How can spectrum of DTFT of sampled band-limited analog signal be in $[-\pi,\pi]$

1. What is spectrum of sampled signal?
2. What is DTFT spectrum?
3. What are the limitations of DTFT?

What is spectrum of sampled signal?

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, f_s. Since the sampled spectrum is periodic, it goes on forever and only a portion of it can be shown.

What is DTFT spectrum?

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.

What are the limitations of DTFT?

Two computational disadvantages of the DTFT are: the direct DTFT is a function of a continuously varying frequency and the inverse DTFT requires integration. The Fourier series coefficients constitute a periodic sequence of the same period as the signal; thus both are periodic.