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A new signal processing algorithm for absolute temperature measurement using white light interferometry has been proposed and investigated theoretically. The proposed algorithm determines the phase delay of an interferometer with very high precision (≪ one fringe) by identifying the zero order fringe peak of cross-correlation of two fringe scans of white light interferometer. The algorithm features cross-correlation of interferometer fringe scans, hypothesis testing and fine tuning. The hypothesis test looks for a zero order fringe peak candidate about which the cross-correlation is symmetric minimizing the uncertainty of mis-identification. Fine tuning provides the proposed algorithm with high precision sub-sample resolution phase delay estimation capability. The shot noise limited performance of the proposed algorithm has been analyzed using computer simulations. Root-mean-square (RMS) phase error of the estimated zero order fringe peak has been calculated for the changes of three different parameters (^{-4} at 31 dB ^{-8}. Also, at 35 dB ^{-3} fringe was obtained. The proposed signal processing algorithm uses a software approach that is potentially inexpensive, simple and fast.

Although fiber optic interferometric sensors offer the possibility of performing measurements with very high sensitivity and resolution [

From the beginning of this research, an all fiber white light interferometry (AFWLI) absolute temperature measurement system as shown in _{C}_{P}_{S}_{R}_{DC}_{SAW}_{P}

This AFWLI for temperature measurement produces two fringe scans, one from the sensing FFPI and another one from the reference FFPI, as shown _{S}_{R}_{S}_{R}_{P}_{P}_{P}_{,}_{S}_{P}_{P}_{,}_{R}_{P}_{,}_{S}_{P}_{,}_{R}^{d}_{P,S}_{P,R}^{d}_{S}^{d}^{d}

Two major classes of signal processing algorithms for WLI are the hardware approach and the software approach. Both approaches have a more or less “tracking zero order fringe peak” feature. Gerges proposed a hardware approach which locks the zero order fringe position of interferometer by a feedback loop [

There are many software algorithms to estimate phase delay Φ^{d}_{S}_{R}_{S}_{P}_{R}_{P}_{d}

While WLI has the potential to identify the interference fringe order from the output pattern of an interferometer [_{min} required to identify the zero order fringe peak [

Representative values of the coherence length of different light sources like white light lamp, light emitting diode (LED), superluminescent diode (SLD), are about 10, 20, and 40 in terms of optical fringes. The _{min} required to identify the zero order fringe peak by amplitude difference Δ_{C}_{min} in

Additionally, once the zero order fringe peak is identified, then for a more accurate sub-sample resolution time delay estimation we will have to use interpolation which is possible by either quadratic interpolation in time domain or frequency domain zero-padding [

Notwithstanding the above mentioned shortcomings, cross-correlation is a still useful tool for time delay estimation as shown that cross-correlation with no pre-filtering is an optimal maximum likelihood estimator to estimate the time delay between two similar signals if the noise processes _{R}_{S}_{R}_{S}

The outcome of the time delay estimation depends on the combined performance of coarse estimation (zero order fringe peak identification) and sub-sample resolution estimation of time delay. In this article, a new signal processing algorithm which can accurately identify the zero order fringe peak of cross-correlation

The proposed signal processing algorithm uses a software approach, which is potentially inexpensive, simple and fast. And, this proposed signal processing algorithm has a low peak mis-identification rate of 3 × 10-^{4} at 31 dB ^{4} fringe as will be shown from the computer simulation results.

The proposed signal processing algorithm consists of five steps applied to sampled signal of WLI fringe scans. They are:

Normalization and cross-correlation

Peak and zero crossing detection

Matched filtering

Hypothesis test

Fine Tuning

Each procedure is explained briefly below.

As a preliminary procedure, the output of photodetector signals _{S}_{P}_{R}_{P}_{S}_{R}_{S}_{R}_{S}_{S}_{R}_{S}_{R}

After normalization, _{S}_{R}_{0}_{d}

In _{C}_{,}_{eff}_{C}_{min} in _{C}_{0}=_{d}

At this stage, all the peaks _{i}_{i}_{i}_{0}=_{d}_{-1} is negative first order fringe peak position, _{1} is positive first order fringe peak position in terms of sample number and so on. Also every zero crossing between peak _{i}_{i}_{j}_{j}

Linear interpolation is the reasonable method because the cosine function crosses the zero essentially as a straight line as shown in

Then the zero crossing period _{j}_{j}_{j}_{+}_{1}_{Z}

A matched filter is the optimum filter to maximize the _{0}_{0} [second]. When input signal _{0}(

In _{0}_{s}_{(}_{-t}_{)}_{h}_{(}_{t}_{)}(_{0}(_{s}_{(}_{-t}_{)}_{h}_{(}_{t}_{)}(_{ss}_{ss}_{ss}_{0}(_{ss}_{ss}

Thus if we process a signal-plus-noise with a matched filter, the largest peak outputs due to the signal will correspond to _{ss}

If we define the _{N}

At this stage, following the concept of matched filter, only one fringe of cross-correlation signal _{i}_{i}_{+}_{1}_{M}

Then, for a given one fringe of cross-correlation signal _{S}

_{i}_{i}_{i}_{i}_{+}_{1}

For the case of a zero order fringe peak _{0}) of _{0} of envelope between _{0} and _{1} is approximately “1” because _{0}) =1 and _{0}) ≈ _{1}). Then, _{0}) is improved by the factor of:

In the above _{i}_{i}_{i}_{i}_{i}_{0}=_{-1}_{1}=_{-2} …(_{i}=_{-}_{i-1}) due to the even symmetric property of _{0}= _{d}

In hypothesis test, signal processing algorithm chooses the nine biggest peaks of _{j}_{j}_{j}_{0}) is expressed as:

Ideally all the values of _{0} is zero (_{i}_{-i-1}_{j}_{0}. But, practically the zero order fringe peak candidate producing minimum |_{j}_{0}. Note that ideally zero order fringe peak _{0} happens at _{d}_{,} first positive order fringe peak _{1} at _{d}_{S}

_{d}_{t}_{d}_{t}

First, fine tuning algorithm assumes that the cross-correlation _{t}_{test}

In _{d}_{d}_{C}^{-}^{1} of its maximum value. And value of _{S}_{S}

Then, the distance between _{d}_{t}_{test}_{d}

In _{sub}

Interpretation of _{test}_{d}_{f}_{f}_{t}_{d}_{test}_{d}_{f}_{t}_{d}_{f}_{t}

Calculating _{t}_{d}_{t}_{d}

Additionally it must be emphasized that the principle of fine tuning in _{i}_{i}_{+1})) and the matched filter _{M}_{i}_{M}

The proposed signal processing algorithm was verified using computer simulations. To see the shot-noise limited performance of the proposed signal processing algorithm, the normalized AFWLI fringe scans, _{S}_{R}_{S}_{R}_{0,S}_{0}_{,}_{R}_{S}_{R}_{S}_{R}_{t}_{0,S}_{0R}_{d}_{d}_{t}_{S}_{R}_{C}

Sample rate _{S}

Effective Coherence length

Size of fine tuning step Δ

In the first simulation miss rate (misidentification rate) of the proposed signal processing algorithm was tested at different shot noise levels. The _{d}_{d}

In

To extrapolate the miss rate beyond the range of 10^{-4} on the BER curve, data points in abscissa in ^{-8} at

After the zero order fringe peak was identified, fine tuning was calculated for resolution enhancement. Phase error Φ_{error}_{,}_{i}_{t}_{0}) and fine-tuned zero order fringe peak _{t}_{0}_{error}_{,}_{i}

^{-3} [fringe] (which is the fine tuning step size) must be greater than 35 dB

In this simulation, the _{0}_{0}

As shown in _{0} is approximately 0.3/_{S}_{0} totally depends on the sample rate. But this RMS error was reduced down to ∼0.0015 fringe when _{0} was fine tuned. Note that RMS error was not sensitive to the sample rate _{S}_{S}_{S}_{S}_{R}_{S}_{sub}

In this simulation, estimated effective coherence length _{C}_{,}_{eff}_{S}_{S}

This is presumably due to the fact that _{test}

This simulation is to show the effect of estimated coherence length _{C}_{C}_{C}_{,}_{eff}_{C}_{C}_{,}_{eff}_{C,eff}_{t}_{d}_{test}

In this section a comparison to the previous literature data regarding the resolution of zero order fringe peak detection is given. Interestingly enough, reference [_{i}_{i})_{i}_{c}

Then, the derivative of ^{2} is given as:

Then, substituting _{0}

As can be seen from the

Increasing the fine tuning range will help to locate the zero order fringe peak correctly and lower the RMS error down to the theoretical limit, although this is time consuming. Optimum fine tuning range of

A new signal processing algorithm for white light interferometry has been proposed. The goal of signal processing algorithm was to find the time delay (phase shift) between two fringe scans which makes it possible to measure the absolute optical path length of a sensing interferometer. This new signal processing algorithm can be used for absolute temperature measure measurement by mapping the zero order fringe peak position of cross-correlation ^{-4} at 31 dB shot noise and the extrapolated miss rate at 35 dB shot noise was 3×10^{-8}. Also resolution of less than 10^{-3} fringe was obtained at 35 dB shot noise (_{C}_{S}_{C}_{S}

The author would like to thank the late Dr. Henry F. Taylor in Texas A&M University for his kind advice and Mr. H. S. Choi who allowed me to use his AFWLI setup.

In this appendix, _{S}_{R}

Here _{P}_{S}_{R}

When _{P}_{P}_{P}_{0}) is a modulation frequency (or angular velocity) of fringe scan. Following the same token,Φ_{S}_{R}_{S}_{R}_{d}

Using the following substitutions:
_{S}_{P}_{R}_{P}

In _{0}/_{f}^{2} and it is assumed that Φ_{R}_{S}_{R}_{S}_{R}(_{0} was set to “1” for normalization in order to make _{S}^{-}^{1} at _{f}

The first part of _{C}_{S}_{R}_{S}_{R}

The first term in _{NF}^{−}^{at2} cos(_{NF}_{d}^{−(}^{t − td}^{)2} cos(_{d}

Well-known Fourier transform relationships useful for the calculation of

If we only consider the envelope of _{NF}^{−at2}_{NF}_{,}_{env}^{−}^{at2} and use the Fourier transform relation:
_{NF}_{,}_{env}_{NF}_{,}_{env}_{d}

Before cross-correlation, from _{S}_{S}^{−}^{1} of its peak value at:

After cross-correlation, from ^{-}^{1} of _{S}_{s}

And the coherence length of the cross-correlation (effective coherence length) has increased by a factor of

The second part of _{d}_{S}_{R}_{d}_{d}

The first term of _{NF}^{−at}^{2} cos(

Then

Again, _{S}_{R}

It can be shown that energy of the (noise-free) interference signal

Useful relations are given by:

Then

But, the second term in

Then, finally _{NF}_{,}_{MAX}_{f}

See

All Fiber White Light Interferometer.

Output of sensing and reference FFPI from AFWLI.

Output of white light interferometer (_{S}_{R}

Linear Interpolation and zero crossing.

Least Square Fit of zero crossing positions.

Matched filtering.

Distribution

Illustration for true zero order fringe peak and discrete sample zero order fringe peak of

Comparison between miss rate and BER.

Extrapolation of miss rate on BER curve.

Change of RMS error as a function of

Change of RMS error as a function of

Comparison between RMS error with fine tuning and RMS error without fine tuning.

Change of RME error as a function of estimation error in zero crossing period.

Change of RMS error as a function of estimation error in light source coherence length.

Comparison of computer simulation and theoretical limit (

Comparison of computer simulation and theoretical limit (

Example of peak and zero crossing table (_{S}

Peak and zero crossing label | _{0} |
_{100} |
_{1} |
_{101} |
_{1} |
---|---|---|---|---|---|

Position | 35 | 39.34 | 45 | 50.13 | 55 |

Miss rate of the proposed signal processing algorithm as a function of

| |||
---|---|---|---|

1 | 0.95 | 17 | 0.46 |

2 | 0.94 | 18 | 0.40 |

3 | 0.93 | 19 | 0.34 |

4 | 0.91 | 20 | 0.30 |

5 | 0.89 | 21 | 0.23 |

6 | 0.87 | 22 | 0.19 |

7 | 0.85 | 23 | 0.14 |

8 | 0.83 | 24 | 0.10 |

9 | 0.80 | 25 | 0.066 |

10 | 0.76 | 26 | 0.041 |

11 | 0.73 | 27 | 0.019 |

12 | 0.69 | 28 | 0.009 |

13 | 0.64 | 29 | 0.004 |

14 | 0.60 | 30 | 0.001 |

15 | 0.56 | 31 | 0.0003 |

16 | 0.51 | 32 | 0 |