2.1

Transmitter

We consider a generic description of a multi-channel ground-to-satellite link which employs temporal, frequency, spatial and site diversity. The transmitter performs error correction coding of an input bitstream and interleaves the resulting FEC symbols in time. The resulting bitstream is mapped to symbols, which are drawn from a (complex) signal constellation, to form a sequence s. Each symbol s_{…}, being an element of s, is chosen in correspondence with a generator matrix G(n,k,o) as described in Section 3. The symbol s_{…} is mapped to a time slot n, carrier frequency f₀ + kΔf, transmit beam b and ground station o, with zero-based indexing. In a generalized way, this can be described by a transmit signal:

where B denotes the number of transmit beams (per OFL), K the number of optical channels, Δf the channel spacing in frequency, f₀ the carrier frequency of the first optical channel, ϕ_k(t) represents the combined initial laser offset and phase noise, P_t the total transmit power (per OFL), mod(·, ·) the modulus operation and g(t) the prototype waveform used for pulseshaping. We assume g(t) to be a non-return-to-zero (NRZ) rectangular function with a symbol duration T_s.

2.2

Channel

After propagation through the (turbulent) channel, the received signal at the o^th telescope of the satellite can be described by the sum of the B simultaneously received, spatially overlapping transmit beams x_o,b(t):

with h_o,b(t) and n_o,b(t) being the complex channel coefficient and (background) noise term for the o^th optical ground station (OGS)and b^th transmit beam, respectively. The complex channel coefficient can be decomposed in a channel gain which equals and a complex phase factor e^jArg(h_o,b). The phase fluctuations are assumed to be uniformly distributed over the interval [0, 2π). L_o is the aggregate static link loss of OFL o which is the product of transmitter and receiver efficiencies, optical amplification stages, telescope gains and free-space path losses. The irradiance fluctuations of each channel I_o,b, for turbulence regimes varying from weak to strong, can be accurately modelled by the Gamma-Gamma distribution:²

where Γ(·) and K_p(·) are the Gamma function and modified Bessel function of the second kind of order p respectively. The characteristic parameters α and β model the small- and large-scale scintillations and are coupled to the scintillation index by . Though one could model the fluctuations due to beam wander and pointing errors as well,⁹ we restrict ourselves to (3) since our primary interest is in diversity techniques rather than the irradiance models.

As this study aims to overcome the irradiance fluctuations by temporal, frequency and spatial diversity it is important to understand the channel’s coherence time T_c, coherence bandwidth Ω_c and coherence distance Z_c. If the time difference, frequency difference and/or spatial separation between two transmitted FEC symbols is sufficiently large, then we can assume that those symbols are affected by a channel response h_o,b which is independent (and identically distributed) and exploit diversity. These coherence metrics are discussed in Section 3 which covers the topic of diversity.

Note, the channel can be characterized as block and flat fading channel as the coherence time T_c and coherence bandwidth Ω_c are considerably larger than the symbol time and channel bandwidth, respectively. Therefore, the channel coefficients h_o,b can be considered to be a complex constant over a number of symbol periods for each optical channel. Consequently, accurate channel state information can be obtained at the receiver, through estimating the channel coefficients with training symbols.

2.3

Receiver

At the receiver, the received signal is demultiplexed into the distinct optical channels. Each channel is then fed to a 3 dB-splitter and a 180°-hybrid to feed the sum and difference terms of the electric fields of the receive signal E_r and the local oscillator (LO) E_LO to a balanced detector with a matched filter. The electric field of the LO is described by with P_LO, f_LO and ϕ_LO(t) being the power, frequency and phase (noise) term of the LO, respectively. The resulting photo current, while excluding noise terms for the sake of clarity, for the k^th optical channel of OFL o becomes:

where R_d is the responsivity of the photodetector and homodyne detection is assumed such that f_LO = f₀+kΔf with a phase locked loop, i.e., ϕ_LO(t) = ϕ_k(t). Our received photo current is affected by several noise mechanisms which include beat terms of both signal and input noise with the LO, shot noise, intensity noise, phase noise and thermal noise. One of the advantages of the coherent receiver is that we can use a high LO power and thereby operate in the shot-noise limited regime and assume the other noise contributions negligible. As the shot noise is dominated by the part associated with the LO power, which is typically large in contrast to the receive signal, we can approximate it as an additive white Gaussian noise (AWGN) source.

A clear advantage of coherent detection is that the photo-current in (4) is dependent on h_{···} rather than as in thermal-noise limited direct-detection systems. This means that the SNR is proportional to the receive power P_r(t) (rather than ). As a consequence, the irradiance fluctuations lead to smaller SNR and BER degradations in the case of shot-noise limited coherent detection in comparison to thermal-noised limited direct detection. This helps to meet our goal of fading mitigation and limit the required temporal interleaving depth and its associated latency.

3.1

Temporal diversity

The coherence time for clear sky and strong turbulence conditions is typically in the order of 0.1-10 ms.³ A measurement campaign for GEO ground-to-satellite optical links has shown that some fades have durations up to 100 ms.¹⁰ In order to exploit temporal diversity we need to assure that (some of) the symbols are separated in time with a temporal separation larger than the coherence time such that the channel coefficients become independent. A mapping function is used π_T(·) such that an interleaved symbol vector is created based on the FEC symbol vector c such that where i is the index of the interleaved symbol vector. By applying an interleaver we essentially spread out the impact of fades over multiple code blocks such that – with sufficient error correction capability – the distributed errors can be handled by the error correction algorithm. With a latency constraint of L for the interleaver and a coherence time of T_c, we can get about L/T_c independent transmissions during a block code.

3.2

Frequency diversity

The ground-to-satellite path is typically characterized by a very short impulse response with hardly any delay spread and negligible inter-symbol interference (ISI). The coherence bandwidth – being the inverse of the RMS delay spread of the channel – is typically in the order of 1-100 THz.¹¹ However, when there is rain, fog or thin (cirrus) clouds – due to scattering – the delay spread increases and the coherence bandwidth can drop to 100 GHz.¹² Assuming a multi-channel OFL the transfer function for optical channels which are spaced at Δf > Ω_c could be regarded as independent (but identically distributed) which provides additional diversity. To exploit frequency diversity we can interleave FEC symbols in a similar way to conventional temporal interleaving. A mapping function π_Ω(·) interleaves symbols from the FEC symbol vector c to form a new vector , with . The aim is to maximize the distance between input FEC symbols, not just in time, but in time-frequency. With a complete bandwidth Ω (e.g., 4.8 THz for the infrared C-band) and a coherence bandwidth of Ω_c we can theoretically get Ω/Ω_c independent transmissions, at the same time instant. Frequency diversity may be very welcome when there is additional loss due to thin clouds, provided that the losses do not become too high to close the link budget.

3.3

Spatial diversity

In addition to the temporal and frequency domain, the spatial domain can be considered for additional diversity. While for an uplink path, the spatial coherence distance at the satellite is many times larger than the probable size of the satellite, the coherence distances, at sea level, is roughly 2–30 cm at visible and IR wavelengths. ¹³ Given the large receive apertures, and small transmit beam waist typically used, one can fit multiple transmit beams in the receive aperture,¹⁴ create multiple independent (but identically distributed) transmit channels provided that the spatial separation, i.e., , between two transmit beams is larger than the coherence distance Z_c. Given the large coherence distance at the satellite, multiple receivers at the satellite will essentially measure the same signal, and not provide additional diversity. Consequently, a multiple-input single-output (MISO) configuration with B transmit beams and a single receiver, is the configuration of interest to attain spatial diversity gain.

3.3.1

Space-time block coding

Alamouti has proposed a very simple and elegant scheme for 2 × 1 MISO systems to exploit transmit diversity. It is characterized by maximum diversity gain, full-rate transmission, maximum-likelihood linear decoding and does not require channel state information at the transmitting end (only at the receiver).¹⁵ Considering a transmitter employing the Alamouti scheme, the symbols s_{α,β} as in (1) are chosen in correspondence to the α^th row and β^th column, with zero-based indexing, of the STBC generator matrix G_2×2(n,k,o):

with (·)* representing complex conjugation and [·] the floor function. The formulation for η allows for STBC in combination with interleaving of the symbols over K optical channels and O OGS sites. A visualization of the interleaving is shown in Table 1.

Table 1.

Visualization of the process of interleaving symbols over the dimensions of time, space, frequency and OFLs sites. This example illustrates a transmitter following the model in (1) employing the STBC generator matrix of (5), and uses arbitrary values {N, B,K,O} for illustrative purposes. The indices of the symbols s ∈ s are shown in the top row of the table with directly underneath the tuples {n, b, k, o} which shows the mapping of the symbol to the specified time, transmit beam, frequency and OFL indices.

Unfortunately, there is not a natural extension from the Alamouti scheme to N × 1 MISO configurations. E.g., it can be proven that there does not exist a full-rate, orthogonal STBC for four transmit beams which achieves full diversity. This leaves us with the choice to either relax the full-rate, the orthogonality or the full diversity constraint. As we usually not operate under very high SNR conditions, we relax the orthogonality constraint at the cost of a small noise penalty, and can create a rotated quasi-orthogonal STBC by :¹⁶

where a rotated constellation is used with , to assure that a full diversity order of 4 is achieved.¹⁶ The maximum-likelihood decoding follows a pairwise minimization as described in. ¹⁶ Note that decoding assumes channel state information at the receiver and accurate synchronization among the different transmit beams. If one of these conditions is not met, then the performance may be severely degraded.

3.3.2

Multi-beam, multi-λ

Dependent on the system configuration, it may not always be possible to meet the requirements for STBC, i.e., accurate timing synchronization and channel state information. As wavelength multiplexing is envisioned to be used for high-throughput OFLs,¹⁴ an alternative is proposed; to assign each transmit beam to a distinct wavelength. The symbols s_{α,β} are chosen in correspondence to the α^th row and β^th column, with zero-based indexing, of the generator matrix Λ_2×2(n,k,o):

where ΔΩ resembles the frequency separation required to demultiplex the two signals. The choice of ΔΩ is dependent on the baudrate, prototype waveform g(t) and (matched) filter capabilities. Note that it is important that the choice of ΔΩ does not lead any of the transmit beams to spectrally overlap with any of the other K optical channels, all to avoid inter-channel interference. The matrix in (8) is easily extended to a 4 × 4 configuration:

We could get the impression that the multi-λ, multi-beam operates at B times the rate of the STBC scheme. However, we need to be aware that also B times as many optical (sub)channels are used. Effectively, both schemes are able to operate at a full rate per single wavelength channel, and it depends on the required diversity gain, wavelength plan and other system constraints which of the two schemes is preferred. A visualization of both STBC and multi-beam, multi-λ is shown in Fig. 1.

Figure 1.

Visualization of multiplexing and diversity in the domains of time (t), frequency (λ), space (x,y) and OGS sites (N) for space-time block coding (left) and multi-beam, multi-λ (right). Both schemes transmit – with the time-frequency-space-site bins shown – a total of 8 (complex) symbols and the transmission of symbols s₁ and s₂ are highlighted.

3.5

On the outage capacities & probabilities

The multi-beam, multi-λ as well as the temporal, frequency and site diversity techniques all rely on some form of interleaving FEC symbols such that the FEC symbols are spread out in space-time-frequency. Consequently, the FEC symbols within a codeword are affected by multiple independent transmissions, which leads to D-fold diversity, with D = O · B · F where O, B, F represent the number of independent optical feeder links, transmitter beams and frequency bands, respectively. Note, that D could include temporal interleaving as well, but is excluded to distinguish the temporal interleaving from the other diversity methods.

To assess the performance of the diversity techniques it is instructive to look at the outage probability, i.e., the probability that a certain desired spectral efficiency R (in bits/s/Hz) is not met by the channel capacity. For a slow fading channel – with coherence time T_c much larger than the symbol time and without interleaving – the outage probability is:

whereby SNR is defined as the average receive signal energy divided by the average noise energy, |h|² SNR resembles the instantaneous SNR and the outage probability is directly dependent on the channel coefficient h. We can considerably improve the outage probability by applying interleaving such that we get multiple independent transmissions. We essentially construct D parallel channels such that the outage probability becomes an average of all the achievable rates given D channel coefficients h(δ):

With the STBC decoders we do not average over independent channels, but receive B space-time coded beams in the decoder, all scaled in power by 1/B. This leads to an outage probability of:¹⁷

with h being the vector with B channel coefficients. The two equations (10) and (11) allow us to compare the B-fold transmit diversity to the D-fold interleaving-based diversity. By Jensen’s inequality for concave functions we know that:

which indicates that the achievable rate for STBC is always equal or higher than the rate for interleaving-based diversity techniques, assuming B = D. Equality is only met when the summands h are all equal, which is – by definition – not the case if we aim for spatial diversity. This will also be shown by simulations in Section 4. It should be said however that despite the better performance of STBC, it still suffers from an SNR penalty by 1/B as shown in (11). If we want to overcome this, then we need channel state information (CSI) also at the transmitter. With full CSI, we can pre-correct the transmit beams (e.g., with an extra phase shift or adaptive optics) to assure that the B beams constructively interfere at the receive aperture, and thereby minimize the outage probability:

As explained in Section 1 accurate CSI at the transmitter may not be available. Therefore, we direct our attention in upcoming sections to diversity techniques requiring no CSI or CSI only at the receiver.

4.1

Simulation model

A ground-to-satellite OFL has been simulated using Mathworks MATLAB, based on the system and channel model sketched in Section 2. We have simulated fading based on a Gamma-Gamma distribution, with exponential covariance function, for moderate turbulence conditions with α = 4.0 and β = 1.9 and τ_c = 0.01 s. The parameter τ_c, related to the coherence time T_c, is defined by the time where the (normalized) exponential covariance function decayed to 1/e,¹⁸ and the required length of the (temporal) interleaver is proportional to this parameter. The fading realizations of the channel model are generated based on the methodology in^{18, 19} and have a duration of 100 s and a sampling interval of 0.1 ms. Each simulation run simulates 400 000 frames, which are either uncoded or Reed-Solomon (n,k) coded, which is either (255,233) or (255,127), where n resembles the block length, k the message length and the coding rate is k/n. Interleaving takes place, not on symbols, but on the Reed-Solomon FEC symbols, as we prefer to have all potential bit errors in a single symbol rather than in multiple symbols. Maximum distance separable codes like Reed-Solomon offer optimal burst erasure protection, which is desired for block fading channels.²⁰ Note that by applying (multiple) diversity techniques, the nature of the error distribution changes from bursty to random, and other coding schemes may offer better performance.²¹ All considered, for our OFL design we have chosen for Reed-Solomon coding as it strikes a good balance between coding gain, computational and power-efficiency, which is important as the envisioned datarates are in the order of terabit-per-second and the heavy lifting – the decoding – takes place on-board of the satellite.

4.2

Simulation results

The top-row plots of Fig. 2 show the BER for QPSK with STBC, for both additive white Gaussian noise (AWGN) and fading channel, with and without uniform interleaving of FEC symbols over a duration of 50 ms, employing RS(255,223) and RS(225,127), and with B ∈ {1,2,4}. No additional spatial, frequency or site diversity is employed. The combined coding and diversity gain without interleaver is 0.5 dB, 8.5 dB, 12.5 dB for 1, 2 and 4 transmitter configurations, for a BER of 1E-5, with RS(255,223). Lowering the coding rate, by using RS(255,127), gives 1.5, 9.5 and 14.5 dB gain, respectively. Though considerable diversity gain is achieved, adding an additional interleaver, constrained to 50 ms, can still provide additional dBs in diversity gain as is visible in the top-right plots of Fig. 2. With an interleaver constrained to 50 ms and RS(255,223), a coding and diversity gain of 6.5 dB, 13.5 dB, 15 dB is achieved, respectively. With RS(255,127) this becomes 12.5 dB, 16.5 dB, 18.5 dB, respectively.

Figure 2.

BER curves for OFL characterized by a Gamma-Gamma fading channel with moderate turbulence (and AWGN as a reference case). The top row plots show various MISO configurations employing STBC whereas the bottom row plots are without STBC, but with several values of the diversity parameter D which resembles the number of independent fading gains – due to frequency, spatial and/or site diversity – in each codeword. The applied coding is Reed-Solomon (255,223) and (255,127) with respective coding rates of 0.87 and 0.50, respectively.

The multi-beam, multi-λ as well as the temporal, frequency and site diversity techniques all rely on interleaving FEC symbols such that essentially D independent transmissions are created, whereby D = O · B · F with O, B, F representing the number of optical feeder links, transmitter beams and frequency bands, respectively. The BER curves for different values of D are shown in the bottom plots of Fig. 2. As the block length is 256 FEC symbols, with D = 256 essentially all the FEC symbols undergo independent fading. Additional temporal interleaving then does not improve the BER curves. For smaller values of D, additional temporal diversity is beneficial and can provide additional dBs in diversity gain as is visible from the bottom-right plots in Fig. 2.

The error probabilities for the cases where B = D show that (QO)-STBC clearly outperforms the interleaving-based diversity techniques. This is in line with the analysis of the outage probabilities in Section 3.5. Note, that the differences are large for small B and D, but in the limit for large diversity orders with sufficiently high coding rates, become comparable. Though the plots show a limit set of combinations, it is important to stress that – as an example – STBC can also be combined with temporal, frequency and site diversity, as described by the generic transmit & diversity model in (1).