In a digital communication system, the overall pulse shape p(t) at the receiver before the sampler has the Fourier transform P(f). If the symbols are transmitted at the rate of 2000 symbols per second, for which of the following cases is the inter symbol interference zero?

This question was previously asked in

GATE EC 2017 Official Paper: Shift 1

Option 2 :

CT 1: Ratio and Proportion

2672

10 Questions
16 Marks
30 Mins

__Concept__:

Nyquist ISI criterium states that for ISI – free response:

\(h\left( {n{T_s}} \right) = \left\{ {\begin{array}{*{20}{c}} {1;}&{n = 0}\\ {0;}&{n \ne 0} \end{array}} \right.\)

i.e.

\(\frac{1}{{{T_s}}}\mathop \sum \limits_{T = - \infty }^{{T_\infty }} H\left( {b - \frac{k}{{{T_S}}}} \right) = 1\forall \;f\)

\(\mathop \sum \limits_{T = - \infty }^{{T_\infty }} H\left( {b - \frac{k}{{{T_S}}}} \right) = T_s = Constant\;\forall \;f\)

__Calculation__:

\(f_s = \frac{1}{{{T_s}}} = 2k\;symbols/sec\)

\(\mathop \sum \limits_{n = - \infty }^{ + \infty } p\left( {f - n\;{f_s}} \right) = {T_s}\)

\(\mathop \sum \limits_{n = - \infty }^{ + \infty } P\left( {f - n\;2k} \right) = 2k\)

We observe that only option (B) satisfies this condition, i.e.

\(\mathop \sum \limits_{n = - \infty }^{ + \infty } P\left( {b - n2k} \right) = Constant\)