Correspondence between time domain and frequency domain of DSP

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Correspondence between time domain and frequency domain of DSP [Copy link]

Below is the correspondence between time domain and frequency domain: Continuous non-periodic signal in time domain ←→ continuous non-periodic spectrum in frequency domain (FT) Periodic signal in time domain ←→ discrete spectrum in frequency domain (FST) Discrete signal in time domain ←→ periodic spectrum in frequency domain (DTFT) Discrete periodic signal in time domain ←→ discrete periodic spectrum in frequency domain (DFT) In DFT, both sides of the transform are discrete, and thus it is a transform pair that can truly be used for digital signal processing using a computer. Both sides are periodic, so the processing can be done in just one cycle, which has two important meanings. First, the processing is limited (this is necessary for computers); second, all information can be retained in just one cycle (this is necessary for accurate processing). In real engineering, the signals we encounter are often continuous and non-periodic. In order to perform DFT, they must be discretized (sampled) and periodized (periodic extension). When sampling the signal, the Nyquist criterion must be followed. The Nyquist sampling frequency fS=1/T=2fm (i.e. the lowest sampling frequency without distortion); the Nyquist sampling interval T=1/2fm (i.e. the maximum sampling interval without distortion). Since the ideal low-pass filter is not achievable, the actual low-pass filter has a rise time and a fall time. Therefore, fS>2.5～3 is often used in engineering. The signal after fm. sampling is discrete in time, but the amplitude is still continuous. It is not a true digital signal, and its amplitude must be quantized, that is, the amplitude at discrete time points is classified into a finite number of levels.