Taylor, J., Clarke, C., Schuettler, M. and Donaldson, N. (2012)
The theory of velocity selective neural recording: a study based
on simulation. Medical and Biological Engineering and
Computing, 50 (3). pp. 309-318. ISSN 0140-0118
Link to official URL (if available): http://dx.doi.org/10.1007/s11517-
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The Theory of Velocity Selective Neural Recording: A
Study Based on Simulation
John Taylor1, Martin Schuettler2, Chris Clarke1 and Nick Donaldson3
1. Department of Electronic and Electrical Engineering, University of Bath, Bath UK
2. Laboratory for Biomedical Microtechnology, Department of Microsystems Engineering –
IMTEK, University of Freiburg, 79110 Freiburg, Germany.
3. Department Medical Physics and Bioengineering, University College London, London WC1 UK.
Corresponding author: John Taylor. Tel +44 (0)1225 38 3910; email firstname.lastname@example.org
Abstract- This paper describes improvements to the theory of velocity selective recording and some
simulation results. In this method, activity is different groups of axons is discriminated by their
propagation velocity. A multi-electrode cuff and an array of amplifiers produce multiple neural signals;
if artificial delays are inserted and the signals are added, the activity in axons of the matched velocity are
emphasized. We call this intrinsic velocity selective recording. However, simulation shows that
interpreting the time signals is then not straight-forward and the selectivity Qv is low. New theory shows
that bandpass filters improve the selectivity and explains why this is true in the time domain. A
simulation study investigates the limits on the available velocity selectivity both with and without
additive noise and with reasonable sampling rates and analogue-to-digital conversion (ADC) parameters.
Bandpass filters can improve the selectivity by factors up to 7 but this depends on the speed of the action
potential and the signal-to-noise ratio.
Keywords: Electroneurogram recording, Simulation, Multielectrode cuffs, Velocity selective recording
Total text words: 5848
Words in the abstract: 159
Number of figures: 6
Number of Tables: 2
Velocity selective recording (VSR) is a technique which should allow more information to be extracted
from an intact nerve with a recording set-up that does not allow action potentials from single fibres to be
seen at spikes [9, 1]. The method is in essence very simple and relies on taking measurements of a
propagating action potential (AP) at two or more points. The distance between the sample points divided
by the delay between two appearance of the AP provides a measure of the propagation velocity.
Conversely, artificially delaying one or more signals before adding them gives a maximum response at
the matched velocity. This simple idea is not new and various researchers have investigated practical
adaptations of it in the past-e.g. [6, 11, 2, 3].
However, at present the idea has not been demonstrated with naturally-occurring nerve traffic though
experimenters have used multi-electrode cuffs (MECs) to observe appropriate outputs from compound
action potentials [5, 6, 11]. Two papers about the theory of VSR have been published by the same
authors [9, 1]. The first presented a spectral analysis of a single axon in an MEC with a tripolar (double-
differential) amplifier system and the signal processing arrangement shown in Figure 1. The bandpass
filter (BPF) that follows the adder was shown to improve selectivity in the velocity domain, however the
filters were ideal and of infinitesimal bandwidth. We envisage that a useful VSR system might actually
have several signal processing units (Figure 1), each matched to one of the propagation velocities of
functional significance .
The second theory paper  considered the thermal noise generated by the detection system and
compared its amplitude to that of the signal resulting from the summation of multiple single fibre action
potentials (SFAPs) which were assumed to occur at random times. This allowed the firing rates required
from various sizes of nerve fibre in a given MEC to provide a signal that could be detected above the
background noise to be calculated.
The current paper presents material which supplements and expands our earlier work [9, 1]. In essence it
is a study (by simulation) of improvements in velocity selectivity obtainable by the use of BPFs,
investigating in particular the limitations of the method with and without noise. The intended outcome of
this work is to achieve a significant improvement in the performance of VSR systems so that they can be
employed to discriminate between populations of naturally-occurring APs. As already noted, the
relatively poor velocity selectivity of the VSR systems tested to date has limited their application to
Deterministic models of nerve signals (i.e. SFAPs) are used exclusively in the study, the effects of
random signal generation being reserved for presentation in a subsequent paper.
2.1 Basic principles
The response of an MEC to a propagating SFAP has been discussed in detail elsewhere and is only very
briefly summarised here [9, 1]. The input to the MEC is a trans-membrane action potential function
(TMAP), Vm(t), with the corresponding spectrum Vm(f). The resulting SFAP is a propagating wave with
the time dependence of the underlying TMAP function, the relationship between the two being explained
in . For the purpose of simulation, we represent the TMAP function and its spectrum by the Fourier
transform pair :
Vm(t) = Atne-Bt
where A, B and n are constants and f is frequency (the symbology has been preserved from ). This
function has been proposed as a suitable basis for the simulation of mammalian ENG [8, 7, 10]. The
output Y(f, v), which is a function of both frequency and velocity, is obtained by treating the MEC as a
linear time-invariant system with transfer function H(f, v):
This equation describes the output of a cuff with N tripoles, electrode pitch d and propagation velocity v.
Ra, the intra-axonal resistance per unit length, has been assumed to be large compared to Re, the extra-
axonal resistances per unit length inside the cuff. The artificial time delays are multiples of (see  for
a full explanation). Equation 2 is the product of the spectrum of the TMAP (Vm(f)), the transfer function
of one tripole (Ho(f,v)), and the transfer function of the delay-and-add block (G(f,v)). At matched
velocities (i.e. where τ = d/v and v = vo in eqn (2)), eqn (2) reduces to:
The output of the system Y(f,v) is a function of two variables and it was pointed out in  that if f is
fixed by passing the output through a bandpass filter (so that f = f0), Y becomes a function of propagation
velocity v only, enabling the velocity selectivity profile (see the tuning curves in ) to be calculated.
We define a velocity quality factor, Qv, by analogy with linear systems in the frequency domain :
where v0 is the matched (i.e. peak) velocity and v3+ and v3- are the velocities at which the output has
fallen to 1/√2 (-3 dB) of the peak value. Close to the matched velocities, the velocity selectivity is
dominated by the function G(f,v) and an approximate formula for Qv was derived in :
Qv and its approximation (eqn (5)) will be used to characterise velocity profiles in this paper.
2.2 The intrinsic velocity spectrum (IVS)
If bandpass filtering is not applied, as noted above, the output will depend on v and on frequency
dependent elements in the system including the spectral properties of the input signal (i.e. the TMAP
function) and the characteristics of the channel. For these reasons, unlike the bandpass filtered version,
the resulting time domain function is difficult to interpret. Simulating in the time domain, Fig 2 is the
time-domain output of the adder in Fig 1 when the system is stimulated with a TMAP resulting in an
SFAP propagating at a velocity of 30 m/s. Two TMAP functions are considered, based on equation (1),
with the parameters given in Figs 2(b) and 2(c) (the scaling parameter A has been adjusted in both cases
so that the peak amplitudes of the functions are normalised to unity). The matched velocity vo is treated
as a parameter leading to the family of curves shown in Fig 2(a). The peak value is reached when the
artificial delays exactly match (cancel) the naturally-occurring delays at which point the output signal
has the same form as a single SFAP, with amplitude multiplied by N .
Fig 2(b) shows the IVS of the system, stimulated by an SFAP generated by TMAP#1 (and repeated in
Fig 2(c) for TMAP#2). This is a plot of the peak values of the output time record (Fig 2(a)) as a function
of propagation velocity after the tripole signals have been subjected to delay and add operations only.
Each curve in the time record shown in Fig 2(a) has three peaks, labelled , and , two positive and
one negative, corresponding to the phases of the tripolar SFAP. Whilst it is possible to calculate the IVS
using the height of any of these peaks as the amplitude, this paper considers only the two larger-
amplitude peaks, and . The resulting spectra peak at the same matched velocity (40 m/s), but have
different selectivities as shown in Fig 2(b), where the two IVS plots are shown together with the values
of Qv calculated from the figure using eqn (4). These values of intrinsic velocity selectivity are used as
baseline references for the enhancements described in the next section.
From the above observations, it is clear that if this method is to be used to separate neural signals in
terms of velocity, certain problems of interpretation arise:
1. The measured IVS depends on the point in the time response of the delayed and summed signal used
to make the measurement;
2. The IVS profile depends on the properties of the TMAP function, in particular, shorter responses
result in larger values of Qv;
3. The actual selectivity obtainable is quite low and declines with increasing velocity. This accords
with the theory presented below (see also ).
These issues are considered in the next section.
2.3 Improved Velocity Selective Recording Using Bandpass Filters (BPFVS)
The theory presented in our earlier papers and summarized above described the operation of a delay-
matched VSR system in the velocity domain. However in order to understand better the generation of the
IVS using this method and especially the enhancements possible using bandpass filters (BPFVS), a time
domain interpretation is very useful. Consider the representation of the tripolar amplifier outputs in Fig 1
by N, equally-spaced rectangular pulses of arbitrary height h and duration δt. The delay between adjacent
tripolar outputs is τ and that this is related to the SFAP propagation velocity v and the inter-electrode
spacing d as follows:
Clearly if the N pulses are summed and δt < τ, the result will be a train of pulses of height h and spacing
τ – δt. If , on the other hand, artificial delays are introduced to cancel the naturally-occurring delays as
shown in the signal processing unit in Fig 1, the summed output will be a single pulse of duration δt and
height N.h. Between these extreme cases lie intermediate states where the externally applied delay is not
exactly equal to τ and hence the delay is not exactly matched. The rapidity of the transition through these
intermediate states determines the velocity selectivity and Qv. As the inserted delay is increased from
zero, the pulses begin to overlap and it is easy to see that the value of inserted delay required for this to
happen depends on both τ and on δt, the width of the pulses themselves. So, in order to produce a
summed output of particular amplitude, the value of the inserted delay is required to be closer to τ than
would be the case for wider pulses (adding more pulses-increasing N-has a similar effect). This is
observed as an increase in intrinsic velocity selectivity for narrower pulses. In summary, we can draw
the following initial conclusions regarding intrinsic delay matched VSR:
1. The intrinsic velocity selectivity, Qv is inversely proportional to the width of the tripolar SFAP
pulses and therefore strongly dependent on the spectral content of the underlying TMAP
2. Qv is proportional to N, the order of the system (number of tripoles).
The proposed approach to reducing the sensitivity to the TMAP is to place a BPF at the output of each
tripolar amplifier of the VSR system. The impulse response of a bandpass filter is a burst of damped
sinewaves whose frequency is the centre frequency of the BPF while the duration of the burst depends
on the filter order and relative bandwidth. For example, Fig 3(a) shows the response of an 8th order
Butterworth BPF of centre frequency 1 kHz and relative bandwidth 20% to a narrow pulse of unit
amplitude. Since the same type of response is elicited by stimulating a BPF with a tripolar action
potential, the ‘delay matching’ process is transformed into matching delayed sinewaves rather than
complex SFAP waveforms as in the intrinsic case. Unlike the SFAP waveform itself, the BPF output has
no dependence on the characteristics of the TMAP except for its amplitude and its exact position in the
time record. The addition of BPFs in this way therefore allows the measurement of velocity selectivity to
be decoupled from the spectral properties of the TMAP and to be controlled by means of the centre
frequency of the filters which is to some extent a free parameter. Note that although the proposed
arrangement requires a BPF at the output of each tripolar channel, in practice, due to the linearity of the
processes involved, the summation and filtering operations can be reversed. This leads to the much
simpler and more practical arrangement of Fig 1, where only a single BPF is required.
In order to determine the velocity spectrum obtained by delay matching with BPFs, it is sufficient to
calculate the peak value of the output as a function of inserted delay. As already noted, the transient
response of a BPF is a burst of damped sinewaves (see Fig 3(a)) and so the overall response of the
system, f(t), close to the point of exact match can be represented as the sum of N sinewaves, a simplified
example of which is shown in Fig 3(b). Each of the three sinewaves shown in the figure represents the
sinewave at the centre of the burst at the output of each BPF. For simplicity, the origin of the time axis
has been placed at the centre of the set of sinewaves (so that when the signals are matched they peak at t
= 0) and the amplitudes of the sinewaves have been normalised to unity. The period of the sinewaves is
T, which is the inverse of f0, where f0 is the centre frequency of the BPF and the offset due to the passage
of the AP along the MEC is τ. The result of adding the N sinewaves where N is odd (similar principles
apply for N even) is:
cos ...cos ...
where τ = d/v0 and ω0 = 2πf0. This function reaches a maximum at t = 0 for any values of N and τ and so,
to consider the selectivity, eqn (7) is rewritten:
So, for example, for N = 3:
for N = 5:
and so on.
After some manipulation, it is possible to show for the general case that:
where the series terminates when the exponent is zero. The sum of this standard finite series is given in
) 2/ sin(
) 2/( sin
Eqn (9) is just the function G from eqn (2) except that, as already noted, the delay (τ) shown in eqn (2)
has been normalised so that it actually represents offsets from the matched value. This is the formula
derived in  by spectral (velocity domain) analysis, showing that the two views of the method are
2.4 The influence of analogue bandwidth and analogue-to-digital conversion
Figure 4 shows the frequency spectrum of a 9-tripole IVS system with an inter-electrode spacing (d) of 3
mm, SFAP derived from both the TMAPs considered in this paper and matched at a propagation velocity
of 30 m/s. The sample rate is 100 ks/s and the resolution of the analogue to digital converter (ADC) is 10
bits, both requirements being realisable in a practical system. In the matched case (i.e. v = v0), as already
noted, the transfer function of the system is just that for a single tripole multiplied by N (see eqns (2) and
/ sin/ sin4,
v fdvv fd
The transmission zeros of this function occur at the following frequencies:
... 3, 2, 1, 0, ;
where the sequence extends in theory to the Nyquist limit (fs/2, where fs is the sampling frequency) but in
practice is likely to be limited by the finite bandwidth of the analogue channels, the constraints of ADC
and noise. Examination of these spectra indicates that the amplitude of TMAP #2 declines much more
slowly with frequency than that of TMAP #1. As a result, due to the wider bandwidth of the TMAP #2
spectrum, the dynamic range is reduced compared to TMAP #1, easing the demands on ADC. Operation
at higher frequencies, approaching the Nyquist limit is therefore possible for TMAP #2 if the analogue
channel bandwidth permits it. This in turn permits much higher levels of selectivity, since this is
proportional to frequency (see eqn (5)). In summary, for the two TMAPs considered, the limiting factor
for TMAP #1 is ADC resolution, whereas in the case of TMAP #2 it is determined by analogue
bandwidth and the Nyquist limit. It should be noted, of course that any selectivity enhancements are
subject to the degrading effects of noise. This aspect is dealt with in Section 3 below.
2.5 The significance of bandpass filter order
Fig 5 demonstrates the effect and importance of filter order on the behaviour of a VSR system. As
described above, the bandpass filter added at the output of an IVSR system is required to produce a burst
of damped sinusoidal oscillation at the centre frequency of the BPF when stimulated with a wide band
signal such as an SFAP. For this to happen, the filter must reject out-of-band signals that would
otherwise corrupt its response. In particular, the ‘baseband’ (borrowing some terminology from
communication systems) SFAP signal generated from TMAP#1 peaks at about 1.5 kHz and has a much
larger amplitude than the ‘harmonics’ further up the frequency scale. The figure shows that this is still
true for a p = 2 Butterworth bandpass filter since the filtered spectral amplitude at frequencies in the
range 0-5 kHz is actually larger than in the passband. This suggests the need for a BPF with p = 4, at
least. The small insets in the corners of the figure show the effect of filter order in the time domain. For
p = 2, for example, the filter output is dominated by the SFAP input signal, which is consistent with the
frequency domain situation. By contrast, for p = 8 the input signal has been suppressed and the output is
a symmetrical burst of damped sinusoids. Note however that p cannot be increased without limit
because, as the filter order is increased, the time domain response becomes longer. As discussed in
Section 2.6, this tends to increase the prevalence of false ‘image’ outputs. In addition, the requirement
for higher filter order tends to increase the ratios of the digital filter coefficients, making realisation more
2.6 Generation of spurious responses
One of the effects of using BPFs to enhance the performance of IVSR is the potential for the generation
of spurious velocity spectral responses (‘images’). These occur because single tripolar SFAP waveforms
are replaced by bursts of damped sinewaves of frequency fo, the centre frequency of a BPF. Therefore,
although a global maximum occurs when all the signals are matched, with or without BPFs, the potential
exists for subsidiary, ‘image’ responses to be generated when the signals are mismatched by integer
multiples of the period of the sinewaves, 1/fo. Images will be generated if and only if:
W ≥ N
where W is the number of cycles of sinewave in each burst and N is order of the system (i.e., the number
of tripolar channels being delay-matched) and that the velocity at which the images are formed is given
where d is the inter-electrode spacing, τ is the naturally-occurring delay and q is an integer. Note that the
velocities at which image responses occur depend on fo, the centre frequency of the BPF, in contrast to
an output resulting from an excited population which will appear at the same velocity irrespective of fo.
In addition, for practical values of the other parameters in eqn (10), the images tend to occur at velocities
that lie outside the band of interest (say 20 – 120 m/s). For these reasons, images are easily distinguished
from genuine outputs, as illustrated in Fig 6.
3.1 Simulated results without noise
In order to demonstrate the effects of adding BPFs to a delay-matched IVS system, the MATLAB
simulations shown in Fig 2 were repeated with a single bandpass filter of centre frequency f0 placed at
the output of the system, as shown in Fig 1. The SFAPs were generated using TMAP#1 and TMAP#2
and the system was noiseless. The filter was an 8th-order digital Butterworth BPF and centre frequencies
of 1 kHz, 2 kHz, 4 kHz, 8 kHz or 16 kHz and relative bandwidth 20% were used. The velocity spectra
are plotted in Fig 6 (the amplitudes of the plots corresponding to the two TMAPs have been normalized)
and show good responses at the matched velocities. It can be shown that it is possible to obtain
satisfactory responses for BPFs with centre frequencies up to the Nyquist frequency (50 kHz in this
case). The responses to SFAPs generated from both the TMAPs are identical in the sense that the values
of Qv measured at the matched velocities are the same in both cases. These results support the assertion
that the bandpass filtered velocity selectivity depends only on N, f0 and v and some physical constants,
not on the characteristics of the TMAP function, as is the case for IVS. The values of Qv are listed in
Table 1 together with values calculated from eqn (4). The calculated values fit the simulated ones very
3.2. Simulated results with additive noise
Zero-mean white Gaussian noise was added to the system in a manner consistent with the approach
adopted in  (i.e., 11 sources of uncorrelated voltage noise were introduced, one at the input to each
monopolar channel). These noise sources represent the total noise present in each channel referred to the
input. As noted in  the total input-referred noise is the sum of several individual sources that are
assumed to be independent and uncorrelated. The signal-to-noise ratio (SNR) was defined as the ratio of
the r.m.s. value of signal (i.e. TMAP) to noise over a sequence of standard length (10.23 ms,
corresponding to 1024 samples), referred to the input.
In order to test the effect of the noise on the system and in particular on the ability of the BPFs to
increase the velocity selectivity compared to IVS, the simulations described in Section A. were repeated
with varying levels of additive white Gaussian noise. The results are presented in Table 2 for three
values of SNR (1, 10, 100) for each of the two TMAPs. The frequency in column A for each value of
SNR gives the maximum frequency (fomax) at which an intelligible output is obtainable from a BPF
centred at that frequency. Once fomax has been determined, the maximum available velocity selectivity
(Qv) can be calculated from eqn (10) (column B) and the enhancement factor found (i.e. compared to
IVS-column C). In general TMAP#2 performs better than TMAP#1, due to the wider bandwidth of the
signal. There is thus more energy at higher frequencies in SFAPs generated from TMAP#2, whilst the
additive noise has the same spectral density at all frequencies and for both TMAPs. In the worst case
considered, with SNR set to unity, there is no enhancement in Qv for TMAP#1, whilst for TMAP#2 a
modest enhancement of about 3.5 is possible. For SNR = 10 the values increase to 2 and 7 respectively
and 4 and 7 respectively for SNR = 100, although in this case, higher values could be obtained for
TMAP#2 if more bandwidth were available.
4.1 Limitations of the theory
The use of bandpass filters to improve the selectivity of VSR systems was first proposed in [9, 1] and the
current paper extends and develops these methods. After briefly summarising the basic theory of the
method, the effects of key filter parameters, in particular centre frequency and filter order are examined.
The proposed system uses digital bandpass filters and for the purposes of this paper these were
implemented in software, using MATLAB. Since it is likely that a real VSR system would use similar
BPFs, whether implemented in software or hardware, the results of the simulations presented in this
paper can be considered to be quite reliable and representative.
4.2 Limits of available selectivity with and without noise
In Section 2.4 it was suggested that the factors limiting available velocity selectivity are (i) the spectral
bandwidth of the TMAP function, (ii) the analogue bandwidth of the signal capture/acquisition system,
(iii) the resolution of the digitisation process (ADC) and, (iv), noise. Furthermore, if digitisation with 10-
bits resolution is assumed, in the absence of noise, the level of available velocity selectivity is limited by
(i) and (ii). It was then demonstrated that in the case of TMAP #1, ADC resolution set the limit whilst
for TMAP #2, with its wider spectrum, analogue bandwidth (which in turn is limited by the sampling
rate/Nyquist frequency) was the limiting factor. However, these considerations are purely hypothetical
and the introduction of noise tends to alter the situation completely. Uncorrelated white Gaussian noise
was added to each monopolar input as suggested in . Since the input-referred noise spectrum is flat in
contrast to those of the TMAPs, the presence of noise has most influence at higher frequencies where the
signal spectral density is lowest. The resulting bandwidth restriction in turn limits the available velocity
selectivity, both IVS and FVS. So, although the effect of bandpass filtering removes the influence of the
TMAP spectrum on the value of Qv at a particular frequency, the maximum available value of Qv will
depend on the TMAP function, because of the effects of noise. In spite of the restrictions imposed by
noise, very useful enhancements in selectivity are nonetheless predicted for both TMAPs, especially
TMAP #2, as detailed in Table 2.
4.3 The effect of image responses
The formation of spurious responses (‘images’) was discussed in Section 2.6. For the example shown in
Fig 6, d = 3mm, τ = 100 μs (i.e. for vo = 30 m/s) and so for the 16 kHz BPF, velocity spectral images
would be expected at 13.3 m/s, 18.5 m/s and 80 m/s. The lower velocity images are clearly visible in
both figures, the higher velocity signal being outside the range shown. As already noted, the amplitude
and range of occurrence of these images depend on N and W, while W depends on the BPF parameters,
especially the filter order and relative bandwidth and can often be quite small in comparison with
genuine outputs from excited populations. Note that there is no possibility of confusing a genuine output
with an image since the former will occur at the same velocity in the spectra calculated at all filter
outputs whilst images will appear at velocities dependent on fo.
In conclusion, this paper presents a practical method to improve the performance of VSR systems
compared to currently available methods. The use of BPFs provides a significant increase in velocity
selectivity which will form the basis of new systems to widen the application of the method beyond the
current state-of-the-art which is restricted to the recording and analysis of CAPs.
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List of Figure Captions
Figure 1. This shows a multielectrode cuff (MEC) connected to a tripolar (double differential) amplifier
array. The N tripolar outputs (where N is typically about 10) are digitised and processed in the signal
processing unit on the right of the figure. For an action potential (AP) propagating symbolically
downwards, voltages are induced sequentially at the electrodes which result in the appearance of tripolar
output signals. The delay between the appearances of successive outputs depends on the propagation
velocity of the AP and the inter-electrode spacing of the MEC and allows a simple calculation of the AP
velocity. The calculation process is carried out in the digital domain using the programmable delays
indicated. Finally, the bandpass filter (BPF) is used to improve the velocity selectivity as described in the
text. We refer to the velocity spectrum calculated at the output of the adder as the intrinsic velocity
spectrum (IVS) and that at the output of the BPF, the filtered velocity spectrum (FVS).
Figure 2. Time and intrinsic velocity domain responses of the system shown in Fig 1 (output of the
adder), with a propagation velocity of 30 m/s. Subplot (a) shows the time domain response of the system,
each curve corresponding to a different matched velocity, the SFAP being generated from TMAP#1. The
three peaks of the waveforms are labelled , & . Subplot (b) shows the intrinsic velocity spectrum
(IVS) for TMAP#1 derived from Fig 2(a) measured at points α (top curve) and β (bottom curve). Subplot
(c) shows the same calculation using TMAP #2. The parameters of the TMAP function (x(t) = Atne-Bt)
Figure 3. Plot (a) shows the time record of the output of an 8th order Butterworth bandpass filter of 20%
relative bandwidth stimulated with a narrow pulse of unit amplitude. Plot (b) shows three sinusoids, each
representing the central half-cycle of the output of a channel BPF. The amplitudes have been normalised
to unity and the delay offset τ is the naturally-occurring inter-channel delay. N sinusoids are summed to
calculate the velocity selectivity of the system.
Figure 4. Frequency domain spectrum (FFT) of a 9-tripole system driven by SFAPs generated by
TMAPs #1 and #2. The propagation velocity is 30 m/s and the system is matched at that velocity. Note
that the spectral amplitude generated by TMAP #1 declines with frequency much more rapidly than that
generated by TMAP #2. The sampling rate is 100 ks/s.
Figure 5. Time domain responses (insets) and corresponding bandpass filtered spectra at the output of a
VSR system (derived from TMAP #1 and matched at 30 m/s-see Fig 4) compared to the unfiltered (IVS)
spectrum. The unfiltered output is passed through Butterworth BPFs with centre frequency (fo) 8 kHz
and order (p) 2, 4 and 8 and with 20% relative bandwidth. For the p = 2 filter the spectral amplitude at
frequencies in the ‘baseband’ range 0-5 kHz is larger than at fo, suggesting significant breakthrough in
the passband and the need for a BPF with p = 4, at least.
Figure 6. Bandpass filtered version of the time domain IVS plot shown in Fig 2 and comparing the
differences between SFAPs generated using TMAP functions #1 (dashed line) and #2 (dotted line) with
a propagation velocity of 30 m/s. The filters are 8th order Butterworth digital units with centre
frequencies f0 are (a) 1 kHz, (b) 2 kHz, (c) 4 kHz, (d) 8 kHz, (e) 16 kHz and (f) 32 kHz. There is no
significant difference between the two responses and the corresponding values of Qv are 1.4, 2.9, 5.7.
The velocity step is 1 m/s and there is no additive noise. Note the formation of spurious ‘image’