JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B. Tech III Year DIGITAL COMMUNICATIONS Previous Question Paper (October - 2020 - R16)

The Sampling Theorem states that a band-limited signal of finite energy, which has no frequency components higher than f_m Hz, is completely described by specifying the values of the signal at instants of time separated by T_s = 1/(2f_m) seconds. This minimum sampling rate, f_s = 2f_m, is known as the Nyquist rate.

In Ideal Sampling (or impulse sampling), the continuous-time signal x(t) is multiplied by a periodic train of unit impulses δ_T(t). The sampled signal x_s(t) is represented as:

In the frequency domain, the spectrum of the sampled signal X_s(f) is the convolution of the signal spectrum X(f) and the impulse train spectrum. This results in a periodic repetition of the original spectrum scaled by 1/T_s:

To reconstruct the original signal without aliasing, the sampling frequency f_s must satisfy f_s ≥ 2f_m. If this condition is met, the original signal x(t) can be recovered by passing x_s(t) through an ideal low-pass filter with a cutoff frequency f_c such that f_m ≤ f_c < f_s - f_m.

The process is visualized as follows:

• The input signal x(t) is multiplied by an impulse train, resulting in a series of weighted impulses at intervals of T_s.
• The frequency spectrum X_s(f) shows copies of X(f) centered at multiples of f_s.
• If f_s < 2f_m, the spectral copies overlap, causing aliasing, which makes perfect reconstruction impossible.
• If f_s ≥ 2f_m, the spectral copies are separated by guard bands, allowing the low-pass filter to extract the baseband signal perfectly.

Quantization is the process of mapping a continuous-amplitude analog signal into a finite set of discrete amplitude levels. The range of the input signal is divided into L levels, each of step size Δ. If the input signal range is V, then Δ = V / L. The difference between the actual input value and the quantized output value is the quantization error, denoted as q, which is uniformly distributed in the interval [-Δ/2, Δ/2].

To derive the Signal-to-Quantization-Noise Ratio (SQNR), we first calculate the quantization noise power Nq, which is the variance of the quantization error:

Since the probability density function f(q) = 1/Δ for the uniform distribution:

Assuming a sinusoidal input signal x(t) = A cos(ωt), the signal power S is:

For a quantizer with n bits, L = 2ⁿ, and the peak-to-peak range is 2A = LΔ, which implies Δ = 2A / 2ⁿ. Substituting Δ into the noise power expression:

The SQNR is the ratio of signal power to noise power:

In decibels, this is expressed as: