For x(t) = sin(10 πt)/πt, what is the condition on the sampling interval T so that x(t) is uniquely represented by the discrete-time sequence x[n] = x(nT) ? 

Nyquist  Theorem:

Nyquist sampling theorem states that “a signal can be exactly reproduced if it is sampled at the rate fs which is greater than or equal to twice the maximum frequency fm of the modulating signal."​, i.e.

The minimum sampling frequency is:

fs ≥ 2 fm

In terms of the time period:

\(\frac{1}{T_s}<2f_m\)

\(T_s<\frac{1}{2f_m}\)

Ts = Nyquist sampling interval

Application:

Given, x(t) = sin(10 πt)/πt

Since, 2πfm = 10 π

fm = 5 Hz

As, \(T_s<(\frac{1}{2f_m}=\frac{1}{10})\)