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10.1 MultiplexedSignal Processing

• single-channel signal processing and the corresponding SNR• array detector signal processing and the corresponding SNR• multiplexed signal processing and the corresponding SNR• the Hadamard transform instrument and math• Hadamard transform examples including Johnson noise and Poisson counting noise

Single-Channel Signal Processing

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rotating prism fixedslit

detector

A scanning emission spectrometer shown above, where the slit passes a band of wavelengths, ∆λ. • the prism is rotated until the beginning wavelength of interest, λstart , passes through the slit• the prism position is held fixed while the detector output is processed for some time, t• the prism is rotated to a new position, λ = λstart + ∆λ , and the detector output processed for the same time, t• the process is repeated until the ending wavelength of interest, λend , is examined

Single-Channel SNR

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Characteristics:• the measurement time per step is t• the total number of steps is given by N = |λstart - λend | / ∆λ• the total time to perform the experiment is given by T = N t. • the signal-to-noise ratio for the scanned spectrum, SNRscanned , is determined by t• 1/f noise is determined by T

The signal-to-noise ratio can only be improved by increasing the time, t, that data is collected for each wavelength. This in turn increases the total time, T. If T gets too large it might be possible to decrease the number of data points by increasing ∆λ, which also increases the signal strength.

Scanning is not an optimum strategy if the spectrum is sparse. Examples might be atomic emission or Raman scatter. With sparse spectra scanning spends most of the time averaging where there is no signal!

Array Detector Signal Processing

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fixed prism detectorarray

λ1λ2

λN

An emission spectrometer with an array detector is shown above. Each element of the array examines a band of wavelengths, ∆λ. • the prism does not rotate• all detectors simultaneously produce their own signal, which ismeasured for a time t

Array Detector SNR

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If the spectrum is measured for the same time t as the scanned experiment:• the signal-to-noise ratio is identical to the scanned spectrum, SNRarray = SNRscanned• the spectrum is obtained N times faster than the scanned spectrum because all wavelengths are measured simultaneously• 1/f noise is decreased because of the reduced measurement time• this strategy is good for signals that occur on a fast time scale (chromatography) • good where high sample throughput is needed (process control)

If the spectrum is measured for time T of the scanned experiment:• the signal-to-noise ratio is increased, SNRarray = N1/2SNRscanned• the SNR improvement is called the multiplex advantage

Multiplexed Signal Processing

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fixed prism multipleslits

The Hadamard transform spectrometer shown above has two unique components - a mask into which is etched an array of slits, and a large area detector. Hadamard transforms are used for spectral regions where detector noise limits the measurement, e.g. infrared.

With multiplex detection the spectrum has to be obtained by a computation. With infrared, NMR, and mass spectrometry the computation involves a Fourier transform. These calculations are too complicated to easily demonstrate the multiplex advantage. Multiplexing will be demonstrated by the Hadamard transform.

Hadamard Transform SNR

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• Assume there are N wavelengths that will be measured, and that each slit has a bandpass, ∆λ. For a large mask, about half the spectrum reaches the detector. • There are N different masks each with a unique arrangement of slits, resulting in N measurements of half the signal. • The N measurements are processed using the mathematics of the Hadamard transform. This de-multiplexing results in a spectrum.

Like array detection, Hadamard detection allows:• the spectrum to be obtained N/2 times faster than a scanned spectrum at the same SNR• the signal-to-noise ratio to be increased by (N/2)1/2 greater than that obtained by scanning

The Hadamard transform provides a half-multiplex advantage, while Fourier transform techniques provide a full-multiplex advantage.

Hadamard Mask and Signal

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λ11

λ71

λ61

λ50

λ41

λ30

λ20

00101117

01011106

10111005

01110014

11100103

11001012

10010111

slit patternmask #

An example mask is shown above. The digit 1 signifies that the mask passes light, while the digit 0 signifies that the mask blocks the light. Seven such masks are required to obtain a spectrum.

The particular sequence of digits in the above mask is called cyclic in that all masks can be obtained by rotating the digits (the rightmost digit of one mask becomes the leftmost digit of the following mask). This property facilitates construction of Hadamard spectrometers based on mechanical movement of the mask.

Hadamard De-Multiplexing

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The equations at the right show how the intensities, Iλ , at each wavelength add to produce the detector output, dmask, for each mask.

4 2 3 4 7

6 2 4 5 6

7 3

1 1 4 6 7

2 1 2 5 7

3 1 2 3 6

5 1 3 4

5 6

5

7

d I I I Id I I I Id I I I Id I I I I

d I Id I I I

I Id I I I

I

I

= + +

= + + += + + += + + +

=+

= + + += + +

+

+

+ +

( )( )( )( )( )( )( )

1

2 1 2 3 4 5 6 7

3 1 2 3 4 5 6 7

4 1 2 3 4 5 6 7

5 1 2 3 4 5 6 7

6 1 2 3 4 5 6 7

7 1 2 3 4

1 2 3 4

5 6

6

7

5 7 4

4

4

4

4

4

4

S

S d d d d d d d

S d d d d d d d

S d d d

d

d d d d

S d d d

d

d d d d

S d d d d d d d

S d d d d d

d d d

d

d d

d

= + + − + − −

= − + + + − + −

= − − + + + − +

= − − + + + −

= − + − − + + +

= − + − − + +

= + − + − − +

The second set of equations show how the data values, dmask , are used to compute an averaged signal, Sλ , at each wavelength.

Note how for S1 the intensity for all wavelengths except I1 are subtracted out! Etc.

Note that I1 contributes to S1 four times, or (N+1)/2. This means the signal-to-noise ratio will improve by the factor

( )1 2 2N N+

Matrix Hadamard Transform

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Let M be the matrix of Hadamard mask ones and zeros. Let S be the signal vector. Let D be the vector of measured intensities. When the data vector is obtained by measurement, the spectrum can be obtained using the inverse of the Hadamard mask matrix.

One of the attractive properties of Hadamard transform methods is the ease in obtaining the inverse mask matrix. • take the transpose of the mask matrix• for the transpose matrix convert every 0 into a -1• normalize each row of the matrix by the number of positive ones• since each row has the same number of positive ones, the normalization can be done outside the matrix

=D MS = -1S M D

14= -1S M D

M MT 4×M-1

1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1

1 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1

1 1 1-1 1-1-1-1 1 1 1-1 1-1-1-1 1 1 1-1 1 1-1-1 1 1 1-1-1 1-1-1 1 1 1 1-1 1-1-1 1 1 1 1-1 1-1-1 1

400 431.5 463 494.5 526

0

0.5

1

wavelength (nm)

ampl

itude

(V)

Peak with Johnson Noise

Reducing Johnson Detector Noise

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0 32 64 96 12814

16

18

20

22

mask number

mul

tiple

xed

ampl

itude

(V)

Mask Outputs with Johnson Noise

A 127-element mask was used to decompose the spectrum into mask outputs. The cyclic mask is from, "Digital Communications with Space Applications," ed. S. W. Golomb, Prentice-Hall, Englewood Cliffs, NJ, 1964, pp. 169-171. The SNR has improved a factor of 8.

400 431.5 463 494.5 5260

0.28

0.55

0.83

1.1

wavelength (nm)

ampl

itude

(V)

Hadamard Transform

Multiplexing Poisson Noise

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Mask Outputs - With and Without Poisson Noise

0 50 1000

1000

2000

3000

wavelength (nm)

mul

tiple

xed

ampl

itude

(cou

nts)

400 450 50050

0

50

100

150

wavelength (nm)

ampl

itude

(cou

nts)

Original and Transformed Data

The same cyclic mask was used as the last slide. For this example, the mask determined total counts. The noise was added to each of the mask counts.

Below are the original spectrum with counting noise (blue), and the spectrum obtained from the mask outputs (red). Note how the peak noise has spread over the entire spectrum.

The Multiplex Disadvantage

400 450 5000

50

100

wavelength (nm)

ampl

itude

(cou

nts)

Noise-Free Sparse Spec trum

Original Spectrum with Poisson Noiseand the Transformed Data

400 450 50050

0

50

100

150

wavelength (nm)

ampl

itude

(cou

nts)

Poisson noise was added to the mask values for the spectrum at the left. The Poisson noise arising from the large peak will begreater than the small peak. The counts due to each mask are dominated by the contribution from the large peak. As a result,the noise due to the counts in the large peak is spread throughout the spectrum. The SNR of the spectrum got worse, which is called the multiplex disadvantage!

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