Noise performance of the complex monopulse ratio
1 Abstract —The paper provides a characterization of the complex monopulse ratio in terms of autocorrelation and power spectral density of its fluctuations during satellite tracking, taking into account the presence of additive noise on sum and difference channels. The considered spectral structure and statistical distribution of the incoming signal is of interest for satellite missions. In particular it is assumed that the signal available at the monopulse processor after frequency down conversion contains a Gaussian term produced by low pass filtering of a constant envelope modulation, plus a monochromatic component representative of a possible residual carrier. The results can be used for optimizing the design of a monopulse tracking system.
Index Terms — autotrack, complex monopulse ratio I. I NTRODUCTION
ATELLITE tracking by ground directive antennas requires accurate pointing to the flying system. For example, the half-power beamwidth of an antenna with a diameter of 35m operating at 32GHz can be as small as 17 to 19 millidegrees, which means that a pointing accuracy in the order of few millidegrees will be required, in order not to penalise the communications link budget. The pointing to the satellite can be achieved either by programming the antenna direction based on the known position of the satellite (blind pointing), or by use of the received signal for tracking purposes. The latter method, which is the subject of this paper, is especially useful when the a-priori position of the satellite is not known with sufficient accuracy, e.g. shortly after launch, at the first acquisition of signal, or during “routine” tracking when the blind pointing leads to a residual slowly varying pointing error. There are various techniques by which a received signal can be used for tracking purposes, however in this context we are interested in monopulse autotrack [1], where the antenna feed system produces error signals based on the signal reception, without conical scanning. In particular a monopulse autotrack system produces two signals, called “sum” and “difference” (or “delta”), which encode the angular misalignment between the antenna boresight and the direction of the satellite: more specifically and with reference to Fig. 1, if 𝜃𝜃 𝑠𝑠 and 𝜑𝜑 𝑠𝑠 are the two angles identifying the satellite direction in the antenna frame, and if 𝜃𝜃 𝑠𝑠 is a fraction of the half-power beamwidth, two complex signals Σ ( 𝑡𝑡 ) and Δ ( 𝑡𝑡 ) will be available at the input of the monopulse processing system after frequency down Marco Lanucara is with the European Space Agency. conversion to complex baseband and low pass filtering, related by the following Δ ( 𝑡𝑡 ) = 𝜃𝜃 𝑠𝑠 𝐾𝐾 𝐹𝐹 𝑒𝑒 𝑖𝑖𝜑𝜑 𝑠𝑠 Σ ( 𝑡𝑡 ). (1) The signal Σ ( 𝑡𝑡 ) is used not only for tracking, its unfiltered version carries the information transmitted by the satellite, suitably coded and modulated onto the downlink carrier. The signal Δ ( 𝑡𝑡 ) carries the tracking information, it vanishes when the tracking is perfect and its amplitude and phase, referred to Σ ( 𝑡𝑡 ) , contain the components of the angular offset. The constant 𝐾𝐾 𝐹𝐹 is a parameter of the antenna system (the so called “tracking slope”), and determines the magnitude of the difference signal for a given angular misalignment. It is to be remarked that the Eq. (1) is ideal in many respects: first of all noise is present in a real system, added to both signals Δ ( 𝑡𝑡 ) and Σ ( 𝑡𝑡 ) ; secondly different time delays can be experienced by the two signals during the frequency down-conversion process, such that a spurious phase term can show up on top of 𝜑𝜑 𝑠𝑠 ; furthermore the error signal may not vanish even while perfectly tracking, and the tracking null may be displaced vs. the antenna boresight direction. All above effects (plus other) result in a degradation of the tracking performance, and will be ignored in this paper, with the exception of the first, the presence of noise, which can never be eliminated. When considering noise, the signals available at the input of the monopulse processing system can be expressed as follows 𝑆𝑆 ( 𝑡𝑡 ) = Σ ( 𝑡𝑡 ) + 𝑛𝑛 Σ ( 𝑡𝑡 ), 𝐷𝐷 ( 𝑡𝑡 ) = 𝜃𝜃 𝑠𝑠 𝐾𝐾 𝐹𝐹 𝑒𝑒 𝑖𝑖𝜑𝜑 𝑠𝑠 Σ ( 𝑡𝑡 ) + 𝑛𝑛 Δ ( 𝑡𝑡 ) (2) where 𝑛𝑛 Σ ( 𝑡𝑡 ) and 𝑛𝑛 Δ ( 𝑡𝑡 ) are complex noise terms. Let’s now consider the simple action performed by the monopulse processor: it forms the ratio between the noisy difference and sum signals 𝑀𝑀 ( 𝑡𝑡 ) = 𝐷𝐷 ( 𝑡𝑡 ) 𝑆𝑆 ( 𝑡𝑡 ) (3) and integrates it over a configurable time interval 𝑇𝑇 , e.g. 𝑀𝑀 𝑇𝑇 ( 𝑡𝑡 ) = 1 𝑇𝑇 � 𝑀𝑀 ( 𝑡𝑡 ′ ) 𝑡𝑡𝑡𝑡−𝑇𝑇 𝑑𝑑𝑡𝑡 ′ . (4) Noise performance of the complex monopulse ratio
Marco Lanucara S 𝑀𝑀 . Indeed, once the spectral properties of 𝑀𝑀 are known, the ones of 𝑀𝑀 𝑇𝑇 may be derived by knowledge of the specific filter transfer function. For such reason, in the following we will focus on 𝑀𝑀 . Fig. 1: definition of antenna coordinate frames and representation of the angular misalignment between antenna boresight and satellite direction through the angles 𝜃𝜃 𝑠𝑠 and 𝜑𝜑 𝑠𝑠 . First of all we notice that, in absence of noise, Eqs. (2) and (3) give ( 𝑀𝑀 ) no noise = 𝜃𝜃 𝑠𝑠 𝐾𝐾 𝐹𝐹 𝑒𝑒 𝑖𝑖𝜑𝜑 𝑠𝑠 , (5) which shows that, after dividing by the known 𝐾𝐾 𝐹𝐹 , the complex ratio contains the complete information about the angular misalignment between antenna boresight and satellite direction, and the error signal, split over its real and imaginary components, could be fed back to a servo system to enable autotrack. In presence of noise however the magnitude of the mean of the complex ratio will decrease, and 𝑀𝑀 will fluctuate around its complex mean. In general, under noisy conditions, one may write 𝑀𝑀 = 𝛼𝛼 ∙ ( 𝑀𝑀 ) no noise + 𝛿𝛿𝑀𝑀 (6) where ≤ 𝛼𝛼 ≤ is a reduction factor and 𝛿𝛿𝑀𝑀 is the fluctuation of the ratio around the reduced mean. The expression of 𝛼𝛼 can be retrieved from literature [2], together with the full expression of the probability density function of the ratio. The subject of this paper is finding the spectral density of 𝛿𝛿𝑀𝑀 , whose knowledge is mandatory for designing an autotrack feedback loop, for a class of signals spectra and statistical distributions of interest for applications. The expression of such parameter is known for purely Gaussian signal and noise (see for example [3] for what concerns the correlation properties of 𝛿𝛿𝑀𝑀 ), however the addition of a residual carrier leads to a more general expression, which is analysed in this paper. Before starting the analysis, some simplifying assumptions are needed. First of all we have to consider that the time integration in Eq. (4) will select only the low frequency components of 𝑀𝑀 , the reason being that its mean value, which is the quantity of interest, is slowly varying because it is linked to a physical misalignment between two directions. For this reason the knowledge of the spectral density of 𝛿𝛿𝑀𝑀 is required only around the zero frequency of the complex baseband. The second consideration is that during tracking, which is the regime of interest, the real and imaginary components of the monopulse ratio fluctuate around a zero mean, which will substantially simplify the computation of the spectral density of 𝛿𝛿𝑀𝑀 . II. S IGNALS MODEL
Due to the variety of possible carrier modulations, it is not possible to establish a priori what kind of spectral structure and statistical distribution could be applied to Σ ( 𝑡𝑡 ) . A constant envelope modulation is desirable in many satellite applications, due to the need of operating the on-board power amplifier near saturation, where its power efficiency is maximum [4]. However low pass filtering is generally applied prior to monopulse processing, to increase the signal to noise ratio at the output of the filter. Due to the filtering, the constant envelope condition will generally be lost, and one could model the signal Σ ( 𝑡𝑡 ) as a complex Gaussian process. Such modelling however would be insufficient when dealing with modulations with residual carrier, still common especially in deep space applications. Instead an adequate modelling of Σ ( 𝑡𝑡 ) could be the following Σ ( 𝑡𝑡 ) = �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖 ( 𝑐𝑐 𝑡𝑡+𝜃𝜃 𝑐𝑐 ) + 𝑥𝑥 ( 𝑡𝑡 ), (7) where 𝑃𝑃 𝑐𝑐 , 𝑓𝑓 𝑐𝑐 and 𝜃𝜃 𝑐𝑐 are the power, frequency and initial phase of the residual carrier and 𝑥𝑥 ( 𝑡𝑡 ) is the realization of a complex zero mean Gaussian process of power 𝑃𝑃 𝑥𝑥 representative of the frequency down-converted and low-pass filtered modulation spectrum. When including the noise on the sum channel as per Eq. (2), we have the following expression for 𝑆𝑆 ( 𝑡𝑡 ) 𝑆𝑆 ( 𝑡𝑡 ) = �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖 ( 𝑐𝑐 𝑡𝑡+𝜃𝜃 𝑐𝑐 ) + 𝑥𝑥 ( 𝑡𝑡 ) + 𝑛𝑛 Σ ( 𝑡𝑡 ), (8) i.e. the sum of a monochromatic signal plus a Gaussian process including noise, of power 𝑃𝑃 Σ , and signal component. In summary, the spectrum of 𝑆𝑆 ( 𝑡𝑡 ) , available to the monopulse processing system may look like the example of Fig. 2. Fig. 2: sample power spectrum at the sum input of the monopulse processor 𝑛𝑛 Δ ( 𝑡𝑡 ) and 𝑛𝑛 Σ ( 𝑡𝑡 ) , we assume them to be realizations of zero-mean Gaussian processes statistically independent between each other and with 𝑥𝑥 ( 𝑡𝑡 ) . It is to be remarked that in other applications like radar, the statistical independence between sum and difference noise may be inadequate, as in those contexts interfering signals are embedded in the noise terms. However in the satellite tracking scenarios under consideration the correlation between sum and difference noise reduces to a weak component linked to the sky brightness, which can generally be neglected. To complete the modelling, we report some properties of the autocorrelation of the involved signals, which we denote generically with 𝜉𝜉 ( 𝑡𝑡 ) until the end of the section. We adopt in this paper the following definition of autocorrelation, valid for a stationary complex or real process (Eq. 10-42 in [5]): 𝑅𝑅 𝜉𝜉𝜉𝜉 ( 𝜏𝜏 ) ≜ 𝐸𝐸 [ 𝜉𝜉 ( 𝑡𝑡 + 𝜏𝜏 ) 𝜉𝜉 ∗ ( 𝑡𝑡 )], (9) where 𝐸𝐸 [ ∙ ] denotes statistical average and ( ∙ ) ∗ complex conjugation. We can decompose the signal 𝜉𝜉 in its quadrature components 𝜉𝜉 ( 𝑡𝑡 ) = 𝜉𝜉 𝑐𝑐 ( 𝑡𝑡 ) + 𝑖𝑖𝜉𝜉 𝑠𝑠 ( 𝑡𝑡 ). (10) As all involved processes are complex envelopes of pass-band processes assumed to be wide sense stationary (WSS), then the following correlation properties hold (Eq. 11-63 in [5]) 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑐𝑐 ( 𝜏𝜏 ) = 𝑅𝑅 𝜉𝜉 𝑠𝑠 𝜉𝜉 𝑠𝑠 ( 𝜏𝜏 ), 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑠𝑠 ( 𝜏𝜏 ) = −𝑅𝑅 𝜉𝜉 𝑠𝑠 𝜉𝜉 𝑐𝑐 ( 𝜏𝜏 ) (11) and consequently, by applying Eqs. (9) and (10) and taking into account Eq. (11), we have 𝑅𝑅 𝜉𝜉𝜉𝜉 ( 𝜏𝜏 ) = 2 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑐𝑐 ( 𝜏𝜏 ) − 𝑖𝑖 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑠𝑠 ( 𝜏𝜏 ). (12) Furthermore, being 𝜉𝜉 𝑐𝑐 , 𝜉𝜉 𝑠𝑠 real, the following holds 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑐𝑐 ( −𝜏𝜏 ) = 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑐𝑐 ( 𝜏𝜏 ), 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑠𝑠 ( −𝜏𝜏 ) = −𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑠𝑠 ( 𝜏𝜏 ). (13) We will make use of the following normalised coefficients 𝜌𝜌 𝜉𝜉 ( 𝜏𝜏 ) = 2 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑐𝑐 ( 𝜏𝜏 ) 𝑃𝑃 𝜉𝜉 , 𝜌𝜌 𝜉𝜉 (0) = 1, 𝜇𝜇 𝜉𝜉 ( 𝜏𝜏 ) = 2 𝑅𝑅 𝜉𝜉 𝑐𝑐 𝜉𝜉 𝑠𝑠 ( 𝜏𝜏 ) 𝑃𝑃 𝜉𝜉 , 𝜇𝜇 𝜉𝜉 (0) = 0, (14) where 𝑃𝑃 𝜉𝜉 is the power of the process, such that the autocorrelation of Eq. (9) is expressed by 𝑅𝑅 𝜉𝜉𝜉𝜉 ( 𝜏𝜏 ) = 𝑃𝑃 𝜉𝜉 �𝜌𝜌 𝜉𝜉 ( 𝜏𝜏 ) − 𝑖𝑖𝜇𝜇 𝜉𝜉 ( 𝜏𝜏 ) � . (15) We also define, for later use, the complex correlation coefficient 𝑟𝑟 𝜉𝜉 ( 𝜏𝜏 ) = �𝑟𝑟 𝜉𝜉 ( 𝜏𝜏 ) �𝑒𝑒 𝑖𝑖𝜑𝜑 𝜉𝜉 ( 𝜏𝜏 ) = 𝜌𝜌 𝜉𝜉 ( 𝜏𝜏 ) − 𝑖𝑖𝜇𝜇 𝜉𝜉 ( 𝜏𝜏 ) (16) with obviously �𝑟𝑟 𝜉𝜉 � = 𝜌𝜌 𝜉𝜉2 + 𝜇𝜇 𝜉𝜉2 ≤ (17) III. M EAN OF THE COMPLEX RATIO
Even though the expression of the mean of the complex ratio is available in literature [2], it is re-computed here to adapt it to the scenarios and terminology considered in this paper. Let’s form the monopulse ratio based on Eqs. (2) and (3) 𝑀𝑀 ( 𝑡𝑡 ) = 𝜃𝜃 𝑠𝑠 𝐾𝐾 𝐹𝐹 𝑒𝑒 𝑖𝑖𝜑𝜑 𝑠𝑠 Σ ( 𝑡𝑡 ) + 𝑛𝑛 Δ ( 𝑡𝑡 ) Σ ( 𝑡𝑡 ) + 𝑛𝑛 Σ ( 𝑡𝑡 ) . (18) When applying the statistical expectation to the above, and after averaging with respect to 𝑛𝑛 Δ ( 𝑡𝑡 ) (assumed to be statistically independent from 𝑛𝑛 Σ ( 𝑡𝑡 ) ), we immediately get 𝐸𝐸 [ 𝑀𝑀 ( 𝑡𝑡 )] = 𝜃𝜃 𝑠𝑠 𝐾𝐾 𝐹𝐹 𝑒𝑒 𝑖𝑖𝜑𝜑 𝑠𝑠 𝐸𝐸 � Σ ( 𝑡𝑡 ) Σ ( 𝑡𝑡 ) + 𝑛𝑛 Σ ( 𝑡𝑡 ) � . (19) When comparing Eq. (19) with Eq. (6) and when taking into account Eq. (5), we have 𝛼𝛼 = 𝐸𝐸 � Σ ( 𝑡𝑡 ) Σ ( 𝑡𝑡 ) + 𝑛𝑛 Σ ( 𝑡𝑡 ) � . (20) From the above it is evident that, under the assumption of statistical independence between sum and difference noise, the reduction factor 𝛼𝛼 applied to the noiseless complex monopulse ratio depends only upon the noise present in the sum channel. By taking into account Eq. (7), Eq. (20) becomes 𝛼𝛼 = 𝐸𝐸 � �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 + 𝑛𝑛 Σ � + 𝐸𝐸 � 𝑥𝑥�𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 + 𝑛𝑛 Σ � , (21) having set 𝑡𝑡 = 0 due to the wide sense stationarity of all involved processes, and with the understanding that 𝑥𝑥 , 𝑛𝑛 Σ are evaluated in the time origin. Let’s tackle the first expectation in Eq. (21), and as a first step we average with respect to 𝜃𝜃 𝑐𝑐 . 𝐸𝐸 𝜃𝜃 𝑐𝑐 � �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 + 𝑛𝑛 Σ � = 12 𝜋𝜋 � 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 + 𝑛𝑛 Σ �𝑃𝑃 𝑐𝑐 𝑑𝑑𝜃𝜃 𝑐𝑐2𝜋𝜋0 . (22) By changing variable in the integral as follows: 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 = 𝑤𝑤 , and by applying the residue theorem it is immediate to verify that 𝐸𝐸 𝜃𝜃 𝑐𝑐 � �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 + 𝑛𝑛 Σ � = � 𝑥𝑥 + 𝑛𝑛 Σ | < �𝑃𝑃 𝑐𝑐 𝑥𝑥 + 𝑛𝑛 Σ | ≥ �𝑃𝑃 𝑐𝑐 . (23) 4 The remaining expectation with respect to 𝑥𝑥 , 𝑛𝑛 Σ immediately leads to 𝐸𝐸 𝑥𝑥 , 𝑛𝑛 Σ �� 𝑥𝑥 + 𝑛𝑛 Σ | < �𝑃𝑃 𝑐𝑐 𝑥𝑥 + 𝑛𝑛 Σ | ≥ �𝑃𝑃 𝑐𝑐 � = 1 − 𝑒𝑒 − 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 +𝑃𝑃 Σ . (24) Let’s now focus on the second expectation of Eq. (21). Again the average with respect to 𝜃𝜃 𝑐𝑐 is easily conducted by use of residue theorem, leading to 𝐸𝐸 𝜃𝜃 𝑐𝑐 � 𝑥𝑥�𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 + 𝑛𝑛 Σ � = � 𝑥𝑥𝑥𝑥 + 𝑛𝑛 Σ | 𝑥𝑥 + 𝑛𝑛 Σ | > �𝑃𝑃 𝑐𝑐 𝑥𝑥 + 𝑛𝑛 Σ | ≤ �𝑃𝑃 𝑐𝑐 . (25) When performing the residual averaging with respect to 𝑥𝑥 , 𝑛𝑛 Σ we obtain the following 𝐸𝐸 𝑥𝑥 , 𝑛𝑛 Σ �� 𝑥𝑥𝑥𝑥 + 𝑛𝑛 Σ | 𝑥𝑥 + 𝑛𝑛 Σ | > �𝑃𝑃 𝑐𝑐 𝑥𝑥 + 𝑛𝑛 Σ | ≤ �𝑃𝑃 𝑐𝑐 � = 𝑃𝑃 𝑥𝑥 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ 𝑒𝑒 − 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 +𝑃𝑃 Σ . (26) When using Eqs. (24) and (26) in Eq. (21) we finally obtain 𝛼𝛼 = 1 − 𝑒𝑒 − 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 +𝑃𝑃 Σ � 𝑃𝑃 Σ 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ � , (27) which is in line with Eq. (72) of [2], which is more general in allowing a statistical dependence between sum and difference noise, and however considers a constant signal term. Fig. 3 visualises Eq. (27) as a function of the following two parameters 𝑆𝑆𝑆𝑆𝑅𝑅 = 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 𝑐𝑐 𝑃𝑃 Σ , 𝜆𝜆 = 𝑃𝑃 𝑥𝑥 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 𝑐𝑐 , (28) which express the signal to noise ratio in the sum channel ( 𝑆𝑆𝑆𝑆𝑅𝑅 ) and the relative proportion between Gaussian and monochromatic components in the signal part ( 𝜆𝜆 = 1 corresponds to a purely Gaussian signal, 𝜆𝜆 = 0 to a residual carrier only). It is evident from Fig. 3 that, for a wide range of signal to noise ratios, the reduction is more pronounced for a pure Gaussian signal than for a pure monochromatic signal. Fig. 3: reduction factor 𝛼𝛼 as a function of the signal to noise ratio in the sum channel. The top curve corresponds to the case a pure monochromatic signal, whereas the bottom one is valid for a Gaussian signal. The curves in between address the simultaneous presence of both components. IV. S PECTRAL DENSITY OF THE COMPLEX RATIO
Let’s now turn our attention to the spectral density of 𝛿𝛿𝑀𝑀 ( 𝑡𝑡 ) , which can be obtained from its autocorrelation 𝑅𝑅 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ( 𝜏𝜏 ) = 𝐸𝐸 [ 𝛿𝛿𝑀𝑀 ( 𝑡𝑡 + 𝜏𝜏 ) 𝛿𝛿𝑀𝑀 ( 𝑡𝑡 ) ∗ ]. (29) Indeed the spectral density 𝑆𝑆 𝛿𝛿𝛿𝛿 ( 𝑓𝑓 ) is obtained by Fourier transform of Eq. (29). As discussed in the introduction however, we are interested in the spectral density around the origin of the complex baseband, which means that we will compute 𝑆𝑆 𝛿𝛿𝛿𝛿 (0) = � 𝑅𝑅 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ( 𝜏𝜏 ) 𝑑𝑑𝜏𝜏 ∞−∞ . (30) As discussed in the introduction we are interested in computing the above quantities during tracking, when 𝜃𝜃 𝑠𝑠 is vanishing (null tracking) and the mean of 𝑀𝑀 is zero, which, when taking into account Eq. (2) and (3), leads to 𝑅𝑅 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ( 𝜏𝜏 )= 𝐸𝐸 � 𝑛𝑛 Δ ( 𝑡𝑡 + 𝜏𝜏 ) Σ ( 𝑡𝑡 + 𝜏𝜏 ) + 𝑛𝑛 Σ ( 𝑡𝑡 + 𝜏𝜏 ) ∙ 𝑛𝑛 Δ ( 𝑡𝑡 ) ∗ Σ ( 𝑡𝑡 ) ∗ + 𝑛𝑛 Σ ( 𝑡𝑡 ) ∗ � , (31) which can be plugged in Eq. (30) to give, when using Eq. (7) and after setting 𝑡𝑡 = 0 due to the wide sense stationarity of all involved processes 𝑆𝑆 𝛿𝛿𝛿𝛿 (0)= � 𝐸𝐸 � 𝑛𝑛 Δ ( 𝜏𝜏 ) �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖 ( 𝑐𝑐 𝜏𝜏+𝜃𝜃 𝑐𝑐 ) + 𝑥𝑥 ( 𝜏𝜏 ) + 𝑛𝑛 Σ ( 𝜏𝜏 ) ∞−∞ ∙ 𝑛𝑛 Δ (0) ∗ �𝑃𝑃 𝑐𝑐 𝑒𝑒 −𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 (0) ∗ + 𝑛𝑛 Σ (0) ∗ � 𝑑𝑑𝜏𝜏 . (32) We multiply and divide the above by the same exponential term 𝑒𝑒 −𝑖𝑖2𝜋𝜋𝑓𝑓 𝑐𝑐 𝜏𝜏 to get 5 𝑆𝑆 𝛿𝛿𝛿𝛿 (0)= � 𝐸𝐸 � 𝑛𝑛 Δ′ ( 𝜏𝜏 ) �𝑃𝑃 𝑐𝑐 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 ′ ( 𝜏𝜏 ) + 𝑛𝑛 Σ′ ( 𝜏𝜏 ) ∞−∞ ∙ 𝑛𝑛 Δ′ (0) ∗ �𝑃𝑃 𝑐𝑐 𝑒𝑒 −𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑥𝑥 ′ (0) ∗ + 𝑛𝑛 Σ′ (0) ∗ � 𝑑𝑑𝜏𝜏 , (33) having defined the following signals. 𝑛𝑛 Δ′ ( 𝑡𝑡 ) = 𝑛𝑛 Δ ( 𝑡𝑡 ) 𝑒𝑒 −𝑖𝑖2𝜋𝜋𝑓𝑓 𝑐𝑐 𝑡𝑡 , 𝑛𝑛 Σ′ ( 𝑡𝑡 ) = 𝑛𝑛 Σ ( 𝑡𝑡 ) 𝑒𝑒 −𝑖𝑖2𝜋𝜋𝑓𝑓 𝑐𝑐 𝑡𝑡 , 𝑥𝑥 ′ ( 𝑡𝑡 ) = 𝑥𝑥 ( 𝑡𝑡 ) 𝑒𝑒 −𝑖𝑖2𝜋𝜋𝑓𝑓 𝑐𝑐 𝑡𝑡 . (34) The above corresponds to have frequency shifted all signal and noise spectra (residual carrier and Gaussian terms) by −𝑓𝑓 𝑐𝑐 , thus positioning the residual carrier in the centre of the complex baseband. In the frame of the analysis we will make reference to the total Gaussian signal in the sum channel, including the signal component 𝑥𝑥 ′ ( 𝑡𝑡 ) and the noise 𝑛𝑛 Σ′ ( 𝑡𝑡 ) 𝑧𝑧 ( 𝑡𝑡 ) = 𝑧𝑧 𝑐𝑐 ( 𝑡𝑡 ) + 𝑖𝑖𝑧𝑧 𝑠𝑠 ( 𝑡𝑡 ) = 𝑥𝑥 ′ ( 𝑡𝑡 ) + 𝑛𝑛 Σ′ ( 𝑡𝑡 ). (35) The process 𝑧𝑧 ( 𝑡𝑡 ) is WSS, as it can be easily verified by computing 𝐸𝐸 [ 𝑧𝑧 ( 𝑡𝑡 ) 𝑧𝑧 ( 𝑡𝑡 ) ∗ ] , and using Eqs. (34) , (35) as well as the fact that 𝑥𝑥 ( 𝑡𝑡 ), 𝑛𝑛 Σ ( 𝑡𝑡 ) are assumed to be zero mean WSS processes. The autocorrelation of 𝑧𝑧 ( 𝑡𝑡 ) can be expressed as 𝑅𝑅 𝑧𝑧𝑧𝑧 ( 𝜏𝜏 ) = 2 𝑅𝑅 𝑧𝑧 𝑐𝑐 𝑧𝑧 𝑐𝑐 ( 𝜏𝜏 ) − 𝑖𝑖 𝑅𝑅 𝑧𝑧 𝑐𝑐 𝑧𝑧 𝑠𝑠 ( 𝜏𝜏 )= 𝑃𝑃�𝜌𝜌 ( 𝜏𝜏 ) − 𝑖𝑖𝜇𝜇 ( 𝜏𝜏 ) � , (36) where 𝑃𝑃 = 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ (37) and with similar properties as presented in section II for the originating Gaussian signal. Furthermore we define also in this case the normalised correlation coefficient as follows 𝑟𝑟 ( 𝜏𝜏 ) = | 𝑟𝑟 ( 𝜏𝜏 )| 𝑒𝑒 𝑖𝑖𝜑𝜑 ( 𝜏𝜏 ) = 𝜌𝜌 ( 𝜏𝜏 ) − 𝑖𝑖𝜇𝜇 ( 𝜏𝜏 ). (38) Due to the assumed statistical independence between the noise on the difference channel and the signal and noise components in the sum channel, Eq. (33) becomes 𝑆𝑆 𝛿𝛿𝛿𝛿 (0) = � 𝑅𝑅 𝑛𝑛 Δ′ 𝑛𝑛 Δ′ ( 𝜏𝜏 ) 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) 𝑑𝑑𝜏𝜏 ∞−∞ , (39) where we have denoted with 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) the autocorrelation of the following complex signal 𝑠𝑠 ( 𝑡𝑡 ) ≜ �𝑃𝑃 𝑐𝑐 ∙ 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑧𝑧 ( 𝑡𝑡 ). (40) From now onwards we move our attention to the “difficult” term in Eq. (39), namely 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) , and we will re-discuss 𝑆𝑆 𝛿𝛿𝛿𝛿 (0) only at the end of the section. In order to simplify the terminology we define subscripts for instances of the same variable separated by a time interval 𝜏𝜏 , define 𝐴𝐴 = �𝑃𝑃 𝑐𝑐 , and re-write 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) as follows 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 )= 𝐸𝐸 � 𝐴𝐴 ∙ 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑧𝑧 𝑐𝑐 , + 𝑖𝑖𝑧𝑧 𝑠𝑠 , ∙ 𝐴𝐴 ∙ 𝑒𝑒 −𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑧𝑧 𝑐𝑐 , − 𝑖𝑖𝑧𝑧 𝑠𝑠 , � . (41) The computation of the above expectation requires the knowledge of the covariance matrix for the Gaussian column vector 𝐳𝐳 = { 𝑧𝑧 𝑐𝑐 , , 𝑧𝑧 𝑠𝑠 , , 𝑧𝑧 𝑐𝑐 , , 𝑧𝑧 𝑠𝑠 , }’ (the superscript ′ denotes vector transposition), which, based on Eq. (36), is simply given by 𝐑𝐑 = 𝑃𝑃 � 𝜌𝜌 −𝜇𝜇𝜇𝜇 𝜌𝜌𝜌𝜌 𝜇𝜇−𝜇𝜇 𝜌𝜌 � . (42) The expectation (41) can be computed as follows 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 )= 12 𝜋𝜋 � � � 𝑝𝑝�𝑧𝑧 𝑐𝑐 , , 𝑧𝑧 𝑠𝑠 , , 𝑧𝑧 𝑐𝑐 , , 𝑧𝑧 𝑠𝑠 , �𝐴𝐴 ∙ 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑧𝑧 𝑐𝑐 , + 𝑖𝑖𝑧𝑧 𝑠𝑠 , ∙ 𝑑𝑑𝑧𝑧 𝑐𝑐 , 𝑑𝑑𝑧𝑧 𝑠𝑠 , 𝑑𝑑𝑧𝑧 𝑐𝑐 , 𝑑𝑑𝑧𝑧 𝑠𝑠 , 𝐴𝐴 ∙ 𝑒𝑒 −𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑧𝑧 𝑐𝑐 , − 𝑖𝑖𝑧𝑧 𝑠𝑠 , � 𝑑𝑑𝜃𝜃 𝑐𝑐 , (43) where 𝑝𝑝 ( ∙ ) is the multivariate Gaussian probability density function associated to the vector 𝐳𝐳 . Before proceeding further let’s apply the following unitary variable transformation to the inner integral 𝐳𝐳 = � cos 𝜃𝜃 𝑐𝑐 − sin 𝜃𝜃 𝑐𝑐 𝜃𝜃 𝑐𝑐 cos 𝜃𝜃 𝑐𝑐 𝜃𝜃 𝑐𝑐 − sin 𝜃𝜃 𝑐𝑐 𝜃𝜃 𝑐𝑐 cos 𝜃𝜃 𝑐𝑐 � ∙ 𝐰𝐰 , (44) such that the inner integral becomes, after simples computations � 𝑝𝑝�𝑧𝑧 𝑐𝑐 , , 𝑧𝑧 𝑠𝑠 , , 𝑧𝑧 𝑐𝑐 , , 𝑧𝑧 𝑠𝑠 , �𝐴𝐴 ∙ 𝑒𝑒 𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑧𝑧 𝑐𝑐 , + 𝑖𝑖𝑧𝑧 𝑠𝑠 , ∙ 𝑑𝑑𝑧𝑧 𝑐𝑐 , 𝑑𝑑𝑧𝑧 𝑠𝑠 , 𝑑𝑑𝑧𝑧 𝑐𝑐 , 𝑑𝑑𝑧𝑧 𝑠𝑠 , 𝐴𝐴 ∙ 𝑒𝑒 −𝑖𝑖𝜃𝜃 𝑐𝑐 + 𝑧𝑧 𝑐𝑐 , − 𝑖𝑖𝑧𝑧 𝑠𝑠 , = � 𝑝𝑝�𝑤𝑤 𝑐𝑐 , , 𝑤𝑤 𝑠𝑠 , , 𝑤𝑤 𝑐𝑐 , , 𝑤𝑤 𝑠𝑠 , �𝐴𝐴 + 𝑤𝑤 𝑐𝑐 , + 𝑖𝑖𝑤𝑤 𝑠𝑠 , ∙ 𝑑𝑑𝑤𝑤 𝑐𝑐 , 𝑑𝑑𝑤𝑤 𝑠𝑠 , 𝑑𝑑𝑤𝑤 𝑐𝑐 , 𝑑𝑑𝑤𝑤 𝑠𝑠 , 𝐴𝐴 + 𝑤𝑤 𝑐𝑐 , − 𝑖𝑖𝑤𝑤 𝑠𝑠 , , (45) which is independent from 𝜃𝜃 𝑐𝑐 , and with 𝑝𝑝 ( ∙ ) being the same multivariate Gaussian probability density function introduced in Eq. (43) . Based on the above we can simply re-write Eq. (43) without the averaging with respect to 𝜃𝜃 𝑐𝑐 , and setting 𝜃𝜃 𝑐𝑐 equal to zero. 6 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 )= � 𝑝𝑝 ( 𝐳𝐳 ) �𝐴𝐴 + 𝑧𝑧 𝑐𝑐 , + 𝑖𝑖𝑧𝑧 𝑠𝑠 , ��𝐴𝐴 + 𝑧𝑧 𝑐𝑐 , − 𝑖𝑖𝑧𝑧 𝑠𝑠 , � 𝑑𝑑 𝐳𝐳 ∞−∞ . (46) The multivariate density 𝑝𝑝 ( 𝐳𝐳 ) is related to its characteristic function 𝑐𝑐 ( 𝐱𝐱 ) by 𝑝𝑝 ( 𝐳𝐳 ) = 1(2 𝜋𝜋 ) � 𝑐𝑐 ( 𝐱𝐱 ) 𝑒𝑒 −𝑖𝑖𝐱𝐱 ′ ∙𝐳𝐳 𝑑𝑑 𝐱𝐱 ∞−∞ , (47) where 𝐱𝐱 = { 𝑥𝑥 𝑐𝑐 , , 𝑥𝑥 𝑠𝑠 , , 𝑥𝑥 𝑐𝑐 , , 𝑥𝑥 𝑠𝑠 , }’, ∙ denotes scalar product and where, for a zero mean Gaussian vector (Eq. (8-57) in [5]) 𝑐𝑐 ( 𝐱𝐱 ) = 𝑒𝑒 −12𝐱𝐱 ′ ∙𝐑𝐑∙𝐱𝐱 . (48) When using Eqs. (47) and (48) into Eq. (46) we have 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) = 1(2 𝜋𝜋 ) � 𝑒𝑒 −12𝐱𝐱 ′ ∙𝐑𝐑∙𝐱𝐱 𝑑𝑑 𝐱𝐱 ∞−∞ ∙ � 𝑒𝑒 −𝑖𝑖𝐱𝐱 ′ ∙𝐳𝐳 �𝐴𝐴 + 𝑧𝑧 𝑐𝑐 , + 𝑖𝑖𝑧𝑧 𝑠𝑠 , ��𝐴𝐴 + 𝑧𝑧 𝑐𝑐 , − 𝑖𝑖𝑧𝑧 𝑠𝑠 , � 𝑑𝑑 𝐳𝐳 ∞−∞ . (49) We focus now on the inner integral 𝐼𝐼 = � 𝑒𝑒 −𝑖𝑖𝐱𝐱 ′ ∙𝐳𝐳 �𝐴𝐴 + 𝑧𝑧 𝑐𝑐 , + 𝑖𝑖𝑧𝑧 𝑠𝑠 , ��𝐴𝐴 + 𝑧𝑧 𝑐𝑐 , − 𝑖𝑖𝑧𝑧 𝑠𝑠 , � 𝑑𝑑 𝐳𝐳 ∞−∞ (50) and perform the following change of variables 𝐯𝐯 = 𝐴𝐴𝐰𝐰 + 𝐳𝐳 , (51) where 𝐯𝐯 = �𝑣𝑣 𝑐𝑐 , , 𝑣𝑣 𝑠𝑠 , , 𝑣𝑣 𝑐𝑐 , , 𝑣𝑣 𝑠𝑠 , � ′ , 𝐰𝐰 = {1,0,1,0} ′ , (52) to get 𝐼𝐼 = 𝑒𝑒 𝑖𝑖𝑖𝑖𝐱𝐱 ′ ∙𝐰𝐰 � 𝑒𝑒 −𝑖𝑖𝐱𝐱 ′ ∙𝐯𝐯 �𝑣𝑣 𝑐𝑐 , + 𝑖𝑖𝑣𝑣 𝑠𝑠 , ��𝑣𝑣 𝑐𝑐 , − 𝑖𝑖𝑣𝑣 𝑠𝑠 , � 𝑑𝑑 𝐯𝐯 ∞−∞ . (53) The integral in Eq. (53) is a 4-dimensional inverse Fourier transform which can be immediately computed, to give 𝐼𝐼 = − 𝜋𝜋 𝑒𝑒 𝑖𝑖𝑖𝑖𝐱𝐱 ′ ∙𝐰𝐰 �𝑥𝑥 𝑐𝑐 , − 𝑖𝑖𝑥𝑥 𝑠𝑠 , ��𝑥𝑥 𝑐𝑐 , + 𝑖𝑖𝑥𝑥 𝑠𝑠 , � . (54) The above result can also be obtained by changing to double polar coordinates in the integral of Eq. (53), by using Eq. (3.338) in [6] for the integration over the angle variables, and and by finally integrating over the radial variables. When plugging Eq. (54) into Eq. (49), after applying the following change of variables ( 𝑗𝑗 = 1,2) 𝑥𝑥 𝑐𝑐 , 𝑗𝑗 = 𝑟𝑟 𝑗𝑗 cos 𝑞𝑞 𝑗𝑗 , 𝑥𝑥 𝑠𝑠 , 𝑗𝑗 = 𝑟𝑟 𝑗𝑗 sin 𝑞𝑞 𝑗𝑗 , (55) with 𝑟𝑟 , 𝑟𝑟 ∈ [0, ∞ ) and 𝑞𝑞 , 𝑞𝑞 ∈ [0,2 𝜋𝜋 ) , and by using Eq. (38) we obtain 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) = − 𝜋𝜋 (2 𝜋𝜋 ) � 𝑒𝑒 −𝑃𝑃4�𝑟𝑟 +𝑟𝑟 � 𝑑𝑑𝑟𝑟 𝑑𝑑𝑟𝑟 ∙ � 𝑒𝑒 − | 𝑟𝑟 | 𝑃𝑃𝑟𝑟 𝑟𝑟 ( 𝑞𝑞 −𝑞𝑞 −𝜑𝜑 ) ∙ 𝑒𝑒 𝑖𝑖𝑖𝑖𝑟𝑟 cos 𝑞𝑞 +𝑖𝑖𝑖𝑖𝑟𝑟 cos 𝑞𝑞 𝑒𝑒 𝑖𝑖 ( 𝑞𝑞 −𝑞𝑞 ) 𝑑𝑑𝑞𝑞 𝑑𝑑𝑞𝑞 . (56) We then use the following expansion exp( 𝑖𝑖𝑖𝑖 cos 𝑥𝑥 ) = � ( 𝑖𝑖 ) 𝑛𝑛 𝐽𝐽 𝑛𝑛 ( 𝑖𝑖 ) exp( 𝑖𝑖𝑛𝑛𝑥𝑥 ) ∞𝑛𝑛=−∞ , (57) where 𝐽𝐽 𝑣𝑣 ( ∙ ) is the Bessel function of order 𝑣𝑣 . By use of Eq. (57) we transform Eq. (56) into the following 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) = − 𝜋𝜋 (2 𝜋𝜋 ) � 𝑒𝑒 −𝑃𝑃4�𝑟𝑟 +𝑟𝑟 � 𝑑𝑑𝑟𝑟 𝑑𝑑𝑟𝑟 ∙ � ( 𝑖𝑖 ) 𝑛𝑛+𝑚𝑚 𝐽𝐽 𝑛𝑛 ( 𝐴𝐴𝑟𝑟 ) 𝐽𝐽 𝑚𝑚 ( 𝐴𝐴𝑟𝑟 ) ∞𝑛𝑛 , 𝑚𝑚=−∞ ∙ � 𝑒𝑒 − | 𝑟𝑟 | 𝑃𝑃𝑟𝑟 𝑟𝑟 ( 𝑞𝑞 −𝑞𝑞 −𝜑𝜑 ) ∙ 𝑒𝑒 𝑖𝑖𝑞𝑞 ( 𝑛𝑛+1 ) −𝑖𝑖𝑞𝑞 ( 𝑚𝑚+1 ) 𝑑𝑑𝑞𝑞 𝑑𝑑𝑞𝑞 . (58) The inner integral of Eq. (58) can be computed easily, by lengthy computations around the integral representation of Eq. (8.431-5) in [6], to obtain � 𝑒𝑒 − | 𝑟𝑟 | 𝑃𝑃𝑟𝑟 𝑟𝑟 ( 𝑞𝑞 −𝑞𝑞 −𝜑𝜑 ) ∙ 𝑒𝑒 𝑖𝑖𝑞𝑞 ( 𝑛𝑛+1 ) −𝑖𝑖𝑞𝑞 ( 𝑚𝑚+1 ) 𝑑𝑑𝑞𝑞 𝑑𝑑𝑞𝑞 = 4 𝜋𝜋 ( − 𝑛𝑛+1 𝐼𝐼 𝑛𝑛+1 � | 𝑟𝑟 | 𝑃𝑃𝑟𝑟 𝑟𝑟 � 𝑒𝑒 𝑖𝑖𝜑𝜑 ( 𝑛𝑛+1 ) 𝛿𝛿 𝑛𝑛𝑚𝑚 , (59) where 𝛿𝛿 𝑛𝑛𝑚𝑚 is the Kronecker delta symbol and 𝐼𝐼 𝑣𝑣 ( ∙ ) is the modified Bessel function of the first kind of order 𝑣𝑣 . When plugging Eq. (59) into Eq. (58) we obtain 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) = � 𝑒𝑒 𝑖𝑖𝜑𝜑 ( 𝑛𝑛+1 ) ∞𝑛𝑛=−∞ � 𝐽𝐽 𝑛𝑛 ( 𝐴𝐴𝑟𝑟 ) 𝐽𝐽 𝑛𝑛 ( 𝐴𝐴𝑟𝑟 ) ∞0 ∙ 𝑒𝑒 −𝑃𝑃4�𝑟𝑟 +𝑟𝑟 � 𝐼𝐼 𝑛𝑛+1 � | 𝑟𝑟 | 𝑃𝑃𝑟𝑟 𝑟𝑟 � 𝑑𝑑𝑟𝑟 𝑑𝑑𝑟𝑟 . (60) The integrals in Eq. (60) are solved by expanding the functions 𝐼𝐼 𝑣𝑣 ( ∙ ) in power series and by using Eq. (6.631) in [6] 7 to resolve the resulting integrals, to finally get, after laborious manipulations 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) = 1 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ � ( 𝜌𝜌 + 𝑖𝑖𝜇𝜇 ) 𝑛𝑛+1 � 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ � 𝑛𝑛 Γ (1 + 𝑛𝑛 ) ∙ � ( 𝜌𝜌 + 𝜇𝜇 ) 𝑘𝑘 Γ ( 𝑛𝑛 + 1 + 𝑘𝑘 ) Γ ( 𝑘𝑘 + 1)( 𝑛𝑛 + 1 + 𝑘𝑘 ) ∞𝑘𝑘=0 ∙ 𝐹𝐹 � 𝑘𝑘 + 𝑛𝑛 , 1 + 𝑛𝑛 , − 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ � + 1 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ � ( 𝜌𝜌 − 𝑖𝑖𝜇𝜇 ) 𝑚𝑚 � 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ � 𝑚𝑚+1 Γ (2 + 𝑚𝑚 ) ∙ � ( 𝜌𝜌 + 𝜇𝜇 ) 𝑘𝑘 Γ ( 𝑚𝑚 + 1 + 𝑘𝑘 ) Γ ( 𝑘𝑘 + 1) ∞𝑘𝑘=0 ∙ 𝐹𝐹 � 𝑘𝑘 + 𝑚𝑚 , 2 + 𝑚𝑚 , − 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ � , (61) where 𝐹𝐹 ( ∙ , ∙ , ∙ ) is the confluent hypergeometric function defined e.g. in Eq. (9.210-1) in [6], and where we have re-introduced 𝑃𝑃 𝑐𝑐 = 𝐴𝐴 and 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ = 𝑃𝑃 . Eq. (61) constitutes the main output of this paper, and can be used in Eq. (39) to recover 𝑆𝑆 𝛿𝛿𝛿𝛿 (0) . We notice that if only Gaussian components are present in the sum channel ( 𝑃𝑃 𝑐𝑐 = 0) , the Eq. (61) reduces [ 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 )] 𝑃𝑃 𝑐𝑐 =0 = − 𝑅𝑅 𝑧𝑧𝑧𝑧 ( 𝜏𝜏 ) ln � − | 𝑅𝑅 𝑧𝑧𝑧𝑧 ( 𝜏𝜏 )| ( 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ ) � , (62) which is equivalent to Eq. (38) of [3], when assuming one target point and no offset. It is important to remark that both Eqs. (61) and (62) diverge for 𝜏𝜏 = 0 , which indicates that, irrespectively of the presence of a monochromatic signal of any finite amplitude, the process 𝑠𝑠 ( 𝑡𝑡 ) of Eq. (40) has infinite power, which of course does not prevent 𝑆𝑆 𝛿𝛿𝛿𝛿 (0) of Eq. (39) from being finite. V. A N APPLICATION
Let’s assume that the Gaussian random process on the sum channel (combination of signal and noise component) has a flat power spectrum over a bandwidth 𝑊𝑊 centred in the zero frequency of the complex baseband, with total power 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ . On the delta channel there is the same spectral structure with power 𝑃𝑃 Δ . The monochromatic component (available only in the sum channel, during null tracking) is positioned at 𝑓𝑓 𝑐𝑐 with power 𝑃𝑃 𝑐𝑐 . According to the procedure built in the previous section, we first shift all Gaussian and monochromatic spectra by −𝑓𝑓 𝑐𝑐 such that the autocorrelation of 𝑧𝑧 ( 𝑡𝑡 ) and of 𝑛𝑛 Δ′ of Eq. (36) and (39) are given by 𝑅𝑅 𝑧𝑧𝑧𝑧 ( 𝜏𝜏 ) = ( 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ ) sin( 𝜋𝜋𝑊𝑊𝜏𝜏 ) 𝜋𝜋𝑊𝑊𝜏𝜏 𝑒𝑒 −𝑖𝑖2𝜋𝜋𝑓𝑓 𝑐𝑐 𝜏𝜏 , 𝑅𝑅 𝑛𝑛 Δ′ 𝑛𝑛 Δ′ ( 𝜏𝜏 ) = 𝑃𝑃 Δ sin( 𝜋𝜋𝑊𝑊𝜏𝜏 ) 𝜋𝜋𝑊𝑊𝜏𝜏 𝑒𝑒 −𝑖𝑖2𝜋𝜋𝑓𝑓 𝑐𝑐 𝜏𝜏 , (63) which according to Eq. (36) implies that 𝜌𝜌 ( 𝜏𝜏 ) − 𝑖𝑖𝜇𝜇 ( 𝜏𝜏 ) = sin( 𝜋𝜋𝑊𝑊𝜏𝜏 ) 𝜋𝜋𝑊𝑊𝜏𝜏 𝑒𝑒 −𝑖𝑖2𝜋𝜋𝑓𝑓 𝑐𝑐 𝜏𝜏 . (64) When inserting the above into Eq. (61) to obtain 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) , after plugging the results into Eq. (39) and when taking into account the expression of 𝑅𝑅 𝑛𝑛 Δ′ 𝑛𝑛 Δ′ ( 𝜏𝜏 ) in Eq. (63) we obtain 𝑆𝑆 𝛿𝛿𝛿𝛿 (0) = 𝑃𝑃 Δ 𝑊𝑊 ( 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ ) 𝜒𝜒 � 𝑃𝑃 𝑐𝑐 𝑃𝑃 𝑥𝑥 + 𝑃𝑃 Σ , 𝑓𝑓 𝑐𝑐 ( 𝑊𝑊 ⁄ ) � , (65) where we have defined the general function 𝜒𝜒 ( 𝑖𝑖 , 𝑏𝑏 ) = 2 𝜋𝜋 � 𝑖𝑖 𝑛𝑛 Γ ( 𝑛𝑛 + 1 + 𝑘𝑘 ) 𝑐𝑐 ( 𝑛𝑛𝑏𝑏 ) Γ (1 + 𝑛𝑛 ) Γ ( 𝑘𝑘 + 1)( 𝑛𝑛 + 1 + 𝑘𝑘 ) ∞𝑛𝑛 , 𝑘𝑘=0 ∙ 𝐹𝐹 (1 + 𝑘𝑘 + 𝑛𝑛 , 1 + 𝑛𝑛 , −𝑖𝑖 ) + 2 𝜋𝜋 � 𝑖𝑖 𝑚𝑚+1 Γ ( 𝑚𝑚 + 1 + 𝑘𝑘 ) 𝑐𝑐 � ( 𝑚𝑚 + 1) 𝑏𝑏�Γ (2 + 𝑚𝑚 ) Γ ( 𝑘𝑘 + 1) ∞𝑚𝑚 , 𝑘𝑘=0 ∙ 𝐹𝐹 (1 + 𝑘𝑘 + 𝑚𝑚 , 2 + 𝑚𝑚 , −𝑖𝑖 ) (66) and where the coefficients 𝑐𝑐 have been defined as 𝑐𝑐 𝑛𝑛 ( 𝑚𝑚 ) ≜ � � sin 𝑦𝑦𝑦𝑦 � 𝑛𝑛 𝑒𝑒 𝑖𝑖𝑚𝑚𝑦𝑦 𝑑𝑑𝑦𝑦 ∞−∞ (67) and can be found according to Eq. (3.836-2) of [6] 𝑐𝑐 𝑛𝑛 ( 𝑚𝑚 ) = 𝑐𝑐 𝑛𝑛 ( −𝑚𝑚 )= ⎩⎪⎨⎪⎧𝑛𝑛𝜋𝜋 𝑛𝑛 � ( − 𝑣𝑣 ( 𝑛𝑛 + 𝑚𝑚 − 𝑣𝑣 ) 𝑛𝑛−1 𝑣𝑣 ! ( 𝑛𝑛 − 𝑣𝑣 )! �𝑚𝑚+𝑛𝑛2 �𝑣𝑣=0 ≤ 𝑚𝑚 < 𝑛𝑛 𝑚𝑚 ≥ 𝑛𝑛 , 𝑛𝑛 ≥ 𝜋𝜋 ⁄ 𝑚𝑚 = 𝑛𝑛 = 1 . (68) It is to be noted that for the case of vanishing monochromatic signal one can use directly the closed form expression of 𝑅𝑅 𝑠𝑠𝑠𝑠 ( 𝜏𝜏 ) of Eq. (62) to obtain the following limit case lim 𝑎𝑎→0 𝜒𝜒 ( 𝑖𝑖 , 𝑏𝑏 ) = − 𝜋𝜋 � ln � − � sin( 𝑦𝑦 ) 𝑦𝑦 � � 𝑑𝑑𝑦𝑦 ∞−∞ ≅ (69) The function 𝜒𝜒 ( 𝑖𝑖 , 𝑏𝑏 ) was computed for various values of 𝑏𝑏 between and ( ≤ 𝑏𝑏 ≤ corresponds to the realistic case of the monochromatic signal being within the noise spectrum), and for a set of values of 𝑖𝑖 between and . After trials verifying proper numerical convergence of the sums in Eq. (66), the range from to was selected for the indexes 𝑛𝑛 , 𝑚𝑚 , whereas 𝑘𝑘 covered the range from to . Furthermore the function 𝜒𝜒 ( 𝑖𝑖 , 𝑏𝑏 ) was also obtained as result of simulations where narrowband noise and signal components were digitally 8 generated, the complex monopulse ratio formed according to Eq. (18), and its power spectrum estimated at the origin of the complex baseband. The results of theoretical computations (black points and dotted line) and simulations (red filled points) are presented in Fig. 4. Fig. 4: computation of the normalized function 𝜒𝜒 ( 𝑖𝑖 , 𝑏𝑏 ) of Eq. (66) for various values of 𝑖𝑖 , 𝑏𝑏 . The red filled points represent simulations. It is to be noted that only values of 𝑏𝑏 up to are of practical interest, as they correspond to a monochromatic signal within the noise spectrum. However the computations and simulations were extended to 𝑏𝑏 up to to verify the agreement between theory and simulations. VI. C ONCLUSIONS
The results of this paper can contribute to the design of closed loop tracking system for satellite missions, in those cases where the incoming signal cannot be modelled, when available at the input of the monopulse processor, as a simple Gaussian process. The Eq. (61), even though complex and involving double sums as well as evaluation of special function, can be practically used for computing the spectral density of the complex ratio fluctuations in the low frequency region. Such fluctuations, filtered by the time integration (4) and ultimately by the transfer function of the closed loop system, translate in a mechanical angular jitter of the antenna system during tracking and have a critical impact on the system performance. A CKNOWLEDGMENT
The author acknowledges the extremely valuable inputs from Dr. Yang Yang, of the National Laboratory of Radar Signal Processing, Xidian University, aiming at improving the content and formal correctness of the manuscript. R
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