Chapter 2
Time Delay Estimation
enhancement through Signal Design
Timing recovery represents the most critical function in every radio-location systems, including those based on satellite positioning. In particular, positioning accuracy depends on the accuracy in TDE between transmitted codes and local replicas. In this scenario, it is apparent that the more accurate the TDE is, the more precise the user position will be. This thesis proposes some criteria to improve TDE accuracy of SS signals, focusing on the properties of the transmitted signal, with particular emphasis on investigating the fundamental limits of tracking performance. The aim of the Chapter is thus to give a close picture of TDE and its maximization through signal design, providing a deep insight into its Cram´er-Rao lower bound (CRB) and filling some gaps of the literature. In particular, after recalling the bases on estimation theory and specifically on the CRB in Sect.2.2 and 2.3, respectively, we will fix the problem of measuring TDE accuracy for different signal formats in Sect.2.4. In fact, specific formulas of the CRB for TDE in an AWGN channel will be discussed in Sect.2.4, while Sect. 2.5 will propose the interpretation of the same CRB formulas as functions of the spectral properties of digitally modulated signal, revealing to be the key-point for optimizing and comparing different signal modulations. Finally, the problem of signal optimization will be further discussed in Sect.2.6 for an AWGN channel, and further in Sect.2.7 for a signal embedded in a channel affected by multipath.
2.1 Motivations
Global navigation satellite systems GNSS are based on the capability of a receiver to estimate the propagation times of a set of spread-spectrum SS signals broadcast by multiple satellites placed at known locations [44]. When at least four propagation times are available, the receiver can unambiguously obtain its own spatial coordinates and the time reference [59].
This thesis is specifically focused on the problem of improving positioning accuracy for GNSS, and thus TDE. In particular, time synchronization can easily be cast into a conventional parameter estimation problem [52], to be tackled with the tools of estimation theory [45]. Although acquisition and tracking issues for spreading codes in the field of satellite positioning are well documented in the literature [12, 56], their fundamental limits are relatively less investigated. Many activities [5,34,37] aiming at enhancing the overall navigation performance are currently ongoing. This is typically performed by designing enhanced signals compared to those available today, e.g., by optimizing the modulation schemes. This can be achieved either introducing novel chip waveforms [5, 34] or combining existing signals, as is taken for the multiplexed binary offset carrier (MBOC) modulation [37], or finally adopting different modulation schemes.
This thesis proposes some criteria to improve TDE accuracy of SS signals [63], focusing on the properties of the transmitted signal. The problem is thus assessed using conventional parameter estimation and signal synchronization tools [45], which makes the proposed analysis suitable for both navigation and communication systems and independent of the particular receiver configuration. In the remainder of the thesis, we focus on satellite positioning, but the results can easily be readapted to wireless communication by replacing the satellite with the communication terminal.
Following these aims, this Chapter poses the theoretical bases on TDE for signal optimization. The fundamental limits in time synchronization are here recalled and further investigated, both in an AWGN channel and in a channel affected by multi- path. The results of this Chapter either comes from the literature or can be easily predicted, but are not easy to be explicitly found in a unique text.
2.2 Estimation theory 27
2.2 Estimation theory
This section recalls the fundamental performance limits on the estimation accuracy of a scalar parameter [45]. Let λ to be the deterministic parameter to be estimated and r to be the random vector of the observable samples (or data or outcomes) that depend on λ. The generic estimation process based on the observation of a realization of r will be denoted hereafter by ˆλ(r) or simply by ˆλ.
An estimator is a function that maps a sample design to a set of sample estimates.
A sample design can be thought of as an ordered pair ( r, p (r, λ) ) where p (r, λ) is the probability density function (pdf). The pdf maps the set of r to the closed interval [0,1], and has the property that the sum (or integral) of the values of p (r, λ), over all elements in r, is equal to 1. The pdf is parameterized by the unknown parameter λ, i.e, there is a class of pdf where each one is different due to a different value of λ [45].
When the pdf is viewed as a function of the unknown parameter (with r fixed), it is termed the likelihood function.
Intuitively the “sharpness” of the likelihood function determines how accurately we can estimate the unknown parameter. To quantify this notion it can be observed that the sharpness is effectively measured by the negative of the second derivative of the logarithm of the likelihood function at its peak. This is the curvature of the log-likelihood function.
As such, ˆλ(r) is a random variable, since it depends on the particular observation r, and thus different observations lead to different estimates. As a random variable, ˆλ(r) is characterized by its statistical properties, whose main definitions and properties are being reported. For all the properties below, the value λ, the estimation formula, the set of samples, and the set probabilities of the collection of samples, can be considered fixed. Yet since some of the definitions vary by sample (yet for the same set of samples and probabilities), we must use r in the notation. Hence, the estimate for a given sample r is denoted as ˆλ(r).
We have the following definitions and attributes.
1. For a given sample r, the error ε of the estimator ˆλ is defined as
ε = ˆλ(r) − λ. (2.1)
Note that the error depends not only on the estimator (the estimation formula or procedure), but also on the sample itself.
2. The mean squared error of ˆλ is defined as the expected value (probability- weighted average, over all samples) of the squared errors; that is,
MSE(ˆλ) = E[(ˆλ − λ)2]. (2.2)
It is used to indicate how far, on average, the collection of estimates are from the single parameter being estimated.
3. For a given sample r, the sampling deviation of the estimator ˆλ is defined as
ˆλ(r) − E(ˆλ), (2.3)
where ˆλ(r) is the estimate for sample r, and E(ˆλ) is the expected value of the estimator. Note that the sampling deviation depends not only on the estimator, but also on the sample itself.
4. The variance of ˆλ is simply the expected value of the squared sampling devia- tions; that is,
var(ˆλ) = E[(ˆλ − E(ˆλ))2]. (2.4) It is used to indicate how far, on average, the collection of estimates are from the expected value of the estimates. Note the difference between MSE and variance.
5. The bias of an estimator ˆλ is defined as
b(λ) = E(ˆλ) − λ. (2.5)
It is the distance between the average of the collection of estimates, and the single parameter being estimated. It also is the expected value of the error, since E(ˆλ) − λ = E(ˆλ − λ). The relationship between bias and variance is analogous to the relationship between accuracy and precision.
6. An estimator ˆλ is an unbiased estimator of λ if and only if b(λ) = 0. Note that bias is a property of the estimator, not of the estimate. Often, people refer to a “biased estimate” or an “unbiased estimate”, but they really are talking about an “estimate from a biased estimator”, or an “estimate from an unbiased estimator”.
2.3 Cram´er-Rao bound (CRB) and Modified Cram´er-Rao bound (MCRB) 29
7. The MSE, variance, and bias, are hence related:
MSE(ˆλ) = var(ˆλ) + (b(λ))2, (2.6) i.e. mean squared error = variance + square of bias.
8. The standard deviation of an estimator of λ (the square root of the variance), or an estimate of the standard deviation of an estimator of λ, is called the standard error or root mean square error (RMSE) of λ.
2.3 Cram´er-Rao bound (CRB) and Modified Cram´er- Rao bound (MCRB)
2.3.1 The Cram´er-Rao lower bound (CRB)
The Cram´er-Rao lower bound (CRB) is a fundamental lower bound on the variance of any estimator [14, 71] and, as such, it serves as a benchmark for the performance of actual estimators [3, 45, 52] .
For a scalar parameter λ, the CRB states that the variance (or covariance) of any estimator of λ with bias function b (λ) is lower bounded by [14, 71]
var(ˆλ) = E[(ˆλ − E(ˆλ))2] 1 CRB (λ) (2.7) where CRB (λ) denotes the true CRB, given by [14, 67, 71]
CRB (λ) = 1 +dλdb2
Er
h
∂ln p(r|λ)
∂λ
i2 . (2.8)
The p(r|λ) is the pdf of the observations r when λ is the true value and Er{·} in (2.7) and (2.8) denotes statistical expectation wrt the pdf p(r).
Obviously, when the estimator is unbiased, the CRB simply reduces to
CRB (λ) = 1
Er
h
∂ln p(r|λ)
∂λ
i2 (2.9)
or, equivalently, to
CRB (λ) = 1
− Ern∂2ln p(r|λ)
∂λ2
o . (2.10)
2.3.2 The Modified Cram´er-Rao lower bound (MCRB)
The CRB is well known and widely adopted for its simple computation, but its close- form evaluation becomes mathematically intractable when the vector of observables contains, in addition to the parameter to be estimated, also some nuisance parameters, i.e., other unknown random quantities whose values we are not interested in (infor- mation data, random chips of the code of a ranging signal etc.), but that concurs to shape the actual values of the observables. To encompass the problem, it has been shown in [16] that in the presence of nuisance parameters, the variance of any unbiased estimator is lower bounded by the so-called Modified Cram´er-Rao lower bound (MCRB), which is much simpler to evaluate than the true CRB. As proven in [16] the MCRB is in general looser than the true CRB, but it has been also demonstrated that in a few specific cases of synchronization parameter estimation, the MCRB is essentially as tight as the true CRB.
To better understand the problem, lets assume that the observable is given by a received waveform in an AWGN channel, whose baseband equivalent (or complex envelope) is
r (t) = x (t) + n (t) , (2.11)
and which is observed over an interval Tobs. In (2.11) x (t) is the information-bearing signal and n (t) represents the complex-valued additive white Gaussian noise with two-sided power spectral density 2N0. If we now assume that the signal is known in most of its basic characteristics (nominal carrier frequency, modulation format, signaling interval and so on), the remaining unknown parameters can be divided into two groups: the group of the parameter/parameters to be estimated and the group of unwanted parameters. Limiting to the case of estimating one parameter, denoted by λ, all other parameters, including the data, are collected in a random vector u having a known pdf p(u) which does not depend on λ. An exact representation of the observed waveform r (t) would require infinite-dimensional vector spaces, but it is realistic to assume that a finite-dimensional vector r can be found to represent r (t) with adequate accuracy. It follows that the observation vector r is thus given by
r = x(λ, u) + w. (2.12)
To compute the CRB as in (2.9) the pdf p(r|λ) is needed. In principle it can be
2.3 Cram´er-Rao bound (CRB) and Modified Cram´er-Rao bound (MCRB) 31
computed from the integral
p (r|λ) =
+∞
Z
−∞
p (r|u, λ) ·p (u) du, (2.13)
where p (r|u, λ), the conditional probability density function of r given u and λ, is easily available, at least for additive Gaussian channels. Unfortunately, in most cases of practical interest, the computation of (2.9) is impossible because either the integration in (2.13) cannot be carried out analytically or the expectation in (2.9) poses insuperable obstacles. It is in this case that the MCRB reveals fundamental.
The MCRB in fact is defined as follows
M CRB (λ) = 1
Er,u
h
∂ln p(r|u,λ)
∂λ
i2 (2.14)
or, equivalently,
M CRB (λ) = 1
Eu
Er|u
h
∂ln p(r|u,λ)
∂λ
i2 (2.15)
and it reveals much easier to compute. In fact, for the Gaussian channel as in (2.12), the pdf is
p(r|u, λ) = exp
− 1
2σ2w|r − x(λ, u)|2
= exp
− 1
2N0|r − x(λ, u)|2
(2.16) and the MCRB reduces to [16]
M CRB (λ) = N0
Eu
∂x(λ,u)
∂λ
2 . (2.17)
Coming back to the problem of computing the CRB for the signal r (t) (2.11), in [71]
it is shown that in the limit, as the number of dimensions of r tends to infinity, a formula like (2.17) does still apply provided that p(r|u, λ) is replaced by the likelihood function
Λ(u, λ) = exp
− 1 2N0
Z
Tobs
|r (t) − x (t)|2dt
, (2.18)
and the expectation over r is replaced by the expectation over the noise process n (t).
With this changes (2.17) becomes
M CRB (λ) = 1
Ew,u
h∂ln Λ(u,λ)
∂λ
i2 (2.19)
and substituting (2.18) into (2.19) one gets after some manipulations [16],
M CRB (λ) = N0
Eu
R
Tobs
∂x(t)
∂λ
2
dt
(2.20)
Note that the MCRB is much simple to evaluate than the CRB.
For completeness, we report the CRB expression when no nuisance parameters are present for the signal (2.11). Starting from (2.9), after some manipulation we get
CRB (λ) = N0
R
Tobs
∂x(t)
∂λ
2
dt
(2.21)
whose numerical value clearly depends only on the type of modulation, on the time of observation and on the parameter to estimate.
2.4 Signal optimization through the CRB of time delay estimation
2.4.1 Motivation
As clearly stated in the Introduction, this thesis deals with topics in signal analysis for TDE, with particular application to satellite positioning. In particular, few different SS modulated signals will be analyzed throughout, as insight to actually adopted GNSS signals and as alternative for future SIS. In fact, as delineated in Sec.1.6.3, the modulation actually adopted is GNSS is a DS-SS linear modulation, thanks to which signals coming from different satellite are univocally determined by the code sequence associated to each satellite. For this reason, in the sequel we will focus on the class of SS modulations, starting from the DS-SS linear modulation and encompassing spread-spectrum continuous-phase-modulation and multicarrier signals. Performance of the different SS modulated signals will be thus discussed in details, focusing on positioning accuracy (and thus TDE accuracy) together with some other parameters of analysis that has to be taken into account in GNSS systems.
As far as TDE accuracy is concern, the CRB will be used as performance bench- mark, since it is independent from the receiver structure, and thus it can be used to characterize different signal modulations, relying only on the signal structure itself.
2.4 Signal optimization through the CRB of time delay estimation 33
2.4.2 Signal model
In this section, we formalize the lower bounds in the TDE for a generic digitally modulated signal.
In particular we focus on a bandpass signal, whose format is xBP(t) = Ren
x (t) ej(2πf0t+ϕ)o
, (2.22)
where Re {·} denotes the real part of a complex-valued argument; f0 and ϕ are the carrier frequency and phase, respectively and x (t) is the complex signal with average transmitted power Px, digitally modulated by the vector a = {an}. The vector a represents the data-modulated spreading code, given by the product of the binary data symbols with the spreading code ranging sequence c = {cn}Nn=0−1 assigned to each satellite.
In the sequel we will adopt this notation, using the subscript BP when referring to the real bandpass signal, while nothing will be added when referring to the baseband equivalent (complex) signal.
In this section, we restrict the study of TDE fundamental limits to a frequency-flat channel, which represents an acceptable approximation for the satellite communica- tion channel [25]. The analysis is further refined in Sect. 2.7 by extending the results to a channel affected by MP.
Hence, assuming ideal coherent demodulation (thus with the realistic assumption that during signal tracking the carrier frequency f0and the carrier phase ϕ are known to a sufficient accuracy), the baseband-equivalent of the received SS signal reduces to r (t) = x (t − τ ) + n (t) , (2.23) where x (t) is the complex-valued SS signal; τ is the group delay experienced by the radio signal when propagating from the satellite to the receiver (as seen in the reference time of the receiver) [44]; and n (t) represents the complex-valued AWGN with two-sided power spectral density 2N0.
It is worth noting that the following analysis is focused on detecting a single satellite during the tracking stage (corresponding to single-user detection in a communication scenario). As a result, although the effects of other users should be explicitly included in a multiuser model, considering those effects in n (t), as done in (2.23), represents a good approximation in view of the central limit theorem.
For the sake of simplicity, the transmitted signal is assumed to be an unmodulated signal, in the sense of data-less signal: a = c. This is equivalent to consider either a pilot signal, or a data signal in which the data modulation is removed prior to the tracking stage. This approach does not reduce the generality of the problem, since the effects of data modulation can be neglected when considering modulated SS signals with high processing gain N , as is typical in the field of satellite navigation.
Moreover, when secondary codes are used, the following analysis can easily be applied considering the vector c as the product of the primary code with the secondary code.
As a result, the received signal (2.23) can be seen as the combination of noise with a
“spreading signature waveform x (t) ” defined as a function
x (t) = f (c, p (t) , Tc) , (2.24) thus depending on the type of modulation f (), code sequence c, chip-rate 1/Tc and pulse shape p (t) adopted.
2.4.3 CRB and MCRB for time delay estimation: signal opti- mization
We can now calculate the CRB, starting from the likelihood function of the scalar parameter to be estimated, i.e., in our case, the delay parameter τ . We can process the observed signal r (t) with some unbiased estimator to derive an estimate ˆτ of τ . Starting from (2.9), the variance σ2τof any unbiased estimator of τ (the so-called jitter variance) is lower bounded by [52]
στ2= Er(τ − ˆτ )2 ≥ CRB (τ ) ,
"
Er
( d ln p(r|τ ) dτ
2)#−1
, (2.25) where r is a vector representation of r (t) on a complete orthonormal basis [45]; p(r|τ ) is the conditional probability density function (pdf) of r for a given τ (the likelihood function of τ ); and Er{·} denotes statistical expectation with respect to (wrt) p(r).
The received signal r (t) and thus r depend on x (t) as in (2.23). Hence, CRB (τ ) is a function of the particular values of the “spreading signature waveform x (t) ” (2.24).
After some manipulations [52], as the ones applied to (2.20), the CRB can be finally expressed as
CRB (τ ) = N0
"
Z
Tobs
dx (t − τ ) dτ
2
dt
#−1
, (2.26)
2.4 Signal optimization through the CRB of time delay estimation 35
where Tobs= N · Tc is the observation interval.
Equation (2.26) gives a practical criterion to numerically assess the performance in terms of position accuracy by TDE accuracy.
It can be shown [34] that the estimation variance of the simple delay locked loop (DLL) [30, 31, 38] attains the CRB. Thus, designing a signal (2.24) that minimizes the CRB (2.25) appears to be a motivated approach for improving positioning per- formance. Therefore, improving TDE accuracy translates into minimizing (2.26).
As we have underlined, the transmitted signal x(t) can be thought as a “spreading signature waveform x (t) ” as in (2.24), thus minimizing (2.26) can be achieved by either modifying the code sequence c or the shaping pulse p (t) separately, or by selecting code sequence and shaping pulse jointly in an optimization process of the entire “spreading signature waveform” (2.24). This process can imply also to use a different modulation format, that is, a different function f (·) in (2.24). In the next chapters, we propose different methods to improve TDE accuracy for future GNSSs based on signal design at the transmitter side, deeply discussing DS-SS linear modulations and encompassing SS-CPM and multicarrier signals. For this reason, in the next chapters the CRB will be specifically re-formulated case by case depending on the modulation format under investigation. Obviously, when the signal model will imply nuisance parameters u, TDE accuracy will be studied making use of the MCRB, that is,
M CRB (τ ) = N0
Eu
R
Tobs
dx(t−τ ) dτ
2
dt
. (2.27)
2.4.4 Considerations on the applicability of the CRB and MCRB
In the previous sections, we have recalled the basics of the estimation theory, giving particular emphasis to the CRB. Particularities of the CRB and MCRB applied to different signal formats, will be discussed in later chapters, as soon as they will be used. We here mention some general characteristics on the existence and applicability of such bounds, as coming from the literature.
First of all, the true CRB is never below the MCRB so that the MCRB is in general looser than the true CRB. However, it was shown in [16] that in a few specific cases of synchronization parameter estimation, the MCRB is essentially as tight as the true CRB at high signal-to-noise ratio (SNR).
Moreover, in [53] it is proven that the MCRB equals the asymptotic CRB at high SNR when the parameter to be estimated is not coupled with the nuisance parameters, finding that for considerably many cases of signal synchronization the asymptotic CRB is essentially the same as the MCRB.
In particular, in [16] it is shown that approximate equality between the CRB and MCRB is found to occur for estimation of τ when the carrier frequency f0, the carrier phase ϕ and data are known.
On the other hand, the CRB and the MCRB are known to yield poor results for small SNR ratios. To this regard, alternative bounds such as the Ziv-Zakai lower bound (ZZB) [15], [76] could be investigated, as they are known to be tighter bounds at low SNR. Anyhow, in the sequel, we will deal with problems for which the SNR values belong to a range of medium-high SNR, for which the CRB/MCRB validity is ensured, and thus no considerations on ZZB will be made.
A second point to be clarified is that the valid application of the CRB (2.9) requires that the signaling waveform be sufficiently smooth [76], since it has to exist the second derivative of the signal. To this regard, we can anticipate that in the cases of practical interest reported in this thesis, this condition results satisfied. Moreover, as it will be highlighted in Sec.2.5, for the few cases in which this condition is not fulfilled, an alternative CRB expression can be used, since it relies on the signal properties in the frequency domain.
Finally, it has to be mentioned that, even if the bounds recalled up till now are applied when estimating a scalar parameter, they can be used also for the estimation of a uniformly distributed random variable, as it is our case of interest. (Simply the reader can note that the tracking process is fulfilled immediately after the acquisition stage and thus the time delay to be estimated is known to belong to a certain time interval). In fact, when the estimation of a random variable is quite accurate, the CRB/MCRB expression for a scalar parameter, can be used as good approximation of the real expressions [71, 76]. In particular, rigourously speaking, if the a priori pdf of the variable τ to be estimated is known, the performance becomes the ones of an estimator of a random variable, showing that the lower bound of the mean squared
2.5 MCRB(τ ) as a function of the signal spectral properties of a digitally
modulated signals 37
estimation error (MSEE) is set to [71]
En
[ˆτ − τ ]2o
> N0
Eτ
( R
Tobs
h∂x(t,τ )
∂τ
i2
dt −∂2ln p∂τ2τ(τ )
) . (2.28)
It follows that the inequality requires the signal being “smooth” (as before), and also the function ∂2ln p∂τ2τ(τ ) existing. When the parameter τ is uniformly distributed, (and consequently the pdf pτ(τ ) holds jump discontinuities), the bound can be reformulated avoiding the discontinuities in pτ(τ ), obtaining [71], [76]
En
[ˆτ − τ ]2o
> N0
R
Tobs
h∂x(t,τ )
∂τ
i2
dt
, (2.29)
valid when the estimation is accurate, that is, for high SNR values. It can be observed that, under such hypothesis, (2.29) coincides with (2.26). In the sequel, we will refer always to formula (2.26), intending either that there is no knowledge on the a-priori distribution of the τ or, if τ is uniformly distributed, that the approximation (2.29) is implied.
2.5 MCRB(τ ) as a function of the signal spectral properties of a digitally modulated signals
In the previous section, we have seen that when focusing on the estimation of the time- delay experienced by a digitally modulated signal, the CRB formulas (2.26) and (2.27) apply. Nevertheless, starting from such formulas it is at times difficult to compare the CRBs of different modulations. Just to cite an example, in the navigation community, it is common practice to characterize the performance of the time of arrival (TOA) estimation accuracy of the DS-SS linear modulations by the inverse of the normalized root mean square (RMS) bandwidth [8] (or, equivalently, of the normalized Gabor bandwidth [37]) of the spectra of the chip pulse signal. This is a direct consequence of the expression of the MCRB of the TOA estimation of a linearly modulated signal.
Following such technique, the performance comparison of different DS-SS linearly modulated signal formats, collapses into the analysis of the shape of the PSD of the ranging signal.
The motivation of a similar analysis, that is particularly expedient when the signal format is different from the classical DS-SS, (as for multicarrier for example) is not easily available in the open literature and is at times given for granted without adequate formalization.
Nevertheless the problem of determines the ultimate performance of TDE dates back over three decades, and plenty of contributions dealing with this problem can be found [51, 69]. A recent formalization of the problem that inspired our work is contained in [3], but with specific assumptions on the ciclostationarity of the signal and on the piecewise constant domain of the parameter to estimate. Similar results are also given by works like [51] and [69] in which it is calculated the variance of some proposed estimators of the TDE of radar signals. Moreover, same CRB expression can alternatively be obtained from the CRB for time-difference-of-arrival (TDOA) estimation for general stationary signals derived in [46]. In fact, the current delay estimation problem can be considered as a TDOA estimation problem with one sensor receiving the clean signal at infinite SNR, leading to the same expression that will be proposed in a moment. We intend here to review and simplify what has already been done to come to a ”clean” formulation of the TDE MCRB in the frequency domain that gives much insight into the opportunities to solve the problem of signal optimization.
2.5.1 MCRB(τ ) in the frequency domain
We focus again on a generic bandpass modulation, whose format is as in (2.22).
The modulation format of the complex baseband equivalent x (t) is generic and it is assumed that it uses a symbol sequence c, modeled as an instance of a random sequence. Without loss of generality, the symbol sequence c can be thought as a random data sequence in a data communication application, or as the ranging code in radio navigation and positioning. Using the nomenclature of ranging codes (symbol=chip), the chip are modeled as binary iid random variables, so that the signal x (t) turns out to be a parametric random process, for which each sample function is a signal ¯x (t) with finite power Px and chip rate Rc = 1/Tc.We recall that the PSD of a parametric random process x (t) like ours is defined to be
Sx(f )=∆ lim
Tobs→∞
Ec| ¯XTobs(f, c) |2 Tobs
(2.30)
2.5 MCRB(τ ) as a function of the signal spectral properties of a digitally
modulated signals 39
where ¯XTobs(f, c) is the Fourier transform of the generic sample function ¯xTobs(t) truncated in the time interval [−Tobs/2; Tobs/2] and thus having finite energy, and where Ec{·} denotes statistical expectation over the code chips. The power of the (band-pass) signal xBP(t) is
Px=
∞
Z
−∞
SxBP(f )df =1 2
∞
Z
−∞
Sx(f )df . (2.31)
The baseband equivalent of the observed signal in a coherent receiver can be modeled as
r (t) = x (t − τ ) + n (t) (2.32)
where τ is the group delay experienced by the ranging signal when propagating from the transmitter to the receiver in the time reference frame of the receiver, and n (t) is complex-valued AWGN with two-sided power spectral density 2N0. The MCRB (τ ) can be calculated starting from the definition of the MCRB (2.27), where the code sequence c is considered as a nuisance parameter:
M CRB (τ ) = N0
Ec
( R
Tobs
∂x(t−τ,c)
∂τ
2
dt ) =
N0
R
Tobs
Ec
∂x(t−τ,c)
∂t
2 dt
= N0
Tobs
∞
R
−∞
1 TobsEc
∂xTobs(t−τ,c)
∂t
2 dt
(2.33)
where the MCRB is expressed as a function of the signal, xTobs(t) = x (t)·rect (t/Tobs), that has finite energy. Using Parseval’s relation we get
M CRB (τ ) = N0
Tobs
Tobs
∞
R
−∞
Ecn
|j2πf |2|XTobs(f )|2o df
= N0
Tobs
∞
R
−∞
4π2f2Ec
n|XTobs(f )|2o
Tobs df
(2.34)
We now adopt a crucial assumption that leads to an accurate approximation of the bound. Specifically, we assume Tobs very large, so that Ec
n|XTobs(f )|2o.
Tobs ∼=
Sx(f ). Under this hypothesis,
M CRB (τ ) = N0
Tobs4π2
∞
R
−∞
f2SX(f ) df
= BeqTc
4π2·NEc
0β2x (2.35) where we also let Tobs= N · Tc (N very large), and where βx is the root second-order moment of the signal spectrum, normalized to the complex signal power
∞
R
−∞
SX(f ) df = 2Px; and defined by
βx2=∆
∞
R
−∞
f2SX(f ) df
∞
R
−∞
SX(f ) df
, (2.36)
Ec = Px· Tc is also the average signal energy per chip, and Beq = 1/2N Tc is the (one-sided) noise bandwidth of a closed-loop estimator equivalent to an open-loop estimator operating on an observation time equal to N Tc. From (2.35)-(2.36), we conclude that the MCRB depends on the second-order moment of the PSD of the complex signal, independent of the type of signal format (modulation, spreading, etc.) that is adopted. In particular, we see that signals with the same PSD have the same MCRB even if generated by different modulations. As a proof of it, in Chapter 4 we will demonstrate the equivalence of the MCRB for a single carrier signal and a multicarrier signal with the same PSD.
2.5.2 Dependence on the center of gravity of the signal spec- trum
Since the CRB is proportional to the inverse of the second order moment of the PSD, some considerations need to be done also on the effects of the center of gravity or center frequency of the signal spectrum.
The center of gravity of the signal spectrum is given by [3]
ωG = 2πfG, 1 2Px
∞
Z
−∞
2πf · SX(f ) df , (2.37)
where fG is the center frequency and we define also the normalized second-order
2.5 MCRB(τ ) as a function of the signal spectral properties of a digitally
modulated signals 41
moment of inertia Mx(2) of the signal spectrum as
Mx(2), (2π)2βx2= 1 2Px
∞
Z
−∞
(2πf )2· SX(f ) df , (2.38)
while the central second-order moment of the signal is
m(2)x = Mx(2)− ω2G= 1 2Px
∞
Z
−∞
(2π)2(f − fG)2· SX(f ) df . (2.39)
The (squared) Gabor bandwidth of signal x [29], is also given by
(BG)2= 1 2Px
∞
Z
−∞
(f − fG)2· SX(f ) df . (2.40)
(
q
Mx(2)−ω2G
2π is the root mean square bandwidth, also denoted as Gabor bandwidth).
Normally, SX(f ) is considered even-symmetric and, as such, fG = 0. In this case, the second order moment of the signal coincides with the central second-order moment and thus the MCRB directly depends on the Gabor bandwidth of the signal.
When the PSD is not even-symmetric, then the center of gravity of the PSD is not null and the normalized second-order moment of inertia Mx(2)is generally greater than the centralized second-order moment m(2)x , following the relation:
Mx(2)= ωG2 + m(2)x , (2.41) or equivalently,
βx2= fG2 + (BG)2. (2.42) As an example, we consider a bandpass signal transmitted from a satellite as in (2.22) and thus seen from the receiver as
rBP(t) = Ren
x (t − τ ) ej(2πf0(t−τ )+ϕ)o
+ nBP(t) , (2.43) nBP(t) being the band-pass AWGN with power spectral density N0/2. The PSD of the (bandpass) received signal is
SrBP(f ) = 1
4Sx(f − f0) +1
4Sx(−f − f0) +N0
2 . (2.44)
Assume also that, as is often the case in practice, Sx(f ) is even-symmetric and strictly bandlimited within [−B, B]. Instead of demodulating the signal wrt the nominal carrier frequency f0, we may use the frequency
fB= f0− B (2.45)
so that the resulting demodulated I/Q complex signal is
r (t) = x (t − τ ) ej(2πB(t−τ )+ϕ)+ n (t) , (2.46) whose spectra is clearly asymmetric even if Sx(f ) is symmetric. In particular, the sig- nal z(t) = x (t − τ ) ej(2πB(t−τ )+ϕ)is an analytical signal with no spectral components on f < 0.
This means that
z(t) = zI(t) + ˇzI(t) (2.47) with ˇzI(t) being the Hilbert transform of zI(t), and also means that both zI(t) and zQ(t) = ˇzI(t) are bandlimited to 2B.
For such situation
βz2= B2+ (BG)2> (BG)2. (2.48) The inevitable conclusion is that z(t) is better than x(t) for delay estimation. The price to be paid is the increased receive bandwidth: the receiver needs twice as much processing bandwidth wrt the case of conventional demodulation (fG= 0). The most striking aspect of this computation is that the transmit bandwidth is unchanged.
In fact, coming back to the general case (2.41), even if the bandwidth of the transmitted signal is unaltered and fixed to 2B, the receiver needs a total processing bandwidth of 2fG+ 2B when the center of gravity ωG 6= 0, thus requiring an extra processing bandwidth of 2fG. GPS HW designers know very well about the classical modes of operation for a GPS receiver: the term (BG)2 applies to conventional code tracking on the symmetric baseband spectrum, whilst fG2 can be though of as relevant to carrier navigation, where delay estimation is performed by tracking the cycles of the carrier on the bandpass signal. On the other hand, it is known that tracking the carrier ensures high precision estimation, but with the drawback of high estimation ambiguity.
In the sequel, when no explicitly mentioned, the PSD of the received base-band signal will be thought as even-symmetric (fG= 0) and the CRB will be expressed as function of the Gabor bandwidth.
2.5 MCRB(τ ) as a function of the signal spectral properties of a digitally
modulated signals 43
It is worth noticing that the condition fG = 0 guarantees also that the estimation problem of estimating the clock phase (τ ) and the carrier phase (ϕ) of (2.22) is decoupled, so that, when estimating the time delay, it does not matter wether the carrier phase is assumed to be known or not [3]. Otherwise, if we have a coupled estimation problem, the bound cited above is still valid although not as tight [3]. In fact, the coupled bound on time delay inversely depends on Mx(2)− ωG2 (and not on Mx(2) as in (2.35) ), thus showing that, if the carrier phase is unknown, the bound on the variance of the time delay (clock phase) estimate is influenced by the centralized second-order moment (m(2)x = Mx(2)− ω2G) of the signal spectrum rather than by the moment of inertia (Mx(2)). Complete explanation is discussed in [3] together with the analysis of the bound on carrier phase. We here only cite that timing errors lead also to carrier phase errors when ωG6= 0, since for coupled estimates the CRB of the carrier phase directly depends on the CRB(τ ) as in [3]
CRB (ϕ) = N0
2TobsPx
+ ω2G· CRB(τ ). (2.49) When the noise bandwidth of the carrier phase synchronizer is much greater than the noise bandwidth of the TDE (which is true in most cases) than the effect is negligible.
Otherwise, if it is not negligible, attention must be paid to the correlation between carrier- and clock phase estimates.
We here mention only some main relation among those parameters:
1. when the carrier phase ϕ of (2.22) is known, the CRB(τ ) inversely depends on the second-order moment Mx(2) of the signal spectrum as in (2.35), thus depending on ωG2
2. when the carrier phase ϕ of (2.22) is not known, the CRB(τ ) inversely depends on the centralized second-order moment (m(2)x = Mx(2) − ω2G) of the signal spectrum, thus being generally greater than the CRB(τ ) calculated for known carrier phase and independent from ω2G
3. when the center of gravity of the received signal is null (ωG= 0), the estimation of ϕ and τ is decoupled, and the CRB(τ ) applies as in (2.35), thus inversely depending on the second-order moment Mx(2) of the signal spectrum
4. the modulation scheme only suggests the value of ωG: the true value is deter- mined by the receiver implementation
5. when the modulation scheme is the same, it can happen that two different receiver structures can lead to different ωG, while the Gabor bandwidth is obviously the same (same m(2)x = Mx(2)− ωG2). In a joint estimation of ϕ and τ , the bound on the variance of TDE will be identical for both structure, whereas the carrier recovery will not because timing errors lead to carrier phase errors if ωG= 0
6. when focusing only at the minimization of the CRB(τ ), receiver structures producing higher ωG induce lower CRB values, at the expenses of an higher receiver processing bandwidth.
2.5.3 Conventional DS/SS linear modulation
When the signal is a classical DS-SS signal, with binary pseudorandom noise (PN) code, we have
x (t) =p2Px
LP AM−1
X
l=0
γlg (t − lTc) (2.50) with γl = {±1}, iid, represent the product between the code element and the data symbols, g (t) is a real-valued shaping pulse with energy Tcand Tobs= LP AMTc. The PSD (2.30) is now calculated as
Sx(f ) = 2Px|G (f ) |2 Tc
(2.51) and thus, substituting (2.51) into (2.35), we obtain the conventional expression [52]
of the MCRB for a PAM signal
M CRB (τ ) = Tc2
4π2· 2LP AM· ENc0ξg
= BeqTc· Tc2 4π2· ENc0ξg
(2.52)
where
ξg= Tg2·
∞
R
−∞
f2|G (f )|2df
∞
R
−∞
|G (f )|2df
(2.53)
is the pulse shaping factor (PSF), an adimensional parameter related to the shape of the Fourier transform G (f ) of the pulse g (t). Tg is the generic symbol spacing, that is, in this case Tg= Tc.
2.6 Signal Optimization in an AWGN channel - Conclusions 45
2.5.4 MCRB(τ ) for filtered signals
Another advantage of the MCRB expressed as a function of the signal spectral properties, is that the approximation in (2.35)-(2.36) is particularly useful when a band-limitation is applied to the signal, since the MCRB can be calculated by simply limiting the frequency integral in (2.36) to the band of interest.
In fact, assuming that the received signal is filtered by a low-pass filter hIF(t) as in r (t) = x (t − τ ) ⊗ h(t) + n (t) , (2.54) the correspondent MCRB is calculated as in (2.35) taking into consideration the filter transfer function |H(f )| as
M CRB (τ ) = N0
Tobs4π2
∞
R
−∞
f2SX(f ) · |H(f )|2df
(2.55)
2.6 Signal Optimization in an AWGN channel - Con- clusions
After the considerations of Sect.2.4 and 2.5, it can be concluded that the ultimate accuracy of a positioning system in an AWGN channel can be optimized by properly designing the “spreading signature waveform x (t) ” (2.24) at the transmitter side that minimizes the CRB(τ ) on a pre-set bandwidth and for a certain SNR. This can be achieved by a proper design of the type of modulation, code sequence, chip-rate and pulse shape adopted.
Using the CRB(τ ) expression as function of the modulated signal properties in the time domain doesn’t allow to make qualitative comparison of different modulation schemes and the quantitative computation of the CRB(τ ) itself can be cumbersome depending on the adopted modulation. Here we solve the problem by looking at the CRB(τ ) expressed as function of the signal properties in the frequency domain. It has been shown that the CRB(τ ) can be formulated in terms of the PSD of the received baseband signal, thus highlighting that it is determined by only four parameters of the signal spectra, independently of the type od modulation adopted:
1. The average power Px
2. The center of gravity ωG of the spectrum
3. The second-order moment of inertia Mx(2) of the spectrum
4. The chip-rate 1/Tc of the signal
The preceding discussion shows that the placement in frequency of the signal spec- trum can heavily affect TDE accuracy and so, positioning accuracy.
The minimization of the CRB(τ ) is fulfilled by signals showing a PSD shifted at the edge of the dedicated band, as it is straightforward from (2.35). This is a well- known result for DS-SS linear modulation, that will be re-analyzed for each type of modulation presented in this thesis.
This concept is generally referred to as the Gabor bandwidth (GB) principle since the maximization of the GB induces the minimization of the CRB. Properly speaking, it would be more precise to refer to the maximization of the second-order moment of the spectra, thus considering also the receiver characteristics contained in fG, as discussed in Sect.2.5.2. Anyhow, the GB is the only parameter related to the transmitted signal, and it can thus be adopted as reference figure of merit when focusing on the optimization of the transmitted signal.
As final remark, TDE accuracy is not the unique parameter to be taken into consid- eration when optimizing signals for GNSS. Just to cite an example, the properties of the autocorrelation function of the signal are important as well, since they determine performance of the acquisition stage. To this regard, it is also known that when the signal energy is concentrated at the edge of the band (GB maximization), its autocorrelation function becomes more oscillatory and as such it is more prone to ambiguity errors (of which the CRB(τ ) is indeed unaware), as it is shown in [74].
For these reasons, in the following Chapters the various modulations will be analyzed taken into consideration various signal constraints, not only positioning accuracy.
Anyhow, in this thesis the maximization of the TDE accuracy will be taken as starting point of the signal optimization for each type of modulation, since it delineates the ultimate accuracy of a positioning system. Then the other signal constraints will be posed on top of it, aiming at guaranteeing at least the same performance as they are provided in today GNSS systems.
