NON-HOMOGENEOUS GAIN STATE OPTIMIZATION FOR TRANSPONDING SATELLITE COMMUNICATIONS

Jim Marshall The MITRE Corporation

Kenneth Y. Jo Satellite Communications Division

Defense Information Systems Agency

ABSTRACT

Recent advances in the design of transponding communications satellites provide the ability to switch transponder sub-bands among multiple up-link and down-link antennas. This feature increases the flexibility of trafic routing and improves spectral re-use characteristics. A side benefit of such a new system is the capability of applying different transponder gains to carriers in different sub-channels within a given transponder. Recently, techniques have been developed for analyzing link supportability and the transponder loading with these non-homogeneous gain stares. However, these new analysis techniques require that the gains be known a-priori. To complement these analysis techniques a method is needed for selecting the best gains to be used in each sub-channel. This paper presents such a method for selecting transponder gains. We define the constraining equations, which ensure that each link can be supported, and propose an optimization metric that maximizes the overall supportability. We then suggest an optimization approach that can be used to select the gains for each sub-channel and the overall transponder operating point. The approach described includes a means of selecting the starting point for the optimization. A numerical example is provided.

1. INTRODUCTION

Transponding communications satellites have been in operation for many years, providing point-to-point connectivity for both military and civilian users. Advances in transponding satellite technology provide for a highly flexible bandswitching capability in the satellite. In this approach “sub-channels’’ in the transponder can be switched among multiple uplink and downlink antennas. This capability, coupled with relative narrow antenna patterns, improves the spectral utilization of the satellite. Having provided the ability to divide the transponder bandwidth into sub-channels, it is a relatively easy extension of the new capability to allow

different gains in each sub-channel. This adds a power management flexibility to the frequency management flexibility associated with the sub-channel approach. This flexibility in managing power and transponder gains is especially important when different size terminals must operate in the same transponder.

Recently, new analysis techniques have been developed to assess the performance of links in a transponder with non-homogeneous gain states. These techniques are an extension of previously developed transponder loading approaches that determine link supportability and the uplink power to use for each link. To complement these new analysis techniques, a method is needed to determine a good set of gains for the suhchannels in the transponder. These gains will depend on the link requirements and the earth terminal EIRP and G/T values. We expect that smaller tenninals will require more subchannel gain, than large terminals.

The purpose of this paper is to provide a method for choosing sub-channel gains. After providing an overview in section 2, we will develop an optimization approach for selecting the gains in section 3. We also suggest a method for selecting a starting point for the sub-channel gain vector in section 4. Section 5 provides a numerical example. Conclusions are given in Section 6.

2. OVERVIEW

In a loading analysis we need to determine if a set of requested links is supportable and what up-link power to use. To collapse these two questions into one, we define supportability in terms of the required up-link effective isotropic radiated power (EIRP). If the required up-link EIRP is less than the available EIRP, we say the link is supportable. Hence, we only need to compute the required up-link EIRP.

In order to analyze performance, we independently model one transponder in the satellite at a time. Regardless of the number of antennas, a single

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transponder is associated with a single HPA, since this is where the non-linearity exists. Figure 1 shows a model for the transponder. Each transmitting earth terminal, i, has an up-link EIRP of E, and an up-link path loss ai. The power at the input to the transponder is labeled xi and the power out of the transponder is called yi. The transponder itself is modeled as having added noise, with density kT,, followed by ideal gain, G, an ideal filter, and finally a zero memory non-linearity. The added noise and non-linearity are common to all sub- channels, whereas the ideal gain and filtering are particular to each sub-channel. All transponder gain, including the HPA gain, is included with the gain of each sub-channel, G., and the transponder non-linearity is assumed to have unity small signal gain. We generally use "n" as the subscript for different sub- channels. Sub-channels are summed prior to going to the HPA non-linearity.

We assume that the transponder non-linearity causes gain compression and intermodulation (Evl) products, but no other degradation. Up-link path loss ai includes the satellite receive antenna gain, and down-link path loss bi includes both satellite transmit and earth terminal receive antenna gains. The noise that is added by the receiving earth terminal has temperature Ti. We assume that the up-link path loss is known, so determining the required up-link EIRP is equivalent to finding the

Satellite Transponder , I Ideal Filter -LF-

Bandw. = Bn

required transponder input power xi.

In order to evaluate the effects of various signal levels on the non-linearity, we need a way to express the drive level of the transponder.

Transponder Non-Linearity

j J - Y i Power

- output

We define a drive level variable, z, in terms of the total power that would be at the output of the transponder if there were no gain compression, normalized by the saturated power, P . Thus we have:

-- Saturated Power

Here, the subscript notation, n(i), indicates the sub- channel used for link i. Also, kT, is the noise density of the transponder and B. is the bandwidth of the n* sub- channel.

It is assumed that there are many signals operating in the transponder simultaneously and that none of these are large relative to the signal composite. This ensures that the gain compression for the signals will be nearly the same [2] and also ensures that any individual signal is not likely to be distorted greatly, other than by gain

I

/Uplink Downlink I / Path

U p l i E1 Ei

Path Loss bi

Transmitting

Terminal I Esrth

'I

Demodulator

Terminal Figure 1. FDMA Channel Model

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compression. Approximating the gain compression as being the same for all links simplifies the analysis considerably and we characterize this gain compression in terms of a function of the operating point and designate the function g(zJ. The net gain through the transponder for any given signal, i, is Gn,ijg(z).

We also approximate the IM products as white in the channel [2,10], and recognize that the IM spectral density is a function of the operating point. The normalized IM characteristics function is designated h(z) and the actual IM spectral density is h(r)P/B. The gain compression and intermodulation products are related to the amplitude modulation (AM) to AM and AM to phase modulation (PM) characteristics of the non-linearity [3, 8, 91.

3. OPTIMIZATION APPROACH

Our optimization approach is as follows. We attempt to minimize the output powers on each of the links, while constraining each link camer to noise density ratio to its required value. Minimizing the output power helps us maintain a low uplink power, hence maximizing supportability by our definition.

Given a vector y = [y l ,..., yM 1 , define a metric or n o m of y as:

U

Z(y) =[E yp1”p i=l

where p 2 1. equation (2) becomes: Z(y) = max [ y, ] .

For both bandwidth-limited and power-limited systems, the satellite loading problem is now formulated as:

Note that, as p approaches infinite,

I L i i M

Minimize I ( Y ) ~ (3)

where q 2 1 and Z(yp is a convex function of I(y) to assure the uniqueness of the optimal solution for the problem. For simplicity we allow q = Up, so that our goal is to:

Minimize 5 yp i =I

(4)

Following the analysis in [I], we constrain the output signal levels as follows:

( 5 )

Let E, = x,a, denote the required terminal power for link (carrier) i. The quantity x, is the received power for the i* link at the transponder. Let b, be the downlink loss for carrier i. Then define the noise and interference density function as:

(6)

This quantity represents the spectral densities of all the interference processes, referenced to the receive station of carrier i. The optimization problem (4) can he formulated as the Lagrangian optimizations problem as:

The Kuhn-Ticker necessary conditions (see [I] and 131) will describe the characteristics of the optimal signal power for each camer i as:

6 -L(y,A) = 0 6yi

Here the bar indicates the upper bound for that variable. Using (6) and (7), the conditions (8.a) can be restated as:

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The optimum yi must satisfy (9). Note that Go(i) can be chosen to form the relationship: yi =X~G,,~) l g ( z ) given xi.

Solving (9) for yi yields:

If the solution y, from (IO) is greater than the upper bound of (8.e). y, will be trivially set at 7,. This upper bound is applied when q is limited by the uplink EIW.

The proposed algorithm will start with h, = 0, then from (lo), it should be

y, = O and G,(,) =_C, the lower bound of G as an initial feasible choice. If Ai is not zero, it must be from (8.b)

Note that the condition (8.a) is for a local minimum point at which the derivatives of the objective function becomes zero. If the solution of (8.a) is out of the range by the upper and lower bounds of yi. the optimality occurs at the upper or lower bound of yi. whichever is closer. The Lagrangian multiplier can now be expressed in term of y, as:

ai = PY (12) (f,(Y)- yivzf(Y))

b , f j 2 (Y)

The natural starting point for a numerical algorithm is zero, i.e., 1, = 0, and this makes y, zero

from (12). However, this solution does not satisfy (Kc), so it is not a feasible solution. The goal of the algorithm is to start from y, = 0 and iteratively find the optimal solution that satisfies the optimality conditions (8.a) - (8.d).

It is assumed that, if all necessary conditions are met, .the optimality occurs at the dual maximum point with property (8.a). So it suffices to find yi that satisfies (8.h) since the multiplier is not zero

From (13) yi is given as:

Properties (13) and (14) satisfy both optimality conditions (8.a) and (8.b).

4. OPTIMIZATION ALGORITHM

Based on the results derived so far, an optimal trandponder-loading algorithm can be introduced. The advantages of using the algorithm based on the Lagrangian multipliers is that the dual problem of maximization will provide the lower hound of the objective function (5)

q y * . n ) 5 i xy* .a* ) 2 u y , a*) (15)

Here, the asterisk indicates the optimum value of the variable. It is known that both yi =O and Ai = O are initially set and will then increase and decrease, respectively, until they satisfy the optimality condition (13). Hence the efficient optimization algorithm will not necessarily go through cumbersome routines of computing partial derivatives to iteratively update (14). The algorithm only updates yi, regardless of any values of wi and p. Steps to compute the optimum yi are given next. For a wide class of satellite loading problems, Ai turns out to be extremely small, i.e., a range of minus 10.’ to 10.”. So using the multipliers will also make the problem numerically unstable. Steps for computing the optimal loading solution is summarized next.

m: Lagrangian multipliers

Set yy’ = 0, 1 5 i 5 M , and set the initial

ay) = o Also set G:::) = G , an initial feasible value

x:’) = o , l i i s ~ k = I (iteration)

(lower bound)

-2:

foranew, l < i l L , For given Ay) and y = l y , , y z ,,., y , ] , solve

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y!"" = b,c,f,"'(y) (16) L(y.A)<L(y' ,A*) gives the lower bound for the minimization problem, so we know how close to the m: Checkif optimum solution during executing each iteration.

5. STARTING POINT

We desire a approximate starting point for the optimization. This can be accomplished as follows.

(17)

If so, optimality is reached go to Step 0. Otherwise go to step 4.

Ignoring spectral overlap, we recall that each link must satisfy the following criteria:

If I y:"" - y y ) I< E , for all i , stop the optimality is reached.

Otherwise, set k = k + 1 and go to Step 2.

Suppose the algorithm uses the Lagrangian multipliers. Then Step 4 can be written as

x:") = minb,"'"g'k'(z")) /GLt:), z]

G,,(j) @ + I ) - - Mn[y,'"g"' ( z ' ~ ) )I x("1

If I A~"" - /I:" I< E , for all i , stop the

Otherwise, set k = k + 1 and go to Step 2.

W ( L )

optimality is reached.

Property (18) implies that xi reaches its upper bound that is decided by the maximum allowable EIRP for the transmitting ground terminal of carrier i. At each iteration, if the camer is not supported or if xi reached its uppers bound, G,,, should be increased by using the relationship (19). To make the algorithm stable, the step size can be limited by an upper bound which is a function of the composite output backoff of the transponder and the distance from the current EIRP and its maximum allowable EIRF' of each terminal. Note that for subchannel n(i) the same gain is applied by comparing the potential gains for all links in subchannel n(i).

The conditions derived in the previous section, an efficient loading algorithm can be defined.

Previously we have taken the quantity xj as the required transponder input power and the quantity G,(,, as the specified gain. However, if we reinterpret xj as the maximum available transponder input power, given the max EIRP of the terminal, we can solve for the desired gain, G, assuming equality in the expression above. Doing so produces:

Here we use the notation xM, to represent the maximum available EIRP at the transponder input, given the earth terminal maximum EIRP. The equation above provides an estimate of the effective desired gain needed in the channel for link i, GED, , remembering that there will be some gain compression. The expression should be applied to the most disadvantaged link in the channel. The values of h(z) and g(z) will need to be estimated to determine the gain to use.

6. EXAMPLE Determining the supportability of a set of links

is an important step. Transponder itself is modeled as having added noise, with density kT,, followed by ideal gain G,, ideal filter, and finally a zero memory nonlinearity. The added noised and non-linearity are common to all subchannels, whereas the ideal gain and filtering are particular to each sub-channel.

There are five subchannels, each of which has 10 carriers. Systems parameters are chosen from WGS specifications. The Lagrangian multiplier is approximately 4x10'" in each case.

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Table 1. Example Results of Optimization

I DataRate I EIRP(dBW) I Sub-channel I (kbk) 512 64 128 128 50

Gain (dB) 55.1 127.2 55.0 120.6 42.4 134.9 66.5 114.2 43.0 135.0

~

The advantage of the algorithm is that one does not have to invert the matrices, just numerically update fly) at each iteration.

7. CONCLUSIONS

We have developed an approach for optimization of the gain states of a transponding communications satellite that has different gain states in each of several sub- channels. This approach includes a method for determining an approximate starting point and an iteration approach that uses Lagrange multipliers for a constrained optirmzation. The optimization approach uses an iterative method that updates the gains and the signal powers, while preventing uplink EIRP from exceeding the maximum for the terminal.

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Knab, J, “Satellite Loading Tools using

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J. Marshall, “SATCOM Loading Analysis with [7] Heterogeneous Gain States,” Proceedings

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Communications Agency, Computer

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