Question

Consider a discrete-time periodic signal

x [ n ]   =   cos ( 0 . 7 πn )

(1) Determine the fundamental period of x[n].

(2) Suppose x[n] is obtained by sampling the continuous-time signal  x ( t )   =   cos   ( πt ), by letting the sampling period to be T s =   0 . 7 and considering the sample values at each n. Is the Nyquist sampling conditions satisfied in this case? Explain and relate this to the answer given before.

(3) Under that conditions would sampling a continuous-time signal x ( t )   =   cos ( Ω 0 t ) give a discrete-time sinusoid x[n] that resembles x(t)? Explain and give an example.

Answer 1

1. It is given that x[n] = cos(0.7 π n) So first, we will have to check whether the signal is a periodic one or an aperiodic one. For this, let us check the conditon of periodicity.

Condition of periodic function= ( 2 π ) / ( W) = Rational number( R)

Here W= 0.7 π. So, ( 2 π)/ ( 0.7 π) = Rational value. This implies the signal is periodic and we can move on to calculating the fundamental period value.

Fundamental period value= (2 π)/ W *k, where k= 0.7

So, N=2 is the required fundamental period.

2. A thorough understanding of the modern interpretation of Nyquist's criterion is mandatory when dealing with sampled data systems. A continuous analog signal is sampled at discrete intervals, t_s = 1/fs, which must be carefully chosen to ensure an accurate representation of the original analog signal. It is clear that the more samples taken (faster sampling rates), the more accurate the digital representation, but if fewer samples are taken (lower sampling rates), a point is reached where critical information about the signal is actually lost.

3. Simply stated, the Nyquist criterion requires that the sampling frequency be at least twice the highest frequency contained in the signal, or information about the signal will be lost. If the sampling frequency is less than twice the maximum analog signal frequency, a phenomenon known as aliasing will occur.