Signal Processing for Cognitive Radios

APPENDIX ANYQUIST SAMPLING THEOREM

Software-defined radios and cognitive radios are digital communications devices that hold and process information in digital domain. The first stage of interface between analog and digital worlds is the process called sampling. Given an analog information waveform x(t), sampling generates a discrete-time sequence x[n]. The important objective is to ensure that there is no information loss in representing the analog waveform x(t) by the sampled sequence x[n]. If this objective is indeed satisfied, then the sequence x[n] must be equivalent to the analog waveform x(t) in the sense that it is possible to reconstruct x(t) from the sequence of samples x[n].

Let ƒ_s denote the sampling rate (in the number of samples per second) so that is the corresponding sampling period. Let x_s(t) denote the sampled version of x(t) as shown in Figure A.1:

(A.1)

(A.2)

(A.3)

FIGURE A.1 A signal x(t) and its sampled version x_s(t).

where (A.1) and (A.2) follow from ...