Statistical behaviour of laser Doppler vibrometer detector signals and application of statistics for signal diversity

The problem of speckle noise and the so-called signal dropouts is well-known by many LDV sensors. Signal power is reduced when less light power is scattered into the detection aperture due to laser speckle. Signal dropouts occur when the carrier-to-noise ratio (C/N) of the photodetector signal falls below the frequency modulation (FM) threshold. This problem can be resolved with signal diversity. However, the influence of the speckle noise and signal dropouts has not been quantitatively analysed the state of the art. We introduce in this paper a new mathematical model of the statistical behaviour of detector signals of LDV sensors. This allows us to calculate the influence of speckle noise and signal dropouts on the detector signals of LDV sensors. Furthermore, we optimize the signal combination method of signal diversity according to the state of the art. Finally, we present some experimental results to prove our mathematical model and signal diversity for reducing speckle noise and signal dropouts.


1.
Introduction The theoretical vibration-amplitude resolution of a laser Doppler vibrometer (LDV) on a mirror is very high (a few femtometer per square root Hertz).However, high resolution is difficult to achieve in many practical applications because many factors can affect the signal-to-noise (S/N) ratio of the demodulated signal.One Key factor is the speckle noise [1].When a laser beam illuminates a rough surface, the high coherence of the light produces a random interference effect known as laser speckle, and the light intensity is distributed non-uniformly [2].When the photodetector of an LDV sensor receives the bright speckle, a strong signal is obtained.Signal power is reduced when less light power is scattered into the detection aperture which is denoted as dark speckle.Signal dropout occurs when the carrier-to-noise ratio (C/N) of the photodetector signal falls below the frequency modulation (FM) threshold.The modulated signal can be demodulated correctly if the C/N is higher than the FM threshold.If the C/N is below the FM threshold (in case of a signal dropout), the demodulated signal contains only noise.This problem is well known by different LDV sensors such as out-of-plane LDVs [3,4,5,6,7,8] and in-plane LDVs [9,10].Speckle noise in out-of-plane LDVs resulting from additional in-plane motion has been widely studied and could be reduced [11,12,13].The most efficient solution among them is signal diversity.Diversity technology was first used to improve signal quality in communications [14,15,16].Radiofrequency (RF) signal is transmitted via several stochastically independent channels, whereby the simultaneous occurrence of signal dropouts in these channels is extremely unlikely.After combining the signals from independent channels, the errors can be drastically reduced.In out-of-plane LDVs, signal diversity was introduced to reduce signal dropouts and speckle noise [17,18,19].We also extended the diversity method to an in-plane LDV in our previous publication [20].However, the signal combination method from the state of the art can still be optimized.The dominant noise sources of a photodetector of an LDV, such as shot noise, thermal noise, and technical noise of an operational amplifier can be considered as additive white Gaussian noise (AWGN).In telecommunication technology, maximum ratio combining (MRC) [21] is known as the optimal combining method for independent AWGN channels [22].We implement in this paper MRC in LDV sensors.Moreover, we improved our mathematical model, which was published in [20].The first result about the statistical behaviours of the detector signal and the signal diversity in [20] considered only the occurrence of signal dropouts.We calculate in this paper not only the probability of signal dropouts but also the influence of speckle noise on the carrier noise ratio (C/N) of detector signals.We provide an explanation of why MRC is better than the combination of the state of the art.This improved model can be theoretically applied to both types of LDVs (out-of-plane and in-plane).Finally, we provide some experimental results to validate this theoretical model.

2.
Theory In this section, we introduce some basic theories and our mathematical model of statistical behaviour of LDV detector signals.

Out-of-plane and in-plane LDV
In this paper, we study the statistical behaviour of detector signals of heterodyne LDV.Generally, there are two types of such LDVs.One is out-of-plane LDV, which measures the out-of-plane movement (shown in figure 1).It is principally a two-beam laser interferometer.The measurement beam is directed to the target.Then the scattered light from the target surface is collected and interfered with the reference beam on a photodetector.The frequency of one of the beams (measurement or reference beam) is shifted (usually by a Bragg cell).This frequency is the carrier frequency   .Doppler frequency shift   (Difference between detector signal frequency and the carrier frequency) is proportional to the velocity to be measured () with () =  •   () .
(2.1)  with amplitude  ̂ of the carrier signal, the carrier frequency   , the modulation index , the velocity to be measured  and the noise term ().The single difference is, that is for an out-of-plane LDV and is for an in-plane LDV.

Statistical behavior of a single detector signal of an LDV
Laser speckles result from a random interference effect.To analyse the influence of speckle noise and the so-called signal dropouts of the detector signals, the statistical behaviour of detector signals must be researched.As shown in figure 3 left, light scattered from a single or several laser speckles passes through an objective and is finally collected by a photodetector of an LDV sensor.We can assume that the light signal is transmitted through multiple paths to the detector.For each path, the signal strength and phase are random variables.To analyse the total signal amplitude on the detector, we can compare this situation to Rayleigh fading [23,24] in telecommunication technology (see in figure 3   In a Rayleigh fading channel, an RF signal reaches the receiver through a multiplicity of paths, where the signal is scattered by multiple objects in the vicinity of the transmitter and receiver.The attenuation of the signal on each path can be considered as a stochastically independent random variable.Thus, the received signal can be considered as the sum of multiple stochastically independent random variables.If there is no dominant path, the phase of the received signal is equally distributed between 0 and 2π.The amplitude of the received signal corresponds to a Rayleigh distribution [25].Similarly, the incident light of an LDV sensor is scattered by laser speckles and via multiple paths detected at the photodetector.The detector signal can also be considered as the sum of several stochastically independent random variables.Here it is taken that the phases are equally distributed [1].Therefore, the distribution of the carrier amplitude   of a single detector corresponds to a Rayleigh function as in Rayleigh fading. 0 is here the expected value of E(  ).
The carrier power   of a single detector signal is proportional to the square carrier amplitude   2 with   =     2 .
(2.7)Here   is a constant equal to inverse of impedance.The probability density function of   is thus given as with the expected value  0 = E(  ) and corresponds to an exponential distribution.
The carrier noise ratio (C/N)  is defined by the carrier power   and the total noise power   with  =   /  .An out-of-plane LDV is commonly shot noise limited by the constant reference-light power [5] for any received measurement-light power level, which means shot noise is dominated.That light power is constant and on usual rough technical surfaces much larger than the light power from the measurement beam.Therefore, the shot noise and the total noise power   are independent of the speckle effect and can be considered constant.For an in-plane LDV, photodetectors only collect the scattered light from the target surface.The received light power is then usually very low (a few μW or even in the nW range).Thus, the main noise source is not shot noise, but thermal noise or technical noise from the operational amplifiers of photodetectors.These noise sources are not affected by the speckle effect and   can be considered constant, as well as by an out-of-plane LDV.Then the probability density function of C/N can be given as ) . (2.10) With the expected value  0 = E() =  0   ⁄ ,   () is given as The cumulative distribution function of C/N of a single detector signal   () can be calculated with   () as ) . (2.12) The probability of signal dropouts  , ( ≤   ), that C/N below the FM threshold at a C/N value of  =   , of a single detector signal can be given as (2.13)

Signal diversity and MRC
The problem of speckle noise and signal dropouts can be well solved by signal diversity.This technology was first used in communications.The signal is transmitted over multiple stochastically independent channels.The probability that the signal dropouts occur simultaneously for all independent signals is much lower for a single channel.Using an optimal signal combination method, the total probability of a dropout over all channels is less than the product of the probability over all individual channels.For laser sensors, wavelength, angular (different apertures for the receive channels) or polarization diversity can produce statistically independent speckle patterns [1] and thus independent detector signals (see figure 4).Here   () is the diversity-combined velocity signal,  , () is the demodulated velocity signal from a single detector signal,  is the number of different diversity channels and   is the weighting factor.  is in [17] determined by with   the C/N of the -th detector signal.However, this combination algorithm can still be optimized.The primary noise sources of an LDV detector, e.g.thermal noise, shot noise, and technical noise of the operational amplifier, can be considered as additive white Gaussian noise (AWGN) with a limited bandwidth.Thus, a single detector signal from an LDV sensor is an AWGN channel.As known in telecommunication technology, maximum ratio combining (MRC) is the optimal combining method for independent AWGN channels, where the S/N of the diversity-combined signal is best [22].MRC is a linear combination method.The combined signal   () (velocity signal) is also given by the formula (2.14 . (2.20) When all detectors are equally adjusted (the mean C/N of all detector signals equal to  ̅̅̅̅ ), the optimal mean C/N of the diversity-combined signal  ̅̅̅̅  is given as  ̅̅̅̅  =  ̅̅̅̅ .
(2.21)For simplicity, we assume in the following text that all individual detectors are equally adjusted.

Statistical behavior of diversity combined signal of an LDV
In section 2.2, we derived the density function of the C/N of a single detector signal as formula (2.11).To determine the density function of the C/N of the combined signal, the moment generating function (MGF) is considered.The MGF   () of a single detector signal is given as ( where   () from formula (2.11) is the density function of the C/N of a single detector signal.The MGF of the sum of multiple independent random variables is the product of the MGF of every random variable [27].The MGF    () of the diversity-combined signal from  different detectors is then expressed as This MGF corresponds to the gamma or Erlang distribution (the shape parameter of a gamma distribution of  is an integer) [22].Thus, the density function of the C/N of the diversity-combined signal    (  ) can be written as The expected value  0 of   is then given as Therefore, it is expected that the average C/N  ̅̅̅̅  of the diversity-combined signal is the sum of the C/N of all individual diversity channels of nCN ̅̅̅̅ 0 .This is consistent with the results of MRC from the formula (2.21).
The cumulative distribution function of C/N of the diversity-combined signal    () can be calculated with    (  ) as with   the FM threshold,  0 the expected value of  from formula (2.11), and  the number of independent diversity detector signals.
The probability of signal dropouts  , ( ≤   ), that C/N below the FM threshold at a C/N value of  =   , of the diversity-combined signal can be given as ) . (2.27) The reduction of the probability of signal dropouts ∆  by using the diversity technique is calculated from the difference between probabilities of dropouts of a single detector signal  , ( ≤   ) from (2.13) and the diversity-combined signal  , (  ≤   ) from (2.27) as ) . (2.28) When  0 (corresponds in practice to the mean C/N of a single detector signal) approaches infinity, the limit of ∆  from (2.28) is expressed as )) = 0 . (2.29) If  0 approaches zero, the limit of ∆  is given by using the rule of de L'Hospital [28] through Therefore, diversity technology can improve signal quality only if at least one channel has sufficient signal strength.Thus, for a very small mean C/N (corresponds to  0 → 0), the noise may be too strong to allow diversity to improve signal quality.Moreover, there is almost no improvement in reducing signal dropouts at very good C/N (corresponds to  0 → ∞) because the single signal is already of good quality.However, the signal quality can still be improved by using diversity, since the C/N of the diversity-combined signal is the sum of the C/N of each single detector signal (see formula (2.21)).

Experimental setup
In section 2, we established a mathematical model for the statistical behaviour of the detector signal of LDV sensors.This model can be applied not only for an out-of-plane LDV but also for an in-plane LDV.
For this purpose, we use an out-of-plane LDV in [29] and an in-plane LDV in [20,30].Meanwhile, this in-plane LDV was built with signal diversity, whereby both photodetectors collected the light from orthogonal polarization states and stochastic independent detector signals were obtained (see in [20]).
An aluminium wheel with a circumference of 0.314 m was used as the measurement target.The surface of the wheel was painted with retro spray to increase the scattered light power.The schematic setup is shown in figure 5.If the wheel is at different heights, the same process can be repeated several times and totally   = 10 4 different speckle positions can be measured over the entire wheel surface.

Statistical behavior of the carrier signal.
As in section 2.2 mentioned, we provided the density function of the carrier amplitude    (  ) in formula (2.6) and power    (  ) in formula (2.9) of a single detector signal from an LDV sensor.The first result is about the statistical behaviour of carrier signals.The histogram of the measured carrier amplitude  , and power  , at   = 10 4 different speckle positions are plotted with the theoretically estimated density functions    (  ) and    (  ) (see figure 7).The expected value  0 in (2.6) and  0 in (2.9) can be determined by the measured mean carrier amplitude  ̅ , of and the measured mean carrier power  ̅ , of The theoretic density functions in the histogram were multiplied by the bin width of the histogram.The measured carrier amplitude  , of an out-of-plane LDV matches the estimated density function    (  ) from formula (2.6) and the measured carrier power  , is also in agreement with the estimated density function    (  ) from formula (2.9). Figure 8 shows the same results but from the inplane LDV.This figure shows that the theoretical density functions are in good agreement with the experimental results.Therefore, our mathematical model can be applied not only to an out-of-plane LDV but also to an in-plane LDV.

Statistical behavior of the C/N.
We provided in sections 2.2 and 2.4 the density function of the C/N   () in formula (2.11) of a single detector signal and    (  ) in formula (2.24) of the diversity-combined signal.Here we compare the measured C/N from the in-plane LDV in [20,30] with the mathematical model.Thus, the theoretical density functions   () and    (  ) also agree with the experimental results   and  , .Because the out-of-plane LDV (see in [29]) used does not have signal diversity.So only the results of signal diversity from the in-plane LDV are presented in this paper.

3.2.3
Probability of signal dropouts.We established in sections 2.2 and 2.4 the theoretical probability of signal dropouts from a single detector signal  , in formula (2.13) and from the diversity-combined signal  , in formula (2.27).In both formulas, we assume that the C/N value at FM-threshold is equal to 1, which means that the total noise power greater than the carrier power defines a signal dropout.In this section, we show the results of the probability of signal dropouts in figure 10.
Both solid lines describe the theoretical probability of dropouts  , and  , .The expected value  0 in both equations is determined by the mean C/N  ̅̅̅̅ as Comparing  ,, with  , and  ,,, with  , , our mathematical model is in high agreement with the experimental results. ,, (blue diamond) lies upon  ,,, (purple square).Therefore, it is demonstrated that diversity technology can significantly reduce the occurrence of signal dropouts.Because of  ,,, (red triangle) above  ,,, (purple square), we can conclude that practically the MRC is also better than the combination method from state-of-the-art.
Figure 11 shows the reduction of the probability of signal dropouts.The red solid line is the theoretical reduction of ∆  in formula (2.28).The blue "+" describes the experimental results given by ∆ , =  ,, −  ,,, , (3.6) with  ,, the experimental relative frequency of dropouts from a single detector signal and  ,,, the experimental relative frequency from the diversity-combined signal with MRC (see figure 10).The orange triangle corresponds to the experimental results expressed as ∆ ,,− =  ,,, −  ,,, , (3.7) with  ,,, the experimental relative frequency from the diversity-combined signal with the signal combination from state-of-the-art (see figure 10).This figure demonstrates that diversity technology can only significantly improve signal quality if at least one channel has sufficient signal strength.Thus, with a very low mean C/N, the noise is too strong to allow diversity to improve signal quality.Furthermore, there is no improvement in reducing dropouts with a very strong C/N because the single signal is already of good quality.These both conclusions are also consistent with the theoretical model from formulas (2.29) and (2.30) (the limit of ∆  converges to zero by  0 to zero or infinity).On the other hand, signal diversity can still improve the signal quality when the average C/N  ̅̅̅̅ approaches infinity ( 0 in (2.29) approaches infinity accordingly).In this case, there is almost no signal dropout in a single detector signal.From formula (2.21), the mean C/N of the diversity-combined signal is the sum of the mean C/N of all the single detector signals.

Conclusion
The paper presents a signal-combination method for different LDV sensors that improves the state of the art.We build a mathematical model to estimate the signal quality (C/N and probability of signal dropouts) of detector signals of LDV sensors.This model can not only be applied to out-of-plane LDVy but also by to in-plane LDVy.The experimental results are in high agreement with the theoretical model and demonstrate that signal diversity can significantly improve the C/N and reduce the occurrence of signal dropouts of the detector signal of an LDV sensor.In this paper, we employed the new method to an in-plane LDV [20] which is not shot-noise limited as the out-of-plane LDV.Therefore, we have not performed experiments of signal diversity by an out-ofplane LDV.Otherwise, similar results (as figure 9, 10 and 11) can be expected also for an out-of-plane LDV, since there is no obvious difference between the statistical behaviours of an in-plane and an outof-plane LDV (see figure 7 and figure 8).

Figure 1 . 3 Here
Figure 1.The optical structure of an out-of-plane LDVThe other sensor type of laser-Doppler vibrometry (LDVy) is the in-plane LDV, which can measure inplane movement.Its schematic is shown in figure2.It is also a two-beam laser interferometer like an out-of-plane LDV.However, both beams superimpose on the target surface at an angle  and create an interference fringe pattern.One of both beams is also frequency-shifted (usually by a Bragg cell) so that the fringes also move at a carrier frequency of   .When a scattering body (laser speckle) on the target surface moves against the fringe, the frequency of intensity modulation of the fringe pattern corresponds to the velocity () perpendicular to the fringes.() can be calculated with () =   () 2 sin  2

Figure 2 .
Figure 2. Optical structure of an in-plane LDV Both detector signals are all frequency modulated (FM) or phase modulated (PM), although both LDVs use different principles.The AC-coupled detector signal can be described as   () =  ̂ cos (2   +  ∫ ( ) d t 0 right).

Figure 3 .
Figure 3.The scattered light transmits onto a photodetector of an LDV (left).The RF signal transmits to the receiver by Rayleigh fading (right).

Figure 4 :
Figure 4: Schematic view of the wavelength (a), angular (b), and polarization (c) diversity.Only two diversity channels are displayed for each method.Then the signals from different diversity channels are combined.The improvement of the C/N and the reduction of the probability of signal dropouts are different for the different signal combination algorithms.The signal combination in LDV sensors from state of the art (CSoA) is performed for example with velocity signals as [17]   () = ∑    , ()

Figure 5 .
Figure 5. Schematic view of the experimental setup.The wheel was driven by a servo motor and rotated at a linear speed of 0.01 m/s.A single speckle position was measured every 31.4ms, corresponding to a distance of 314 μm between adjacent speckle positions.Thus, there were total 1000 speckle positions equally distributed on the target surface.The width of the image area of the photodiode on the target surface was 300 μm.The distance between two speckle positions was larger than the image area, thus the adjacent speckle positions were not overlapped.The measurement time for each speckle position was 0.03 ms.The in-plane motion of the measurement surface during this time is 0.3 µm (0.1 % of the speckle size).The speckles can be assumed

Figure 6 .
Figure 6.Configuration of the speckle positions on the surface of the rotating wheel.

Figure 7 .
Figure 7. Histogram of the measured carrier amplitude  , and the theoretical density function    (  ) from formula (2.6) (left).Histogram of the measured carrier power  , and the theoretical density function    (  ) from formula (2.9) (right).Data were measured with the out-of-plane LDV.

Figure 8 .
Figure 8. Histogram of the measured carrier amplitude  , and the theoretical density function    (  ) from formula (2.6) (left).Histogram of the measured carrier power  , and the theoretical density function    (  ) from formula (2.9) (right).Data were measured with the in-plane LDV.
Figure 9 shows the histogram of the measured C/N with the theoretical density functions.The expected value  0 in both formulas is determined by the measured mean C/N  ̅̅̅̅ of   the measured C/N from   = 10 4 different speckle positions.

Figure 9 .
Figure 9. Histogram of the measured C/N (   of a single detector signal and  , of the diversity-combined signal) and the theoretical density function (  () from formula (2.11) of a

3 . 5 )Figure 10 .
Figure 10.Probability of signal dropouts.The measured relative frequency of signal dropouts from a single detector signal  ,, , from the diversity-combined signal with the signal combination from state-of-the-art  ,,, , and the diversity-combined signal with MRC  ,,, .The theoretical probability of signal dropouts from a single detector signal  , in formula (2.13) and from the diversity-combined signal  , in formula (2.27).

Figure 11 .
Figure 11.Reduction of the probability of signal dropouts.∆  the theoretical reduction in formula (2.28), ∆ , the experimental results in formula (3.6) and ∆ ,,− the experimental results in formula (3.7).
).But the weighting factor   is calculated other than the formula (2.15).For a certain time,  , is the amplitude of the carrier signal, and  ̅  2 =  , is the mean noise power in the i-th diversity channel.The carrier power of the diversity-combined signal can be expressed as  , the total noise power of the diversity-combined signal and   the C/N of the i-th detector signal.Because the noise can be considered as AWGN, we can assume , = ∑(   ̅  ) 2The equality holds exactly if the terms    ̅  and  , / ̅  are linearly dependent (  ~,  ̅  the weighting factor   of the i-th detector signal is given by 6 with