frequencies, Nyquist). A single sine with an integer number of cycles in the window gives a narrow peak; a pulse spreads energy across bins—time–bandwidth intuition before the continuous Fourier transform.

Key equations

X[k] = Σ_{n=0}^{N−1} x[n] · e^{−2π i kn/N}

|X[k]| = √(Re² + Im²) · k = 0 … N/2

Frequently asked questions

Why does the sine preset give a tall bar at one bin?: When the sine completes an integer number of full cycles across exactly N samples, its energy aligns with a single DFT basis vector (up to the conjugate symmetry mirror at N−k). That produces a concentrated peak. If you painted a sine-like wave with a non-integer number of cycles, energy would spread across neighboring bins—an example of leakage tied to implicit rectangular windowing.
Why is the impulse preset’s spectrum spread out instead of a single spike?: A single nonzero sample is the discrete analogue of a very short pulse. In the frequency domain, short time localization forces a broad spectral footprint. Conversely, a narrowband tone in frequency requires many oscillations across the window—illustrating the reciprocal relationship between duration and bandwidth.
Are the green bars power, amplitude, or something else?: They show the magnitude |X[k]| = √(Re(X[k])² + Im(X[k])²) of each complex coefficient. The display scales bars by the largest magnitude so peaks are easy to compare; absolute numbers appear in the sidebar for DC, the strongest interior bin, and Nyquist.
Does this replace a full FFT library for real engineering work?: No. It is a teaching FFT with a fixed length, no window functions beyond the implicit rectangle, no zero-padding controls, and no phase plot. Production spectrum analyzers add windowing, overlap, calibration, and often use hardware-accelerated transforms.

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