Timing-Error Minimization of Optoelectronic Pulse Signals

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Timing
-
Error Minimization of Optoelectronic Pulse
Signals


Božidar Vojnović

Laboratory for Stochastic Signals and Processes Research, Division of Electronics,

Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia

e
-
mail:vojnovic@irb.hr


Abstract



In optical distance measurements, using time
-
of
-
flight techniques, associated
time interval is defined by start and stop pulse signals. Measurement error depends
strongly on timing
-
error of stop optoelectronic pulse signal. The pulse processing and
puls
e timing circuits, based both on charge
-
controlled properties of step
-
recovery diode
properties were optimised in a way to achieve minimum timing
-
error. The system is
very suitable to achieve picosecond resolution in timing pulses having longer rise
-
times.


1.

INTRODUCTION

In optical (laser) distance measurements, using time
-
of
-
flight techniques,
associated time interval is defined by start and stop pulse signals. In the analysis
it was supposed that the start signal is fully deterministic, very sharp and
p
recisely defined in time. The stop signal in general case is stochastic, described
by parameters; amplitude, rise
-
time and associated additive noise.

The problem of high
-
resolution timing of stochastic pulses has been treated
extensively in many scientific

and technical areas: nuclear spectroscopy, radar
ranging, optical (laser) distance/displacement measurements, laser altimetry,
optical communications etc. [1,2,3,4,5]. These analysis were focused mostly on
two main separate topics: (1) influence of the st
ochastic pulse parameters on
pulse timing
-
error, assuming the ideal timing discriminator and (2) development
of new timing
-
discriminator circuits and their optimisation depending on the
application and desired time resolution.

In this paper an attempt was

made to optimise previously described method [6],
which functionally integrate detecting and processing circuits with timing
discrimination circuit. The analysis was made under the following presumptions:
(1) Optical pulse signal is stochastic, but its ar
rival
-
time is considered jitter free;
(2) Gaussian, white noise characterizes optical and electronic noise sources; (3)
Signal processing circuits belong to the class of linear time
-
invariant filters; (4)
System output pulse has to be bipolar, and time dis
crimination occurs at signal
zero
-
crossing point; (5) Timing discriminator does not belong to the class of
regenerative circuits, thus avoiding the effect of triggering charge (energy)
sensitivity.


2. PULSE PROCESSING
-
DISCRIMINATION SYSTEM ANALYSIS

2.1. C
riteria for timing
-
error minimization

Pulse signal timing is a part of wider area of stochastic pulse arrival
-
time (time
-
delay) measurements, which belong to the class of stochastic signal parameters
estimation problems. Different criteria were considered
here to study estimation
error minimization.

(1) Maximum likelihood (ML) criterion was widely applied in stochastic signal
amplitude and time
-
delay estimation [4, 8]. The idea was to use ML criterion to
find out the impulse
-
response function h(τ) of the op
timum filter (optimum
estimator), which minimizes pulse timing
-
error. The problem was analysed for
the case of signal accompanied with white, Gaussian noise, as well as for the
case where the signal is non
-
stationary Poisson process. In both examples it wa
s
assumed the pulse width is mach shorter than the observation time. The
minimum error was obtained at the time point, which maximizes signal output
from the optimum filter. This method thus is not very applicable for
nanoseconds pulse widths and required
sub
-
nanosecond time resolution.

(2) Minimum error
-
rate approach attempts to find out the filter that
)
minimizes
the mean
-
number of zero
-
crossings of the noise component of the stochastic
signal. This method could be interesting in the case when the “spike”

noise is
predominant. Theoretically white (current) noise contains the series of random
positive and negative pulses q
i
δ(t
1
), where δ(t) is Dirac delta
-
function, q
i

and t
i

are stochastic variables (charge content and arrival
-
time of each pulse). The
mean number of the noise zero
-
crossings is [11] is defined as:





























0
2
0
2
2
0
0
1






d
h
d
h
h
C
R
R
N
h
F
yy
yy





(1)

where: c is a constant, h(τ
) and h


(τ) are the impulse response function of the
noise filter and its second derivative, R
yy
(τ) and R
yy


(τ) are the auto
-
correlation
function of the noise at the filter output and its second derivative respectively.
Minimum of the functional F[h(τ)]
gives us h(τ) that minimizes
N
, which is
well known “cusp” shaped function. Because h(τ) is unipolar function, as it is
also pulse signal, zero
-
crossing timing is not possible and timing is amplitude
sensitive. The optimal timing poin
t is not explicitly defined and depends on the
point where the noise
-
to
-
signal slope ratio has a minimum.

(3) Minimum square
-
error criterion [10] leads to maximization of signal
-
to
-
noise ratio, which in the timing case means minimization of signal variatio
n
(noise)
-
to
-
signal slope ratio. This criterion is more appropriate approach if the
signal variations (noise) are much slower than the signal rise
-
time. The analysis
results in optimum (matched) filter that minimizes pulse timing
-
error against
signal varia
tions for assumed ideal timing discriminator circuit. To find
optimum impulse response function h
0
(τ) the following expression should be
minimized





2
2
2
m
T
t
s
t
t
S
dt
d
t





















(2)

where T
m

is a timing point and σ
s
2

is the input signal variance which

includes
effects of statistical processes of optical signal generation, propagation and
detection. The denominator in (2) represents squared average signal slope. From
expression (2) a functional G=G[h(τ)] could be made and minimized, which
gives us h
0

) of the optimum timing filter. The general solution, which is of our
interest, has the form







m
T
t
t
S
d
t
S
d
h

















1
0








(3)

where α is a factor indicating relative noise at the system input.

2.2. Theory of system operation

The system will be analysed u
nder the condition

»S(T
m
-
τ), which is valid for
larger optical distance measurements. In this case response of the filter has the
form h
0
(τ) = S

(T
m
-
τ). The same result [5] was obtained replacing signal variance
by noise variance in equation (2). The timi
ng measurement chain contains the
following essential parts: (1) optical signal detector (sensor); (2) electronic pulse
preamplifier; (3) signal pre
-
processing circuit and (4) filter
-
discriminator circuit.
Optical detector and preamplifier have to convert
optical signal into electronic
one with minimal response time and timing
-
jitter as well as with maximum
signal
-
to
-
noise ratio. As the preamplifier noise is usually “coloured”, the pre
-
processing filter is aimed to “whiten” this noise, making simpler synthe
sis of
filter
-
discriminator circuit to achieve minimum timing
-
error of the output pulse.
Impulse response function of the whole optimum filter is thus convolution of
impulse response functions of all parts in the chain. Detector signal, containing
charge Q
, w
as considered as current impulse, i(t) = Qδ(t). The input noise has
two components: voltage and current white noises, having power spectral
-
densities N
e
/2 and N
i
/2 respectively. We consider detector
-
preamplifier part as a
low
-
pass, single
-
pole circuit, thu
s the whitening filter is pole
-
zero cancellation
circuit. Output of the filter could be approximated by the signal v(t)=Q/C
T
e
-
t/

c

and white voltage noise with N
e
/2, τ
c
2
=C
T
2
N
e
/N
i
, where τ
c

is the “noise
-
corner”
time constant, C
T

is the total input capacita
nce of the detector
-
preamplifier
circuit. The impulse response function of the system (optimum filter) is first
derivative of “cusp” shaped function, having time constant τ and shifted in time
by T
m

[5,9]. The impulse response function h
s
(τ) of the propose
d system (Fig.1)
with step
-
recovery diode represents, under certain, realizable conditions, good
linear approximation of the h
0
(τ) of the theoretical (unrealisable) optimum
timing filter. Step
-
recovery diode circuit is based on switching properties of
step
-
recovery diode, a P
-
I
-
N junction structure, that acts as a charge
-
controlled
switch. Diode forward current, static or transient, injects the charge into the
diode. When this charge is being removed by the reverse pulse current, the diode
continues conduct
ing (low impedance state) until all the charge is removed. At
the point when the total amount of charge in the diode becomes zero, it stops
conducting (high impedance state) very abruptly. In our system the input current
signal injects the charge into unbi
ased diode. The same pulse inverted and
amplified by a factor k, starts after well
-
defined delay
-
time t
d

to remove the
stored charge. At the moment T
m

when the total charge becomes zero, we get the
sharp edge pulse having rise
-
time less than 100 ps, which
gives us the time
reference relying very strongly on pulse arrival
-
time. In the analysis it is
assumed that the diode is “ideal” current integrator and charge zero
-
crossing
discriminator, because the following requirements are met: (1) Injected charge is
c
ompletely removed during reverse recovery phase; (2) The leakage charge due
to reverse diode

capacitance is negligible; (3) The minority carrier life
-
time
could be chosen long compared to diode conducting time; (4) The diode does
not exhibit charge (energy
) triggering sensitivity because it does not belong to
the class of regenerative circuits.


















Figure 1. Impulse response function of the
system





















Figure 2. Impulse response function of
DL
-
SRD circuit


2


1

k
-
1


w


1


1

h
D
(

)

1/

1



h
s
(

)

T
m


2


1




w

1

-
1


1


That is important e
specially for slower pulse discrimination, because there is no
time
-
walk due to different signal slope at the point of discrimination; (5) The
diode is true time zero
-
crossing discriminator, thus avoiding the problem of
initial discrimination level adjustm
ent and stability.

The best ultra
-
fast, low noise preamplifiers have rise
-
time in the order of
nanosecond, integrating time
-
constant could be some milliseconds and τ
c

is
usually in the range 10
-
100 nanoseconds. Under these conditions, the proposed
system could be realized si
mply by combination of pulse amplifiers, delay lines
(DL) and step
-
recovery diode (SRD) with minority carrier life
-
time constant τ
mc

greater than 100 ns. As the diode acts as an integrator, the DL
-
SRD circuit has
impulse response function as on Fig.2. If
the relation τ
c
»T
m

is valid, the input of
DL
-
SRD circuit sees the step voltage of the amplitude V
s
= Q/C
T

and white
voltage noise with N
e
/2.

The timing
-
error variance is, consequently





1
2
1
2
1
2
2
2
2






k
V
N
Q
C
N
k
w
s
e
w
T
e
t










(4)

2.3. Discussion of results

Calculation
of timing
-
error with some realistic example of signal and circuits
parameters (τ
w
=20ns,
e
N
=1nV/
Hz
, Q=10pC, C
T
=100pF, k=5), gives us σ
T
=
0.5ps. This result was obtained under the assumption of idealized circuits, as

indicated before and without calculation of SRD noise contribution to timing
error. Nevertheless, we can say that the proposed method is capable to achieve
picosecond timing
-
resolution against the noise influence. Regarding the
amplitude and rise
-
time var
iation as well as triggering charge sensitivity, we can
conclude the following;(1) the method is insensitive to pulse amplitude
variations;(2) For linear signal
-
rise at the DL
-
SRD timing circuit input, system
is fairly insensitive to rise
-
time variations;(
3) Timing discriminator does not
exhibit triggering charge sensitivity, because SRD is non
-
regenerative
discriminator circuit. The simplified version of DL
-
SRD circuit was tested in
two types of measurements;(1) static, measuring time
-
walk ΔT against pulse

amplitude and rise
-
time, and (2) dynamic, measuring statistical distribution of
output pulse time
-
delay. Obtained results gave us ΔT =04ns, with amplitude and
rise
-
time dynamics of 10:1, (20
-
200ns). Standard deviation of timing
-
error was
calculated from m
easured time spectra (pdf of delay
-
times), and we got 120ps,
220ps and 300ps in three measurements with input pulse rise
-
times of 100ns,
220ns and 500ns respectively. The pulse amplitude dynamics was 10:1.

3. CONCLUSION

The proposed DL
-
SRD timing circuit
(charge balanced discriminator) is very
suitable for applications in longer range optical distance measurements.
Theoretical analysis and experimental results have shown that the method
assures picosecond time
-
resolution even with wider pulse signals of se
veral
hundreds of nanoseconds, exhibiting significant insensitivity against signal
amplitude and rise
-
time variations.

REFERENCES

[1.]

E. Gatti, V. Svelto, “Optimum Filter for Timing
Scintillation Pulses”,
Nuclear Instruments and Methods,
Vol. 39, pp. 309
-
313, 1966.

[2.]

J. Llacer, “Optimum Filter for Determination of the Position of an
Arbitrary Waveform in the Presence of Noise”,
IEEE Transactions On
Nuclear Science,
Vol. NS
-
28, No.1, pp.

630
-
633, 1981.

[3.]

E. Gatti, S. Donati, “Optimum Signal Processing for Distance
Measurements with Laser”,
Applied Optics,
Vol. 10, No.11, pp. 2446
-
2451,
1971.

[4.]

G. Lee, G. Schroeder,

Optical Pulse Timing Resolution”,
IEEE
Transactions On Information Theory,
pp
. 114
-
118, 1976.

[5.]

B. Vojnović, “Resolution Improvement of Stochastic Pulse Arrival
-
Time
Determination”,
Ph.D. Dissertation
, Faculty of Electrical Engineering,
University of Zagreb, Zagreb 1973.

[6.]

B. Vojnović, “A Subnanosecond Timing Circuit Using Snap
-
off Diode”,
Compte rendu
s du Colloque sur l'Electronique nucleaire,

pp. 59
-
1 to 59
-
7,
Versailles, 10
-
13 Sept. 1968., Paris

[7.]

B. Vojnović,
Private communication

[8.]

R.N. Donough, A.D. Wahlen, “Detection of Signals in Noise”
,

2
nd

Edition,
Academic Press, New York 1995.

[9.]

M. Konrad,

Detect
or Pulse Shaping for High Resolution Nuclear
Spectroscopy”,
IEEE Transactions on Nuclear Science,

NS
-
15, No.1, pp
-

268, 1968.

[10.]

V. Radeka, N. Karlovac
,
“Least
-
Squared Error Amplitude Measurements

of
Pulse Signals in Presence of Noise
”, Nuclear Instruments an
d Methods
,
Vol.52, pp.86
-
92, 1967.

[11.]

J.S. Bendat, A.G. Piersol, “ Random Data, Analysis and Maesurements
Procedures”, 2
nd

Edition, John Wiley & Sons, New York, 1986.