Page 111 - Photodetection and Measurement - Maximizing Performance in Optical Systems

System Noise and Synchronous Detection

104 Chapter Five

tering. Once signal and reference are converted in analog-to-digital converters
(ADC), all multiplications and integrations in the Fourier analysis can be
carried out free from errors using digital hardware and software techniques. By
taking this approach, great flexibility and richness of functions can be provided.
It is, for instance, hardly more complex to compute the power in many har-
monics of the signal, each in a different bandwidth, or even the full Fourier
spectrum of the signal. With the relentless attack of digital solutions on areas
once considered solely analog, the increasing use of digital signal processing in
synchronous detection must be expected to continue. Low-frequency signals and
those recorded for off-line analysis have long been dealt with this way. Even the
humble radio receiver is being reduced to an aerial, ADC, and digital process-
ing chip.

5.6 Walsh Demodulators

The ease of construction of digital (±1) multipliers hides a disadvantage of
increased noise. This is easily seen from the frequency-domain representations
of signal and reference. The passbands in the frequency domain formed by
modulation and synchronous demodulation are obtained by convolving the
frequency-domain representation of the reference signal with the double-
sided passband obtained from the postdemodulation low-pass filter. The Fourier
analysis of a sine-wave reference is just a single amplitude, so multiplication of
the input signal by a sine wave at f mod delivers a filter with a single passband,
width ±1/2pRC centered on f mod . The Fourier decomposition of a bipolar square-
wave reference clock is a set of odd harmonics of the fundamental frequency.
This produces passbands at f mod , 3f mod , 5f mod , etc., with relative amplitudes of 1,
1/3, 1/5, etc. (Fig. 5.11). Hence square-wave references open up not just a single
passband, but an infinite series of passbands. If the input has signal, noise, or
interference power at these harmonic frequencies, they will be detected and con-
tribute to the overall output. It is possible to roll off the receiver to give a
reduced power at three times the modulation frequency, but the suppression
will not be complete. When a single RC filter only achieves an ultimate cutoff
slope of -6dB/octave (-20dB/decade), it is difficult to get much suppression at
3f mod . Hence these harmonic responses can be a problem and must be kept in
mind if strange interference effects are seen.
A partial solution to the harmonic responses can be obtained by approxi-
mating the sine function needed as reference waveform for the synchronous
detector. If we can choose a small number of binary waveforms to approximate
the sine wave, these can then be applied to weighted binary multipliers and
summed together for an overall response similar to that of a sine wave refer-
ence. Just as Fourier synthesis can be used to approximate an arbitrary func-
tion using a small number of sine/cosine terms, we will use Walsh synthesis to
approximate the sine wave with a small number of Walsh functions.
Walsh functions are two-parameter binary (±1) functions that form an orthog-
onal series. They can be used just like the sine and cosine series of Fourier analy-

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