Evaluation of an auditory model for echo delay accuracy in

Evaluation of an auditory model for echo delay accuracy in
Evaluation of an auditory model for echo delay accuracy
in wideband biosonar
Mark I. Sandersona)
Department of Neuroscience, Brown University, Providence, Rhode Island 02912
Nicola Neretti
Brain Sciences, Brown University, Providence, Rhode Island 02912
Nathan Intrator
Institute for Brain and Neural Systems, Brown University, Providence, Rhode Island 02912
James A. Simmons
Department of Neuroscience, Brown University, Providence, Rhode Island 02912
共Received 27 November 2002; revised 23 May 2003; accepted 16 June 2003兲
In a psychophysical task with echoes that jitter in delay, big brown bats can detect changes as small
as 10–20 ns at an echo signal-to-noise ratio of ⬃49 dB and 40 ns at ⬃36 dB. This performance is
possible to achieve with ideal coherent processing of the wideband echoes, but it is widely assumed
that the bat’s peripheral auditory system is incapable of encoding signal waveforms to represent
delay with the requisite precision or phase at ultrasonic frequencies. This assumption was examined
by modeling inner-ear transduction with a bank of parallel bandpass filters followed by low-pass
smoothing. Several versions of the filterbank model were tested to learn how the smoothing filters,
which are the most critical parameter for controlling the coherence of the representation, affect
replication of the bat’s performance. When tested at a signal-to-noise ratio of 36 dB, the model
achieved a delay acuity of 83 ns using a second-order smoothing filter with a cutoff frequency of 8
kHz. The same model achieved a delay acuity of 17 ns when tested with a signal-to-noise ratio of
50 dB. Jitter detection thresholds were an order of magnitude worse than the bat for fifth-order
smoothing or for lower cutoff frequencies. Most surprising is that effectively coherent reception is
possible with filter cutoff frequencies well below any of the ultrasonic frequencies contained in the
bat’s sonar sounds. The results suggest that only a modest rise in the frequency response of
smoothing in the bat’s inner ear can confer full phase sensitivity on subsequent processing and
account for the bat’s fine acuity or delay. © 2003 Acoustical Society of America.
关DOI: 10.1121/1.1598195兴
PACS numbers: 43.80.Lb, 43.64.Bt 关WA兴
I. INTRODUCTION
It is widely accepted from behavioral and physiological
evidence that echolocating bats determine the distance to objects, or target range, from the time delay of frequencymodulated 共FM兲 echoes 共Grinnell, 1995; O’Neill, 1995;
Schnitzler and Henson, 1980; Simmons, 1973, 1980, Simmons and Grinnell, 1988; Sullivan, 1982兲. In two-choice or
yes–no discrimination tests, big brown bats 共Eptesicus fuscus兲 can distinguish differences in echo delay as small as
50–100 ␮s 共equivalent to 1–2 cm of target range; Moss and
Schnitzler, 1995; Simmons and Grinnell, 1988兲. These behavioral thresholds are roughly consistent with the lower
limits 共⬃100–300 ␮s兲 for the accuracy of echo-delay registration by response latency in single neurons of the big
brown bat’s auditory midbrain, and the sharpness of delay
tuning in individual forebrain neurons 共Dear et al., 1993;
Feng et al., 1978; O’Neill and Suga, 1982; Pollak et al.,
1977兲. However, big brown bats can detect much smaller
changes in delay in a different behavioral procedure where
echoes jitter in delay from one broadcast to the next 共Moss
a兲
Electronic mail: mark – [email protected]
1648
J. Acoust. Soc. Am. 114 (3), September 2003
and Schnitzler, 1995兲. In several versions of the jitteringecho experiment, conducted in two different laboratories,
bats were able to detect changes as small as 0.5–1 ␮s
共Menne et al., 1989; Moss and Schnitzler, 1989; Moss and
Simmons, 1993兲. The bat’s threshold for jitter was measured
to be 10–15 ns when an apparatus was developed to test
even smaller delay jitter 共Simmons et al., 1990兲. At a controlled echo signal-to-noise ratio of 36 dB, big brown bats
can detect delay changes as small as 40 ns, which actually is
possible from an information-theoretic perspective if the bats
used several successive broadcasts to judge whether echoes
jittered in delay 共Simmons et al., 1990兲. Nevertheless, the
degree of temporal precision required to support this performance has been described as virtually impossible for the auditory system to achieve, so the ‘‘10-ns result’’ is widely
assumed to be due to an artifact, most likely spectral in nature, rather than perception of such small changes in time
共Beedholm and Mohl, 1998; Menne et al., 1989, Pollak,
1993; Schnitzler et al., 1985兲. The work reported here concerns whether models of peripheral auditory transduction and
coding are in fact unable to support submicrosecond delay
perception at levels achieved by echolocating bats.
0001-4966/2003/114(3)/1648/12/$19.00
© 2003 Acoustical Society of America
FIG. 1. Time waveforms and their filterbank output for different signal-tonoise ratios. A pulse 共90–20 kHz, linear FM兲 was followed by an echo at
6-ms delay. The pulse and echo are
identical, except that bandlimited
noise 共90–20 kHz兲 was added to the
signal 2 ms after the pulse. The signalto-noise ratio decreases until the echo
is no longer visible, which in these examples occurs at 10 dB. The scale for
the time waveforms at top is the same
for all signal-to-noise ratios. The filterbank output for each waveform is plotted in the bottom row 共see the text for
details兲. The scaling for the four bottom plots varies to display the full data
range in each plot.
This paper describes an auditory filterbank model of the
bat’s auditory periphery and the accuracy of this model for
determining echo delay compared to standard sonar signalprocessing techniques 共cross correlation兲 as well as a previously used filterbank model of the bat’s auditory periphery
共Menne and Hackbarth, 1986; Hackbarth, 1986兲. We simulated the essential signal-processing steps performed by the
bat’s inner ear in multiple-trial tests to evaluate model performance for detecting jitter in echo delay. These tests allowed us to identify the values of model parameters required
to account for the bat’s submicrosecond acuity and the 10-ns
result. One goal is to guide future physiological experiments
by determining whether the bat’s performance really is impossible, or whether there is a range of physiologically testable model parameters that makes this performance possible.
The question of the bat’s delay acuity has been addressed previously for cross-correlation receivers 共Menne
and Hackbarth, 1986兲 and for a simple filterbank receivers
共Hackbarth, 1986; Menne, 1988兲. Although fine delay acuity
is possible with cross correlation, it cannot be achieved with
the filterbank model as it was constituted originally. We
chose to reexamine this question because several aspects of
the Hackbarth 共1986兲 filterbank model were unrealistic in
auditory terms and several critical parameters have never
been measured in bats. In particular, the design of the filterbank’s smoothing 共low-pass兲 filters removed all phase information from the envelopes to be processed for echo delay.
Because several behavioral experiments have shown that
bats may be able to detect changes in relative phase 共Altes,
1984; Simmons et al., 1990, Moss and Simmons, 1993兲, we
decided to revisit the question of echo-delay accuracy and
focus our attention on the role of the smoothing filter, which
had been singled out as a critical stage for replicating the
bat’s performance 共Simmons, 1980兲. A companion paper
共Neretti et al., 2003兲 addresses the related question of the
delay resolution achievable by a similar auditory model. It
should be clearly stated that resolution and accuracy refer to
separate features of an echo-delay estimator, and the terms
should not be used interchangeably 共see Schnitzler and Henson, 1980兲. The accuracy of a delay estimator is the uncertainty in its estimate for the arrival time of echoes from a
single reflecting point. In contrast, delay resolution refers to
J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
the minimum separation between two nearly simultaneous,
overlapping echoes where the two reflections are assigned
their own delay estimates instead of being interpreted as a
single echo.
II. METHODS
A. Biosonar signals
The characteristics of the model are shown in the left
panel of Fig. 1. The procedure for testing the model simulated the bat’s biosonar emission, or pulse, in an acoustic
environment that contained a single point-like reflector at a
target range of ⬃1 m. For simplicity, the bat’s pulse was
modeled with a single harmonic, linear frequency-modulated
共FM兲 chirp that swept from 90 to 20 kHz in a total duration
of 1.5 ms. The test echo was a copy of the pulse shifted in
time by an appropriate delay for a reflector at a range of
1.032 m 共6 ms兲. The pulse and test echo were placed together
in a single time record 12 ms long 共top traces in Fig. 1兲.
This simulation assumed, as have other models, that the
bat acquires a noise-free estimate, or template, of its outgoing pulse 共see Menne and Hackbarth, 1986, Matsuo et al.,
2001, Saillant et al., 1993兲. Bandlimited Gaussian noise
共90–20 kHz兲 was added to the time record beginning 2 ms
after the offset of the pulse and continued over the entire
epoch in which the echo was embedded 共Fig. 1, the 30-dB
panel, shows an example of the onset of this noise兲. The echo
signal-to-noise ratio was determined according to the following equation 共Menne and Hackbarth, 1986兲:
Signal-to-noise ratio共 dB兲 ⫽20 log10共 冑 d 兲 ,
共1兲
d⫽2E/N 0 ,
共2兲
where E is the echo energy flux, or the integral of squared
amplitude over the duration of the signal, units are in
pascals2s; N 0 is the noise power per unit Hz, or mean-square
amplitude divided by bandwidth, units are in pascals2s.
Foraging bats in their natural habitats deal with acoustic
environments that consist of more than just a single reflector
and Gaussian noise. Instead, bats encounter non-Gaussian
atmospheric effects and considerable clutter—other bats’
emissions, multiple insects, background vegetation, etc. To
Sanderson et al.: Echo ranging accuracy in wideband sonar
1649
be clear, the noise added in these simulations was not meant
to simulate noise in the bats’ real-world acoustic environment. The added noise simulated only the controlled white
noise the bats encountered in psychophysical experiments
with jittering echoes 共for details on the experimental signalto-noise ratio, see Simmons et al., 1990兲. Bats flying in their
natural environments often deal with signal-to-noise ratios
lower than the 30–50 dB. Nevertheless, our intention was to
understand how a signal-processing model performed in a
situation similar to that seen by the bat in experiments from
which researchers have obtained precise, informative measures of their sonar capabilities.
The echo was just a single delayed reflected replica of
the broadcast; Doppler shifts and those scattering properties
of the target that affect the echo spectrum were not considered here 共See Neretti et al., 2003, for the resolution of a
model that deals with multiple echoes.兲
The pulse–echo time waveforms for four signal-to-noise
ratios are shown in Fig. 1 共top row兲. For simplicity, the pulse
and echo were set to have equal energy, while noise power
increases from the left to the right in successive plots. Below
the time traces in Fig. 1 are the 22 parallel filterbank outputs,
which consist of envelopes that trace the FM sweep in the
pulse and the echo in a spectrogram-like format. The location
of the echo is obvious in the time waveform and filterbank
output for high signal-to-noise ratios 共e.g., signal-to-noise
ratios 40, 30 dB in Fig. 1兲. At lower signal-to-noise ratios,
however 共e.g., 10 dB兲, the noise swamps the presence of the
echo in both the waveform and filterbank displays. Our
simulations of echo-delay determination assessed how different versions of the filterbank channels performed in varying
levels of noise. Note that this model does not consider the
effect that the receiver’s internal noise has upon delay accuracy. That is, the filterbank itself and subsequent processing
steps are assumed to be noise-free; only the external signalto-noise ratio is included here.
TABLE I. Values for various parameters of the filterbank model and delay
estimation methods.
Sample rate
2 MHz
Signal
Duration
Sweep type
Harmonics
Frequency start
Frequency end
Rise/fall time
1.5 ms
Linear
1
90 kHz
20 kHz
0.15 ms, cosine2 ramp
Bandpass filter
Number of filters
Design
Ripple in passband
Order
Bandwidth3dB
22; see Table II
Chebyshev type 1 共IIR兲
6 dB
8
4 kHz
Nonlinearity
Half-wave rect. y⫽x
y⫽0
for x⭓0
for x⬍0
Low-pass filter
Design
Order
Corner3dB freq.
Chebyshev type 1 共IIR兲
2, 5
1, 2, 4, 8, 125 kHz
Other
A priori window
⫾1000 ␮s
Filter center frequencies were restricted to integer multiples of the 0.5-␮s sample period 共listed in Table II兲. This
step was to minimize interference between the simulation’s
sample rate 共2 MHz兲 and filter CF. This restriction provided
better digital approximations of the equivalent analog filter
impulse responses. Otherwise, if this step was not taken, the
outputs of filters with certain CFs exhibited interference effects with the sample rate in the peak region of the impulse
TABLE II. Period and center frequency of the 22 bandpass filters used in the
filterbank.
B. Filterbank
␮s
kHz
We modeled the bat’s cochlea with a filterbank composed of 22 channels, each of which had three components
connected in series: a bandpass filter, a half-wave rectifier,
and a low-pass filter. The frequency tuning of the ‘‘cochlea’’
was implemented with a series of overlapping bandpass filters 共Chebyshev IIR filters, constant 4-kHz bandwidth兲. We
chose the Chebyshev design because it allows a narrower
bandwidth than the Butterworth design of the same order.
Simulation of the transduction and capacitance of the inner
hair cell was implemented with a half-wave rectifier and
low-pass filter 共Chebyshev IIR filters兲. The resulting output
from the low-pass filter corresponded to the probability of
neurotransmitter release at the base of the inner hair cell and
represented the excitation function delivered to auditorynerve fibers for coding as spikes. The model’s filterbank parameters are listed in Table I. The center frequencies 共CFs兲
for the bandpass filters were chosen to 共1兲 cover the 20–90kHz frequency range of the big brown bat’s hearing 共Koay
et al., 1997兲, and 共2兲 to overlap as closely as possible with
their neighbors at their 3-dB down points.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
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J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
50
44
39.5
35.5
32.5
29.5
27
25
23
21.5
20
19
18
17
16
15
14
13.5
13
12.5
12
11.5
20.000 000 0
22.727 273 0
25.316 456 0
28.169 014 0
30.769 231 0
33.898 305 0
37.037 037 0
40.000 000 0
43.478 261 0
46.511 628 0
50.000 000 0
52.631 579 0
55.555 556 0
58.823 529 0
62.500 000 0
66.666 667 0
71.428 571 0
74.074 074 0
76.923 077 0
80.000 000 0
83.333 333 0
86.956 522 0
response envelope. If these filters were used to estimate delay the resulting estimates would be biased away from the
true peak in the impulse response. Restricting the filter CFs
to those in Table II ensured that the digitally implemented
impulse responses for all filters had the same envelopes. This
minimized any potential bias in the delay estimates in individual channels.
The primary question was how the filterbank receiver’s
delay accuracy compares against a cross-correlation receiver.
Here, we adopted a similar trial-by-trial approach to that
used previously 共Menne and Hackbarth, 1986; Hackbarth,
1986兲. The task was to estimate the delay between a pulse
and a single echo at a fixed delay for multiple occurrences of
the pulse and the echo. We implemented four delay estimation methods for comparison purposes.
C. Delay estimation methods
Three methods 共‘‘analog sum,’’ ‘‘Saillant,’’ and ‘‘Hackbarth’’ methods兲 were used to estimate the delay between
events corresponding to the pulse and echo generated by the
filterbank. Our interest was focused on how the filterbank
design parameters, not the different delay estimation methods, affected delay estimation accuracy. The estimation
methods described below were not intended to realistically
model the full range of physiological processes that the bat
auditory system uses to estimate pulse–echo delay subsequent to auditory transduction by the inner ear. Instead, the
aim was to measure, with the best precision possible, the
temporal intervals between events in the filterbank channels.
1. Cross correlation
To obtain an optimal 共matched-filter兲 delay estimate, we
first computed the cross-correlation function between the
noise-free pulse and the echo embedded in noise for each
noise iteration at each signal-to-noise ratio. Then, the peak in
the cross-correlation function was located within an a priori
window ⫾1000-␮s window around the true delay 共as in
Menne and Hackbarth, 1986 and Hackbarth, 1986兲. Finally,
the three points around the peak were fit with a quadratic
function in order to determine the peak’s precise position. By
this interpolation procedure, delay estimates could be ‘‘recorded’’ with a precision greater than the 0.5-␮s sample period we used to generate our pulse and echo signals.
The time-estimation process was repeated for each of
the 400 trials with independent noise samples in a Monte
Carlo procedure 共see below兲. The delay estimate for each
trial can be compared against the theoretical accuracy possible for a cross-correlation receiver 共Burdic, 1968兲
␴ ⫽ 共 2 ␲ Bd 兲 ⫺1 ,
共3兲
where d is from Eq. 共2兲, and B is the sonar emission’s noncentralized root-mean-square bandwidth, which for this
simulation equaled 57.79 kHz. This value was slightly higher
than the typical value, ⬃55 kHz, estimated for signals recorded in psychophysical experiments 共Simmons et al.,
1990兲. The cross-correlation approach did not use the output
from the filterbank, whereas the next three methods did.
J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
2. Analog sum method
The second method for estimating delay, which will be
called the ‘‘analog sum’’ approach, used the ‘‘cochlear’’ filterbank to process broadcasts and echoes. First, the output of
all the filter channels was dechirped to remove the slope of
the FM sweep by aligning the peaks of the outputs for the
emitted pulses. Peaks for the broadcast in individual channels were located by finding the highest value in the channel
output over the time window containing the broadcast 关see
Fig. 2共C兲兴 and sliding each time series signal by a corresponding amount to align the peaks in different channels to
the same reference time value. Then, the dechirped output
was summed across all frequency channels 关Fig. 2共B兲兴, and
the position of the peak in the resulting sum was located
within an a priori window ⫾1000 ␮s around the true delay.
Finally, using a quadratic fit, the interpolated peak location
was determined and stored as the overall estimate for echo
delay. Thus, as for cross correlation, delay estimates have a
higher precision than the 0.5-␮s sampling interval. This process was repeated for each of the 400 trials with independent
noise samples in a Monte Carlo procedure 共see below兲.
3. Hackbarth method: Single peak detection followed
by interval measurement
This approach, described by Hackbarth 共1986兲, also used
the filterbank to process broadcasts and echoes. First, the
location of the peak in the output from each filter channel for
the noise-free pulse was located inside a window from time
zero to 3.5 ms 共Fig. 1兲. Then, the peak for the echo was
located inside the a priori window of ⫾1000 ␮s centered on
6 ms after the peak for the broadcast 关Fig. 2共C兲兴. Following
this initial procedure, the method produced single triggered
events for the pulse and for the echo. The method
共a兲
共b兲
共c兲
共d兲
共e兲
fit the three points around the pulse and echo peak
samples with quadratic functions in order to determine
their exact positions, which were marked by single
pulses 关this was performed separately for the pulse and
echo response in each frequency channel, Figs. 2共D兲
and 共E兲兴;
measured first-order intervals by subtracting the time of
the interpolated pulse peak from the interpolated echo
peak in each frequency channel 关Fig. 2共F兲兴;
generated a point process by projecting these intervals
across filterbank channels onto a single time axis 关Fig.
2共G兲兴;
estimated the density of the point process along the
time axis by convolving the points with a Gaussian
kernel ( ␴ ⫽1 ␮ s, shift step⫽0.25 ␮ s 关Fig. 2共H兲兴; and
located the peak of the density function resulting from
convolution by fitting a quadratic to the peak sample
and its two neighboring samples, as before, and stored
this interpolated value as the overall delay estimate for
that trial.
This process was repeated for each of the 400 trials with
independent noise samples in a Monte Carlo procedure 共see
below兲.
Sanderson et al.: Echo ranging accuracy in wideband sonar
1651
FIG. 2. Three methods for estimating delay from the filterbank output. The schematic at top illustrates how the biosonar signal passes through the filterbank,
the output of which is processed by three different estimators. The cross-correlation estimator operates directly on the pulse–echo signals. 共A兲 Dechirped
output from a filterbank with a second-order low-pass filter cutoff set to 8 kHz, and a signal-to-noise ratio of 36 dB. 共B兲 The summed output, across frequency
channels, from panel 共A兲. The peak position is the ‘‘analog sum’’ method’s echo delay estimate. 共C兲 Output from the filterbank’s third channel (CF
⫽25.3 kHz). 共D兲 Expanded view of the pulse 关corresponds to left gray bar in panel 共C兲兴. Spikes are triggered by the maximum peak for the Hackbarth method
and for the local peaks above a threshold for the Saillant method. 共E兲 Expanded view of the echo with triggered spikes for both methods at bottom. 共F兲
Interpolated delay estimates from the Hackbarth method. 共G兲 Histogram of peaks from 共F兲. 共H兲 Smoothed histogram from 共G兲, using Gaussian kernel with
␴ ⫽1 ␮ s. The interpolated peak is the Hackbarth method’s final delay estimate. 共I–K兲 Same as 共F兲–共H兲 but for the Saillant method. Note that x axes for
共F兲–共K兲 range from ⫾100 ␮s, whereas 共A兲 and 共B兲 range from ⫾1000 ␮s relative to actual echo delay.
4. Saillant method: Detection of multiple peaks
followed by all-order interval measurement
This final method created a series of triggered spike
events for the pulse and then for the echo using a procedure
similar to the ‘‘peak-detection’’ approach of Saillant et al.
共1993兲. First, the peak position in the noise-free window for
the pulse in each filterbank channel was located 共as above兲,
and the corresponding sample times across channels were
used to dechirp the filterbank output 共each channel’s time
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J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
axis was realigned so the peak pulse time corresponded to
time zero兲. Then, the model generated spikes within two a
priori windows: one centered on the pulse and one on the
echo. The pulse a priori window spanned the noise-free time
window from time zero to 3.5 ms 共Fig. 1兲. As before, the
echo a priori window spanned ⫾1000 ␮s and was centered
in time at the true echo delay.
For each filterbank channel, separate thresholds were set
for the pulse and echo to define the time windows inside
Sanderson et al.: Echo ranging accuracy in wideband sonar
FIG. 3. Results from 400 Monte Carlo trials with echo
signal-to-noise ratio fixed at 36 dB. 共A兲 The crosscorrelation receiver’s estimates. 共B兲, 共C兲, and 共D兲 Results for each of the three filterbank models are plotted
in separate columns. As the low-pass filter cutoff frequency increased, the variability of the delay estimates
decreased 共1-kHz results not shown兲. Because the x axis
is scaled so that only the central 1 ␮s of the full ⫾
1000-␮s a priori window is visible, some of the estimates are not visible 共especially for the 1- and 2-kHz
low-pass filter conditions兲. The accuracy of each
1
method was estimated by taking 2 of the 68th percentile
of the distribution of delay estimates, and is summarized in Fig. 4.
which multiple peaks in the filter output would registered as
corresponding multiple spikes. For the pulse, the spikewindow threshold was set at the value of 67% of the peak
amplitude within the previously defined pulse a priori window. For the echo, the first step was to establish a noise
threshold located 2 standard deviations above the mean noise
level in the filter output over a time window containing just
noise. In the second step, the echo spike-window threshold
was set at the value 2/3 of the way between this noise threshold and the amplitude of the largest peak within the echo a
priori window. Spike events then were generated for every
local peak above the pulse-window and echo-window thresholds 关see Figs. 2共D兲 and 共E兲, respectively兴, and the location
of each local peak was identified using interpolation 共as before, a quadratic was fit to each local peak and the immediate
neighboring samples兲. At this juncture, the channel-bychannel filterbank output is converted into two sets of spike
events corresponding to all the local peaks within the pulse
and echo windows 关one channel’s output shown in bottom
panel of Figs. 2共D兲 and 共E兲兴.
To obtain a single estimate for pulse–echo delay from
this multiple-spike representation, the method
共a兲
共b兲
共c兲
共d兲
calculated the all-order intervals 共Cariani and Delgutte,
1996兲 for pulse versus echo spikes within each channel
关Fig. 2共I兲兴;
generated a point process by projecting these resulting
intervals onto a single time axis 关Fig. 2共J兲兴;
estimated the density of the point process along the
time axis by convolution with a Gaussian kernel 关 ␴
⫽1 ␮ s, shift step⫽0.25 ␮ s; Fig. 2共K兲兴; and
located the peak of the resulting density function by
fitting a quadratic to the peak sample and its two neighboring samples and stored it as the overall delay estimate for that trial.
This process was repeated for each of the 400 trials with
independent noise samples in a Monte Carlo procedure 共see
below兲.
J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
D. Monte Carlo trials
Our goal was to examine how the use of different filterbank parameters affected the variance of the delay estimation
procedure. To estimate this variance we adopted a Monte
Carlo procedure, a method that uses many independent trials,
each with a different noise instantiation, in order to build a
distribution of the delay estimates. The variance of the delay
estimation procedure is then measured from this distribution.
The number of trials used in the Monte Carlo simulations do
not improve the accuracy of the model because there is no
memory from trial to trial in the Monte Carlo method. Our
results were identical using 100 or 400 trials. It follows that
the number of Monte Carlo trials bear no relation to how
many emissions the bats actually use in a single trial of the
jitter task.
Pulse–echo delay estimation was repeated for 400 different realizations of noise added to the echo at a fixed
signal-to-noise ratio 关Fig. 3兴. The true echo delay, 6 ms, was
subtracted from each of the 400 estimates to form the error
distribution. To estimate the mean and variance of the accuracy of this distribution we used a bootstrap procedure. Four
hundred samples were drawn, with replacement, from the
error distribution, and the 68th percentile was calculated. The
resulting value was divided by 2 in order to match the standard deviation for a uniform distribution in a 1000-ms window 共⬃683 ␮s兲. This sampling was then repeated 128 times
to compute the bootstrapped estimate for the mean and standard deviation of accuracy.
The bootstrapped accuracy estimates are plotted in Fig.
4 against the range of signal-to-noise ratios tested. This
method for estimating the estimates’ variability was used for
comparison with other models 共Hackbarth, 1986, Menne and
Hackbarth, 1986兲 and the analytical standard deviation calculation 关see Eq. 共3兲 above兴.
E. Additional constraints
The width 共␴兲 of the Gaussian kernel chosen for the
smoothing procedure affected the estimate of filterbank acSanderson et al.: Echo ranging accuracy in wideband sonar
1653
FIG. 4. Filterbank accuracy with different low-pass filter settings. The accuracy of the cross-correlation Monte Carlo simulations is plotted as a heavy gray
line on each panel for comparison. 共A兲 The accuracy of the analog sum method when applied to the filterbank output. Five different settings for the filterbank’s
low-pass filter were used 关see the legend in panel 共C兲兴. Increasing the low-pass filter cutoff value shifted the analog sum method’s accuracy much closer to
that of the cross correlation. The values for Eq. 共3兲 are plotted as a dashed line. 共B兲 Same as 共A兲 except delay estimates were generated by applying the Saillant
method to filterbank output. 共C兲 Same as 共A兲 except delay estimates were generated applying the Hackbarth method to filterbank output. 共D兲–共F兲 These three
plots show the 400 trial-by-trial estimates for the analog sum, Saillant, and Hackbarth methods. Note that for the Hackbarth method, many estimates occur at
the edges of the a priori window.
curacy 共step e of the Hackbarth method and step d of the
Saillant method兲. If the Gaussian was too wide relative to the
‘‘true’’ accuracy of the filterbank, all of the trial-by-trial estimates fell within a single bin. This underestimated the effective accuracy of the filterbank. On the other hand, if the
Gaussian width was too narrow, the smoothing step resulted
in multiple local peaks with equal heights and so failed to
yield a single delay estimate. After testing of sample data
with several different values for ␴ of the Gaussian kernel, we
chose the value that yielded the maximum accuracy ( ␴
⫽1 microsecond).
The size of the a priori window also has a significant
effect on the accuracy of each receiver design, as shown
previously in Menne and Hackbarth 共1986兲 and Hackbarth
共1986兲. However, the a priori window size only affects the
accuracy of the results within a fixed range of signal-to-noise
values less than 15 dB, where the accuracy curve ‘‘breaks’’
or undergoes an abrupt decline due to ambiguity effects
caused by the emergence of prominent sidelobes 共Menne and
Hackbarth, 1986兲. We chose the value of ⫹/⫺1000 microseconds for our a priori window in order to compare our
results with theirs. The use of a priori windows with respect
to biosonar experiments is not unreasonable: the bat certainly
knows when it vocalized its pulse, and in behavioral paradigms learns fairly quickly that most echoes return within a
fixed time window.
and jittered between two values on the other side. The bat
estimated each returning echo’s delay for comparison with
the delay of the next echo to then choose which side had the
jittering echoes.
We simulated this experiment, following the method of
Menne and Hackbarth 共1986兲, in which the virtual bat discriminated a jittering target from a nonjittering target. On
one ‘‘side’’ the virtual bat received two echoes embedded in
noise at a fixed delay. On the other ‘‘side’’ the virtual bat
received two echoes in noise with a temporal offset, or jitter,
added to each echo. The data for the experiment had in fact
been simulated under different conditions in the Monte Carlo
simulations. We could therefore simply draw four delay estimates, without replacement, from these simulations and a
temporal offset (⫹⌬t,⫺⌬t) was added to the two delay
estimates on the jittering ‘‘side.’’ On a single trial the virtual
bat had to decide which side had the jittering echoes. The
jitter experiment was simulated with 100 such trials, and the
results of the simulation were expressed as the percentage of
correct decisions for those 100 trials for different values of
⌬t 共the amount of jitter, ⌬t, was varied from 0 to 20 microseconds兲. This entire procedure was then repeated 128 times
to compute a bootstrap estimate of the variability for these
percent-correct values.
III. RESULTS
A. Monte Carlo results
F. Simulation of the jittering-echo experiment
In the jittering-echo experiment, the bat is trained to sit
on a small platform and emit sonar sounds into microphones,
and its task is to determine which of two loudspeakers returned echoes of those sounds that alternated in delay from
one broadcast to the next 共Simmons et al., 1990兲. In individual jitter trials, the bat necessarily emitted at least two
sounds to the jittering stimulus channel and two sounds to
the nonjittering channel. A single stimulus echo was returned
for each emission, the delay of which was fixed on one side
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J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
The results consist of a series of estimates of delay accuracy obtained with each of the four model designs on 400
Monte Carlo trials at each signal-to-noise ratio. To illustrate
the nature of the simulation results, the 400 trial-by-trial delay estimates obtained with the four methods are plotted for
one signal-to-noise ratio 共36 dB兲 and different low-pass
smoothing frequencies 共1, 2, 4, 8, and 125 kHz兲 in Fig. 3. To
establish a baseline for comparison, the performance of the
cross-correlation procedure is show in Fig. 3共A兲, while the
results from the three filterbank methods are plotted in sepaSanderson et al.: Echo ranging accuracy in wideband sonar
rate columns 关Figs. 3共B兲–共D兲兴. Each point on one of the
graphs represents a single delay estimate for a pulse–echo
pair with one iteration of independent noise added to the
echo. All four methods generated well-behaved delay estimates across the Monte Carlo trials, and in each case the
variability of the estimates changed with the low-pass cutoff.
With increasing low-pass cutoff frequency, the variability of
each distribution decreases appreciably 共except for the Hackbarth method, for which the results at 8 and 125 kHz were
very similar—that is, once the low-pass cuttoff was as high
as 8 kHz, no further improvement in accuracy could be obtained兲. The 125-kHz low-pass filter condition was explicitly
included in the simulations to observe what happened when
all the available phase information was allowed to pass
through the model and made available for processing 共this
was our method for the effective removal of the low-pass
filter兲. As such, the analog sum method with a 125-kHz lowpass filter yielded the tightest distribution of delay estimates,
a distribution that was indistinguishable from the results for
the optimal delay estimator, the cross-correlation receiver,
plotted in Fig. 3共A兲.
The summary and quantification of the variability of all
delay estimates across a wide range of signal-to-noise ratios
is plotted in Fig. 4. These curves show the effects of the
low-pass filter parameters on the accuracy of delay estimates,
and allow comparison of the performance for different estimation methods.
1. Cross-correlation accuracy
Because the cross-correlation receiver’s accuracy was
the lower bound to be expected for the various estimation
methods, it is shown on each plot in Fig. 4 as a baseline. The
cross-correlation results from the Monte Carlo trials fit the
theoretical accuracy 关Eq. 共3兲, dashed line兴 for high signal-tonoise ratios 共⬎15 dB兲. Between signal-to-noise ratios of 15
to 10 dB, the cross-correlation receiver accuracy falls off
sharply compared to what theory predicts. The nature of this
break in the cross-correlation performance was explored previously by Menne and Hackbarth 共1986兲. They showed that
Eq. 共3兲 was applicable only when the signal noise ratio was
above 15 dB, and our results are the same.
2. Filterbank model accuracy: Analog sum method
The analog sum method yielded results almost identical
to the cross correlation when the entire signal bandwidth
passed the low-pass filter 共i.e., cutoff frequency⫽125 kHz).
This shows that bandpass filtering, rectification, and peak
picking had no deleterious effects on accuracy. However, as
high-frequency phase information was progressively removed from the filterbank output by the low-pass filter, accuracy degraded precipitously. In the linear region of the
results 共signal-to-noise ratio ⭓30 dB兲, the accuracy of delay
estimation degraded by 2 orders of magnitude as the lowpass cutoff frequency was decreased from 125 to 1 kHz.
Between signal-to-noise ratios of 20 and 15 dB, the accuracy
curves break sharply from the linear case and eventually converge upon the standard deviation for the whole a priori
window, 680 ␮s.
J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
FIG. 5. Comparison of different filterbank delay estimation methods when
using a second-order low-pass filter set at 8 kHz. Of the filterbank estimators, the analog sum method provided the best performance. There was an
orderly arrangement to where each method ‘‘broke,’’ or underwent a sharp
transition in accuracy: the cross-correlation accuracy broke between 15 to
10 dB, the analog sum method between 20 and 15 dB, the Saillant method
between 25 and 20 dB, and the Hackbarth method between 30 and 25 dB.
The Hackbarth and Saillant methods were identical for signal-to-noise ratios
⭓30 dB.
3. Filterbank model accuracy: Saillant method
The Saillant method used one or more spikes to mark
the occurrence of the pulse and echo in each channel 关Figs.
2共D兲 and 共E兲兴. As such, it uses less of the filterbank output’s
wave structure, compared to the analog sum method, to estimate pulse–echo delay and so would be expected to do
worse at any given signal-to-noise ratio. In Fig. 4, the performance of the Saillant method is displaced upward relative
to the analog sum method to slightly worse performance
across most signal-to-noise ratios. This is more clearly
shown in Fig. 5, which plots the performance of all four
methods at an 8-kHz low-pass cutoff.
4. Filterbank model accuracy: Hackbarth method
In each frequency channel the Hackbarth method used
only one spike to mark the occurrence of the pulse and one
spike for the echo 共see Fig. 2兲. At signal-to-noise ratios ⭓30
dB, the accuracy results for the Hackbarth method were very
similar to those of the Saillant method. The exception was
the 125-kHz low-pass filter condition, which for the Hackbarth method did not show any appreciable difference from
its 8-kHz result.
Unlike the Saillant method, we observed that for signalto-noise ratios below 25 dB the a priori window size interacted with the low-frequency nature of the filterbank output.
The width or duration of the ‘‘excitation’’ in each frequency
channel was a function of the signal sweep rate, integration
time, and low-pass filter time constant. Because the excitation half-width was about 250–300 ␮s 关Fig. 2共D兲兴, the outputs of the filters were such slowly varying signals that they
could almost be considered as linear when viewed within a
small a priori window. Consequently, the signals often
reached their maximum at one of the window edges 关Figs.
3共D兲, 共E兲, 共F兲兴. Because of this edge effect, we discarded any
Sanderson et al.: Echo ranging accuracy in wideband sonar
1655
accuracy measurements if a conspicuous number 共⬎10%兲 of
trials resulted in delay estimates at the edges of the a priori
window.
Figure 4共F兲 shows the trial-by-trial results for the Hackbarth method at a low signal-to-noise ratio. Many of the data
points for delay estimates fall on top of the gray bands that
mark the edges of the a priori window. The analog sum and
Saillant methods do not suffer from this problem 关Figs. 4共D兲,
共E兲兴. Although this edge effect was not reported in Hackbarth
共1986兲, she also only plotted results for signal-to-noise ratio
values above 20 dB, so we assume she had the same problem
in her analysis. Because the Saillant method triggered spikes
from multiple local peaks above threshold 关e.g., Fig. 2共E兲兴, it
did not suffer from this edge effect when the low-pass cutoff
frequency was ⬎2 kHz.
5. Comparison of the three filterbank methods
Figure 5 shows, for a single 8-kHz low-pass filter condition, the performance of the three filterbank methods on
the same plot. From Fig. 5 it was clear that 共1兲 the analog
sum method provided the best accuracy 共other than the crosscorrelation method兲, and 共2兲 the points where the slope of the
curve changed sharply for each method were separated along
the horizontal signal-to-noise axis by about 5 dB.
6. Steeper roll-off for the low-pass filter
We also investigated how the severity of low-pass filtering affected the accuracy of the three filterbank delay estimation methods. The second-order Chebyshev low-pass filter
had a roll-off of 22 dB/decade. This was rather shallow compared to the 100-dB/decade value estimated for the mammalian smoothing filter 共Weiss and Rose, 1988兲. Therefore, we
repeated the simulations using a fifth order Chebyshev lowpass filter that provided attenuation of 62 dB/decade. This
more severe low-pass filtering removed most of the ripple
riding on each frequency channel’s envelope. Consequently,
the filterbank performance decreased by 25 dB for all three
methods 共Fig. 6; Hackbarth method not shown兲. In addition,
the loss of any significant ripple meant that the Saillant
method no longer could trigger multiple spikes for the pulse
or echo. Consequently, the Saillant and Hackbarth methods
converged on using the same single local peak for the pulse
and for the echo to generate their spike events, and they
yielded similar results.
FIG. 6. Increasing the severity of low-pass filtering drastically reduced filterbank accuracy. 共A兲 When the order of the low-pass filter was increased
from 2 to 5, the analog sum method’s results shifted by about 25 dB. 共B兲
Same as 共A兲, but for the Saillant method.
signal-to-noise ratio of 36 dB, is also plotted for comparison.
At this signal-to-noise ratio, the best filterbank method 共analog sum with a second-order low-pass filter兲 had a jitter
threshold of 82.9 ns. The analog sum method required a
rather high SNR, 50 dB, in order to achieve 16.6-ns jitter
detection performance based on the two pairs of echoes as
simulated stimuli.
B. Simulation of the jittering-echo experiment
The performance of the analog sum filterbank model in
the jitter experiment is shown in Fig. 7共A兲. At a signal-tonoise ratio of 36 dB, as in the original behavioral experiment
共Simmons et al., 1990兲, the curves for percent correct show
better performance 共shift to the left兲 with increasing cutoff
frequency for the low-pass filter. The threshold for jitter detection, ⌬t 75 , was taken at the value that resulted in a performance of 75% correct 共actual value was identified using
cubic spline interpolation兲.
The thresholds for the cross correlation, analog sum, and
Saillant method are shown in Fig. 7共B兲. The 40-ns threshold,
which was measured in the behavioral experiments at a
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J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
IV. CONCLUSIONS
These modeling results show that low-pass filtering
strongly affects the accuracy of a filterbank receiver. For
bats, this means that achieving 40-ns jitter acuity at a signalto-noise ratio of ⬃36 dB requires that cochlear bandpass and
low-pass filtering properties adhere to the following two constraints: First, the bandpass filters’ integration time should
match the sweep rate of the FM emission 共discussed previously in Menne, 1988兲. Second, the effective low-pass filtering that takes place before auditory-nerve spike generation
should have, compared to typical mammalian values, a rather
high cutoff frequency and shallow slope.
Sanderson et al.: Echo ranging accuracy in wideband sonar
The simulated jitter experiment assumes the model bat
uses the minimum number of emissions necessary for the
task. In the actual behavioral experiment, the bats certainly
took ‘‘multiple looks’’ at each side, using anywhere between
6 and 15 or 20 emission per side before making a decision
共Simmons et al., 1990兲. Currently, there is limited knowledge of how bats integrate information across multiple emissions. If the model bat was allowed to average delay estimates across six emissions, accuracy could improve by a
factor of 2.45 共in the simulated jitter task, six emissions per
side results in three jitter estimates; therefore, the model bat,
assuming it had perfect memory storage of those estimates,
would see an improvement of 冑 3). Applied to the analog
sum 共second-order兲 results in Fig. 7 at 36 dB, this would
improve the accuracy from 82.9 to 47.9 ns, which is similar
to the observed behavioral threshold. At higher signal-tonoise ratios, use of multiple sounds led to comparable improvements in performance.
ditional assumptions and parameters that would have to be
tested. Internal noise 共such as the variability in when or how
many action potentials are generated by auditory neurons, or
variability in the memory/decision process兲 has been considered elsewhere. For example, Suzuki and Suga 共1991兲 assessed the theoretical accuracy for a topographically arranged pool of cortical neurons with variable delay-tuned
responses. Their best decoder of cortical activity achieved an
echo delay acuity of 800 ns. Another model used to test
echo-delay accuracy, developed by Wotton et al. 共2002兲,
used a population of midbrain, thalamic, and cortical neurons
imbued with the latency variability and delay tuning response uncertainty observed in neurophysiological experiments. Similar to Suzuki and Suga’s 共1991兲 results, this
population model also had a jitter acuity of ⬃1 ␮s. Palakal
and Wong 共1999兲 also developed a model that used cortical
delay-tuned neurons to estimate pulse–echo delay, but did
not systematically explore its accuracy beyond reporting that
the typical error was about 2%, or ⬃20 microseconds, of the
tested target range. The distribution or variance of these errors was not reported. If possible, future models should incorporate both the effects of internal and external noise on
echo delay estimation precision for comparison to the published behavioral data.
This study was motivated in part by an earlier study that
documented the low accuracy of a filterbank in the echodelay estimation task 共Hackbarth, 1986兲. Those earlier results are worse than our filterbank with 1-kHz low-pass filtering 共not shown兲. Several caveats must be mentioned
before comparing those data to our simulation results. First,
the pulse and echo signals were constructed using a recorded
Eptesicus broadcast, which had a smaller effective signal
bandwidth 共it spanned 90–30 kHz兲 than our test signals. Second, among several differences in the earlier filterbank design, the most significant probably resides in the smoothing
procedure. After bandpass filtering, Hackbarth calculated the
envelope in a two-step process: she 共1兲 computed the amplitude envelope by taking the absolute value of the signal’s
Hilbert transform 共essentially a full-wave rectification兲, and
共2兲 low-pass filtered the result at 5 kHz. No further details
were provided about this low-pass filter.
The low-pass filter’s effect was most deleterious for cutoff frequencies less than the bandpass filter bandwidth 共i.e.,
below 4 kHz兲. When the low-pass filter cutoff was ⬍4 kHz,
the low-pass stage not only removes the ac component from
the envelope, but more importantly begins to smear the effective integration time established by the bandwidth of the
bandpass filter. For a low-pass cutoff above ⭓4 kHz, some of
the ac component passes, and the effective integration time
of the whole channel is unchanged, remaining at its minimum value 共⬃250 ␮s, which originates in the bandpass filters’ fixed bandwidth of 4 kHz兲. This surviving ac phase
information is crucial for pushing echo-delay accuracy closer
to the cross-correlation result 共Fig. 4兲.
A. Implementation details for filterbank models
B. Low-pass filtering and inner hair cells
We did not include the effects of internal 共receiver兲
noise upon pulse–echo delay accuracy because modeling the
bat’s nervous system and cognitive state requires many ad-
The results shown in Fig. 4 reveal the need to obtain
constraining empirical data on the bat’s inner hair cell membrane filtering properties. In order to move beyond the de-
FIG. 7. Jitter threshold for single echo delay. Filterbank delay estimates
from the Monte Carlo trials were used to simulate the Simmons et al. 共1990兲
single echo jitter experiment 共see the text兲. 共A兲 The performance of the
simulated bat in choosing the jittering echo at a signal-to-noise ratio of 36
dB is plotted as percent correct 共⫾ st. dev.兲. The low-pass filter’s cutoff
frequency is indicated next to each curve. Threshold was taken where the
curves crossed the 75%-correct level. 共B兲 The threshold for jitter detection
for several different methods. The filterbank models used low-pass filters
with an 8-kHz cutoff frequency. The behavioral threshold measured at a
signal-to-noise ratio of 36 dB, from Simmons et al. 共1990兲, is indicated by
the gray symbol.
J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003
Sanderson et al.: Echo ranging accuracy in wideband sonar
1657
velopment of the present peripheral auditory filterbank
model we need to know both 共1兲 the shape of the effective
integration time and 共2兲 the frequency at which ac ripple is
no longer detectable in the inner hair cell membrane potential. The available data come from guinea pigs and cats 共e.g.,
Palmer and Russell, 1986兲, not from bats. Recordings of the
inner hair cell receptor potential evoked by pure tones typically show two components: an ac component equal in frequency to the pure tone, and a dc component. The ac component decreases in amplitude with increasing test-tone
frequency, while the dc component increases reciprocally.
The falloff in the ac amplitude, which for guinea pigs occurs
between 0.5–2 kHz 共Palmer and Russell, 1986兲, is due to the
inner hair cell’s membrane time constant 共Russell and Sellick, 1978兲. Because acquiring data from hair cells is technically difficult, many researchers have addressed this question
instead in the auditory nerve. Although the relationship between receptor potential and auditory-nerve spike triggering
has not been well studied in a direct manner, for the most
part it is reasonable to study phase sensitivity in the auditory
nerve because, if it does not exist in the spikes traveling to
the cochlear nucleus, then it cannot be detected by the auditory system. In the guinea pig, at least, the decrease in the
receptor potential ac component correlates with the decrease
in auditory-nerve spike phase locking 共Palmer and Russell,
1986兲. For guinea pig and chinchilla auditory-nerve fibers,
phase locking begins to fall off at 0.6 kHz and is negligible
by 3.5 kHz 共Harrison and Evans, 1979; Palmer and Russell,
1986兲. Cat auditory-nerve fibers show a gradual falloff that
begins around 1–2 kHz and is near zero above 4 –5 kHz
共Johnson, 1980兲. Barn owl auditory-nerve fibers show significant phase locking up to 9 kHz 共Koppl, 1997兲. Which of
these values is appropriate for FM bats such as Eptesicus has
yet to be determined. We are not aware of any relevant studies in bats except for one by Suga et al. 共1971兲, in which
two-tone ‘‘beat’’ stimuli evoked phase-locked activity up to 3
kHz in the auditory nerve of Pteronotus parnelli.
C. Physiological experiments
Because the neurons of interest for bats have BFs well
above 10 kHz 共beyond the phase-locking limit seen in birds兲,
phase locking to cycles of pure tones cannot be measured.
Instead of using long pure tones to measure phase locking, a
better test for echolocating animals is one of phase sensitivity. For bats, important acoustic events are extremely brief
FM sounds that typically evoke an average of just 1 spike per
stimulus 共Pollak et al., 1977; Ferragamo et al., 2002; Sanderson and Simmons, 2000兲. What matters are the relative
latencies of these single spikes across neurons, each of which
is evoked by a pulse or echo after a relatively brief period of
quiet. Do neurons show latency shifts when the phase of a
brief tone burst or FM sweep changes?
Thus far, we have conducted a series of experiments in
the brainstem and inferior colliculus of Eptesicus fuscus to
test this question. Local field potentials do show sensitivity
共changes in shape兲 to changes in pure-tone burst starting
phase 共Ferragamo et al., 2002兲. This sensitivity to phase can
be observed for pure tones up to 16 kHz. Results from these
physiological experiments address findings from experiments
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in Eptesicus that imply 共behavioral: Menne et al., 1989; Simmons et al., 1990; Moss and Simmons, 1993兲 or require
共computational: Peremans and Hallam, 1998; Matsuo et al.,
2001; Neretti et al., 2003; Saillant et al., 1993兲 phase sensitivity in the ultrasonic range.
ACKNOWLEDGMENTS
Grants from NSF 共BES-9622297兲 and ONR 共N0001499-1-0350兲 to J.A.S., an NIH Training grant, and a
Burroughs-Wellcome grant to the Brown University Brain
Sciences Program supported this work.
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Sanderson et al.: Echo ranging accuracy in wideband sonar
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