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The HADAMARD principle is known in optics as a multiplex technique. It describes the mode with the most advantageous increase of the signal-to-noise ratio (SNR) in terms of scanning (Fellget advantage). The maximum increase of SNR, we call it gain, is (n+1)/(2On), where n is the number of multiplexing. It is valid in the case of pure detector noise. The multiplex encoding Hadamard pattern in case of n = 7 is 1110100, whereby 1 stands for a switched on channel performed by a field selector. The signals of all (switched on) channels are detected by a single detector. n measurement steps with a cyclic change of the pattern is necessary to perform the Hadamard transformation and to get the result of each individual channel. In case of n = 7 the theoretical gain is 1.51.
For all possible multiplex pattern (1100000, 1110000 and so on) the gain is theoretically investigated. A multiplexing advantage (gain > 1) is reached only by the Hadamard pattern, the inverse Hadamard pattern and for (0111111)-pattern (gain=1.08). Most of the multiplex pattern are disadvantageous. The reason for maximum gain of the HADAMARD transformation is analysed theoretically.
Signal fluctuations during the measurement caused by fluctuations of the illumination or by the object under test, reduce the multiplex gain, too. So the limits for realizing a gain are estimated theoretically. Essential is the transformation procedure and its influence on the error propagation.
The results could be verified by experiments with array spectrometeres. Requirements are derived by numerical simulation concerning the stability of the signals to be multiplexed. It is simulated the needed stability of the signals with increasing order of multiplexing. So the increase of the multiplex gain is limited by signal fluctuations. A realized 96 channel spectral reader is presented as a modern application of an optical multiplexing arrangement.
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! R. Riesenberg, A. Wuttig, B. Harnisch, "Optical MEMS Technology for Multiplexing in High-End Micro-Scpectrometers", Proc. SPIE 4928, 6-14, 2002
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! R. Riesenberg, “MicroMechanical Slit Positioning System as a transmissive spatial Light Modulator”, Proc. SPIE 4457, pp.197-203, 2001
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! R. Riesenberg, U. Dillner, "HADAMARD Imaging Spectrometers“, Proc. SPIE 3753, pp. 203-213, 1999
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