Page 248 - Photodetection and Measurement - Maximizing Performance in Optical Systems

Multiple Channel Detection

Multiple Channel Detection 241

ence clock. If no references are available, then both sine and cosine transforms
will have to be computed and added as sums-of-squares, in order to resolve the
phase ambiguity. This is then essentially a digital or software implementation
of the two-phase lock-in amplifier. Where large numbers of channels are to be
monitored, this flexibility of digital spectral analysis is very enticing.

11.6 Sequency Analysis:
Orthogonal Binary Coding
The idea of labelling a number of optical signals so that they can be sent through
a single channel and separated at the end is closely related to the problem of
coding and transmission for communications uses. Why can’t we use all the
available techniques from that field for instrument use? What is wrong, for
example, with serial coding such as the RS232/V24 standard, where we can
make use of all the UARTs and other cheap digital chips for coding and decod-
ing? We could have the red-wavelength channel transmit a string of “A” char-
acters, the blue channel a string of “B”s. Let’s look at this actual example. The
character “A” in ASCII is represented by 65 dec (01000001 bin ) “B” by 66 dec
(01000010 bin). These could be transmitted simultaneously, and we could look for
just those sequences at the output. We need to multiply the aligned received
signal by the binary number we expect, and add up the result. The problem is
that “A” and “B” are not orthogonal, so an “A” will give some output from the
“B” channel and vice versa. We can see this by translating 0s to -1, and multi-
ply the “A” and “B” binary numbers:
 [ A - ( 1 111111 1 ) ¥ B - ( 1 11111 11
-----
- )] = 4.
----
On the other hand, if we had chosen the characters “U” (85 dec , 01010101 bin ) and
“i” (105 dec, 01101001 bin), we would have obtained a zero sum. These two char-
acters are, in this representation, orthogonal. It is clear that not all characters
are equal. If we choose the characters carefully, the electronics of serial trans-
mission systems could in principle be used for instrumentation. This leads us
back into the analysis of Walsh functions introduced in Chap. 5.
The sine/cosine series has been the basis of coding theory and signalling for
the past century. This is partly because these functions form a complete, orthog-
onal set. By complete, we mean that with increasing number of terms, the
mean-square error in approximation of a function approaches zero. By orthog-
onal, we mean that the integral of products of pairs of terms vanishes. However,
sine waves are in many situations more difficult (i.e., more expensive) to gen-
erate than binary digital waveforms. Hence there is great interest in sets of
binary waveforms which are also orthogonal. The Walsh functions form one set,
which we have already used to synthesize a sine wave. These are binary func-
tions which take on values of ±1, and are written WAL(n, N), where N takes
the role of the time-base ordinate, and n the harmonic number. N is also the
number of waveforms in the set, which should be a power of 2. We looked at a

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