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This paper is a manuscript of a paper (doi:10.1016/j.measurement.2013.07.038) published
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http://www.sciencedirect.com/science/article/pii/S0263224113003370
Subsample interpolation bias error in time of flight
estimation by direct correlation in digital domain
Linas Svilainis*, Kristina Lukoseviciute, Vytautas Dumbrava, Andrius
Chaziachmetovas
Signal processing department, Kaunas University o f Technology,
Studentu str. 50, LT-51368 Kaunas, Lithuania,
* Tel.: +370 37 300532; fax: +370-37-352998; E-mail: linas.svilainis@ktu.lt
Abstract
This work presents an investigation of the bias error introduced in time of flight
estimation realized by subsample interpolation in digital domain. The time of flight
estimation is accomplished based on the evaluation of the peak position of the cross
correlation function. In order to cope with the discrete nature of the cross-correlation
function, subsample estimation exploits three time domain interpolation techniques:
parabolic, cosine, Gaussian and frequency domain interpolation using phase angle. An
empirical equation relating the maximum value of the bias error to sampling frequency
and signal parameters (center frequency and envelope bandwidth) has been derived. It is
found that the maximum value of the bias error is in inverse cubic relation to sampling
frequency and in quadratic relation envelope bandwidth for cosine interpolation. The
maximum value of the bias error is in inverse cubic relation to sampling frequency and
in quadratic relation to center frequency and envelope bandwidth for parabolic
interpolation. The coefficients related to the approximation technique are given. Results
can be applied for bias errors estimation or correction when fast subsample interpolation
is used and application of phase domain interpolation is unacceptable due to processing
speed limitations. The equations for minimum required sampling frequency are derived
by balancing the interpolation error against Cramer-Rao lower bound.
Keywords
Time of flight estimation; time delay estimate; matched filter; subsample
interpolation; bias error.
1. Introduction
The time-of-flight (ToF) measurement is a recently used procedure for estimating
the signal propagation time [1,2]. Ultrasound application for imaging and measurement
is popular since it is the only technique offering the direct interaction with test media [3-
5]. Majority of measurement systems use the signal delay: non-destructive testing [3],
food processing [4], thickness [5], flow [6], temperature meters [7], biomedical [8],
plate wave analysis [9] or load measurement [10]. One of the tasks carried out by such
system is to estimate the ToF of the ultrasonic signal or find the time delay estimate
(TDE). Usually matched filter (correlation processing) is used for signal-to-noise (SNR)
improvement. The cross-correlation maximum can be used as ToF estimate. The lowest
attainable random errors for analog domain have been analyzed by Cramer and Rao in
[11,12]. Ultrasonic applications gained popularity thanks to the small size of the
required equipment, environmental safety and cost-effectiveness. The digitization of the
ultrasonic signals is offering a further increase in system functionality: flexible signal
processing, reduction of electronics noise and EMI [13]. But signal conversion into
digital form introduces additional errors; therefore Cramer-Rao lower error bound has to
be corrected for errors introduced in digital space [22]. Corrections for analog-to-digit
conversion noise and clock jitter noise have been already suggested in [14]. One more
error source remains: the ToF estimator is discrete and interpolation has to be used in
order to obtain the subsample ToF estimates [15,22]. In ideal case the interpolation
should use sinc function. But in order to adapt for capabilities of the digital signal
processing and reduce the amount of interpolation operations, the truncated versions of
sinc function are used. This introduces bias error in the estimation which can not be
compensated if subsample delay value is not known. Our aim was to investigate the bias
error introduced by interpolation when using the sinc function approximates.
2. ToF estimation procedure
The received signal sR(t) can be treated as a delayed and attenuated version of the
transmitted signal sT(t) with additive white noise component:
)()()( tnToFtstAts TR
, (1)
where A(t) is the attenuation function and n(t) is an non-correlated additive white
Gaussian noise (AWGN) with power spectral density N0. The goal of the ToF
measurement is to find an estimate of the true position of signal arrival by using this
corrupted signal. In simple applications, the ToF is computed using the thresholding
technique [9]. This technique offers a low cost and a simple solution [16], but has a poor
accuracy even if high sample rate [17] is used since does not explore all energy of the
signal. Using several zero crossings or application of some integral estimate like center
of area [18] allow for results improvement, yet lacks the matched filter property to align
the frequency components phases.
Three TOF estimation techniques have been indicated in [19]: the direct correlation
maximization, the L1 norm minimization and the L2 norm minimization. The direct
correlation technique uses the peak position of the cross-correlation function (CCF or
matched filter output) x as the signal arrival time (and so the ToF) estimate:
)(maxarg
xToFDC
, (2)
where x (black curve in Fig.1) is:
dttstsx TR
. (3)
The direct correlation technique possesses optimal filtering properties and broad
theoretical analysis has been done on ToF estimation variance. Therefore matched filter
output maximum estimate has been chosen for this analysis. The variance of ToF
standard deviation is defined by Cramer-Rao lower error bound [11,12]:
SNRF
TOF
e
2
1
,
0
2
N
E
SNR
, (4)
where E is the energy of signal sT(t) and Fe its effective bandwidth. The energy of the
signal is obtained either from its temporal signal representation s(t) or from its spectral
density S(f):
0
22 2dffSdttsE T
. (5)
The effective bandwidth is obtained as:
2
0
2
2fFe
,
E
dffSff
2
2
0
2
,
2
2
2
2
0E
dffSf
f
, (6)
where
is the envelope bandwidth and f0 is the center frequency.
If matched filter is applied in digital space, then conversion into digital domain will
introduce aliasing errors, quantization errors and jittering errors. Then, N0 in Eq.(4) has
to be modified to account for digital space [14]:
JQA NNNN 0000
, (7)
where N0A is the noise related to the analog domain, N0Q is the quantization noise and
N0J is the jitter noise.
3. ToF interpolation procedure
For high SNR values, accuracy below the sampling period is expected. In order to
estimate the ToF between the samples, interpolation is applied. The peak in the
correlation function is found by interpolating the discrete version of the cross-
correlation function xm (Fig.1 insert):
K
kkmkm srstx1
, (8)
where stk and srk are the discrete versions of transmitted and received signals at discrete
time instances k.
In ideal case sinc function should be used for the ToF estimation between the
samples. But usually in real time application processing speed is required so several
truncated interpolation techniques exist [20,21,22]. All these techniques locate the peak
position m (Fig.1) using discrete signal and then use some interpolation function to
estimate the subsample delay.
17µ 18µ 19µ 20µ 21µ 22µ 23µ 24µ
-1.0
-0.5
0.0
0.5
1.0
m-1
m
m+1
Arrival time estimate
Estimate: RDC (a.u.)
Time (s)
Fig. 1. Subsample interpolation of the correlation function peak for ToF estimation.
Increase of the sampling frequency fs will reduce the time axis granularity [23]. But
this could increase the system complexity (if physical oversampling is used) or prolong
the processing time (if Fourier domain resampling is used). Therefore it is important to
estimate the truncated interpolating function capabilities to estimate the subsample ToF
value. If bias errors introduced by truncated sinc interpolation can be predicted, then
optimal sampling frequency can be chosen. Three time domain interpolation techniques
have been investigated: parabolic, Gaussian and cosine interpolation. These techniques
were compared against frequency domain interpolation which supposes to have
significantly lower bias error.
Usually the cross-correlation function (CCF) near the peak exhibits a quadratic-like
behavior. This explains the past success of fitting three points, xm-1, xm and xm+1, to a
parabola as approximation of discrete xm peak as an analytical function xp:
2
)(
abcxp
. (9)
The estimated subsample shift is given by [22]:
)2(22 11
11
mmms
mm
s
pxxxf xx
af
b
ToF
. (10)
Although simple to use and computationally efficient, this technique produces
biased estimates of time delays [14].
Search for a better than parabola model of CCF given the same number of model
parameters suggests that a Gaussian function could have a reasonable merit:
2
exp)( cbaxG
. (11)
The reasoning is that the envelope of the interval around the CCF peak for a
bandlimited signal is similar to Gaussian function (see Fig.1). The estimated subsample
shift is given by:
11
11 ln2ln2ln4lnln
mmms
mm
s
Gxxxf xx
f
c
ToF
. (12)
The next model for time delay estimation could be cosine fitting. Cosine curves have
been used extensively in the past to interpolate the discrete CCF and thus estimate
subsample displacement. The reasoning is that the interval around the CCF peak for a
relatively narrowband signal is the harmonic function (see Fig.1). The estimated
subsample shift is given by following expression [22]:
,
0
cos
s
f
ToF
m
mm xxx 2
arccos 11
0
,
0
11 sin2
arctan
m
mm
xxx
. (13)
This technique produces smaller bias and variance than parabolic interpolation [22].
Delay can also be estimated from the phase shift in frequency domain [14]. It is
using the assumption that the matched filter aligns the phases of all frequency
components thanks to multiplication by complex conjugate (*). That is, Eq. (8) can be
rewritten for frequency domain processing as:
mmmmmm XDFTSrStDFTsrDFTstDFTDFTx 1
*
1
*
1
, (14)
where DFT and DFT-1 are forward and inverse discrete Fourier transforms; Stm* is the
complex conjugate of result of DFT for stm; Srm is the result of DFT for srm; Xm is the
frequency domain version of CCF. If
mm Stj
mm
Stj
mm eStSteStSt
*
, (15)
and
mmmmm StToFStj
mmm
ToFStj
mm eStSrSteStSr
2
*
, (16)
then the phase of Xm components should carry the information on time domain shift
ToF:
M
X
ToFToFX
M
m
m
mm
1
. (17)
In order to avoid phase wrapping, signal is first circularly shifted in time domain using
rough ToF estimate obtained by Eq. (8) and Eq. (2):
),(
0DCm
mToFxcircshiftx
. (18)
Then, subsample ToF remainder ToF estimation can be done in frequency domain:
M
X
ToF
M
mm
m
1
0
. (19)
To avoid the noise influence, power spectrum Xm 2 weighted mean can be applied:
M
m
M
mm
mm
w
XM
XX
ToF
1
2
0
1
2
00
. (20)
Though consuming large amount of the processing power, this technique should
have the significantly lower bias error.
4. Numerical experiment for bias error investigation
Numerical experiment was carried out to investigate the ToF bias error. Signal was
modeled as:
ToFtFToFtAts c
2cos/exp 2
5.0
, (21)
where 0.5 is the pulse duration and Fc its center frequency. Signal delay was varied in a
range -0.5Ts to +0.5Ts, where Ts is the sampling period. For every new ToF value the
ToF bias error
(ToF) was obtained, by subtracting the introduced delay from delay
estimated by the subsample interpolation of CCF obtained by Eq.(8):
ToFToFToF ee )(
, (22)
where ToFe is the estimate obtained by the interpolation technique (with subscript index
replaced to p for parabolic, G for Gaussian and cos for cosine) and ToF is the “true”
(unbiased) value.
An example of the ToF bias error value (22) versus subsample shift when using a
2μs duration pulse with center frequency 0.5 MHz is presented in Fig.2, where Eq. (10)
(12) (13) and (20) were applied for subsample estimation.
-0.5 0.0 0.5
-50f
-40f
-30f
-20f
-10f
0
10f
20f
30f
40f
50f
max((ToFcos))
max((ToFp))
max((ToFG))
FFTP
Cosine
Gaussian
Parabolic
Fs=200MHz
Fc=0.5MHz
0.5=2us
(ToF) (s)
Subsample (a.u.)
Fig. 2. ToF estimation bias error vs. subsample shift for signal simulated by Eq. (21).
Maximum bias error deviation was noted (color arrows in Fig.2) for every technique
and used for further analysis. Several Fc values (0.5 MHz, 1 MHz, 2 MHz, 5 MHz,
10 MHz, 20 MHz) were used. Sampling frequency was varied from 5 MHz to 200 MHz.
Duration 0.5 used was 0.5 s, 1 s and 2 s. No antialiasing filter was used; neither the
noise nor amplitude quantization effects were studied. Combinations where sampling
frequency was not four times the center frequency plus bandwidth (1/0.5) were not
evaluated. Results of the maximum bias error versus sampling frequency are presented
in Fig. 3 for a 2 s duration pulse and center frequency 0.5 MHz and 10 MHz where Eq.
(10) (12) and (13) were applied for the subsample estimation.
5M 20M 40M 60M 80M 100M 120M 140M 160M 180M 200M
10f
100f
1p
10p
100p
1n
Fc=10MHz
Fc=0.5MHz
Interpolation:
Parabolic
Gaussian
Cosine
Max((ToF)) (s)
Sampling frequency fs (Hz)
Fig. 3. Maximum ToF bias error vs. subsample shift for signal simulated by Eq. (21).
Numerical modeling results indicate that cosine interpolation ensures lowest ToF
estimate bias error compared to Gaussian and parabolic. Though, all techniques have
some estimation bias error, Gaussian interpolation gives worst performance and is
unstable at subsamples beyond 0.4, therefore it was excluded from further investigation.
Simulation of the signal as per Eq. (21) does not allow easy bandwidth and center
frequency control. Furthermore, signal in Eq. (21) does not possess spread spectrum
property. Therefore, another set of numerical experiments was carried out, aiming to
have separate control over bandwidth and center frequency. Two types of signal were
used: unity pulse and 3 s duration chirp. Both have more or less uniform spectrum.
Unity pulse has stable spectrum spreading over whole frequency range; chirp is a
frequency modulated signal with linear modulation in a range f1 to f2. Both signals were
filtered by bandpass IIR filter. Fourth order Butterworth filter was used, using zero-
phase digital filtering (Matlab© filtfilt command). Both signals were initially created at
1 GHz sampling frequency, filtered by bandpass filter (refer Fig.4 for sample signals)
and then decimated down to desired experiment frequency fsnew using Matlab©
command decimate. Such approach mimics real conditions when significant antialiasing
filtering is used before sampling. Procedure decimate contains antialiasing filter which
is an 8-th order Chebyshev Type I lowpass filter with cutoff frequency 0.8fsnew.
4.0µ 5.0µ 6.0µ 7.0µ 8.0µ 9.0µ
-10.0m
-5.0m
0.0
5.0m
10.0m
Amplitude (V)
Time (s)
2.0µ 3.0µ 4.0µ 5.0µ 6.0µ 7.0µ
-1
-800m
-400m
0
400m
800m
1
Amplitude (V)
Time (s)
Fig. 4. Example of pulse (left) and chirp (right) signals used in simulation for
fs=100 MHz, f0=5 MHz, B=2.5 MHz.
Generated signal was stored as reference for Eq. (8) and delay was introduced to
simulate the received signal. Reversed Eq. (19) was used for subsample shift of the
signals:
ToFSIFFTToFts mm
exp
. (23)
Subsample delay was varied in a range -0.5Ts to +0.5Ts, where Ts is the sampling
period. Obtained signals were used for ToF estimation using frequency domain
subsample estimation, Eq.(20), parabolic interpolation, Eq.(10) and cosine interpolation,
Eq.(13). The ToF bias error Eq.(22) was obtained, by subtracting the introduced delay
from delay estimated by subsample interpolation of CCF obtained by Eq.(8). Maximum
bias error deviation was noted (color arrows in Fig.2) for every technique and used for
further analysis. Three sets of experiments were carried out: i) only sampling frequency
varied; ii) only center frequency varied; iii) only bandwidth varied. Results for
maximum bias error
when sampling frequency fs was 16.6 MHz, 20 MHz, 25 MHz,
33 MHz, 50 MHz, 100 MHz, 200 MHz and 400 MHz, with center frequency fixed at
5 MHz and bandwidth fixed at 4 MHz (80 % bandwidth) are presented in Fig.5.
15M 100M 500M
10e-24
100e-24
100f
1p
10p
100p
1n
FFT estimate
Cosine
Parabolic Signal:
:chirp :pulse
max((ToF)) (s)
fs (Hz)
Interpolation technique:
FFT phase estimate
cosine interpolation
parabolic interpolation
Fig. 5. Maximum interpolation bias error vs. fs.
It can be seen that results for both signals (filtered pulse and chirp) produce similar
error bias. There is a difference of approximately 6 ps between chirp and pulse signals
for cosine interpolation. Actually, it is the same size difference as for parabolic
interpolation but swamped by much larger bias error for this interpolation. The reason is
very simple:
(envelope bandwidth) for pulse signal is approximately 770 kHz but it is
640 kHz for chirp. This 130 kHz difference is the cause for mentioned discrepancy.
Frequency domain estimate of subsample delay has significantly lower bias error which
does not depend on sampling frequency.
Results for maximum bias error
when sampling frequency fs was fixed at
100 MHz, with center frequency varied (2.5 MHz, 5 MHz, 10 MHz, 15 MHz, 20 MHz,
25 MHz and 30 MHz) and bandwidth fixed at 1.5 MHz are presented in Fig.6.
2M 10M 40M
1e-24
10e-24
100e-24
100f
1p
10p
100p
1n
FFT estimate
Cosine interpolation
Parabolic interpolation
Signal:
:chirp :pulse
max((ToF)) (s)
f0 (Hz)
Interpolation technique:
FFT phase estimate
cosine interpolation
parabolic interpolation
Fig. 6. Maximum interpolation bias error vs. center frequency f0.
Again bias error produced by ToF subsample interpolation for both signals is
similar. Bias error for frequency domain estimate is significantly lower and does not
depend on center frequency f0.
Results for maximum bias error
when sampling frequency fs was fixed at
100 MHz, with center frequency fixed at 20 MHz and bandwidth varied (3 MHz,
4 MHz, 6 MHz, 10 MHz, 14 MHz, 20 MHz, 22 MHz and 26 MHz with resulting
envelope bandwidth
from 1 MHz to 9.5 MHz) are presented in Fig.7.
1M 10M
10e-24
100e-24
100f
1p
10p
100p
1n
FFT estimate
Parabolic interpolation
Cosine interpolation
Signal:
:chirp :pulse
max((ToF)) (s)
(Hz)
Interpolation technique:
FFT phase estimate
cosine interpolation
parabolic interpolation
Fig. 7. Maximum interpolation bias error vs. envelope bandwidth
.
Here, bias error produced by ToF subsample interpolation for both signals is similar.
Again, bias error for frequency domain estimate is significantly lower and does not
depend on the envelope bandwidth
.
It should be noted that all techniques under investigation posses a bias error which is
not present when estimate is located exactly on the sampling points. In spite of the
above, errors of the FFTP technique are several orders lower, which can be used as a
proof that unbiased subsample estimation can be obtained.
5. Deriving the equation for maximum bias error estimation
We are interested in predicting the maximum bias error, therefore, if equation for
bias error is available, that would allow predicting the error bias and balancing it against
random errors in Eq.(4). In [22], authors assumed that peak of cross correlation function
can be approximated by cosine and arrived at equation describing the bias error for
parabolic subsample interpolation:
ss00
0s0 T
ToF
Tcos-1ToFcos2 ToFsinTsin
p
ToF
. (24)
Such equation is valid only for narrowband signal, otherwise, as our further
investigation has revealed, envelope bandwidth is influencing the bias error (refer
Fig.8).
-0.5 -0.4 -0.2 0.0 0.2 0.4 0.5
-0.04
-0.02
0.00
0.02
0.04
f0=20MHz, BW=14MHz (70%)
Prediction by [22]
Experiment data
(ToF) (samples)
Subsample offset ToF (samples)
-0.5 -0.4 -0.2 0.0 0.2 0.4 0.5
-0.04
-0.02
0.00
0.02
0.04
f0=20MHz, BW=26MHz (130%)
Prediction by [22]
Experiment data
(ToF) (samples)
Subsample offset ToF (samples)
Fig. 8. Maximum interpolation bias error: experimental vs. predicted by [22].
Fig.8 demonstrates that there is additional influence on bias error, which is not
predicted by Eq.(24) from [22]. Furthermore, we are only interested in the maximum
value of bias error over subsample range (color arrows in Fig.3). Eq.(24) is not suitable
for arriving into simple solution. We also need equation for cosine interpolation errors
estimation. As indicated in [22], such solution is not easy to derive.
Instead of that, we have decided to develop and empirical equation able to estimate
the maximum bias error, basing the simulation results. Numerical modeling results were
processed to develop the analytical function for the ToF estimation bias error. Empirical
equation obtained was a function of components constituting the effective bandwidth Fe
(envelope bandwidth and center frequency), and the sampling frequency fs. For cosine
interpolation we suggest to estimate the peak value of the bias error as:
3
2
)(max
s
cf
ToF
. (25)
For parabolic interpolation the maximum bias error value can be estimated as:
3
2
0
3
2
0
3
2
2
)(max
2
)(max
s
c
s
s
pf
f
ToF
f
f
f
ToF
. (26)
Obtained equations show few essential findings:
i) contrary to Eq.(24) there is no center frequency influence on bias errors in case of
cosine interpolation, which can be derived from Fig.6 too;
ii) bias error for cosine interpolation is defined by envelope bandwidth;
iii) for parabolic interpolation, bias error is defined mostly by center frequency:
usually center frequency is larger than envelope bandwidth (Fig.7);
iv) primary influence is by sampling frequency: both equations contain 1/fs3.
Now, with equation for bias error available, we can balance it against random errors
in Eq.(4). If the ToF value is unknown, then the interpolation bias error behaves
randomly and increases the random error part. In order to avoid the bias error influence
it is enough to have it’s RMS three times lower than the RMS of the random part
defined by Eq.(4). Assuming that the bias error shape is close to sinusoidal, RMS is a
peak value divided by the square root of two. Then the minimum necessary sampling
frequency in case of cosine interpolation of the CCF peak is:
323
cosmin__ 18
3
)( SNRFf
TOF
ToFRMS esc
. (27)
Parabolic interpolation has higher bias error so we do not recommend using this type
of approximation. If parabolic interpolation of the CCF peak is used, minimum
sampling frequency is:
32
2
0min__ )2(3 SNRFff epars
. (28)
Equations (27) and (28) give engineers a tool which can be used to estimate the
minimum sampling frequency that has to be used in order to keep the interpolation error
below the random error value. It is not necessary to sample the signal at the frequency
indicated: if Nyquist criteria is satisfied, signal can be upsampled to required frequency
using Fourier transform interpolation.
6. Experimental verification of the equation for the bias error estimation
Using the analytical functions for ultrasonic signal description allows for better
control of signal properties, but do not reflect real signal properties. Experiment was
carried out with real signals. Signals were recorded, averaged, stored as reference and
later artificially shifted to investigate the subsample interpolation errors.
Real signals were recorded using 5 MHz wideband (130 %) composite transducer
C543 and 10 MHz C544 from Panametrics Ndt. (Olympus). Standard delay line
(7.8 mm diameter, 11.84 mm long) was attached to transducer and reflection from delay
line end was recorded. Bipolar pulser [24] was used for excitation. Excitation amplitude
was 3 V for 5 MHz transducer, and 14 V for 10 MHz transducer. Low noise variable
gain amplifier AD8331 with 12.7 dB gain was used. 3-rd order Butterworth was used as
antialiasing filter with 30 MHz cut-off frequency. Signal was sampled at 100 MHz
using 10 bit A/D converter. In order to keep the jitter error low (Eq.(7)), reference
oscillator with a jitter of 5 ps was used. To remove the DC component and excitation
ringing, 2-nd order Butterworth HP filter with 0.5 MHz cut-off was used. Four types of
excitation signals were used: wideband (WB) chirp, covering the 200 % of the
bandwidth; chirp, covering the 100 % of the bandwidth; narrowband continuous wave
(CW) burst with frequency corresponding to transducer frequency and a rectangular
pulse with duration optimal for transducer frequency. Signals used in the experiment are
listed in Table 1. First four signals were used with 5 MHz transducer, the remaining
were applied on 10 MHz transducer.
Table 1
Signals used in experiment.
Signal type
Polarity
Received signal BW (@-6dB/-20dB)
WB chirp 1-10 MHz, 3 s
Bipolar
5/8.5 MHz
Chirp 4-7 MHz, 3 s
Bipolar
2.2/5 MHz
CW burst 5 MHz, 3 s
Bipolar
0.4/0.6 MHz
Pulse 100 ns
Unipolar
4/6 MHz
WB chirp 2-20 MHz, 3 s
Bipolar
5.3/15 MHz
Chirp 6-15 MHz, 3 s
Bipolar
4.6/10 MHz
CW burst 10 MHz, 3 s
Bipolar
0.5/0.6 MHz
Pulse 50 ns
Unipolar
7.8/10 MHz
One hundred A-scans were averaged to remove the noise component. All A-scans
were aligned before the averaging in order to remove the signal position variation due to
temperature. Time position values were obtained using first A-scan as reference for
Eq.(8) cross-correlation and Eq. (2) and (20). Eq. (23) was used for subsample
alignment of the signals. Averaged signal was then stored as reference. Spectrum of the
signals generated with the 5 MHz C543 and the 10MHz C544 transducers are presented
in Fig.9 and Fig.10 respectively.
0.0 2.0M 4.0M 6.0M 8.0M 10.0M 12.0M
-40
-20
0
20
40
Pulse
WB chirp Chirp
CW burst
Magnitude (dB)
Frequency (Hz)
WB chirp 1-10MHz
chirp 4-7MHz
CW 5MHz burst
pulse 100ns
Fig. 9. Signal spectrums obtained with the 5 MHz transducer.
Sampling frequency of the signal was altered in a range 25-500 MHz. Original
sampling frequency was 100 MHz. In order to reduce the sampling frequency, signal
was decimated, using decimate procedure in Matlab©. Divisor used was 2, 3 and 4,
corresponding to 50 MHz, 33 MHz and 25 MHz sampling frequency. In order to
increase the sampling frequency Matlab© command interpft was used. This procedure
is using sampling frequency increase by zero-padding in frequency domain. Such
approach corresponds to sinc interpolation which should produce ideal result. Sampling
frequency 200 MHz, 400 MHz and 500 MHz was obtained.
0.0 5.0M 10.0M 15.0M 20.0M
-40
-30
-20
-10
0
-6dB
WB chirp 2-20MHz
chirp 6-15MHz
CW 10MHz burst
pulse 50ns
Normalised magnitude (dB)
Frequency (Hz)
Fig. 10. Signal spectrums (normalized) obtained with the 10 MHz transducer.
Obtained signal was stored as reference for corresponding sampling frequency.
Artificial delay, simulating ToF, was applied to the signal using Eq.(23). Subsample
delay range was -0.5Ts to +0.5Ts, where Ts is the sampling period. Resulting signal was
then processed to obtain the ToF using parabolic Eq.(10) and cosine Eq.(13)
interpolation. Bias error was obtained by using Eq.(22): the used subsample delay was
subtracted from the obtained ToF. Results comparison with theoretical error estimation
using Eq.(25) and Eq.(26) is presented in Fig.11 for 5 MHz transducer and in Fig.12 for
10 MHz transducer.
20M 100M 500M
100f
1p
10p
100p
1n
2n , : Experiment; , : Theory
, : cosine; , : parabolic;
, WB chirp
, Chirp
, CW burst
, Pulse
max((ToF)) (s)
fs (Hz)
Fig. 11. ToF bias error vs. sampling frequency for 5 MHz transducer.
20M 100M 500M
100f
1p
10p
100p
1n
2n
, : Experiment; , : Theory
, : cosine; , : parabolic;
, WB chirp
, Chirp
, CW burst
, Pulse
max((ToF)) (s)
fs (Hz)
Fig. 12. ToF bias error vs. sampling frequency for 10 MHz transducer.
Solid lines represent the bias error value obtained by the analytical model and scatter
is for experimental data representation. Experimental results confirm that maximum
bias error can be predicted using the developed equation. Results are also summarized
in Table 2.
Table 2
Summary of the experiment results.
Signal type
CRLB
(ToF cos)
@100Ms/s
(ToF par)
@100Ms/s
fs_min_cos
fs_min_par
WB chirp 1-10 MHz, 3 s
12ps
6.5ps
24ps
101MHz
147MHz
Chirp 4-7 MHz, 3 s
9.5ps
1.4ps
20ps
68MHz
150MHz
CW burst 5 MHz, 3 s
9.5ps
0.5ps
16ps
42MHz
141MHz
Pulse 100 ns
115ps
6.6ps
14ps
41MHz
55MHz
WB chirp 2-20 MHz, 3 s
9.1ps
14ps
59ps
144MHz
220MHz
Chirp 6-15 MHz, 3 s
6.8ps
9.5ps
61ps
136MHz
244MHz
CW burst 10 MHz, 3 s
6ps
2.3ps
65ps
97MHz
262MHz
Pulse 50 ns
84ps
12ps
37ps
67MHz
90MHz
Expected random errors can be predicted by Cramer-Rao lower error bound using
Eq.(4). Signal parameters f0 and
were obtained from averaged signal record. Received
signal gain was provided by the low noise amplifier AD8331 which output voltage
noise density entot at low gains is constant, 180 nV/Hz. This value can be used for noise
power density (N0) calculation. Obtained Cramer-Rao lower error bound (CRLB) is
presented in Table 2 for every signal used. For the high energy signals (chirp and CW
burst) random errors are 9-12 ps for the 5 MHz transducer and 6-9 ps for the 10 MHz
transducer. But interpolation errors vary significantly: from 0.5 ps (5 MHz transducer,
narrowband CW burst) to 14 ps (10 MHz transducer, WB chirp) for cosine
interpolation, and 16 ps to 65 ps for parabolic interpolation at 100MHz sampling
frequency. Pulse signal has low energy so its CRLB is high. Therefore low sampling
frequency (41 MHz for 5 MHz transducer and 67 MHz for 10 MHz transducer) is
demanded for these signals. Since CRLB is small for high energy signals, required
sampling frequencies are high for parabolic interpolation (141 MHz to 262 MHz). On
the other hand, cosine interpolation is less sensitive to center frequency so required
sampling frequency is lower: for narrowband CW burst it is only 41 MHz, same as for
low SNR signal (pulse, 42 MHz). Again, we would like to stress that it is not necessary
to sample the signal at the frequencies indicated in Table 2. It is enough to sample
satisfying the Nyquist sampling criteria. Later, signal can be upsampled to required
frequency using frequency domain zero padding.
7. Conclusions
Bias errors are introduced when truncated sinc functions are used for subsample
delay estimation. The aim of the investigation was to derive the equation for these errors
estimation. Two functions were selected as truncated sinc candidates suitable for
subsample interpolation: cosine and parabolic functions. Series of numerical
experiments were used to derive the empirical equation, describing the maximum value
of the bias error using just a few signal parameters and sampling frequency. Our
investigation indicates that cosine interpolation bias error depends only on envelope
bandwidth and sampling frequency: it is in inverse cubic relation to sampling frequency
and in quadratic relation to envelope bandwidth. Parabolic interpolation bias error has
additional dependence on center frequency: it is in inverse cubic relation to sampling
frequency and in quadratic relation to center frequency and envelope bandwidth.
Obtained equations can be used to estimate the minimum required sampling
frequency so subsample interpolation errors are significantly smaller than random errors
defined by Cramer-Rao lower bound. Corresponding equations were derived which give
engineers a tool which can be used to estimate the minimum sampling frequency that
has to be used in order to neglect the interpolation error. It is not necessary to sample
the signal at the frequency indicated: if Nyquist criteria is satisfied, signal can be
upsampled to required frequency using Fourier transform interpolation.
An experimental study was carried out to confirm the finding in real signals. Results
confirm that maximum bias error can be predicted using the developed equations.
Acknowledgement
This research was funded by a Grant (No. MIP-058/2012) from the Research
Council of Lithuania.
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