Question

A signal m(t) = 3 cos (6000 πt) + 4 cos (7000 πt) + 5 cos (10000 πt) to be sampled. What is the minimum sampling rate for bandpass sampling for truth full representation:
1. 2000 Hz
2. 3000 Hz
3. 5000 Hz
4. 10000 Hz

Answer 1

Correct Answer - Option 4 : 10000 Hz

Concept:

Sampling Theorem:

1) A continuous signal band limited to f_m hertz can be converted to its sample equivalent without information loss provided f_s ≥ 2 × f_m samples/sec.

2) Aliasing is a form of overlapping between signal components that occurs when f_s < 2 × f_m.

3) The minimum sampling rate and maximum sampling rate interval allowed to avoid aliasing is called the Nyquist rate.

Calculation:

m(t) = 3 cos (6000 πt) + 4 cos (7000 πt) + 5 cos (10000 πt)

f_m1 = 3000 Hz

fm2 = 3500 Hz

fm3 = 5000 Hz

Minimum Sampling Rate (S):

(S) = 2 × max.(fm1,fm2,fm3)

(S) = 2 × 5000

(S) = 10000 Hz.

Consider a modulating signal:

m(t) = (cos(2πf₁t)  ×  cos(2πf₂t)) + (cos(2πf₃t)  ×  cos(2πf₄t))

The maximum sampling rate will be:

(S) = 2 × max((f₁ + f₂), (f₃ + f₄))