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Modulation and Coding to Improve Power and Bandwidth Efficiency of Small Earth Stations


Abstract


The objective of this project is to maximise the power and bandwidth efficiency of small satellite earth communication terminals whose main limitation to satellite capacity is a non linear transmitting power amplifier. A form of envelope equalisation is presented in which regular QPSK modulation is used together with instantaneous data magnitude modulation so as to produce a near constant envelope signal when filtered with root 5% raised cosine filtering. The algorithm for deriving the magnitude values is presented along with results using a data look-up table approximation. Bit error rate figures are presented for envelope equalisation combined with a (406,4) binary product code and MAP decoding. It is shown that at 10-6 BER the is .5 dB some 0.6 dB from the linear system. The results are compared to the Shannon bound for code performance obtained by solving recursively Shannon’s equations. The product code is within 1.5 dB of the Shannon limit for a code with the same block size and code rate. When used with a non linear power amplifier the benefit of envelope equalisation is approximately dB. The power spectrum is identical to that of the linear system with no spectral spreading and no increase in adjacent channel interference.


Introduction and Background


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The aim of this project is to maximise the power and bandwidth efficiency of small satellite communication terminals whose main limitation to communication capacity is a non linear [1] transmitting power amplifier. Any attempt to transmit at maximum output power with such an amplifier produces amplitude and phase distortion plus spectral spreading. The approach that is taken here is to firstly linearise the amplifier by predistorting the amplifier input and then to control the envelope variation of the signal present at the input of the amplifier by magnitude modulation of the input data stream. We have termed this process envelope equalisation. A powerful (406,4) binary product code with iterative MAP decoding is used to produce an error rate performance of 10-6 at an ratio of .5 dB (with 10 iterations) and it is shown that this is within 0.6 dB of the performance of the system when using a linear power amplifier. The code performance is compared to the Shannon theoretical bound [5] which is obtained using a recursive method in order to obtain the exact solution to Shannon’s equations. The product code is within 1.5 dB of the Shannon limit for the same code rate and block size. The net overall benefit of the magnitude modulation system with the non linear power amplifier, compared to regular QPSK modulation with the same product code, is dB plus typically 1 dB output back off to prevent excessive distortion. The overall benefit of dB means that the same satellite terminal with magnitude modulation can transmit at twice the data rate as the conventional system and with the same bandwidth efficiency.


The Linear Power Amplifier and Magnitude Modulation (sspa)


A typical solid state power amplifier has an AM to AM characteristic shown in Fig. 1. By predistortion it can be converted into a linear characteristic plus a saturated region also shown in Fig. 1. Clearly with predistortion any non linear amplifier characteristic can be converted in this way to a linear characteristic plus saturated region. This is taken as our generic non linear amplifier model and assumes predistortion is employed. The data pulse stream is to be bandlimited for bandwidth efficiency and typically 5% root raised [] cosine filtering is the preferred choice in satellite communication systems today and this is used by way of example. The effect in the time domain is to cause envelope variations and it turns out that the peak excursions in envelope are some 4. dB above the average power level (increasing to 4.7 dB for 0% roll-off).


Fig. 1 SSPA AM to AM Characteristic and Linearised Characteristic After Predistortion


This would necessitate backing off the linearised amplifier by an output back off of 4. dB to prevent clipping. This in turn implies the data rate be reduced by the factor .% to maintain the same satellite link power margin. This factor represents the data capacity cost of bandwidth efficiency because without bandwidth constraints, filtering could be used with square constant amplitude pulses thereby operating the amplifier at maximum power.


Magnitude modulation is carried out using a data look up table to determine the instantaneous magnitude as shown in the modulator block diagram of Fig. . As the modulation format is QPSK rather than BPSK, two look up tables are necessary.


Fig. Magnitude Modulated QPSK Modulator


The result of the magnitude modulation clearly limits the peak envelope variation as shown by the state space diagrams of Fig. a and b below


Fig. a State Space Diagram for Magnitude Modulated QPSK with 5% Root Raised Cosine Filtering


Fig. b State Space Diagram for QPSK with 5% Root Raised Cosine Filtering


The magnitude modulated QPSK signal after amplification by the sspa produces the state space diagram shown in Fig. a. There is no overall distortion and the power spectrum is identical to that of the linear system with no spectral spreading and no increase in adjacent channel interference. The method to determine the magnitude modulation tables is described below.


Magnitude Modulation Algorithm


It is assumed that the amplifier is linearised up to its maximum output power and that it saturates beyond a normalised drive level of ± 1. The design problem is then to modify the drive signal such that it delivers maximum output power without driving the amplifier into saturation. Saturation results in spectral spreading, giving loss of bandwidth efficiency and causing adjacent channel interference. The approach in this paper is to magnitude modulate each data pulse prior to pulse shaping with the objective of generating a constant amplitude, bandlimited data stream at the input to the drive amplifier.


Consider a random binary antipodal stream where T is the signalling interval and . When this is applied to a QPSK modulator with the usual root-raised cosine pulse shaping filter (impulse response ), the bandlimited QPSK output has a non-constant envelope. For 5% roll-off, random data can increase the peak power level some 4. dB relative to the average power level, requiring back-off in order to avoid saturation. The back-off can be significantly reduced by performing envelope equalisation via adaptive amplitude modulation of the data. The magnitude modulated samples are easily determined by a recursive algorithm. Considering a baseband system for simplicity; in the ith recursion, the shaped data signal is given by


(1)


Suppose that is limited to a drive amplitude of ± 1 and matched filtered []. The signal prior to the matched filter is then


()


The corresponding error signal magnitude due to clipping is and the matched filter output is


()


This is then sampled to give a new impulse train . Note that is not a true matched filter whilst undergoes non-linear processing. Since clipping reduces signal energy at each iteration, progressively reduces and converges. The approach is readily extended to bandpass systems. For example, a QPSK signal with envelope equalization is given by


(4)


Note that, although the in phase and quadrature data streams are different, common envelope equalisation is used for both streams in order to avoid incidental phase modulation. Coefficients are solved iteratively, as before. Once sequence has been solved for all N cases of an N � bit data sequence, real-time amplitude modulation can be performed using a look-up table.


The impulse response of the root raised cosine filter has to be truncated to perform the envelope equalization by using a look-up table. We have investigated the necessary size of the look-up table in order to obtain a waveform close enough to that obtained with the iterative procedure. The reduction in envelope peak value as a function of the size of the look-up table is shown in Figure 4 for different table sizes (a table size of zero corresponds to the standard unequalised system). It can be observed that very good results are achieved with a 11 � 15 bit (index) look-up table. The transmitted waveform for the envelope equalised system using a 15 bit look-up table is compared to that of a linear unequalised system in Fig. 5.


Fig. 4 Look-up Table Performance


Fig. 5 Amplitude Equalised and Linear System Waveforms


The Probability Density Functions (PDF) for the envelope equalised system with the 15 bit lookup table and the linear system are shown in Fig. 6. It can be seen that a 15 bit look-up table produces near perfect envelope equalisation for 5% root raised cosine filtering. Sharper roll-off factors would generally necessitate larger look-up tables.


Fig. 6 Channel Waveforms PDF


The envelope equalisation results in some data pulses being smaller than others and having a higher probability of error. Powerful Forward Error Correction (FEC) is necessary in order to avoid excessive degradation. In selecting the best choice of code, code rate and block length it is useful to refer to the theoretical limits of coding performance which are given by the Shannon bound. This has been evaluated exactly for short code block lengths by Dolinar, S., Divsalar, D., and Pollara, F. [4] but not for long block lengths due to overflow problems in the numerical evaluation of the Shannon equations [5]. A recursive evaluation of the Shannon bound for long block lengths is described below.


Shannon Limit for Finite Block Lengths


An important bound in the theory of communication is Shannon’s sphere packing bound (or Shannon limit). The sphere packing bound can be used to calculate the needed to achieve a given word error probability for a code having information length k bits, coded block length n symbols, and rate R=k/n. The above quantities are linked by the following formulae [4]


(5)


where and is the solution of the following (solid angle) equation


(6)


Given R, n, and , the is obtained by solving the following equation for A


(7)


It is possible to directly obtain numerical solutions for the equations above provided that n is small, say less than 100. Larger values of n produce numerical overflow/underflow. In order to minimise these problems, a recursive relationship has been derived for the inner integral in equation (7).


Define


(8)


so that equation (7) becomes


()


A recursive relationship can be obtained for J as shown below


(10)


Upon integrating the first term by parts it is readily shown that


(11)


hence


(1)


This equation has been used to calculate the sphere packing bound exactly for an information block size k up to1000, for different code rates and . For values of k higher than 1000 the bound has been evaluated by using Shannon’s approximation as presented in [4]. A comparison between the exact solution and the approximation indicates errors of less than 0.01 dB for block length codes greater than 50. The bounds are presented in Figure 7 for and code rates r =1/, r =/4 and r =7/8.


Fig. 7 Shannon’s limit for Pw=10-6 and different code rates


Simulation


Simulation of the algorithms has been carried out for baseband and bandpass systems. Oversampling of the amplitude modulation sequence is required in order to approximate to continuous-time signals and , followed by sub sampling to extract . By definition for normalised data, the energy per bit of conventional root-raised cosine data is unity, whilst for the modulated data stream is given by


(1)


where M is the over sampling factor and N is the total number of samples over the measurement window. Typically corresponding to a power level .4 dB below the maximum peak amplifier power. There is no amplifier output back-off required. In the standard QPSK unequalised scheme, the average is 4.4 dB below the maximum peak amplifier power plus an output back-off of typically 1 dB. Thus, envelope equalisation results in approximately dB gain in for 5% roll-off QPSK . The magnitude modulation of the data values does however produce a variation of the energy levels of the received bits which for some bits causes considerable reduction in noise margin and a potential increase in bit error rate.


Coding has to be used to compensate for the higher bit error rate of these bits. One of the most powerful binary codes [6] available today is the (406,4) product code based on the extended (64,57) Hamming code with soft decision decoding using iterative MAP decoding. The error rate performance of this code when used with the envelope equalisation system is shown in Fig. 8. Also shown in Fig. 8 is the performance of the code with a linear QPSK system and the Gallager bound for binary codes of this block size and code rate [,4].


The product code is a powerful code because at 10-6 BER the code with regular QPSK is within 0.4 dB of the Gallager bound for binary codes and within 1.5 dB of the Shannon bound for general codes. The envelope equalised performance is some 0.6 dB degraded relative to QPSK showing the efficiency of the product code in absorbing the variation due to magnitude modulation.


Fig. 8 System Coded Performance Compared to the Linear System and the Gallager Bound


Conclusions


An algorithm has been presented to provide magnitude modulation of a binary data stream resulting in envelope equalised transmission such that the output amplifier is never driven into saturation. The output power spectrum is identical to that of the linear QPSK system.


The net effect is approximately dB increase in for the specific case of 5% root raised cosine pulse shaping and QPSK modulation. The authors are currently exploring the benefits of the technique applied to the M-QAM system.


Acknowledgements


This work has been carried out under grant GR/R14606/01 awarded by the Engineering and Physical Sciences Research Council (EPSRC).


References


[1] A.H. Aghvami and I.D. Robertson, “Power Limitation and High-Power Amplifier Non Linearities in On-board Satellite Communications Systems”, IEE Electronics and Communication Engineering Journal, Vol. 5, No. , pp 65-70, April 1.


[] J.G. Proakis, “Digital Communications”, McGraw Hill, 15.


[] R.G. Gallager, “Information Theory and Reliable Communication”, Wiley, New York, 168.


[4] S. Dolinar, D. Divsalar and F. Pollara, “Code Performance as a Function of Block Size”, TMO Progress Report 4-1, http//tmo.jnl.nasa/tmo/progress_report, May 18.


[5] C.E. Shannon, “Probability of Error for Optimal Codes in a Gaussian Channel”, BSTJ, Vol. 8, pp 611-656, 15.


[6] R. Pyndiah, “Near Optimum Decoding of Product Codes Block Turbo Codes”, IEEE Trans. Comm., Vol. 46, No. 8, Aug. 18.





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