0.2 The six elements  (Page 6/10)

The third element: samplers

Since part of any digital transmission system is analog (transmissions through the air, across a cable, oralong a wire, are inherently analog), and part of the system is digital,there must be a way to translate the continuous-time signal into a discrete-time signal and vice versa. The process of sampling an analogsignal, sometimes called analog-to-digital conversion, is easy to visualize in the time domain. [link] shows how sampling can be viewed as the process of evaluating a continuous-time signal ata sequence of uniformly spaced time intervals, thus transforming the analog signal $x\left(t\right)$ into the discrete-time signal $x\left(k{T}_s\right)$ .

One of the key ideas in signals and systems is the Fourier series: a signalis periodic in time (it repeats every $P$ seconds), if and only if the spectrum can be written as a sum ofcomplex sinusoids with frequencies at integer multiples of a fundamental frequency $f$ . Moreover, this fundamental frequency can be written in terms ofthe period as $f=1/P$ . Thus, if a signal repeats 100 times every second ( $P=0.01$ seconds), then its spectrum consists of a sum of sinusoids with frequencies $100,200,300,...$ Hz.

Conversely, if a spectrum is built from a sum of sinusoids with frequencies $100,200,300,...$ Hz, then it must represent a periodic signal in time that has period $P=0.01$ . Said another way, the nonzero portions of the spectrum areuniformly spaced $f=100$ Hz apart. This uniform spacing can be interpreted as a sampling (in frequency) of an underlyingcontinuous-valued spectrum. This is illustrated in the top portion of [link] , which shows the time domain representation on the left and the correspondingfrequency domain representation on the right.