Question

A digital communication system transmits through a noiseless bandlimited channel \([-W, W]\). The received signal \(z(t)\) at the output of the receiving filter is given by: \[ z(t) = \sum_n b[n] x(t - nT), \] where \(b[n]\) are the symbols and \(x(t)\) is the overall system response to a single symbol. The received signal is sampled at \(t = mT\). The Fourier transform of \(x(t)\) is \(X(f)\). The Nyquist condition that \(X(f)\) must satisfy for zero intersymbol interference at the receiver is:

• A. \(\sum_{m = -\infty}^\infty X\left(f + \frac{m}{T}\right) = T\)
• B. \(\sum_{m = -\infty}^\infty X\left(f + \frac{m}{T}\right) = \frac{1}{T}\)
• C. \(\sum_{m = -\infty}^\infty X\left(f + mT\right) = T\)
• D. \(\sum_{m = -\infty}^\infty X\left(f + mT\right) = \frac{1}{T}\)

Hint:

The Nyquist criterion is fundamental for avoiding intersymbol interference (ISI) in digital communication systems. Ensure proper pulse shaping and sampling to meet this condition.

Answer 1

The Correct Option is A

Step 1: Understand the Nyquist Criterion.
The Nyquist criterion is essential to eliminate intersymbol interference (ISI) in sampled signals. It ensures that the signal \( z(t) \) at sampling instances \( t = mT \) does not experience overlapping contributions from neighboring symbols. Step 2: Express the Nyquist Condition in the Frequency Domain.
For \( x(t) \), the system's impulse response, its Fourier transform \( X(f) \) must satisfy the Nyquist condition for zero ISI: \[ \sum_{m=-\infty}^{\infty} X \left( f + \frac{m}{T} \right) = T \] This condition guarantees that the system response does not overlap at multiples of the symbol duration \( T \). Step 3: Explain the Summation Property.
The summation in the frequency domain corresponds to periodic sampling in the time domain. The condition \( \sum X(f + m/T) = T \) ensures correct pulse shaping and prevents overlapping symbol contributions, enabling precise recovery of the transmitted signal. Step 4: Identify the Correct Option.
Among the options provided, the correct condition for zero ISI is: \[ \sum_{m=-\infty}^{\infty} X \left( f + \frac{m}{T} \right) = T \] Therefore, the correct answer is option (1). Final Answer: \[ \boxed{{(1) } \sum_{m=-\infty}^{\infty} X \left( f + \frac{m}{T} \right) = T} \]