Spline function approximation for velocimeter Doppler frequency measurement

892 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 38, NO 4, AUGUST 1989

Spline Function Approximation for Velocimeter Doppler Frequency Measurement

ANDREAS E. SAVAKIS, STUDENT MEMBER, IEEE, JOHN W. STOUGHTON, MEMBER, IEEE,

AND SHARAD V. KANETKAR, MEMBER, IEEE

Abstract-A spline function approximation approach for measuring the Doppler spectral peak frequency in a laser Doppler velocimeter system is presented. The processor is designed for signal bursts with mean Doppler shift frequencies up to 100 MHz, input turbulence up to 20 percent, and photon counts as low as 300 photons. The frequency- domain processor (FDP) employs a bank of digital bandpass filters for the capture of the energy spectrum of each signal burst. The average values of the filter output energies, as a function of normalized frequency, are modeled as deterministic spline functions which are linearly weighted to evaluate the spectral peak location associated with the Doppler shift. The weighting coefficients are chosen to minimize the mean square error.

Performance evaluation by simulation yields average errors in estimating mean Doppler frequencies within 0.5 percent for poor signal- to-noise conditions associated with a low photon count of 300 photons/burst.

I. INTRODUCTION LASER DOPPLER velocimeter (LDV) is an instru- A mentation system which provides nondestructive

measurements of velocities at points inside a flow field such as a wind tunnel [1]-[3]. The flow is not disturbed since only laser light is transmitted to the point where the measurement is taking place. The LDV system can also provide spatial resolution from 20 to 100 pm which cannot be obtained with any other method.

A typical LDV schematic diagram is shown in Fig. 1. Laser light is focused at the control volume, the point where the flow velocity is to be measured. Particles that are injected in the flow pass through the control volume and scatter the laser light. The scattered light is directed to the cathode of a photomultiplier tube (PMT) where it is converted to an electronic signal whose intensity is proportional to the number of photons at the cathode. Ideally, the PMT signal for a photon burst may be characterized by the expression

s ( t ) = F1 exp ( - - a 2 t 2 ) [ 1 + F2 cos (21rft + 4 ) ] = F1 exp ( -u2t2) + FIF2 exp ( -u2t2)

- cos ( 2 x 8 f 4 ) and is shown in Fig. 2.

Manuscript received May 29, 1988; revised January 24, 1989. This work was supported by the NASA-Langley Research Center, under Master Agreement NAS 1-17993-1.

A. E. Savakis is with the Department of Electrical Engineering, North Carolina State University, Raleigh, NC 27695.

J . W. Stoughton is with the Department of Electrical and Computer En- gineering, Old Dominion University, Norfolk, VA 23529-0246.

S . V. Kanetkar is with Engineering Programs,University of Massachu- setts, Boston, MA 02125.

IEEE Log Number 8928365.

1 IAperlure I Aperture Mirror

50150 FLOW SIGNAL

1 1 IPROCESSORI

Fig. 1. Laser doppler velocimeter system.

s ( t

Fig. 2. Idealized photomultiplier tube response.

The PMT signal contains a Doppler shift in laser frequency due to the particle velocity. The frequency trans- form of the ideal PMT signal, shown in Fig. 3, is composed of a low frequency component called the pedestal function and a Gaussian shaped high frequency component centered around the modulating Doppler frequency, f o e

The problem of measuring the flow velocity reduces to estimating fo, since fo is directly proportional to the particle velocity [l]. The modulating frequency can be as- sumed constant throughout the duration of a signal burst. However, it is different for different signal bursts due to the presence of turbulence, and thus should be measured for each signal burst.

The design problems associated with LDV signal processors include signal detection, signal corruption due to noise, frequency variations due to turbulence, and a wide range of mean input Doppler frequencies. Presently, prominent processor designs include the frequency tracker [6] , [7], the high speed burst counter 131-[SI, and the photon correlator [8].

0018-9456/89/0800-0892$01 .OO O 1989 IEEE

SAVAKIS er al . : SPLINE FUNCTION APPROXIMATION 893

PHOTO PULSE DETECTION MULTIPLIER - AND

TUBE CAPTURE CIRCUITRY

BAND PASS DIGITAL ESTIMATES

0.06 Is - $

The frequency tracker operates on the same principle used in the radio reception of FM signals. Its major lim- itation is that a signal must be present at least one percent of the time, otherwise the system drifts out of control. This restricts its use to liquid flows where sufficiently large particle concentration can be achieved.

The high speed burst counter is the processing system that is primarily used today. The counter requires a signal-to-noise ratio of at least 15 dB, and it has a minimum measure of turbulence of 0.5 percent due to time quanti- zation. In order to remove the low frequency component due to the pedestal function, the input signal is highpassed through filters whose parameters are set by the user. If these parameters are not set properly, the Doppler signal could be attenuated, resulting in errors in estimating the Doppler frequency.

The photon correlator measures the Doppler frequency of the signal by measuring the autocorrelation function of the arriving photons. It is primarily used in situations where the scattered light intensity is very low, and the signal is composed of individual photons. The photon correlator can only calculate the mean of the input frequency, and it cannot be used in real time due to the amount of computations needed for the calculation of the autocorrelation function.

In an effort to overcome the limitations of the existing processors a frequency-domain processor (FDP) was pro- posed by Meyers and Stoughton [9]. The FDP is designed to operate in real time, without user intervention. It ideally provides accurate measurements for mean input frequencies from 1 to 100 MHz, Doppler frequency variations due to turbulence up to 20 percent from the mean, and signals with as low as 150 photons per burst. The estimation of the Doppler frequency in [4] is done using curve fitting and counting techniques which were heuris- tically developed. The present work introduces an alter- nate, and more analytical basis for estimating fo. This method is derived from a spline function model which has found much application in the approximation of functions [lo]. The processor organization and signal model are presented in the following sections.

11. FREQUENCY-DOMAIN PROCESSOR (FDP) OVERVIEW

to a constant level by an AGC network, sampled, and quantized by a two bit A/D converter, and shifted through a 256-bit shift register. The two bit conversion was empirically determined to permit sufficient burst character- ization with minimal digital processing hardware. The signal burst is detected when the integral of the shift register contents reaches a preset threshold value. Upon detection of the signal burst, the shift register contents are parallel loaded to temporary registers to await processing. Thus only that portion of the signal that contains useful information is processed.

The second processing stage provides the estimation of the Doppler frequency from the signal burst. The estimation of fo, based on the frequency-domain characteristics of the input signal, is done by determining the frequency that corresponds to the peak of the high frequency component of the energy spectrum. The system employs a bank of 7 digital bandpass filters. The energy at the output of each filter provides an estimate of the signal energy around the filter center frequency. The Doppler frequency is then located by estimating the peak of the discrete energy distribution with respect to the distribution of the individual center frequencies.

The primary advantage of the filterbank approach is that the same filter coefficients can be used to pass different input frequencies by simply adjusting the sampling rate. This way the digital filterbank may be readily adapted to different mean input frequencies relative to the sampling frequency, fs, which may range up to 1 GHz. In addition, the digital filter structure allows separate but concurrent processing of the discrete energy spectrum.

111. SPLINE FUNCTION APPROXIMATION MODEL The input Doppler frequency, which is not observable,

is approximated by linearly weighting the filter output energies, which are observable. The estimation of the Doppler frequency, fo, from the output energies of the 2k + 1 bandpass filters is approached as a linear approximation problem using spline functions. That is, let

i = -k, * * * , 2 , - l , O , 1 , 2 , , k (1)

The FDP block diagram is shown in Fig. 4. The processing of each input signal burst is carried out in two stages. During the first stage, the input signal is amplified

where wi is a scalar chosen to minimize the least mean square error of the approximation [ I l l , [12]. The spline function Pi( * ) is the energy transfer characteristic (ETC),

894 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 3 8 . NO. 4. AUGUST 1989

as a function of the Doppler frequency, of the ith filter centered at A. Since the ETC amplitude also depends on the energy of the input signal and the energy of the noise present at the filter input, normalization is required before using this quantity into the estimation process. One way of normalizing the ETC's is by subtracting the energy due to noise and then dividing by the total energy of the signal burst. Since these two quantities are not accessible, the noise energy is approximated by the smallest energy cap- tured in the filterbank, and the signal burst energy is approximated by the sum of three filter energies at and around the largest filter energy.

For notation purposes define the normalized ETC of the ith filter as Ri ( - ), the normalized Doppler frequency as x, where x = f o / f s , and its approximation as x*. If the approximation is performed over the interval ( a , b ) , the functions R i ( x ) can be viewed as spline functions which are linearly weighted to approximate x in ( a , 6 ) [lo]. Then

k

x* = c a i R i ( x ) , a < x < b (2) i = 1

where a, are the weighting coefficients. The error in the estimation of x is

e = x - x*. ( 3 ) The total mean square error, E, in ( a , b ) is

(4)

I , r - i - 1 . -

150

> U

L 100 W

50

0

0.06 0.08 0 10 0 12 O.1L

FREOUENCY/I,

Fig. 5 . Wide-band filterbank energy transfer characteristics

j -6 0 1 ---I_L ______ L L _____ 3 J _____. 1 _.._._I ___.._ L _____

-3.0 -

0.07 0.09 0.11 0 13 0.15

FREOUENCY/ I ,

0.05

Fig. 6 . Doppler frequency measurement error function, wide-band f i l - ters-40 percent overlap.

Given the functions R i ( x ) , E can be minimized with respect to the coefficients ai (as shown in the Appendix). Once determined, the ai ' s are used in (2) to estimate the normalized Doppler frequency.

IV. FILTERBANK DESIGN The design of the filters in the digital filterbank is gov-

erned by the system design objectives, the frequency estimation algorithm, and the processor cost. For adequate capture, assume that the mean Doppler frequency of in- terest is found at 0.1 fs, where f, is the sampling frequency. The digital filter bank should pass frequencies up to three standard deviations above and below the mean, where 20 percent is the maximum deviation due to turbulence. Thus the range that should be spanned by the filterbank is from 0.04 to 1.6fs. Seven filters were chosen centered at 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, and 0.16 f,. These values represent a compromise between accuracy which requires many filters and cost. For a given signal burst, three of the seven filters contain significant information, so the three corresponding Ri's are linearly weighted for the estimation of the input frequency. All filters are designed as eighth-order Butterworth in order to exhibit monotonic characteristics in the passband, and thus yield smooth ETC's. Lower order filters were examined, but they exhibited poorer spline characteristics. The amount of overlap between adjacent passbands is 40 percent, as empirically determined with the aid of an error

function. The error function is evaluated as follows. First the ETC of each filter is obtained by simulation after averaging the normalized filter output energies over 30 consecutive inputs, each comprised of 750 photons. The input frequency is varied from 0.04 to 0.161, at increments of 0.001 fs. Then the weighting coefficients that minimize the mean square error of the approximation are calculated and stored. The value of the error function at each point is the difference between the actual input frequency and the frequency estimate obtained from linearly weighting the spline functions. Fig. 5 is an illustration of this process, showing the ETC's of the three center filters after averaging 30 consecutive outputs. Fig. 6 shows the error functions obtained for the entire filterbank with passband overlap of 40 percent.

In cases where the input turbulence is much less than 20 percent the frequency range spanned by the filters is too conservative. For input turbulence intensities up to 5 percent a set of filters with narrower bandwidths is used which provides better resolution and more accuracy in the frequency estimation. Three filter sets with seven filters in each set are designed in order to cover the entire range of processor operation. The center frequencies of the filters are at 0.068, 0.076, 0.084, 0.092, 0.100, 0.108, and 0.116f, for the first set, at 0.076, 0.084, 0.092, 0.100, 0.108, 0.116, and 0.124 fs for the second set, and at 0.084, 0.092, 0.100, 0.108, 0.116, 0.124, and 0.132 &

SAVAKIS et al.. SPLINE FUNCTION APPROXIMATION

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for the third set. The narrow filters are again eighth-order Butterworth with adjacent passbands overlapping by 40 percent. Fig. 7 shows the error function obtained from the narrow filters which is significantly smaller than the error function of the wide filters.

DEVIATION

I I I I I I I 1 0 0 1 5 15 20 10

V. PERFORMANCE EVALUATION

A processor simulation program was used for the evaluation of the processor performance. The parameters varied during the simulation testing are the input frequency, the input turbulence intensity, and the number of photons per signal burst. The input turbulence takes values from 0 to 20 percent, and the input signals are comprised of 1500 photons per burst (good signal-to-noise ratio) and 300 photons per burst (poor signal-to-noise ratio). The results presented below are for mean input frequency mapped at 0.095 fs in the normalized spectrum. The results obtained for other values of mean input frequency are very similar.

The processor performance of laminar flow (zero turbulence intensity) is examined initially. The average error and the standard deviation of the error as a function of photon count are shown in Fig. 8. The average error is confined to within 0.1 percent for photon counts as low as 300. Below 300 photons the burst characteristics are poorly defined thus resulting in larger error values.

The error in the mean frequency estimation and the standard deviation for 300 and 1500 photons per burst is shown in Figs. 9 and 10, respectively. As expected from the error function characteristics, the processor performs better for input turbulence less than 5 percent, where the set of narrow filters is used. Finally, the input turbulence estimation is shown in Fig. 11 over the whole range (0- 20 percent) of input turbulence for a photon count of 1500. The solid line represents the ideal measurement. From these data the processor can measure a minimum turbulence of 0.2 percent for good signal-to-noise ratios ( 1500 photons/burst). Similar data at 300 photons per burst indicated that a minimum turbulence measurement of 0.5 percent was obtainable.

LK 0 LK [L W

w 0 0 4 [L W >

A AVERAGE

STANDARD DEVIATION

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0

m < 1.0

-

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0 5 F1 rn a W 0 W

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Fig. 8. Error mean and standard deviation versus photons per burst, normalized frequency = 0.095, no turbulence.

t STANDARD DEVIATION

-1.0 ' I I I I ' 0 0

0 5 10 15 20

INPUT TURBULENCE-%

Fig. 9. Error mean and standard deviation versus input turbulence, normalized frequency-0,095, 1500 photons per burst.

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VI. CONCLUSIONS An analytical spline function approach for measuring

the Doppler spectral peak in LDV signals has been presented. This approach is based on linear weighting of the output energies of bandpass filters. The weighting coefficients are chosen to minimize the mean square error of the approximation. The merits of the approach is that the realization is simple and is suitable for the high-speed throughput requirements in a LDV system employing a FDP. Supporting simulation results indicate that the estimation method will yield both Doppler frequency and

896 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 38. NO. 4, AUGUST 1989

0 5 10 15 20

INPUT TURBULENCE-%

Fig. 11. Measured turbulence versus input turbulence, normalized frequency = 0.095, 1500 photons per burst.

turbulence estimation within the resolution standards of other contemporary methods, such as the high-speed burst counter. In particular, an average error in mean frequency estimation around 0.1 percent was found for situations of laminar flow. Minimum measurable turbulence was between 0.2-0.5 percent, depending on photon per burst levels.

APPENDIX In this appendix the mean square error E of the linear

approximation of (4) is minimized with respect to the coefficients ai. The mean square error is given by the expression

I b 2 E = - J a (x - x*) dx b - a

To minimize E with respect to thejth coefficient let

a aaj - ( E ) = 0

or

k

- 2x c a&(.) dx = 0. i = 1

The variables ai and x are independent of one another, hence, the order of integration and differentiation can be interchanged, or

Since x does not depend on aj :

a - ( x 2 ) = 0. aaj

Similarly

because the aj’s are independent of ai for i = j . Also:

i = I

T k 1 = 2 c U i R i ( X ) R j ( X ) .

l i = I 1 Therefore,

or

or k 1: [ zl aiRi(x) R j ( x )

or

ai sb Ri(x ) R j ( x ) dx = i = I 0

is the equation that is obtained if E is minimized with respect to ai. If E is minimized with respect to all ai, k equations will be obtained. For simplicity in notation let

r b

and r b

Zxj = 3 xRj(x) dx. U

Then the k equations are k c UjZil = ZXI

i = I

k c a i 4 2 = zx2 i = I

a - 2x- c a,R, (x) dx = 0.

aaj [ i : l ] k

SAVAKIS et a l . : SPLINE FUNCTION APPROXIMATION 897

These equations can be expressed in matrix form as: Simulation Facilities, von Karman Institute for Fluid Dynamics, Rhode-Saint-Genese, Belgium, June 21-23, 1971.

[4] P. D. Iten and J. Mastner, “A laser Doppler velocimeter offering high spatial and temporal resolution,” presented at Symp. on Flow-Its Measurement and Control in Science and Industry, Pittsburg, PA, May

[5] J. A. Asher, “LDV systems development and testing,” presented at the Workshop on the Use of the Laser Doppler Velocimeter for Flow Measurements, Purdue Univ., West Lafayette, IN, March 9-10, 1972.

[6] R. M. Huffaker, C. E. Fuller, and T. R. Lawrence, “Application of laser Doppler velocity instrumentation to the measurement of jet turbulence,” presented at the Int. Automotive Engineering Congress, SAE, Detroit, MI, Jan. 13-17, 1969.

[7] 3. D. Fridman, K. F. Kinnard, and K. Miester, “Laser Doppler instrumentation for the measurement of turbulence in gas and fluid flows,” Electro-Optical System Design Conf., New York, NY, Sept. 16-18, 1969.

10-14, 1971. (A. 1 )

[;:: . - * ;:I[ = [;:I I2k I3k ’ I k k Ixk

or

C A = e --

where _C is (kxk). It can be shown that C is nonsingular and thus the elements of A can be determined by A_ = - C-’ B. The proof is beyond the scope of this paper and may be obtained from the authors.

REFERENCES

[ I ] L. E. Drain, The Laser Doppler Technique. New York: Wiley, 1980. [2] F. Durst, Principles and Practice of Laser Doppler Anemometry.

New York: Academic, 1981. [3] A. E. Lennert, F. H. Smith, Jr., and H. T. Kalb, “Applications of

dual scatter, laser Doppler velocimeters for wind tunnel measurements,” presented at 4th Int. Congress Instrumentation in Aerospace

[8] E. R. Pike, “The application of photon-correlation spectroscopy to laser Doppler velocimeters,” presented at the Workshop on the Use of Laser Doppler Velocimeter for Flow Measurements, Purdue Univ., West Lafayette, IN, Mar. 9-10, 1972.

[9] J. F. Meyers and J. W. Stoughton, “Frequency domain laser velocimeter signal processor,” presented at Third Int. Symp. on Appli- cations of Laser Anemometry to Fluid Mechanics,” Lisbon, Portu- gal, July 7-9, 1986.

Reading, MA: Addi- son-Wesley, 1964.

New York: Wiley, 1975.

New York: Wiley, 1969.

[IO] J. R. Rice, The Approximation of Functions.

[ l l ] P. M. Prenter, Splines and Variational Methods.

[I21 D. G. Luenberger, Optimization by Vector Space Methods.

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Spline function approximation for velocimeter Doppler frequency measurement