What happens when to the spectrum of a signal when it is sampled?
Sampling produces a larger number of frequency components not in the original spectrum, even components having negative frequency. The sampled signal has a spectrum that is periodic at the sampling frequency (20 Hz) and has an even symmetry about 0.0 Hz, as well as symmetry about the sampling frequency, fs.
What is the spectrum of DTFT?
The spectrum of the DTFT is continuous, so either f or ω can be used. The common choice is ω, because it makes the equations shorter by eliminating the always present factor of 2π. Remember, when ω is used, the frequency spectrum extends from 0 to π, which corresponds to DC to one-half of the sampling rate.
How is the DTFT related to the spectrum?
In the DTFT (Discrete Time Fourier Transform) the spectrum is periodic with period of 2π . A continuous signal when sampled has a spectrum which is a repeated version of its original spectrum before sampling with a period of sampling frequency.
What is the relationship between sampled and discrete time signals?
This is an important property of sampled or discrete-time signals, and that’s also why the two variables Ω and T are commonly combined into a single variable ω = Ω T = 2 π f / f s. Note that sampling results in a periodization of the spectrum, and this is why information can be lost. This is true for both spectra X s ( j Ω) and X ( e j ω).
How is CTFT related to discrete time Fourier transform?
You’ve shown that the (continuous-time) Fourier transform (CTFT) of a sampled continuous-time signal equals the discrete-time Fourier transform (DTFT) of the corresponding discrete-time signal. In both cases, the spectrum is periodic. The independent variables are related by
How is the periodicity of the spectra related to the sampled signal?
The periodicity of the spectra implies that the fundamental frequency interval f ∈ [ − f s / 2, f s / 2], or, equivalently, Ω ∈ [ − π / T, π / T] or ω ∈ [ − π, π] contains all information of the sampled signal, and all frequencies outside that interval are redundant.