Interpolation and Sampling - Signal Processing for Communications - page 232

9.1 Preliminaries and Notation

l g r , y i d . , © , L s

Interpolation. Interpolation comes into play when discrete-time signals
need to be converted to continuous-time signals. The need arises at the in-
terface between the digital world and the analog world; as an example, con-
sider a discrete-time waveform synthesizer which is used to drive an analog
amplifier and loudspeaker. In this case, it is useful to express the input to the
amplifier as a function of a real variable, defined over the entire real line; this
is because the behavior of analog circuitry is best modeled by continuous-
time functions. Wewill see that at the core of the interpolation process is the
association of a physical time duration T s to the intervals between samples
of the discrete-time sequence. The fundamental questions concerning in-
terpolation involve the spectral properties of the interpolated function with
respect to those of the original sequence.

Sampling. Sampling is the method by which an underlying continuous-
time phenomenon is “reduced” to a discrete-time sequence. The simplest
sampling system just records the value of a physical variable at repeated
instants in time and associates the value to a point in a discrete-time se-
quence; in the following, we refer to this scheme as the “naive” sampling
operator. Other sampling methods exist (and we will see the most impor-
tant one) but, in all cases, a correspondence is established between time in-
stants in continuous time and points in the discrete-time sequence. In the
following, we only consider uniform sampling , in which the time instants
are uniformly spaced T s seconds apart. T s is called the sampling period and
its inverse, F s is called the sampling frequency of a sampling system. The
fundamental question of sampling is whether any information is lost in the
sampling process. If the answer is in the negative (at least for a given class of
signals), this means that all the processing tools developed in the discrete-
time domain can be applied to continuous-time signals as well, after sam-
pling.

Tabl e 9 .1 Notation used in the Chapter.

Name Description Units Relations

T s Sampling period seconds T s = 1 / F s

F s Sampling frequency hertz F s = 1 / T s

Ω s Sampling frequency (angular) rad / sec Ω s = 2 π F s = 2 π/ T s

Ω Nyquist frequency (angular) rad / sec Ω =Ω / 2 = π/ T s

N N s