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If $\omega$ is the frequency of a continuous time sinusoidal signal, $-\infty < \omega < \infty $.

But when it comes to discrete time sinusoidal signals, the range becomes limited : $-\pi \le \omega_{discrete} \le \pi $.

Does that mean discrete sinusoidal signals cannot represent all frequencies possible ? What is the true picture here ?

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It is important to realize that in the discrete-time case, $\omega$ is normalized by the sampling frequency $f_s$:

$$\omega=2\pi f/f_s$$

The inequality in your question is basically the sampling theorem, which says that a discrete-time signal can only represent frequencies lower than half the sampling frequency. All higher frequencies will be folded back into the frequency band $[0,f_s/2]$ (this is called aliasing). If you need higher frequencies then you need to increase the sampling rate.

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  • $\begingroup$ Essentially, once we select a sampling rate, it constrains the representable frequency range. Correct ? $\endgroup$ – curryage May 19 '14 at 9:59
  • $\begingroup$ @curryage: Right. $\endgroup$ – Matt L. May 19 '14 at 10:00

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