Sampling
Consider a continuous-time signal x : Reals ®
Complex. Computers, and digital hardware in general, have difficulty dealing
with such signals. An integral part of what it means to be digital is that
actions occur in discrete steps, not as a continuous evolution. Computers, however,
can manipulate a closely related signal that is constructed by sampling
the continuous-time signal.
A sampler is a system
SamplerT : [Reals ®
Complex] ® [Integers ®
Complex]
where T is the sampling period (it has units of seconds/sample).
The sampling frequency is fs = 1/T, in units
of samples/second (or sometimes Hz, cycles/second). If y = SamplerT
(x) then for all n in Integers,
y(n) = x(nT).
Exercise: Verify that SamplerT is linear
but not time invariant.