- Classical signal processing (studied
by every second year Electrical Engineering student) generalizes
Fourier's Theorem:
- Any signal can be represented by a
vector in an abstract space, with an infinite number of dimensions. The
"dimensions" consist of a family of signals called an "orthogonal basis
set"
- The signal can therefore be decomposed
into a unique linear
combination of the signals in the basis set
- The set of all sine waves (at all
possible frequencies, phases, and amplitudes) qualifies as an
orthogonal basis set
- Therefore, every signal has a unique
Fourier transform which represents the combination of sine waves
necessary to reproduce it
- A radio
receiver (or any device which receives signals from a shared
communications medium) works by:
- Starting
with everything it "hears"
- Rejecting
portions which fall outside the desired subset of the basis set (noise
and interference)
- Decoding
what remains (the signal)
- The
Shannon-Hartley Theorem (AKA Shannon's Law) dictates the maximum amount
of information that can be transmitted without error if noise
falls within the desired subset of the basis set or if filtering is
imperfect
- The
instructor usually mentions -- but only in passing -- that the set of
all sine waves is only one possible orthogonal basis set. There are an
infinite number of others, including:
- Square
waves (Walsh Transform)
- "Sampling"
functions (e.g. sinc(t))
- Wavelets
and wave packets
...all of
which
leads to two very vitally important but generally overlooked
revelations....
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