1
Pilotbased TIADC Mismatch ErrorCalibration for IRUWB Receivers
C. A. Schmidt, J. L. Figueroa and J. E. CousseauInstituto de Investigaciones en Ingenier´ıa El´ectrica  CONICETUniversidad Nacional del Sur, Bah´ıa Blanca, ArgentinaEmails: [cschmidt,ﬁgueroa,jcousseau]@uns.edu.arA. M. TonelloInstitute of Networked and Embedded SystemsAlpenAdriaUniversit¨at Klagenfurt, AustriaEmail: andrea.tonello@aau.at
Abstract
—In this work, we ﬁrst provide an overview of thestate of the art in mismatch error estimation and correction fortimeinterleaved analog to digital converters (TIADCs). Then, wepresent a novel pilotbased online adaptive timing mismatch error estimation approach for TIADCs in the context of an impulseradio ultra wideband (IRUWB) receiver with correlationbaseddetection. We introduce the developed method and derive theexpressions for both additive white Gaussian noise (AWGN) andRayleigh multipath fading (RMPF) channels. We also derive alower bound on the required ADC resolution to attain a certainestimation precision. Simulations show the effectiveness of thetechnique when combined with an adequate compensator. Weanalyze the estimation error behavior as a function of signal tonoise ratio (SNR) and investigate the ADC performance beforeand after compensation. While all mismatches combined causethe effective number of bits (ENOB) to drop to 3 bits and to6 bits when considering only timing mismatch, estimation andcorrection of these errors with the proposed technique can restorea close to ideal behavior. We also show the performance loss at thereceiver in terms of bit error rate (BER) and how compensationis able to signiﬁcantly improve performance.
I. I
NTRODUCTION
IRUWB technologies have become an interesting solutionfor many applications such as localization, power line communications (PLC), high datarate and lowrange communications, sensor networks, among others [1], [2], [3].As a consequence of its wide spectrum, UWB communication systems can have bandwidths of up to several GHz.There exists therefore an ever increasing demand for highperformance, highspeed analogtodigital converters (ADCs)in order to comply with the sampling requirements of IRUWB(and other modern wideband communications systems andstandards like LTE, 5G, optic transceivers, and cognitive radio)[4], [5], [6], [7]. However, there is also traditionally a tradeoff between the achievable resolution and sampling speed [4],[8]. Among the alternatives, TIADCs are a promising solutionthat has become a trend and an active research topic, as theyare key to sample the signals at the required rates.A TIADC is an array composed of several (say
M
) ADCsworking in parallel with timeshifted sampling clocks such thatthe overall effective sampling rate is proportionally increased.However, due to inaccuracies inherent to the manufacturingprocess that prevent the component ADCs from being exactlyequal to each other, there are speciﬁc mismatches that canseverely deteriorate the performance of the whole system.Thus, addressing three typical mismatches: gain, offset andtiming skew, estimation and correction is required [4], [8],[9], [10]. Gain mismatch errors occur when the amplitude ratiobetween analog input and digital output differs for each ADC,whereas offset mismatch is due to different DC values at theiroutput (even when the input is set to zero). Finally, timingmismatch errors cause the output signal to be periodically butnonuniformly sampled [11], [12]. Unlike offset and gain mismatches, which are static, distortion due to timing mismatch isdynamic (i.e. signal dependent) and requires additional signalprocessing with higher computational complexity. While offsetand gain mismatches are quite straightforward to estimateand cancel [13], [11], background online timing mismatchestimation remains a challenge and motivates active research[6], [14]–[19].
A. Stateofthe Art on Mismatch Compensation
In order to reduce the mismatch errors distortion effects,two tasks must be performed. First, an accurate estimation of the mismatch errors must be obtained. Then, this informationcan be used to apply an adequate compensation technique.The mismatch error estimation phase is the most critical part,i.e. the estimation error must be as low as possible. Severalalternatives have been recently proposed to tackle the problem[14], [15]. For instance, signal processing can be performedin the digital domain, or in both analog and digital domains(mixed mode solutions) [6], [16]. In [6], the mean squarederror (MSE) of detection is measured in the digital domain,while delays on the clock paths to each ADC are adjustedin the analog domain through adjustable delay paths until theMSE is minimized. The main drawback of solutions involvinganalog processing is that they require additional hardware,whose precision set a bound to their estimation and correctioncapabilities. On the other hand, fully digital techniques onlyrequire additional digital processing and can thus be adaptedto any ADC [16].Another important distinction can be made based on weatherthe ADC operation must be interrupted or not during the
2
parameter estimation process. Foreground (offline) calibrationrelies on an additional training signal known a priori that isused to estimate the compensator parameters [16], [11]. Thisstrategy results in more accurate estimates of the parametersinvolved, but operation must be (periodically) stopped. On thecontrary, background (online) calibration techniques are capable of directly using the input signal during normal operationof the ADC for estimation and correction of distortion [17].They can also be divided in blind or pilotbased methods.Blind methods do not require any information on the inputsignal. Instead, they use either an additional reference ADCor elaborated algorithms exploiting some system knowledge todesign suitable cost functions and minimize them adaptively(normally using gradientbased methods) [17], [18]. While theadvantage is the resulting great ﬂexibility that can be achieved,their computational complexity can be very high, and theyresult in lower accuracy. For instance, in [18], the distortionpart of the ADC output signal is estimated by doing severalHilbert transforms, frequency shifting and folding operationsand an LMS algorithm for each channel ADC to minimizea cost function. In [19], estimation of timing mismatch isobtained through derivative ﬁlters, where an LMS algorithmusing Taylor approximations is used for the cost function. Inthis case, the method is tested with a sampling frequency of 3.6GHz and timing mismatch in the order of a few picoseconds,which implies that the method is accurate when the timingmismatch is low, in accordance with the results in [11], whereit is shown that linear interpolation and spline interpolation aresufﬁcient in this case. However, for larger timing mismatch,higher order interpolation techniques are needed. Finally,background online (adaptive or not) estimation and correctioncould also be obtained by means of pilotbased methods.A pilotbased online calibration method could gather theadvantages of both blind and foreground techniques. Pilotsignals are transmitted signals known at the receiver that areused in systems and standards such as OFDM, IRUWB, etc.,for different tasks such as channel estimation, synchronizationor carrier frequency offset estimation [20], [21], [22]. The ideabehind this approach is to use these pilot tones or symbolsalong with any particular knowledge on the system itself to get online accurate estimates of the model parameters.While pilotbased estimation and its characteristics have beenwidely studied in certain applications, and particularly in thecontext of channel estimation and synchronization [20], [21],[22], to the best of our knowledge this approach has not yetbeen implemented for estimation and correction of mismatcherrors in TIADCs. In this work, we propose a pilotbasedestimation technique for TIADC mismatch error calibration,and analyze its performance. We show that it enables lowercomputational complexity and higher accuracy when compared to blind methods, as well as tracking capabilities tochanges in the parameters while maintaining normal operationof the sampling stage as opposed to foreground calibration.
B. Paper Contribution
In this paper, we propose a novel pilotbased online mismatch error estimation algorithm in TIADCs for an IRUWBreceiver with reduced complexity. In particular, we propose touse an average of the received pulses to estimate the TIADCmismatch parameters. Then, we use the mismatch estimatesto implement the compensation method in [11], which hasalready been tested both by simulations and experimental veriﬁcation. Finally, correlation based data detection is performedbetween the compensated signal and a clean template. Weanalyze the results in terms of system performance metrics,such as bit error rate (BER), as well as ADC performance metrics, as signal to noise and distortion ratio (SINAD), showingthat after compensation a signiﬁcant improvement can be obtained. We previously showed in [11] both by simulations andmeasurements that the correction method presents excellentresults in compensation performance provided that estimationaccuracy of the mismatch parameters is good enough. In thiswork, we show that highquality estimates can also be obtainedby the newly proposed method.The manuscript provides the following contributions:
•
We propose a novel low complexity online estimationmethod for mismatch errors in a timeinterleaved ADCsuitable for IRUWB receivers. Unlike foreground offline estimation methods with sinewave training signals(and thus, computationally intensive sinewave ﬁttingalgorithms), the novel estimation method is based onthe received signal itself (avoiding the use of externalcalibration signals).
•
The proposed method is able to cope with changes in themismatch parameters through tracking. Hence, periodicalestimation is not needed and the ADC operation is notinterrupted. In addition, we derive the expression on thelower bound of the required ADC resolution in order toattain a certain estimation precision.
•
We analyze different aspects of sampling for an IRUWB receiver and their effect on detection performance,including ADC resolution, and ADC induced distortionconsidering typical wireless communication channels. Weshow that the proposed estimation method gives accurateresults that are capable of restoring adequate performancewhen combined with an efﬁcient compensation block.This work is organized as follows. A brief model of the IRUWB receiver is introduced in Section II, where the transmitted signal, the channel model, and pulse waveform used aredescribed. In Section III, the TIADC structure is presentedalong with gain and offset mismatch estimation methods. Anovel background timing mismatch error estimation techniqueis presented in Section IV, where the method is introduced anda lower bound on the required ADC resolution is also derived.Compensation using the obtained estimates is brieﬂy describedin Section V. Section VI provides a comparative analysis withcurrent state of the art proposals. Simulation results of theADC and receiver performance in terms of SINAD and BERare shown in Section VII, where the effect of ADC resolutionand ADC induced distortion for the case of a TIADC areconsidered. A postcompensator for mismatch errors followingthe approach used in [11] is used to test the accuracy of theestimation technique. Finally, some conclusions are given inSection VIII.
3
Analog filtering
ADC +compensation
Channelestimation
Synchronization
DigitalcorrelationPulsetemplateDetection Antenna
Fig. 1. Block diagram of the receiver.
II. IRUWB
RECEIVER MODEL
Fig. 1 shows the block diagram of an IRUWB receiver andthe processing chain. The signal is received at the antenna andﬁltered in the analog domain. Then, it is digitized by the ADC,which includes a compensation stage at its output to reducedistortion. After channel estimation and time synchronization,a digital correlation with the pulse template is performed fordata detection.IRUWB transceivers transmit a train of ultrashort (andtherefore ultrawideband) pulses, which are then correlatedwith the pulse shape at the receiver side for detection. Thetransmitted signal is
x
(
t
) =
N
F
−
1
k
=0
b
k
p
(
t
−
kT
r
)
(1)where
b
k
∈ {−
1
,
1
}
are the information bits,
p
(
t
)
is thetransmission pulse,
T
r
is the pulse period (bit period), and
N
F
is the number of bits in the data frame. In this work, weuse gaussian pulses as in [2],
p
(
t
) =
K
0
√
2
T
0
e
−
π
2
t
−
t
0
T
0
2
(2)where
K
0
is proportional to the pulse energy and
T
0
deﬁnesthe bandwidth.The performance study that we present involves two different channel models, an additive white gaussian noise (AWGN)channel, and a Rayleigh multipath fading (RMPF) channel.We consider these two channels to serve as best and worstcase scenarios to get insight in the performance of the system.The receiver structure used for the case of an AWGN channelis a matched ﬁlter (MF). When transmitting over a RMPFchannel, which is representative of many radio communicationscenarios when reﬂections and scattering are present, we usea rake receiver with maximum ratio combining (MRC) [23],[24]. The received signal is
r
(
t
) =
x
(
t
)
∗
h
(
t
) +
η
(
t
)
(3)where
η
(
t
)
is AWGN and
h
(
t
)
is the impulse response of thechannel. In the case of an AWGN channel,
h
(
t
) =
δ
(
t
)
. Fora RMPF channel
h
(
t
) =
L
−
1
l
=0
c
l
(
t
)
δ
(
τ
−
l/W
)
(4)where
L
=
T
m
W
+ 1
is the number of taps of the tappeddelay equivalent channel,
W
is the signal bandwidth,
T
m
is thechannel multipath spread, and
c
l
(
t
)
are the baseband complextime varying channel coefﬁcients with Raylegh distributedamplitude and uniform distributed phase. We assume a pulserepetition time
T
r
> T
m
in order to avoid intersymbolinterference (ISI), with a guard time
T
g
=
T
r
−
T
m
. Then,since we have a resolution of
1
/W
in the multipath delayproﬁle, we have
L
resolvable paths. Hence, a MRC receivershould achieve the performance of an equivalent
L
th orderdiversity communication system [24]. The received signal ina time window covering the transmission pulse repetitionrate
T
r
, assuming synchronization, channel estimation, andconsidering the
k
th transmitted bit is
r
k
(
t
) =
b
kL
−
1
l
=0
c
l
(
t
)
p
(
t
−
l/W
) +
η
k
(
t
)
(5)Then, the decision variable using a rake receiver with MRC is
U
(
kT
r
) =
ℜ
T
r
0
r
k
(
t
)
v
∗
(
t
)
(6)where
ℜ
[
x
]
denotes the real part of
x
. Expanding
v
∗
(
t
) =
L
−
1
l
=0
c
∗
l
(
t
)
p
∗
(
t
−
l/W
)
we get
U
=
ℜ
L
−
1
l
=0
T
r
0
r
k
(
t
)
c
∗
l
(
t
)
p
∗
(
t
−
l/W
)
dt
(7)If we consider discretetime processing [25], the receivedsignal is sampled at the output of the RF frontend analogﬁlter at sampling rate
T
c
=
T
r
/N
S
, such that
N
S
=
ML
samples are available to digitize the pulse and the receivedreplicas, we get
r
k
(
nTc
) =
b
kL
−
1
l
=0
c
l
(
nT
c
)
p
(
nT
c
−
l/W
) +
η
k
(
nT
c
)
(8)with
n
= 0
,
···
,N
S
−
1
. Then, the decision variable becomes
U
=
ℜ
L
−
1
l
=0
N
S
−
1
n
=0
T
c
r
k
(
nT
c
)
c
∗
l
(
nT
c
)
p
∗
(
nT
c
−
l/W
)
(9)and the estimated bit is
ˆ
b
k
= sign[
U
]
(10)where the
sign
function is deﬁned as
sign(
x
) =
−
1
, x <
00
, x
= 01
, x >
0
(11)If the sampling frequency is
f
c
= 1
/T
c
= 2
W
, it is theNyquist samplig rate
f
N
. Otherwise, if
f
c
> f
N
there isoversampling, and more terms are added to the decision metric. As a consequence, the correlation with the pulse template
p
(
nT
c
)
is more robust and the equivalent quantization noise isreduced, leading to an improvement in BER performance. Asimilar analysis is possible for the AWGN channel replacing(4) by
h
(
t
) =
δ
(
t
)
. Figure 2 shows two received bits affectedby a multipath channel, where
T
r
,
T
m
and
T
c
are depicted.
4
0 20 40 60 80 100 1200.60.40.200.20.40.60 2 4 6 800.51
TTT
mcr
Fig. 2. Received signal in a multipath channel for
L
= 5
.
III. M
ISMATCH ERRORS IN A
TI ADCUnfortunately, due to fabrication process inaccuracies, anyTIADC architecture presents gain, offset, and timing mismatch errors between the different ADC channels, whichlead to a non ideal behavior [8]. Offset mismatch is causedby different DC levels at each ADC ouput, whereas gainmismatch results when the gain from the analog input to thedigital output is different for each ADC in the interleavedarray. Timing mismatch is due to a static phase shift
∆
t
m
in the clock of ADC
m
, which results in deviation of thesampling instant (and hence an amplitude error in the sampletaken). This error is more critical since it depends on thedynamics of the input signal, and its compensation requiressome extra digital signal processing.Let
r
(
t
)
be the received signal and
T
c
the overall highspeed sampling period,
T
=
MT
c
the sampling period at eachchannel ADC, and
τ
I m
=
mT
c
the ideal sampling shift for the
m
th ADC. Then, the ideally sampled signal without mismatcherrors is
r
I m
(
kT
) =
r
I
(
mT
c
+
kMT
c
) =
r
I m
(
kMT
c
) =
r
I
(
nT
c
)
(12)with
m
= 0
,
1
,
···
,M
−
1
and
k,n
∈
Z
, where
r
I m
(
kMT
c
)
is the polyphase decomposition of the signal [26], and
nT
c
=
mT
c
+
kT
. If we add the gain, offset and timing mismatcherrors
G
m
,
O
m
and
∆
t
m
, we get
r
m
(
kT
) =
G
m
(
kT
)
r
(
mT
c
+
kMT
c
+∆
t
m
)+
O
m
(
kT
)
(13)If we deﬁne
G
m
(
kT
) =
G
(
mT
c
+
kT
) =
G
(
nT
c
)
O
m
(
kT
) =
O
(
mT
c
+
kT
) =
O
(
nT
c
)
,
(14)then
r
m
(
kT
) =
G
m
(
kT
)[
r
I m
(
kT
) + ∆
r
m
(
kT
)] +
O
m
(
kT
)
(15)Considering that
nT
c
=
mT
c
+
kT
, we can write the outputof the TI ADC as,
r
(
nT
c
) =
G
(
nT
c
)[
r
I
(
nT
c
) + ∆
r
(
nT
c
)] +
O
m
(
nT
c
)
(16)From here on, we use the notation
r
m
[
k
]
for
r
m
(
kT
)
and
r
[
n
]
for
r
(
nT
c
)
.Estimation of mismatch errors is required as a ﬁrst stepin order to compensate them. In addition, the quality of theestimates should be as high as possible while keeping lowcomplexity for the compensation to be effective and feasible.According to [13], offset and gain errors can be adaptivelyestimated online by taking the mean and variance of theindividual ADC outputs, respectively.Then, considering a balanced source of information (i.e.,the amount of transmitted ones and zeros is roughly equal),the input signal to the ADC has zero mean and the
offset
mismatch can be calculated by directly averaging
K
samplesat the output of each channel ADC,
ˆ
O
m
= 1
K
K
k
=1
r
m
[
k
]
(17)As for the
gain
mismatch, it can be computed as the varianceof each ADC output,
ˆ
G
m
= 1
K
K
k
=1
(
r
m
[
k
]
−
µ
)
2
(18)where
µ
= 0
if we consider the offset has been previouslycanceled. In that case, (18) reduces to
ˆ
G
m
= 1
K
K
k
=1
(
r
m
[
k
])
2
(19)otherwise,
µ
in eq. (18) can be computed using eq. (17).Note that both (17) and (18) can be adaptively updated (online) by adding the next sample
r
m
[
k
+1]
to the calculations.This enables not only online background estimation but alsotracking capabilities to changes in the parameters that couldarise due to temperature variations, aging, etc.IV. N
OVEL ON

LINE TIMING MISMATCH ERRORESTIMATION
In this section, we propose an online background estimationmethod for timing mismatch errors for the IRUWB receiverunder consideration.
A. Problem statement and formulation for AWGN and RMPF channels
Let us begin with the case of a general communicationschannel under the assumptions in Section II and with a certainimpulse response
h
(
t
)
. Then, according to (1) and (3), the
k
threceived bit can be expressed as
r
k
(
t
) =
b
k
p
(
t
−
kT
r
)
∗
h
(
t
) +
η
k
(
t
)
(20)If we sample (20) with an ideal TIADC and a sampling period
T
c
, we get
r
k
(
nT
c
) =
b
k
p
(
nT
c
−
kT
r
)
∗
h
(
nT
c
) +
η
k
(
nT
c
)
(21)with
n
= 0
,
···
,N
S
−
1
. We assume
N
S
=
ML
where
L
is thenumber of taps of the taped delay equivalent channel, such that
M
samples are taken from each pulse replica, i.e. L samples
5
for each channel in the TIADC. If we assume that gain andoffset mismatch have already been estimated and compensatedfor, we can express the sampled signal with timing mismatchas
r
k
(
nT
c
) =
b
k
p
(
nT
c
−
kT
r
+ ∆
t
n
)
∗
h
(
nT
c
) +
η
k
(
nT
c
)
(22)We now consider the case where a pilot training signalcomposed of
K
P
bits is available, which may be the same usedfor channel estimation and time synchronization. Assuming
L
resolvable paths, the estimated channel coefﬁcients are usedto recover the phase and amplitude of each pulse replica suchthat
h
(
nT
c
)
can be replaced by an impulse train
˜
h
(
nT
c
) =
L
−
1
l
=0
δ
(
t
−
lT
m
/
(
L
−
1)) =
L
−
1
l
=0
δ
(
t
−
l/W
)
(23)We can thus add averaging and a multiplication by the known
b
k
s such that
ˆ
r
(
nT
c
) = 1
K
P K
P
k
=1
b
2
k
p
(
nT
c
−
kT
r
+∆
t
n
)
∗
˜
h
(
nT
c
)+
η
k
(
nT
c
)
(24)As
b
2
k
= 1
, and
η
k
is a white Gaussian noise processes (withzero mean), the averaged pulses can be approximated as
ˆ
r
(
nT
c
)
∼
=
p
(
nT
c
+ ∆
t
n
)
(25)For the case of an AWGN channel, replacing
h
(
t
) =
δ
(
t
)
in (20), the
k
th received bit can be expressed as
r
k
(
t
) =
b
k
p
(
t
−
kT
r
) +
η
k
(
t
)
(26)In this case, sampling (26) with an ideal TIADC, (21)becomes
r
k
(
nT
c
) =
b
k
p
(
nT
c
−
kT
r
) +
η
k
(
nT
c
)
(27)with
n
= 0
,
···
,N
S
−
1
. Here,
L
= 1
and then
N
S
=
M
.Then, assuming that the gain and offset are already canceledas in (22), the sampled signal with timing mismatch is
r
k
(
nT
c
) =
b
k
p
(
nT
c
−
kT
r
+ ∆
t
n
) +
η
k
(
nT
c
)
(28)By averaging and multiplicating by the known
b
k
s, (24)becomes
ˆ
r
(
nT
c
) = 1
K
P K
P
k
=1
b
2
k
p
(
nT
c
−
kT
r
+ ∆
t
n
) +
η
k
(
nT
c
)
(29)So the averaged pulses can be approximated as in (25).These results can be extended for a RMPF channel withimpulse response
h
(
t
)
given by (4) as follows. As stated inSection II, the received signal in a time window coveringthe transmission pulse repetition rate
T
r
is deﬁned by (5),assuming synchronization and channel estimation. In addition,the ideallysampled received signal at the output of the RFfrontend analog ﬁlter at sampling rate
T
c
is given by (8).Now, let us assume
N
S
=
ML
such that we have
M
samplesper pulse replica in the received
k
th frame. We now assumethat gain and offset mismatch have already been estimatedand compensated for, and that a good estimation of thechannel impulse response is available. Considering the timingmismatch as in the AWGN channel case, the received bit is
r
k
(
nTc
) =
b
kL
−
1
l
=0
c
l
(
nT
c
)
p
(
nT
c
+∆
t
n
−
l/W
−
kT
r
)+
η
k
(
nT
c
)
(30)Then, we can multiply each replica in (30) by the complexconjugate of the channel coefﬁcient
c
l
(
nT
c
)
in order to recoverthe phase of the pulse, and then divide by
c
l
2
=
c
l
c
∗
l
tonormalize the amplitude. Then, including averaging over
K
P
bits of a training (pilot) signal and using a delay line with atime step
1
/W
, we get the averaged pulse as
ˆ
r
(
nT
c
) = 1
K
P K
P
k
=1
b
2
k
L
L
−
1
l
=0
p
(
nT
c
+∆
t
n
−
l/W
−
kT
r
)+
η
k
(
nT
c
)
(31)Again, this can be approximated as (25), where we now have
K
P
L
averaged pulses instead of only
K
P
because we caninclude the pulse replicas in the estimation.As UWB channel models can be essentially described as acertain number of exponentially decaying clusters (say
C
) of multipath components with a certain number of exponentiallydecaying resolvable paths (say
L
), they can also be consideredin this analysis and the main difference is the addition of another summation over
C
in equations (30)(31) to add thecontribution of the different clusters [23].Note from (31) that as long as
b
i
is known, more pulses canbe added to the average calculation to improve the estimatorperformance. Therefore, once the bits have been detected, theycan be used in (31) to update the timing mismatch estimationwhich also allows to track changes in the different
∆
t
s.
B. Timing mismatch estimation
In the equation describing the gaussian pulse (2),
t
0
is acausalization constant such that the center peak of the gaussianbell occurs at
t
=
t
0
. If we consider that the pulse bandwidth
W
is 10 dB below the maximum of its Fourier transform [2],the we can compute
T
0
as
T
0
=
ln(10)4
πW
2
∼
= 12
.
35
W
(32)For a unitary pulse amplitude,
K
0
=
√
2
T
0
. With thiscalculations, we can write
p
(
t
)
as
p
(
t
) =
e
−
π
2
t
−
t
0
T
0
2
(33)which is a gaussian bell with mean
t
0
and standard deviation
σ
=
T
0
. Then, if we deﬁne a minimum sampling periodslightly higher than the Nyquist rate
T
c
=
T
0
∼
= 1
/
(2
.
35
W
)
and consider
N
samples per pulse, then
t
0
= (
N/
2)
T
c
.Let’s consider we want to sample the gaussian pulse with adeviation up to
4
σ
from the center peak
t
0
. After
N
samples,
t
N
=
NT
c
=
t
0
+ 4
σ
. Replacing
σ
=
T
c
and
t
0
= (
N/
2)
T
c
,we can solve the equation for
N
, which gives
N
= 8
.Then, we can estimate
t
=
nT
c
+ ∆
t
n
from (25) and (33)for the
n
th ADC averaged output samples as
ˆ
t
=
t
0
±
−
2
T
20
π
ln(ˆ
r
)
(34)