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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. Y, JUNE 2011 1
Interpolation and fitting less sub-sample time delay
estimation
C´
esar Teixeira, Wagner Pereira, and Maria da Graca Ruano
Abstract—In this paper we propose a new method for sub-
sample time delay estimation that operates just over the discrete
signals, without the need for interpolation or fitting. The new
approach codifies the sub-sample delay in the amplitude of
a difference of auto- and cross-correlation sequences and has
two main steps. In first place the discrete delay is computed
just by computing the cross correlation sequence of a pair of
signals. In a second phase the sub-sample delay is computed by
synchronizing the signals, i.e., by eliminating the discrete delay,
and then by computing the auto- and cross-correlation involving
the synchronized signals. It was found that the maximum of the
difference between the auto- and cross-correlation is proportional
to the sub-sample delay, and can be easily converted to a time
difference.
In this paper we compare the proposed method with a method
based on spline fitting, and that is referred to outperforms
other methods over a broader set of conditions [1]. The new
approach, besides presenting a low computational cost, seemed to
outperforms the spline approach in situations of high noise levels.
Concerning other aspects, such as the correlation coefficient
between signals and central frequency of the simulated signals,
the proposed method present performance levels close to the
performance of the spline method.
Index Terms—Sub-sample time delay estimation.
I. INTRODUCTION
TIME delay estimation (TDE), i.e. the assessment of the
amount of time that two signals are delayed, have been
used in several applications. Applications include radar and
sonar [2], [3], speech processing [4], medical ultrasound, etc.
Doppler blood flow estimation [5], tissue elasticity assessment
[6], non-invasive temperature estimation [7]–[9] are examples
of TDE applied to medical ultrasound.
The straightforward procedure to determine the discrete
time delay between two signals is by computing a pattern
matching function, as for example the cross-correlation func-
tion, and determine its maximum (or minimum depending on
the algorithm) [10]–[12]. Methods based on: adaptive filters
[13], Fast Fourier transform [14], Hilbert transform corre-
lation [15], combinations among autocorrelation and cross-
correlation functions [16], analog quadrature detectors [17],
and many others have been proposed.
The applications that used TDE usually demand delay
precisions that goes beyond the sample period, i.e., methods
for sub-sample TDE (SS-TDE) are required. SS-TDE can
be achieved by interpolating the original signals. However,
the interpolation of the source signals can result in a high
computational cost, depending on the interpolation rate. In
this approach TDE precision is improved, but still limited
Manuscript received July 31, 2011; revised xxxxx.
by the sample period of the interpolated signals. Instead of
interpolating the original signals, other methods achieve SS-
TDE by operating over the patter-matching functions, resulting
in a low computational cost and improved TDE estimates. In
this case some samples of the pattern matching function are
used to find the parameters of a continuos time function. Then
by analytical methods a continuos TDE estimate is achieved.
The simplest and mostly used method for continuos time
delay estimation consists on the computation of the correlation
function, by fitting a parabola around its peak, and finally by
determining the zero of the derivative of the fitted parabola
[10], [11]. Other types of fitting functions were proposed, such
as cosine [12]. It is referred that as these methods are based
on a pattern-matching function that is sampled at the original
frequency, they usually suffer from relatively high bias and
variance [1]. Alternatively, continuous TDE estimation can
be started at the original signals, instead of at the computed
pattern-matching function. Viola and Walker [1] proposed a
method that produce a continuous time representation of one
of the original signals. For this representation the authors
apply a fitting by using cubic splines. Using the continuos
representation and the discrete delayed signal it is possible to
obtain an analytical pattern-matching function, which is then
subject to analytic operations aiming to find the continuous
delay estimate. In [1] the pattern-matching function is the sum
of the squared errors (SSE). Through simulation results the
authors found that the algorithm outperforms other algorithms
in terms of jitter and variance over a broad range of conditions.
It is clammed that this improved performance was obtained at
a reasonable computational cost.
In this paper a new algorithm for SS-TDE is proposed,
based only in the auto-correlation and cross-correlation of
the involved signals, without the need of any interpolation or
fitting.
II. PRO PO SE D ALGORITHM
In this section we describe the proposed algorithm by
considering two identical discrete time signals s1[n]and s2[n]
with length N, sampled at fs = 1/T s, and delayed by some
amount of time (δ). The first stage of the algorithm is the
determination of the discrete time delay, as presented in Fig.
1. This is achieved by computing the unnormalized and biased
cross-correlation between the original signals (ˆrs1,s2(τ)):
ˆrs1,s2(τ) = PN−τ−1
k=0 s1[k+τ]s2[k]if τ≥0
r∗
s2,s1(−τ)if τ < 0.(1)
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. Y, JUNE 2011 2
Cross.&
Corr.&&
Synchroniza /on&
Diference&
Auto&
Corr.&
Discrete&Sample&Es/ma/on&
Ts#
A&
B&
C&
D&
E&
F&
G&
H&
Sync&
Sub?Sample&Es/ma/on&
€
SSD =max(r
s1ss1s−r
s1ss2s)⋅Factor
€
DD =argmax
Lag
rs1s2(Lag)
[ ]
€
δ
^
=DD⋅Ts +SSD
€
δ
Fig. 1. Proposed method. (A) and (B) original signals. (C) cross-correlation
function used to determine the discrete time delay. (D) and (E) synchronized
original signals. (F) autocorrelation of the synchronized s1signal. (G) Cross-
correlation between the synchronized signals. (H) Difference between the
autocorrelation of s1and the cross-correlation between s1and s1.
The discrete sample delay estimate (DD) is given by
DD = argmax
τ
rs1,s2(τ).(2)
Using the discrete sample delay it is possible to align the
signals, i.e., to synchronize them. This is the first step of
the second phase of the algorithm, which aims to assess the
sub-sample delay between the signals. The synchronization
consist in eliminating DD samples from both signals so that
they become apparently in phase. The process to account
for the sub-sample delay involves the computation of the
unnormalized and biased autocorrelation of one of the syn-
chronized signals (the reference signal), for example s1[k], the
computation of the unnormalized and biased cross-correlation
between the synchronized signals and the difference between
these computed functions. Mathematically:
ˆrss
1,ss
1(τ) = (PN−DD−τ−1
k=0 ss
1[k+τ]ss
1[k]if τ≥0
ˆr∗
ss
1,ss
1(−τ)if τ < 0
(3)
ˆrss
1,ss
2(τ) = (PN−DD−τ−1
k=0 ss
1[k+τ]ss
2[k]if τ≥0
ˆr∗
ss
2,ss
1(−τ)if τ < 0
(4)
rd(τ) = ˆrss
1,ss
1(τ)−ˆrss
1,ss
2(τ).(5)
!"#$%"&'%!(&$%)"*("+,&%
-"./01%210"3%4-"./0156%
Fig. 2. Maximum and minimum variation in function of the sample delay.
Where ss
1and ss
2are the synchronized original signals with
N−DD samples, and rd(τ)is the difference between the
reference signal autocorrelation (ˆrss
1,ss
1(τ)) and the synchro-
nized signals cross-correlation (ˆrss
1,ss
2(τ)). It was found that
the maximum or the minimum values of rd(τ)are proportional
to the sub-sample delay. Fig. 2 shows the maximum and
minimum variation as function of the sub-sample delay. This
figure shows a clear linear relationship.
The total delay between the original signals can be found
by summing the discrete delay DD with the maximum or the
minimum of rd(τ)multiplied by a conversion factor. In this
paper we consider always the maximum:
ˆ
δ=DD ∗T s +max(rd(τ)) ∗f actor. (6)
The conversion factor can be computed a-priori by inducing
an artificial delay on the reference signal. The process to
compute the factor is presented in Fig. 3. In first place the
reference signal is artificially delayed by one sample. Then the
autocorrelation of the original reference signal is computed
as well as the cross-correlation between the original and
delayed reference signal. Then the functions are subtracted
and the maximum of the difference found. As the signals were
artificially delayed we know exactly the delay that correspond
to the maximum. So, the factor in units of time can be found by
dividing the sample period by the maximum. To mention that
for multiple delay estimations involving the same reference
signal this factor needs to be computed only one first time.
III. PERFORMANCE ASSESSMENT
A. Simulated signals
In this paper ultrasonic echoes were generated to evaluate
the proposed TDE method and compare it with the method
proposed in [1]. The ultrasonic echoes were generated based
on a Gaussian pulse described by:
sr=e−t2
2˙
tv sin(2πfct).(7)
Where fcis the central frequency, and tv is the variance
described by:
tv =−8 log r
4π2B2f2
c
.(8)
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. Y, JUNE 2011 3
!"#$%&'()*)+',-(.)/.('0)
!1234)
53""6))
!
Factor =Ts
max(rref ref "rref refd)
!)
7)
5)
/)
8)9)
Fig. 3. Method to compute the factor. (A) and (B) original and delayed
reference signal. (C) autocorrelation of the original reference signal. (D)
Cross-correlation between the original and artificially delayed reference signal.
(E) Difference between the autocorrelation and the cross-correlation. (F) The
conversion factor is computed by dividing the artificial delay by the maximum
of the difference between the functions.
TABLE I
DEFAU LT PARAM ETE RS U SED F OR S IMU LATI ON TH E ULTR AS OUN D
SI GNA LS
Parameter Value
Centre Frequency (fc) 5 MHz
Sampling Frequency (fs) 50 MHz
Correlation Coeficient (ρ) 0.99
Fractional Bandwidth (B) 0.5 (50%)
SNR 30 dB
Window length 1 µs
Where Bis the fractional bandwidth, and ris:
r= 10−BW R
10 (9)
BW R is the fractional bandwidth reference level. The pulse
bandwidth is 100 ∗Bpercent as measured at a level of BW R
dB with respect to the normalized signal peak. In this paper
BW R was always considered as -6dB.
Pairs of reference and delayed signals with known noise
levels and correlation coefficients, were generated by using
the approach described in [18], [19] and used in [1]. In order
to evaluate the methods across multiple sup-sample delays,
for each pair of signals one of them were delayed by [0, 0.05,
0.1,...,0.95] samples. So, for each generated pair of signals
20 signals were available for performance validation. Delayed
signals were obtained by imposing a linear phase shift in the
Fourier domain. To improve statistics, 1000 pairs of signals
were generated for each analyzed noise level, correlation
coefficient and central frequency, i.e., 20000 delays were
evaluated for each tested value. The default values used for
the signal simulation are listed in Table I.
1) Results: The performance were evaluated based on the
standard deviation of jitter errors. Jitter occurs when signal
!"
#"
$"
%"
Fig. 4. Results for different central frequencies (A), correlation coefficients
(B) and noise levels (C and D).
decorrelation, noise, and finite window lengths cause a slight
displacement of the true peak of the cross correlation function
[18]. The standard deviation of the jitter errors is given by:
σ=v
u
u
t
1
20000
20000
X
i=1
e2
i,(10)
where eiis the error between the estimated and true delay.
Fig. 4 present the achieved results obtained when different
central frequencies (Fig. 4 A), correlation coefficients (Fig.
4 B), and noise levels (Fig. 4 C and D) were considered.
The results show that the proposed method presents improved
performance when the signals were corrupted by high noise
levels. For noise levels inferior to 20 dB of SNR the proposed
method presents a superior performance. Fig. 4 C and D, on
one hand, one can infer that the proposed method presents
a exponential decay in the standard deviation of jitter errors
as the noise level decreases. On the other hand, for the spline
method the standard deviation of jitter errors tends to decrease
linearly with the increase on SNR. Looking at Fig. 4 A
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. Y, JUNE 2011 4
Fig. 5. Average time to compute a time delay between two signals.
the difference between the proposed method and the Spline
method tends to decreases as the central frequency increases.
The difference is maximum at 1 MHz being 0.5 ns which is
approximately 0.025 samples. Concerning the correlation coef-
ficient analysis one can see a decrease of the standard deviation
of jitter errors, for both methods, as the correlation coefficient
between signal pairs increases. The spline method presents
improved performance, being the maximum difference in the
standard deviation of jitter error of 0.13 ns (0.0065 samples).
Concerning the computational cost, i.e., the time to compute
a time delay, the proposed method outperforms the spline
method. Fig. 5 presents the average time to compute one
subsample delay for the proposed and spline method. The
proposed method presents an average execution time of ap-
proximately 1.5 ms, while the spline method of approximately
5 ms.
B. An application: Temperature induced echo-shifts
In order to assess the performance of the proposed method
in real signals, in this section temporal echo-shifts induced
by temperature changes are assessed. As the temperature in a
given medium changes it expands or contracts and the speed
of sound on it also changes. When the medium is irradiated
by ultrasound waves, these changes can be viewed in the
backscattered echoes as variations on the time-of-flight, i.e.,
as temporal echo-shifts [7], [20]. For this paper we used
data acquired from the experimental setup published in [9].
The experiment consisted in a multi-layered phantom heated
by therapeutic ultrasound. During heating and cooling tem-
perature and backscattered ultrasound signals were collected
simultaneously at intervals of 10 seconds. Temperature from
sites located at middle layers and interfaces was saved. In this
paper we consider a trial where the therapeutic device was
setup to irradiate the medium with a power of 0.3 W/cm2,
i.e., a very low intensity. Small temperature changes allow us
to analyze how sensible are the time-shifts computed by the
proposed and by the spline methods. The experiment lasted
35 minutes, during the first five minutes baseline temperature
values were acquired, in the next 15 minutes the therapeutic
device heated the medium and in the remaining 15 minutes the
medium was allowed to cool down. The temperature measured
by the thermocouple that is located in the middle of the
!"
#"
Fig. 6. (A) Temperature change measured in a multilayered phantom and
(B) induced temporal echo-shifts.
phantom is presented in Fig. 6 A. The computed temporal
shifts for the echo originated by the middle thermocouple
are presented in Fig. 6 B. As expected the temporal shift is
monotonic with the temperature. Comparing the result of both
methods one can see that they return approximately the same
shifts, i.e., small differences are observed.
IV. DISCUSSION & CONCLUSIONS
In this paper we propose a new method for subsample
time delay estimation. The performance of the new method
is compared to the performance of the spline method that is
referred to outperforms other algorithms concerning jitter and
bias over a broader set of conditions [1]. The results showed
that the proposed algorithm outperforms the spline method in
situations were the signals are highly corrupted by normally
distributed white noise. These improvement in performance
is because of the difference operation expressed in Eq. 5.
Theoretically, the autocorrelation of a white noise sequence
is a Dirac function, i.e., a unique component at lag zero. As
the correlation is a linear operation, the autocorrelation and
cross-correlation computed in the proposed method exhibits
two fundamental parts; the components related to the signal
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. Y, JUNE 2011 5
contents and a component related to the noise. The noise
components are located around the lag zero in both auto- and
cross-correlations functions and are reduced by the difference
operation. In fact, the remaining noise component in the
difference function can be completely removed by forcing
the zero lag to be zero. This is a valid operation because
for optimal noise free synchronized signals, the difference
function will be always null at lag zero.
Although the proposed method do not outperforms the
spline method concerning the different central frequencies and
correlation coefficients considered, the difference in perfor-
mance is very small (less than 3% of the sample period).
In real signals both methods return approximately the same
temporal-echo shifts, validating the proposed method for real-
word problems.
Besides the good performance, the proposed method
presents a lower computational cost, because it is an inter-
polation and fitting less approach.
ACKNOWLEDGMENT
The authors would like to thank...
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