2.2.1. GMSK

The GMSK modulation used in AIS belongs to the family of Continuous Phase Modulations (CPM). A CPM-modulated signal can be written as followsFootnote ¹ (Sykora, Reference Sykora2003):

(1)$$s(t)=\sqrt{\displaystyle{2E_{{\rm S}} \over T_{{\rm S}}}} e^{\,{\rm j}\phi (t,{{\bf d}})}$$

where d = {d _n}_n∈ℤ is a vector of data symbols from the alphabet { ± 1, ± 3, …, ± (M − 1)}, E _S is the average energy per symbol of the real modulated signal, T _S is the symbol period, ${\rm j}=\sqrt{-1}$ is the imaginary unit, and ϕ(t, d) is the instantaneous phase:

(2)$$\phi ( t,{\bf d})=2\pi\kappa \sum_n \,d_{n}\beta (t-nT_{{\rm S}}).$$

The parameter κ in Equation (2) is commonly referred to as the modulation index. β (t) is the phase response of the modulator, which is related to its frequency response, μ (t), by:

(3)$$\beta (t)=\int_{-\infty}^t {\mu (s)\,ds}$$

For GMSK, M=2, κ=1/2 and the frequency response is obtained as a convolution of a rectangular pulse and a Gaussian-shaped pulse (hence the name of the modulation); the frequency response is given by the following expression (Mengali and D'Andrea, Reference Mengali and D'Andrea1997):

(4)$$\mu (t)=\displaystyle{Q\left( \displaystyle{{2}\pi B}{\sqrt{\ln 2}}\lpar t-\displaystyle{L+1}{2}T_{{\rm S}} \rpar \right)-Q\left(\displaystyle{2\pi B}{\sqrt{\ln 2}} \lpar t-\displaystyle{L-1}{2}T_{{\rm S}} \rpar \right) \over 2T_{{\rm S}}}$$

where B (bandwidth parameter) and L (correlation length) are system design parameters and Q(x) is the complementary cumulative distribution function of the standard Gaussian distribution:

(5)$$Q(x)=\displaystyle{1 \over \sqrt{2\pi } }\int_x^{\infty}e^{-y^{2}/ 2}\,dy$$

The AIS specification (ITU, 2014) requires that BT _S = 0 · 4; however, it does not appear to state any requirements on the correlation length, L. Johnson and Swaszek (Reference Johnson and Swaszek2014a) state that, for GMSK, typically L=4 or 5; therefore L=4 is assumed in this document.

2.2.2. PSK/QAM

The Pi/4-QPSK, 8-PSK and 16-QAM modulations used in VDES all belong to the class of linear modulations; therefore, the modulated signal has the form:

(6)$$s(t)=\sum_n q_{n} h( t-nT_{{\rm S}})$$

where q _n are complex-valued channel symbols and h(t) is a suitable modulation pulse, as discussed further below.

In the case of the Pi/4-QPSK modulation, the channel symbols can be determined from the data symbols, d _n, and internal states of the modulator, σ_n, as follows (IALA, 2018):

(7)$$\eqalign{q_{n} &= d_{n}e^{\,{\rm j} \displaystyle{\pi \over 4} \sigma_{n}} \cr \sigma_{n} &= \left\{\matrix{1, \hfill & n=0 \hfill\cr (\sigma_{n-1}+1)\hbox{mod } 2, \hfill & n \gt 0 \hfill}\right.}$$

where $a\ \hbox{mod}\ b$ denotes the remainder of a after division by b. The data symbols are taken from the alphabet:

(8)$$d_{n}\in \left\{e^{\,{\rm j}\displaystyle{\pi \over 2}i}\right\}_{i=0}^{3}$$

and consequently, the channel symbol alphabet (also referred to here as the constellation) has the following eight members:

(9)$$q_{n}\in \left\{ e^{\,{\rm j}\displaystyle{\pi \over 4}i}\right\}_{i=0}^{7}$$

In other words, the channel symbols are alternately drawn from two QPSK constellations, rotated with respect to each other by π/4 rad (which gives the modulation its name). This arrangement reduces the peak-to-average power ratio of the modulated signal by ensuring that its envelope never collapses to zero. A low peak-to-average power ratio is desirable as it allows the use of more efficient transmitter power amplifiers (Raab et al., Reference Raab, Asbeck, Cripps, Kenington, Popovic, Pothecary, Sevic and Sokal2002).

For 8-PSK, the channel symbols are equal to the data symbols and are given by:

(10)$$\eqalign{& d_{n}\in \left\{ e^{\,{\rm j}\displaystyle{\pi \over 4} i}\right\}_{i=0}^{7}\cr & q_{n}=d_{n},}$$

For 16-QAM, the symbols are given by:

(11)$$\eqalign{& d_{n}\in \{ ( 2i_{1}-5)+{\rm j}(2i_{2}-5)\}_{i_{1}=1,i_{2}=1}^{4,4}\cr & q_{n}=d_{n}}$$

Arguably the most common modulation pulse, h(t), used in digital communication systems is the Root-Raised Cosine (RRC) pulse, which is defined in the frequency domain as followsFootnote ²:

(12)$$H(f) = {\cal F}\{ h(t)\} = \left\{\matrix{\sqrt{T_{\rm S}}, \hfill & \vert\,f \vert \lt \displaystyle{1-\alpha \over 2T_{{\rm S}}} \hfill \cr \sqrt{T_{\rm S}} \cos \left( \displaystyle{\pi T_{{\rm S}} \over 2\alpha} \left( \vert\,f \vert -\displaystyle{1-\alpha \over 2T_{{\rm S}}} \right)\right), \hfill & \displaystyle{1-\alpha \over 2T_{{\rm S}}}\le \vert\,f\vert \lt \displaystyle{1+\alpha \over 2T_{{\rm S}}} \hfill \cr 0,\hfill & \displaystyle{1+\alpha \over 2T_{{\rm S}}}\le \vert\,f\vert \hfill}\right.$$

The coefficient α, 0<α<1, is referred to as the “roll-off factor” and is a measure of the excess bandwidth, that is, the bandwidth occupied by the modulated signal beyond the Nyquist bandwidth of 1/(2T _S). The VDES specification (IALA, 2018) does not explicitly specify the type of the modulation pulse; however, it does specify the roll-off factors for the different modulations, implying the RRC pulse should be used. The values of α stated in the specification (and used in this document) are summarised in Table 2.

Table 2. RRC roll-off factor values used in VDES.

As will become clear later, the two variants of PSK and the QAM modulation are equivalent as far as the ranging performance is concerned; therefore these modulations will be treated in the following sections as one case (referred to as ‘PSK/QAM’).

4.1.1. GMSK (AIS)

For a GMSK signal (see Section 2.2.1), the denominator in Equation (15) can be written as:

(16)$$\eqalign{&{\rm E}_{{{\bf d}}}\left\{\int_{0}^{T_{{\rm o}}} {\left\vert \displaystyle{\partial \over \partial \tau }\sqrt{\displaystyle{2E_{{\rm S}} \over T_{{\rm S}}}} e^{\,{\rm j}\phi \lpar t-\tau, {{\bf d}} \rpar } \right\vert^{2}dt}\right\}\cr &\quad ={\rm E}_{{{\bf d}}}\left\{ \int_{0}^{T_{{\rm o}}} {\left\vert \displaystyle{\partial \over \partial \tau }\sqrt{\displaystyle{2E_{{\rm S}} \over T_{{\rm S}}}} e^{\,{\rm j}\pi \sum_n {d_{n}\beta \lpar t-\tau -nT_{{\rm S}} \rpar } } \right\vert^{2}dt} \right\} \cr &\quad =\displaystyle{2E_{{\rm S}} \over T_{{\rm S}}}E_{{{\bf d}}}\left\{ \int_{0}^{T_{{\rm o}}} {\left\vert e^{{\rm j}\pi \sum_n {d_{n}\beta \lpar t-\tau -nT_{{\rm S}} \rpar } }{\cdot {\rm j}}\pi \sum_n {d_{n}\mu \lpar t-\tau -nT_{{\rm S}} \rpar } \right\vert^{2}dt} \right\}\cr &\quad =\displaystyle{2{\pi^{2}E}_{{\rm S}} \over T_{{\rm S}}}\int_{0}^{T_{{\rm o}}} {E_{{{\bf d}}}\left\{ \left\vert \sum_n {d_{n}\mu \lpar t-\tau -nT_{{\rm S}} \rpar } \right\vert^{2} \right\}dt,}}$$

where $\mu (t)=\displaystyle{\partial \beta \over \partial t}$ is the GMSK frequency response, as defined in Section 2.2.1.

Using the assumption of uncorrelated data symbols (see Section 2.5), the linearity of the expectation operator and the fact that, for GMSK, E{|d _n|²} = 1, the expression above can be simplified to:

(17)$$\displaystyle{2{\pi^{2}E}_{{\rm S}} \over T_{{\rm S}}}\int_{0}^{T_{{\rm o}}} \,\sum_n {\mu^{2}\lpar t-\tau -nT_{{\rm S}} \rpar } dt.$$

Further, by using the Poisson summation formula (Pinsky, Reference Pinsky2008) and assuming that the observation interval, T _o, is an integer multiple of the symbol interval, T _o = L _oT _S, L _o ∈ ℕ, the expression can be rewritten as:

(18)$$\displaystyle{2{\pi^{2}E}_{{\rm S}}L_{{\rm o}} \over T_{{\rm S}}}\mathop{\underbrace{\int_{-\infty}^{\infty}\mu^2(t)\,dt.}}\limits_{\xi}$$

By substituting into Equation (15), the MCRB can now be expressed as:

(19)$$var \lcub \hat{\tau } \rcub = \displaystyle{T_{{\rm S}} \over 2\pi^{2}{\xi L}_{{\rm o}}\displaystyle{E_{S} \over N_{{0}}}}$$

Note that this result is a factor of two smaller than the expression presented in the ACCSEAS AIS R-mode study (Johnson and Swaszek, Reference Johnson and Swaszek2014a). The reason appears to be that the ACCSEAS study modelled the modulated signal as a real signal (centred on a Radio Frequency (RF) carrier), instead of using the complex envelope representation assumed by the MCRB expression given by Equation (15).

The variance of the pseudorangeFootnote ⁶ estimation error is given by:

(20)$$var\lcub \hat{\rho } \rcub =c^{2}\cdot var\lcub \hat{\tau } \rcub =\displaystyle{c^{2}T_{{\rm S}} \over 2\pi^{2}{\xi L}_{{\rm o}}\displaystyle{E_{S} \over N_{{0}}}}$$

where c is the signal propagation speed (assumed to be equal to the speed of light in free space in this paper). Denoting:

(21)$$\eta_{{\rm GMSK}}\equiv \displaystyle{c^{2}T_{{\rm S}} \over 2\pi^{2}\xi}$$

the expression above can be rewritten as:

(22)$$var \lcub \hat{\rho } \rcub =\displaystyle{\eta_{{\rm GMSK}} \over L_{{\rm o}}\displaystyle{E_{{\rm S}} \over N_{{0}}}}$$

The standard deviation of the pseudorange estimation error is then given by:

(23)$$\sigma_{\hat{\rho},{\rm GMSK}}=\sqrt{var\lcub \hat{\rho}\rcub } =\sqrt{\displaystyle{\eta_{{\rm GMSK}} \over L_{{\rm o}}\displaystyle{E_{{\rm S}} \over N_{{0}}}}}$$

Alternatively, the bound can be expressed in terms of the Carrier-power-to-Noise-density ratio, C/N ₀, using the identity E _S = CT _S, where C is the carrier power (equal to the average power of the real modulated signal):

(24)$$\sigma_{\hat{\rho },{\rm GMSK}}=\sqrt{\displaystyle{\eta'_{{\rm GMSK}} \over L_{{\rm o}}\displaystyle{C \over N_{{0}}}}}$$

For the AIS waveform as defined in Section 2.2.1, η_GMSK is equal to 3 · 13 · 10⁸ m² and η′_GMSK = 3 · 00 · 10¹² m²/s. Suitable values for L _o were provided in Table 4, Section 2.4.

As may be intuitively expected, the ranging error decreases with increasing number of data symbols used in the R-mode transmission and increasing Signal-to-Noise Ratio (SNR).

4.1.2. PSK/QAM (VDES)

For a PSK or QAM signal (as defined in Section 2.2.2), the denominator in Equation (15) can be written as:

(25)$${\rm E}_{{{\bf d}}}\left\{ \int_{0}^{T_{{\rm o}}} {\left\vert \displaystyle{\partial \over \partial \tau }\sum\limits_n q_{n} h\lpar t-\tau -nT_{{\rm S}} \rpar \right\vert^{2}dt} \right\}=\int_{0}^{T_{{\rm o}}} {{\rm E}_{{{\bf d}}}\left\{ \left\vert \sum\limits_n q_{n} p\lpar t-\tau -nT_{{\rm S}} \rpar \right\vert^{2} \right\}dt}$$

where $p(t)\equiv \displaystyle{dh(t) \over dt}$.

Using the assumption of uncorrelated data/channel symbols (see Section 2.5) and the linearity of the expectation operator, the expression above can be simplified to:

(26)$$\int_{0}^{T_{{\rm o}}} D\sum_n {p^{2}\lpar t-\tau -nT_{{\rm S}} \rpar } dt,$$

where D ≡ E{|d _n|²}.

By using the Poisson summation formula (Pinsky, Reference Pinsky2008) and assuming that the observation interval, T _o, is an integer multiple of the symbol interval, T _o = L _oT _S, L _o ∈ ℕ, the expression can be simplified to:

(27)$$DL_{{\rm o}}P_{2}(0)$$

where $P_{2}(f)\equiv {\cal F}\{ p^{2}(t)\}$. Using the following identities:

(28)$${\cal F}\lcub p^{2}(t) \rcub =\int_{-\infty }^{\infty} {P\lpar u \rpar P\lpar \,f-u \rpar du}$$

(29)$$P\lpar \,f \rpar ={\cal F}\left\{\displaystyle{dh(t) \over dt} \right\}={\rm j}2\pi fH\lpar \,f \rpar $$

in which $P\lpar \,f \rpar \equiv {\cal F}\lcub p(t) \rcub $ and $H\lpar \,f \rpar \equiv {\cal F}\lcub h(t) \rcub $, and considering that h(t) ∈ ℝ and therefore H( − f) = H*( f), the Fourier transform P ₂(0) can be expressed as:

(30)$$P_{2}\lpar {0} \rpar = {\cal F}\lcub p^{2}(t) \rcub \vert _{f=0}=\int_{{-\infty }}^{\infty } \left( \displaystyle{dh(t) \over dt}\right)^{2}dt=4\pi^{2}\int_{-\infty}^{\infty } {f^{2}\vert H\lpar \,f \rpar \vert ^{2}df}$$

Substituting for P ₂(0) in the expression for the denominator gives:

(31)$$DL_{{\rm o}}4\pi^{2}\int_{-\infty}^{\infty} {f^{2}\vert H\lpar \,f \rpar \vert ^{2}df}$$

For further analysis, it is desirable to express the MCRB in terms of the SNR. It can be shown that the average energy per symbol of the linear modulations defined in Section 2.2.2, under the assumptions made in Section 2.5, is given by (Mengali and D'Andrea, Reference Mengali and D'Andrea1997):

(32)$$E_{\rm S} = \displaystyle{D \over 2} \int_{-\infty}^{\infty} \vert H(f) \vert^{2}df$$

The expression for the denominator can therefore be rewritten as follows:

(33)$${8}\pi^{2}L_{{\rm o}}E_{{\rm S}}\displaystyle{\int_{{-\infty }}^{\infty } {f^{2}\vert H\lpar \,f \rpar \vert ^{2}df} \over \int_{{-\infty }}^{\infty } {\vert H\lpar \,f \rpar \vert ^{2}df} }=\displaystyle{{8}\pi^{2}L_{{\rm o}}E_{{\rm S}}\zeta \over T_{{\rm S}}^{2}}$$

where:

(34)$$\zeta \equiv T_{{\rm S}}^{2}\displaystyle{\int_{{-\infty }}^{\infty } {f^{2}\vert H(\, f ) \vert^{2}df} \over \int_{{-\infty }}^{\infty } {\vert H\lpar \,f \rpar \vert ^{2}df} }$$

is a dimensionless factor related to the shape of the modulation pulse. For an RRC modulation pulse with a roll-off factor α (see Section 2.2.2), ζ can be shown to be equal to (Mengali and D'Andrea, Reference Mengali and D'Andrea1997):

(35)$$\zeta =\displaystyle{1 \over 12}+\alpha^{2}\left(\displaystyle{1 \over 4}-\displaystyle{2 \over \pi^{2}} \right)$$

By substituting into Equation (15), the MCRB can then be expressed as:

(36)$$var\lcub \hat{\tau } \rcub =\displaystyle{T_{{\rm S}}^{2} \over {8}\pi^{2}{\zeta L}_{{\rm o}}\displaystyle{E_{{\rm S}} \over N_{{0}}}}$$

Denoting $\eta_{\rm PSK-QAM}\equiv \displaystyle{c^{2}T_{{\rm S}}^{2} \over 8\pi ^{2}\zeta }$, the variance of the pseudorange estimation error can be written as:

(37)$$var \{\hat{\rho}\} =c^{2} {\cdot} var \{\hat{\tau} \} = \displaystyle{c^{2}T_{\rm S}^{2} \over 8\pi^{2} {\zeta L}_{\rm o}\displaystyle{E_{\rm S} \over N_0}} = \displaystyle{\eta_{\rm PSK-QAM} \over L_{\rm o} \displaystyle{E_{\rm S} \over N_0}}$$

Finally, the standard deviation of the pseudorange estimation error is given by:

(38)$$\sigma_{\hat{\rho},{\rm PSK-QAM}} = \sqrt{var\{ \hat{\rho}\}} = \sqrt{\displaystyle{\eta_{\rm PSK-QAM} \over L_{\rm o} \displaystyle{E_{\rm S} \over N_0}}} = \sqrt{\displaystyle{\eta_{\rm PSK-QAM}^{\prime} \over L_{\rm o} \displaystyle{C \over N_0}}}$$

The values of $\eta_{\rm PSK-QAM}$ and $\eta'_{\rm PSK-QAM}$ for the different modulations and symbol rates used in VDES are provided in Table 5. Suitable values for L _o were provided in Table 4, Section 2.4.

Table 5. Ranging error coefficients for different VDES waveform configurations.

Note from Section 2.2.2 that the PSK and 16-QAM modulations are defined by practically identical mathematical expressions, differing only in the data/channel symbol alphabets. Although the two modulation types have a different mean square value of the data symbols, E{|d _n|²} = D, this term does not appear in the MCRB expressions derived above; therefore, all other waveform and channel parameters being equal, the PSK and 16-QAM modulations are expected to provide equivalent levels of ranging performance.

4.4.1. Received power versus distance

The average received signal power at a given distance from the base station, equivalent to the carrier power C in the preceding section, can be modelled as follows (all quantities are expressed using a logarithmic scale):

(39)$$C=P_{{\rm TX}}-L_{{\rm t,TX}}{\rm}+{\rm }G_{{\rm TX}}-L_{{\rm B}}\lpar d,h_{{\rm TX}},h_{{\rm RX}},\ldots\rpar +D_{{\rm RX}}$$

In the above equation, P _TX is the base station transmitter output power, L _t,TX represents the transmission line loss incurred at the transmitter, G _TX is the transmitting antenna gain, D _RX is the receiving antenna directivity (that is, the gain, G _RX, plus the antenna circuit loss), and L _B is the basic path loss which is a function of the transmitter-receiver distance, d, antenna heights above the sea level h _TX, h _RX and other factors (see also Figure 5).

There are multiple path loss models for the VHF band available in the literature. In this study, the method described in Recommendation ITU-R P.1546-5 (ITU, 2013) is used to estimate the path loss for an all seawater path, free of obstructions. This method is based on interpolation/extrapolation from a set of empirically derived field strength curves as functions of distance, transmitting antenna height, operating frequency, propagation environment type and percentage time for which the stated signal strength is exceeded. The estimated field strength values are then corrected for the height of the receiving antenna and converted to path loss. A summary of assumptions made when modelling the VDES signal strength is provided in Table 7. Figure 4 then shows the predicted received power as a function of distance for the two representative transmitting antenna heights given in Table 7.

Table 7. Assumptions related to received power modelling.

Figure 4. Received power vs. distance for a 12·5 W ERP VHF transmission, using a receiving antenna with a gain of 2·15 dBi (0 dBd).

4.4.2. Radio noise

The radio noise experienced by an R-mode receiver has two principal components: (1) external, or environmental noise received through the antenna from sources external to the receiving system; (2) internal noise generated by the receiving system itself. Assuming the receiver is installed on a vessel, the environmental noise can further be sub-divided into man-made noise generated by sources external to the vessel, and the vessel's topside noise generated by onboard machinery and electronic systems. This section aims to quantify the contributions of each of the mentioned categories of noise.

The amount of noise generated by a component in a radio system is commonly specified in terms of the noise factor, f. The noise factor is defined as the ratio of the noise power available at the output terminals of the component to the portion thereof attributable to thermal noiseFootnote ⁷ in the input termination at a reference temperature of 290 K. Equivalently, it can be defined as the ratio of the input signal-to-noise ratio to the output signal-to-noise ratio when the input termination is at the reference noise temperature of 290 K. The noise factor expressed in decibels is referred to as the noise figure, F=10log f. The concept of noise factor/figure can also be applied to the receiving antenna, as discussed below.

The International Telecommunication Union (ITU) provides statistical data on various types of external radio noise in Recommendation ITU-R P.372 (ITU, 2015). The recommendation specifies the noise levels in terms of the external noise figure, defined by:

(40)$$F_{{\rm a}}=10\log \displaystyle{p_{{\rm a}} \over kT_{{0}}B},$$

where p _a is the noise power available from a reference antenna when placed in a given type of environment, k is Boltzmann's constant, T ₀=290 K is the reference temperature, and B is the measurement bandwidth.

According to Recommendation ITU-R P.372, man-made noise levels are expected to vary depending on the type of environment and the frequency band. For a given environment type, the median value of the external noise figure associated with man-made noise, F _am,m, was found to be a linear function of the logarithm of frequency, f _c:

(41)$$F_{{\rm am,m}}=c-d\cdot \log f_{{\rm c}}$$

With f _c expressed in MHz, the constants c and d take the values given in Table 8.

Table 8. Parameters of the linear model of the median man-made noise (ITU, 2015).

For f _c = 162 MHz, the median man-made noise is estimated to be 15·6 dB, 11·3 dB and 6·0 dB above the thermal noise floor (kT ₀B) for city, residential and rural environments, respectively. For coastal navigation, the rural environment value, F _am,m = 6 · 0 dB, appears appropriate from the standpoint of low density of potential sources of man-made noise while at sea and will be used in this study.

The vessel's topside noise is expected to vary depending on the type of vessel and voyage phase. Recommendation ITU-R M.1467 (ITU, 2006) gives representative figures of the topside noise, P _a,v, for three types of vessel as follows (see also Table 9): the Australian Advisory Group for Aeronautical Research and Development (AGARD) figure represents a naval vessel under normal cruise conditions, while the Department of Defense (DOD) figure represents the maximum noise level under battle conditions. The figure adopted by the Ionospheric Prediction Service of the Australian Department of Industry (IPS) is generally accepted as representing the noise level encountered on container vessels, pleasure cruisers and utility ships, and is also the value that will be used in this document.

Table 9. Parameters of the linear model of the vessel's topside noise.

The values of P _a,v given in Table 9 are referenced to an operating frequency of f _c = 3 MHz and a bandwidth of 1 Hz. The table also shows the corresponding external noise figure values, F _a,v, at this operating frequency. Assuming that the topside noise follows the same linear model used for the man-made noise above, with the parameters c and d given in Table 9, these figures can be converted to the VHF band. The values of c in the table were calculated from the value of F _a,v at f _c = 3 MHz assuming that d (that is, the slope of the noise figure versus log-frequency curve) takes the same value as for the man-made noise curves provided in Recommendation ITU-R P.372. (ITU, 2015) For the IPS vessel and f _c = 162 MHz, the topside noise can then be estimated to be 14·0 dB above the thermal noise floor.

Combining the topside noise with the median man-made noise value for rural areas gives an external noise figure of $F_{{\rm a}}=10\log {\lpar {10}^{F_{{\rm a,v}}/10}+{10}^{F_{{\rm am,m}}/10} \rpar =14{\cdot}6\,{\rm dB}}$.

The noise figure values in Recommendation ITU-R P.372 (ITU, 2015) (and presumably in Recommendation ITU-R M.1467 (ITU, 2006) too) are referenced to a (hypothetical) short, lossless, vertical monopole antenna above a perfectly conducting ground plane (referred to in the following as the ITU monopole antenna). At VHF, a half-wave dipole antenna may be used rather than a monopole; in that case, the external noise figure values may need to be corrected. Recommendation ITU-R P.372-12 (ITU, 2015) and the report by Skeie and Solberg (Reference Skeie and Solberg2016) published by the Norwegian Defence Research Establishment suggest that the external noise figure values should be increased by C _a = 3 · 4 dB if a half-wave dipole is used instead of the reference monopole antenna.

The noise levels seen by the receiver are further influenced by the receiving antenna circuit loss, L _c,RX, (representing the conductive and dielectric losses in the antenna) and transmission line loss, L _t,RX. It is assumed here that a tuned antenna is used and therefore the antenna circuit loss is negligible L _c,RX ≈ 0 dB. The receiver transmission line loss is assumed to have the same value as on the transmitting side, L _t,RX = L _t,TX = 1 · 2 dB.

The noise contribution from the receiver itself is given by the receiver's noise figure, F _RX. A value of F _RX = 10 dB is assumed based on reference (True Heading, 2018).

With reference to Figure 5, the definition of the noise factor and the Friis formula (Friis, Reference Friis1944), the overall median noise factor of the receiving system, including the effects of the external/environmental and receiver internal noise, can be expressed as follows:

(42)$$f=\mathop{\underbrace{(f_{\rm am,m}+f_{\rm a,v})}}\limits_{f_{\rm a}} \cdot c_{{\rm a}}+(f_{{\rm t,RX}}-1)+l_{{\rm t,RX}}\cdot (f_{{\rm RX}}-1)$$

where: $f_{{\rm a}}={10}^{F_{{\rm a}}/10}$ is the external noise factor referenced to the ITU monopole antenna; $f_{{\rm am,m}}={10}^{F_{{\rm am,m}}/10}$ is the component of the external noise factor associated with man-made noise; $f_{{\rm a,v}}={10}^{F_{{\rm a,v}}/10}$ is the component of the external noise factor associated with the vessel's topside noise; $c_{{\rm a}}={10}^{C_{{\rm a}}/10}$ is the external noise conversion factor for the half-wave dipole antenna; f _t,RX is the noise factor associated with the transmission line loss; $l_{{\rm t,RX}}={10}^{L_{{\rm t,RX}}/10}$ is the loss factor corresponding to the transmission line loss and $f_{{\rm RX}}{=10}^{F_{{\rm RX}}/10}$ is the receiver noise factor.

Figure 5. Noise-equivalent model of the system.

It can be shown that if the actual physical temperature of the transmission line is equal to the reference temperature T ₀ = 290K, then f _t,RX = l _t,RX, and the above expression can be simplified to:

(43)$$f=\lpar \,f_{{\rm am,m}}{\rm}+{\rm }f_{{\rm a,v}} \rpar \cdot c_{{\rm a}}-1+l_{{\rm t,RX}}f_{{\rm RX}}$$

Substituting for the variables in Equation (43) (assuming F _am,m = 6 · 0 dB, F _a,v = 14 · 0 dB, C _a = 3 · 4 dB, L _t,RX = 1 · 2 dB and F _RX = 10 dB, as discussed above) yields an overall system noise factor f=75·8, corresponding to a system noise figure F = 18 · 8 dB.

Although taking a different approach, the ACCSEAS AIS R-mode study (Johnson and Swaszek, Reference Johnson and Swaszek2014a) arrived at a similar conclusion with respect to the radio noise levels on ships. The study argued that the AIS requirement of 20% Packet Error Rate (PER) at the minimum signal strength of −107 dBm stated in the AIS technical specification (ITU, 2014) is equivalent (under certain simplifying assumptions) to requiring E _S/N ₀ = 9 · 8 (or 9·9 dB) at the output of the receiver when a signal with the minimum signal strength is applied at its input. In the presence of thermal noise only, the minimum input signal level corresponds to an E _S/N ₀ of 27·2 dB at the receiver input, which implies a receiver noise figure of F _RX = 27 · 2 − 9 · 9 = 17 · 3 dB. In the ACCSEAS study, this figure was effectively used as an estimate of the system noise figure, F. It should be noted that a receiving system using a receiver that just meets the AIS requirement will have a system noise figure that is somewhat higher than 17 · 3 dB due to the transmission line loss and external noise, as discussed above.

For the purpose of the current study, the overall system noise figure (including the effects of the environmental noise) will be assumed to be F=19 dB. This corresponds to a (one-sided) noise power spectral density of $N_{{0}}=kT_{{0}}{\cdot}10^{\displaystyle{F \over 10}}=3{\cdot}18\cdot 10^{-19}\,{\rm W/Hz}$ (or −155 dBm/Hz), which is the value that will be used in calculating the received carrier-power-to-noise-density ratio, C/N ₀.

4.4.3. Maximum range for data reception

As discussed in Section 2.5, it is assumed that in order to be able to measure the range to a VDES base station, the user must be located within the data coverage area of the station. It will therefore be useful to determine the maximum communication ranges for the different VDES waveform configurations.

Table 10 shows the minimum received power (also referred to here as sensitivity), C _min, required to demodulate each of the five waveforms considered in this study, allowing a Bit Error Rate (BER) of at most 10⁻⁶. It can be shown that this level of BER is sufficient to ensure the successful reception of four statistically independent VDES messages with a probability of greater than 99%, which is consistent with the 99% service availability requirement for a GNSS back-up stated in IALA Recommendation R-129 (IALA, 2012). The assumption is made here that the lowest-order modulation and lowest Forward Error Correction (FEC) rate available within the given VDES subsystem, as defined in the draft VDES specification (IALA, 2018), are used. If a higher-order modulation or weaker FEC coding is used, the required power will be increased, and maximum ranges reduced compared to the values given in Table 10.

Table 10. Estimated maximum transmitter-receiver range for successful data demodulation (BER of 10⁻⁶).

The minimum required power in dBm was calculated as follows:

(44)$$C_{{\rm min}}=\gamma_{{\rm b,min}}-G_{{\rm c}}+10\log \lpar {k\cdot R}_{S}\cdot N_{0} \rpar +30$$

where: γ_b,min is the minimum energy-per-bit-to-noise-power-density ratio, expressed in dB, required for the demodulation of the given waveform with a BER of 10⁻⁶, ignoring gains from FEC; the values shown in Table 10 were determined from BER curves published in Middlestead (Reference Middlestead2017); G _c is the coding gain of the FEC code in dB, obtained from Bronk et al. (Reference Bronk, Mazurowski, Rutkowski and Wereszko2016); k is the number of bits per symbol for the given waveform (1 for GMSK, 2 for Pi/4-QPSK) and the remaining terms have been defined before. The last two columns of Table 10 then show the corresponding maximum communication ranges for two representative base station antenna heights, as determined from Figure 4.

4.4.4. Ranging error versus distance

Using the theoretical bounds on the ranging performance derived in Section 4.1, the received power versus distance curves provided in Section 4.4.1, and the R-mode receiver noise floor established in Section 4.4.2, it is now possible to estimate the VDES ranging error as a function of distance from the VDES R-mode base station.

Sample plots are provided below, showing the estimated ranging error for the five VDES waveform configurations and two representative base station antenna heights (18 m versus 57 m) considered previously. Figure 6 shows the error for one-slot R-mode transmissions and Figure 7 assumes that the R-mode receiver combines five successive one-slot transmissions during the pseudorange estimation process (other combinations of antenna heights, transmission and receiver configuration are omitted due to lack of space). Maximum range limits were applied to the plots as per Table 10.

Figure 6. Ranging error vs. distance; one-slot transmissions; no receiver integration; base station antenna height: 18 m.

Figure 7. Ranging error vs. distance; one-slot transmissions; receiver integration over 5 slots; base station antenna height: 57 m.

Caveat: the ranging error predictions shown in this section do not include the effects of transmitter synchronisation error and jitter, multipath propagation and propagation delay biases due to tropospheric effects and terrain topography.