Nyquist Sampling Rate and Nyquist Interval

The theoretical minimal sampling rate at which a finite bandwidth signal can be sampled to retain all information and reconstructed from its sample without any distortion is called the Nyquist sampling rate (f_n). The other term for the Nyquist rate is the minimum sampling rate. That means when the sampling rate becomes exactly equal to 2f_m samples per second then is called as Nyquist sampling rate. For a bandwidth of span C, the Nyquist frequency is just 2C.

The significance of the Nyquist rate is that it helps to avoid the aliasing problem which is the overlapping of two signals.

Nyquist Rate, f_{n = 2fm}

Where f_m is the maximum frequency component that is present in the signal.

1. If the signal is sampled at a rate greater than Nyquist rate (f_n) then the signal is known as oversampled.
2. If the signal is sampled at a rate less than Nyquist rate (f_n) then the signal is known as under-sampled.

Nyquist Interval

The maximum sampling interval is called the Nyquist interval. That means the rate of sampling is equal to the Nyquist rate, then the time interval between any two adjacent samples is represented as the Nyquist interval. The Nyquist interval can be represented by,

  1/f_n=1/2fm

Now to determine the Nyquist rate and Nyquist interval for the following signal.

x(t)=(2000πt) + 3sin(6000πt) + 8cos(12000πt)

Nyquist Rate:

Now, we have to find the three frequencies:

Highest frequency component in term 2000πt = ω_m1t  is  2000π

fm = ωm1/2π
  = 2000π/2π
  = 1000 hz

  The highest frequency component in the term sin 6000πt = ω_m2t  is  6000π

fm = ωm2/2π
  = 6000π/2π
  = 3000 hz

 The highest frequency component in the term cos 12000πt =ω_m3t  is  12000π

fm = ωm3/2π
  = 12000π/2π
  = 6000 hz

Therefore, the maximum frequency component present in the signal is 6000 hz. Thus, the minimum sampling rate to avoid aliasing i.e., Nyquist rate is  f_n= 2f_m

 = 2 × 6000 hz
 = 12000 hz

 Nyquist interval                                

Now, the Nyquist interval is = 1/f_n

=  1/12000
=   8.3 sec