УДК 620.179.16
MAXIMUM FRACTIONAL CROSS-CORRELATION SPECTRUM FOR TIME OF ARRIVAL ESTIMATION OF ULTRASONIC ECHOES
Wei Liang1, Liang Chen1, Feng-xing Zhou2, Zhi-Hua Ge1, Guo Ding1 1 School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, P.R. China E-mail: liangwei@uestc.edu.cn 2 School of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan, 430081, P.R. China
Abstract. In this paper, a maximum fractional cross-correlation spectrum (FCCS) parameter estimation method is proposed to estimate the time of arrival (TOA) and/or time of flight (TOF) of the ultrasonic echoes with a high resolution. The FCCS is the integration of the fractional cross-correlation of a matched filter and the truncation function of the ultrasonic signal with respect to each time delay, and the range of the integration is limited in the essential-bandwidth which is the bandwidth of the fractional autocorrelation of the ultrasonic echo. If the matched filter matched the truncation function of the ultrasound signal, a local maximum magnitude of the FCCS can be obtained. It can simultaneously produce a measure of TOA of the ultrasonic echoes. The use of FCCS in estimate the TOA or/and TOF of the ultrasonic echoes can eliminate/reduce the noise in an effective manner, even under the very noisy condition, the curve of the FCCS is still smooth with the prominent local extremums. The maximum FCCS parameter estimation method can be repeated with different fractional Fourier transform (FRFT) in order to obtain multiple estimates of TOA of the ultrasonic echoes. These multiple estimates can further be averaged to obtain more robust estimates of the TOA. Numerical simulations and experimental results make evidence the good performance of the proposed approach which has good effect of noise suppression and results in much improved accuracy in the estimation of echo arrival time.
Keywords: fractional cross-correlation spectrum (FCCS), fractional fourier transform (FRFT), time of arrival (TOA), ultrasonic testing.
МАКСИМУМ ЧАСТИЧНОГО СПЕКТРА КРОСС-КОРРЕЛЯЦИИ ДЛЯ ОЦЕНКИ ВРЕМЕНИ ПРИХОДА УЛЬТРАЗВУКОВЫХ ИМПУЛЬСОВ
Вей Лианг1, Лианг Чень1, Фенг-цинь Жоу2, Жи-Хуа Ге1, Гуо Динг1 1 Школа механотроники, Китайский университет электронных наук и технологии, Ченгду, 611731, КНР 2 Школа информационных наук и техники, Вуханьский университет науки и технологии, Вухань, 430081, КНР
Предложен метод измерения максимума частичного спектра кросс-корреляции (FCCS) для оценки времени прихода (ТОА) и/или времени распространения (TOF) у.з. эхоимпуль-сов с высоким разрешением. Метод FCCS заключается в интегрировании частичной кросс-корреляции согласованным цифровым фильтром и построении функции прерывания у.з. сигнала для определения задержки в каждый момент времени. Процедура интегрирования ограничена полосой пропускания для автокорреляционной функции у.з. сигнала. Если фильтр согласован с функцией прерывания у.з. сигнала, то может быть получен локальный максимум FCCS. Использование FCCS для оценки ТОА и/или TOF у.з. эхосигналов может исключить или уменьшить шумы. Даже при большом уровне шумов кривая FCCS вполне гладкая с выраженными локальными экстремумами. Метод оценки максимума FCCS может быть применен повторно к фурье-преобразованию с целью получить дополнительные оценки ТОА у.з. эхосигналов, которые можно усреднить для итоговой (более устойчивой) оценки. Численные расчеты и экспериментальные результаты подтверждают действенность предложенного метода, который продемонстрировал значительное подавление шума и привел к улучшению точности времени прихода импульса.
Ключевые слова: частичный спектр кросс-корреляции (FCCS), частичное преобразование Фурье (FRFT), время прихода (ТОА), у.з. контроль.
1. INTRODUCTION
Many ultrasonic testing applications are based on the estimation of the time of arrival (TOA) and/or time of flight (TOF) of the ultrasonic echoes. It produces
a peak at the TOA of the received signal. Unfortunately, the received signals are also contaminated by noise originating from both the measurement system and specimen. The noise embedded in useful signals, sometimes even very heavy, places a fundamental limit on the detection of small defects and the accuracy of measurement. The amplitude of the flaw echo can be quite small, and buried in coherent noise, leading to some difficulty in detecting them with an acceptable signal to noise ratio (SNR) [1, 2]. The methods used to estimate TOA and TOF in ultrasonic applications are mainly based on the cross-correlation method by finding the extremum of this cross-correlation. The time delay estimation (TDE) have been also proposed in the literature [3—6] for involving either the time-domain signals or some transformed version of them using various signal transforms, e. g., the conventional Fourier transform and the Hilbert transform. However, these estimators suffer from severe degradation in performance at low values of the SNR. With the development of the fractional Fourier transform (FRFT), the delay estimation based on the FRFT has received much attention in the signal processing community. Pei and Ding applied discrete fractional correlation to determine the same object located in a different place for the pattern recognition [7]. Sharma and Joshi proposed several estimators of the TDE using the FRFT to enhance the performance of the time delay [8]. Furthermore, Tao presented the mathematical expressions of the output SNR and the accuracy for TDE associated with the FRFT [9].
In this paper, we have presented a maximum fractional cross-correlation spectrum (FCCS) parameter estimation method that can estimate the TOA or/and TOF of the ultrasonic echoes with a high resolution. The use of FCCS in estimate the TOA or/and TOF of the ultrasonic echoes can eliminate/reduce the noise in an effective manner, even under the very noisy condition, the curve of the FCCS is still smooth with the prominent local extremums. The maximum FCCS parameter estimation method can be repeated with different FRFT order to obtain multiple estimates of TOA of the ultrasonic echoes. These multiple estimates can further be averaged to obtain more robust estimates of the TOA.
The remainder of this paper is organized as follows. Section 2 is devoted to the study of the estimation method of the TOA. In Sections 3 and 4, results obtained from the analysis of numerical and actual ultrasonic signals are presented. Conclusions are given last.
2. ESTIMATION OF THE TIME-OF-ARRIVAL
2.1. The fractional Fourier transform and fractional cross-correlation
The FRFT uses a transform kernel which essentially allows the signal in the time-frequency domain to be projected onto a line of arbitrary angle a. The FRFT of a signal f(t) with an angle a is defined as [10]
f
«(„) = { f (t) Ka(t, и) dt,
(1)
where KT(t, и) can be expressed as
K„(t, u) =
1- j cota
-cot a - jut csc a
2% 5 (t - и) 5 (t + и)
a Фпп
a = 2n% , a = 2n% + %
(2)
where n is an integer. The FRFT with a = %H corresponds to the conventional Fourier transform, and it reduces to the identity operator for a = 0. The FRFT orderp can also be used in place of FRFT angle. The relationship between FRFT order and angle is given by
2a
P = —. (3)
n
The fractional cross-correlation of two continuous-time signals ft) and s(t) is given by
(fa ® sa)(u) = f ®a s) (u) = eJ2n("V2)cosasina J f (t) s* (t - u cos a) e"J2ntusinadt, (4)
where the superscript * denotes the complex conjugate.
The fractional cross-correlation can be used along with fast FFT and FRFT [12, 13] algorithms to efficiently implement a discrete approximation of the fractional convolution.
fa ® sa)(u) = (F-Kl2{f (rt/2)+a(h(rt/2)+a)*})(u), (5)
where F-rt/2 denotes the inverse Fourier transform, f(rt/2)+a and h(rt/2)+a are the fractional Fourier transform with an angle (rc/2) + a off and h, respectively.
2.2. Maximum FCCS for estimation of the time-of-arrival
In ultrasonic non-destructive testing, the measured ultrasound signal y(t) can be modeled as [11]
N
y(t) = £ph (t - t,.)+e (t), (6)
i=i
where h(t) represents the transmitted wavelet which is influenced by the propagation as well as the transducer impulse response, and which is limited in time [-T, T]. N is the number of ultrasonic echoes. The term e(t) accounts for the measurement noise and can be characterized as white Gaussian noise (WGN). Tj giving the time of the spikes which is related to the location of the reflector as the distance from the transducer over the speed of ultrasound in the propagation path, and giving their amplitudes which is primarily governed by the impedance, size, or orientation of the reflector.
Consider the fractional cross-correlation of two ultrasonic echoes h(t - x) and h(t), it can expressed as
(h ®a hx)(u) = e2 n(u 2/2cos a sin a)J h (t) h* (t - u cos a) e -J 2 ntu sin a dt. (7)
According to Eq. (7), the maximum magnitude of the fractional Fourier spectrum appears at x = u cos a. The maximum magnitude can be expressed as
I f I l2
d(x) = J \ h (t^ e
- j2%tx tan a
(8)
It can be seen from Eq. (8) that the bigger the term xtana, the faster the term e-PKn tana to be attenuated.
Consider a matched filter s(t) which is limited in time [-T, T] for the measured ultrasound signal y(t). Let yTz(t) denote the truncation function in time [-T+t, T+t] ofy(t)
y?(t) = y (t) • ^ ^2t] = XPh (t - ^) • rec^ ^^] + e (t) • ^^] =
= Xß,h ( - x, )• recti\ — \ + ezT(t )
(9)
where
rectx
t \ [1 -T + x<t<T + x
2T j 10 otherwise
(10)
The fractional cross-correlation of the matched filter s(t) and the truncation function yTx(t) can be expressed as
(h ® yJ)(u) = e
j 2 n(u2/2)cos a sin a
J ylT (t) 5* (t - u cos a) e-j2ntusinadt. (11)
When s(t) = h(t), d(x) = |(s ® yTT)(i)\ can be expressed as
d(u) = |(5 ®„ y*)(u)\ =
T
J Ут (t) h* (t - и cos a) e-j2ntusinadt
J
£ß,h (t - T, )+ eT(t )
h* (t - u cos a)
j 2 ntu sin a
dt
(12)
To maximizes the integra
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