Frequency analysis of discrete sequences.
DTFT X(e^{jω})=Σx[n]e^{-jωn} gives a continuous, 2π-periodic spectrum of a discrete sequence. Used for theory and filter design; practical computation uses the DFT, which samples the DTFT. Properties mirror continuous Fourier transform.
The DTFT of a moving-average filter is a sinc-like low-pass response.
The discrete-time Fourier transform analyses sampled sequences, which is needed for understanding digital filters and sampled data spectra.
Frequency analysis of discrete sequences.
The discrete-time Fourier transform analyses sampled sequences, which is needed for understanding digital filters and sampled data spectra.
Sampling makes the spectrum repeat every sampling frequency, so the useful range is one period, usually from minus to plus half the sampling rate.
The DTFT is continuous in frequency and infinite in time, while the DFT is discrete in frequency and works on a finite block of samples.
Typical applications include finding the frequency response of a digital filter from its coefficients, describing the spectrum of a sampled audio signal and explaining why the spectrum of a sampled signal repeats every sampling frequency.
Study "Quantisation" first (topic 25 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
Periodic in ω. Frequency response of digital filters. Discrete convolution↔multiplication.
The DTFT of a moving-average filter is a sinc-like low-pass response.
The next topic is "Discrete Fourier transform", which builds directly on this one.
It is classed as Beginner in Signals, Control & Comms (topic 26 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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