Finite-length frequency analysis that computers calculate.
DFT X[k]=Σ_{n=0}^{N-1}x[n]e^{-j2πkn/N} converts N samples to N frequency bins spaced f_s/N apart. Inverse reconstructs the samples. Assumes periodic extension, so leakage appears with non-integer cycles. Zero-padding interpolates the spectrum.
A 1024-point DFT at 8 kHz has 7.8 Hz bin spacing.
The DFT is the form computers use for spectrum analysis, so it sets the practical limits of frequency resolution and leakage.
Finite-length frequency analysis that computers calculate.
The DFT is the form computers use for spectrum analysis, so it sets the practical limits of frequency resolution and leakage.
The sampling rate divided by the number of samples, so longer records give finer frequency steps.
It treats the finite block as one period of a repeating signal, which causes leakage when the block does not end at the same value it started.
Typical applications include computing the spectrum of a recorded vibration block in software, estimating the harmonic content of a sampled waveform and finding the dominant frequency in a sensor record.
Study "Discrete-time Fourier transform" first (topic 26 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
Frequency resolution Δf=f_s/N. Real signals have symmetric spectra. Circular convolution.
A 1024-point DFT at 8 kHz has 7.8 Hz bin spacing.
The next topic is "Fast Fourier transform", which builds directly on this one.
It is classed as Beginner in Signals, Control & Comms (topic 27 of 101). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
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