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———————————————— SOURCE: Signal Processing Group, Department of Telecommunications, NTNU

Rate-Adaptive Coding and Modulation with LDPC Component Codes

Ola Jetlund, Geir E. Øien, Kjell J. Hole, and Vidar Markhus Department of Telecommunications Norwegian University of Science and Technology N-7491 Trondheim NORWAY Phone: +47 73 59 27 47 Fax: +47 73 59 26 40 Email: [email protected] B˚ ard Myhre SINTEF Telecom and Informatics N-7465 Trondheim NORWAY

Rate-Adaptive Coding and Modulation with LDPC Component Codes Ola Jetlund

Geir E. Øien

Kjell J. Hole

Vidar Markhus

B˚ ard Myhre

Abstract Wireless mobile channels suffer from time-varying channel conditions, and are commonly modelled as multipath fading channels. Systems using adaptive coding and modulation (ACM) have been introduced to increase spectral efficiency on such channels. An ACM system can be designed by segmenting a (slow) fading channel into time slots, and using an additive white Gaussian noise (AWGN) channel as an approximation within each given time slot. Then, coding schemes for AWGN channels of varying qualities can be temporarily multiplexed on the channel, based on channel state information fed back to the transmitter. In this document we take a closer look at the consequences of introducing block codes into an ACM system. One key issue is the bit-error rate (BER) performance of the error correcting codes used in the scheme. We have simulated the BER performance for the promising lowdensity parity check (LDPC) codes. We also make some simple considerations regarding the mobility constraints of such a scheme.

1

Introduction

A signal travelling in an environment with both natural and man-made objects is scattered, reflected, and diffracted, resulting in multipath transmission, and thus a composite received signal. Changes in the environment due to movement of objects, receiver, and transmitter introduce time-varying fading on the communication channel. How fast the communication channel and thus the amplitude and phase of the received signal vary depend on the velocity of objects, receiver, and transmitter. A wireless channel can be modelled as a multipath channel with either frequency-selective or frequency-flat fading distribution. When the delay spread [1, Eq. (2.146)] of the channel is short compared to the symbol duration, the channel is said to be frequency-flat, since all frequency components are affected in the same way by the channel. In other words, the coherence bandwidth is large compared to the signal bandwidth [1, pp. 75-79]. This is the case e.g. if we are using a multi-carrier technique like orthogonal frequency division multiplexing (OFDM), and the channel in question is a sub-channel in the OFDM scheme [1, pp. 175-177]. We shall restrict ourselves to the frequency-flat case. For a more comprehensive treatment, of time-varying multipath fading channels we refer to [1]. Frequency-flat fading is usually modelled by a complex channel gain with a certain probability distribution. In addition the signal is disturbed by additive white Gaussian noise (AWGN). This is shown by the model in Fig. 1. The received signal, y(t), in the figure is represented by the complex baseband model y(t) = α(t) · x(t) + n(t), (1)

where x(t) is the transmitted signal, α(t) is the complex channel gain, and n(t) is AWGN. If a line-of-sight (LoS) component is present between transmitter and receiver, the received signal is said to exhibit Ricean fading [1, Section 1.2.2]. Rayleigh fading is often used to describe a radio environment with no such component [1, Section 1.2.1]. The Nakagami distribution can approximate both these distributions (e.g. when choosing the Nakagami parameter m = 1 the distribution becomes Rayleigh). It is often used since it in many cases is a closer fit to empirical data [1, Section 1.2.3], and is also easier to manipulate analytically than the Ricean distribution. α(t)

n(t)

? ? x(t) -@ m - my(t) -

Figure 1: Fading channel. The instantaneous received channel signal-to-noise ratio (CSNR) is denoted γ(t), and is defined as |α(t)|2 · P γ(t) = , (2) N0 B where P [W] is the average transmit power, N0 [W/Hz] is the noise power spectral density, and B [Hz] is the transmission bandwidth. We restrict P to be constant. If the channel is assumed to be wide sense stationary, the expected CSNR is γ¯ =

ΩP N0 B

(3)

where Ω = E[|α(t)|2 ], E[X] denoting the expected value of the random variable X. In the case of Nakagami fading, |α(t)| is Nakagami distributed and γ (we omit the time index t from now on to simplify notation) is Gamma distributed with the probability density function [1]  m m−1   m γ γ fγ (γ) = exp −m , γ > 0, (4) γ¯ Γ(m) γ¯ where m is the Nakagami fading parameter and Γ(·) is the Gamma function Z ∞ Γ(x) = tx−1 e−t dt for x > 0.

(5)

0

For a non-adaptive transmission scheme, the spectral efficiency is defined as the number of information bits transmitted per second per unit available bandwidth. For adaptive schemes, the average spectral efficiency (ASE) is defined as the expected spectral efficiency over all available adaptation modes. The maximum average spectral efficiency (MASE) for the Nakagami fading channel is the expected channel capacity in the Shannon sense, i.e., the ultimate upper theoretical limit for the ASE, and it is given by [2, Eq. (23)] MASE(¯ γ ) = em/¯γ log2 (e)

m−1 X k=0

m γ¯

k

  m Γ −k, γ¯

[bits/s/Hz],

(6)

where the Nakagami parameter m is assumed to be an integer and Γ(·, ·) is the complementary incomplete gamma function [3, Eq. (8.350.2)]. The formula in Eq. (6) holds under the assumption that the receiver is able to estimate the channel state information (CSI) perfectly and transmit it back to the transmitter on a noiseless zero-delay return channel.

2 2.1

Rate Adaptive Coding and Modulation Adaptive coding and modulation - Previous works

Several adaptive transmission schemes have been investigated in recent years (see e.g. [2] and [4]). The motivation is to be able to transmit with an ASE as close to the MASE as possible, at an average bit-error rate (BER) which fulfills the desired quality requirements. The schemes are based on dividing a flat fading channel into time slots where the channel introduces impairments that can be closely approximated by AWGN. The channel is periodically evaluated at the receiver and an estimate (prediction) of the future channel state is sent back to the transmitter on a separate channel (the return channel). The transmitter adapts to the instantaneous channel quality by choosing between different available transmission schemes of varying spectral efficiencies. When using code and modulation techniques designed for AWGN channels of different CSNRs, adaptive schemes may behave in a fashion that allows transmission at an ASE close to the MASE [4].

2.2

Adaptation strategy

In this report we use a rate-discrete transmission scheme which approximates the Optimal Rate constant power Adaptation (ORA) in [2]. This implies that the average transmit power is held constant, and the adaptation is done changing the channel code and modulation constellation, and thus the spectral efficiency, according to the channel condition. The constraint used is that the overall codec should have a BER lower than a certain target bit error rate, BER0 , for any given CSNR, except for very low CSNR values, when the channel will not be used for transmission. The CSNR γ can take on all values larger than zero. We divide γ ∈ {0, ∞} into N + 1 regions (indexed with n ∈ {0, 1, ..., N }) as shown in Fig. 2a. The CSNR will at any given time a)

∞ -

0 γ1

γ2

γ3

γN −1 γN

b) Information bits

Adaptive encoder/ modulator

-

Frequency-flat fading channel

-

6 Channel state information, n

Coherent detection and adaptive decoding

Decoded Information bits

Figure 2: a) Fading regions b) Transmission system.

fall into one of these regions (often referred to as fading regions). The estimator at the receiver determines the CSI, and sends a message to the receiver indicating which region n the CSNR is most likely to fall into during the next transmission (see Fig. 2b). For each region there is assigned a component codec, i.e., a proper chosen channel coding and modulation technique. When the estimated index is n = 0, the channel is in such a bad state that it should not be used for transmission. The number of component codecs to use (or equivalent, the number of fading regions) should be so high that the AWGN assumption is a good one over each fading region. However, it should be low enough for the ACM system to be able to adapt reliably between the codecs as the channel varies. The performance of these component codecs, which must of course be known, is measured in BER as a function of CSNR. The performance can be approximated by closed form expressions using curve fitting techniques on simulated data. This is of interest when designing—and analyzing the average BER performance of—ACM schemes. Several of the ACM systems that have been presented so far apply trellis coded modulation (see e.g. [5]). ACM systems using trellis codes are not very sensitive to channel variations over each transmitted channel symbol block, since only a short sequence of channel symbols are transmitted between each channel estimate [6]. In this report however we consider the use of block codes, in particular low-density parity check (LDPC) codes, which have shown very promising performance on AWGN channels [7]. A more comprehensive description of LDPC codes are given in Sec. 3. For now it it is sufficient to say that LDPC codes are block codes, whose promising behavior is based on transmission of relatively long sequences or blocks of channel symbols, combined with soft decision iterative decoding of relatively low complexity. We shall combine LDPC codes with quadrature amplitude modulation (QAM) and phase shift keying (PSK) to produce codewords of multilevel channel symbols. To fully utilize the error correcting properties of the LDPC codes we must ensure that the channel stays within one and the same fading region for the time period, T [s], used to transmit each codeword. For the BER-CSNR relationship of LDPC codes, MacKay et.al. have found a good closed form approximation [7]. We have used a slightly modified version of this equation (the modification being the constant a, which in [7] was set to 1) BER(γ) =

a (1 +

ec(γ−b) )d

,

(7)

where a, b, c, and d are constants that can be found by using some curve fitting technique. The thresholds {γn } are dependent on the target BER and can be calculated by demanding that BER(γn ) = BER0 for code n, and then using the inverse of the approximation used to describe the simulated data points !  1/dn an 1 γn = ln − 1 + bn . (8) cn BER0 We also define γN +1 = ∞. The performance of the ACM system described above is measured by the ASE, which is then compared to the MASE to see how far from optimum the system is operating. The ASE is given by N X ASE = Rn · Pn [bits/s/Hz], (9) n=1

where Rn is the spectral efficiency of the code used in component codec n, and Pn is the probability of that codec being used, i.e., the probability that the CSNR is falling into fading region n. This probability is given by [8] Pn = P (γn 6 γ < γn+1 ) =

mγn+1 n Γ(m, mγ ) γ ¯ ) − Γ(m, γ ¯

(m − 1)!

(10)

where m is the Nakagami parameter.

3

Low Density Parity Check Codes

LDPC codes are well known for their extremely good performance on AWGN channels [7]. As indicated by Shannon’s Channel Coding Theorem, the best performance is achieved when using a code consisting of very long codewords [9, Ch. 8]. In practical systems the codewords have to be short enough for complexity, buffering, and delay not to represent a problem. Also, if LDPC codes with different rates are to be used in a rate adaptive system, the block length must be restricted to the time over which we can reasonably assume that the CSNR stays within one fading region.

3.1

The concept and properties of LDPC codes

LDPC (or Gallager) codes where invented in the 1960s by R. J. Gallager [10]. As opposed to trellis codes which are built on convolutional codes, LDPC codes are block codes. LDPC codes constitute a large family of codes (the family also include Turbo Codes). As with all binary block codes, the codewords are generated by multiplying (modulo 2) binary information words with a binary generator matrix. Gallager’s invention was to construct the parity check matrix H, which is used in the decoding, such that it has a low density of 1s. This ensures that the code can be decoded with a relatively low complexity decoder, while also allowing good error protecting properties. The parity check matrix is made sparse by making sure that no row in the matrix has a Hamming weight larger than tr and no column has weight larger than tc , where tr andtc are small compared to the corresponding matrix dimensions. The rate of a LDPC code is r = k/q = (q − u)/q (information bits per code bit), where k is the length of the sequence of information bits, q is the length of the codeword, and u is the number of parity bits per codeword. s - LDPC t - QAM/PSK x- y- Demod- ˆt - LDPC ˆs ⊕ Encoder Modulator ulator 6 p Decoder n

Figure 3: System model for LDPC coded information transmitted on an AWGN channel using 2-dimensional channel symbols. Fig. 3 show the system model for transmission of information encoded with a LDPC code and subsequently QAM or PSK modulated on an AWGN channel. A k-dimensional sequence of information bits s is coded into a q-dimensional codeword t (column vector) by the LDPC encoder. The coded vector is split up into sub-vectors of size log2 (S), where S is the size of the

modulation constellation used. The vector of channel symbols, x is generated by modulating each of these sub-vectors. The received vector becomes y = x + n, where n is a complex AWGN vector. The vectors y, x, and n all have length M = q/ log2 (S). Since x and n are uncorrelated and the modulation is memoryless, the received vector can be demodulated symbol-by-symbol producing a hard decision, ˆt, on the received information vector t. Probabilities of the decision being correct is also calculated by the demodulator. All this information is fed to the decoder which iteratively decodes the entire received code block. Probabilities of the hard decisions in ˆt being correct are calculated on a symbol-by-symbol basis. For each received channel symbol yj ∈ y (j = 1, 2, . . . , M ), the demodulator calculates log2 (S) probabilities (which are concatenated into a vector p of length q = M log2 (S)) as follows: For channel symbol yj the demodulator calculates the probability of each of the symbols, ζi (i = 1, 2, . . . , S) in the constellation alphabet. The probability that ζi is the symbol transmitted is given by g(ζi , yj , ) Pr(ζi sent|yj , σ) = PS , (11) l=1 g(ζl , yj ) √ where σ = N0 B is the standard deviation of the AWGN vector, which is assumed known by the demodulator, and   (yj,I − ζi,I )2 + (yj,Q − ζi,Q )2 1 g(ζi , yj ) = √ exp − , (12) 2σ 2 2πσ where the indexes I and Q are used to separate the real and imaginary parts of channel symbols. Now, the probability of each information bit tˆh , h = 1, . . . , log2 (S), in the demodulated version of yj being correct can be calculated by ph =

S X

Pr(ζl sent|yj , σ) · Pr(tˆh |ζl ),

(13)

l=1

where Pr(tˆh |ζl ) is defined to be 1 if bit number h in the demodulated version of ζl is equal to tˆh , and 0 otherwise. The decoder uses the well-known fact that the syndrome c = Hˆt = Hz, where z represents the effect of the noise after demodulation, to find a noise vector ˆ z that results in the syndrome vector Hˆ z = c. To determine ˆ z the decoder uses a message passing algorithm called the sumproduct algorithm (also known as belief propagation) [11]. The decoder can be viewed as a bipartite graph built from the parity check matrix [11]. The graph consists of two types of nodes: noise-nodes, zˆj , holding the estimated value of zj ∈ z, and check nodes, ci , holding the syndrome (check) values in c. Nodes zˆj and ci are connected when Hij 6= 0, an example of such a graph is shown in Fig. 4. For each iteration all noise-nodes pass a message to all the check nodes they are connected to. Every single message contains information on the state of the noise-node and the node’s belief in whether this is the correct state or not. Initially this information is equal to the probabilities produced by the demodulator. Each of the check nodes calculate the syndrome, and passes messages back to the noise-nodes on how likely it is that the noise-node is in the correct state, given messages received from the other noise-nodes. This information is used in the next iteration to reevaluate the state of each of the noise-nodes. The decoding ends when the syndrome Hˆ z = c, or when a maximum number of iterations is reached. For a more in-depth description of the LDPC decoding process, see [11].

Noise Nodes m ... zˆm z ˆ zˆm zˆm 1 2 3 q  BJ B@  J J  BJ B @  B J B @ J @ J B J B  @ J B J B  @J  B J  B @J  B  JB @ JB J  B @ ... cm cm cm 1 q 2 Check Nodes

Figure 4: Example of a bipartite graph built from the parity check matrix. Message passing is done in both directions of a branch.

3.2

LDPC codes in rate adaptive coded modulation—Previous work

In [12] Myhre, Markhus, and Øien presented an ACM system employing LDPC codes. The software used to generate the results in [12] is the basis for the software used to find the results presented in this report. However, we did some major modifications to improve the confidence in the results, including replacing a single random number generator with the very short period of 107 , with 3 random number generators with periods ranging from 1016 to 2250 ≈ 1075.3 . To be able to simulate very low bit error rates (BER < 10−5 ) with much higher confidence, the software was also rewritten to enable parallel processing on multiple computers. Although the results in [12] in reality did not have the sufficient confidence the results were very promising, and introduced the use of LDPC codes in an ACM system.

4

Mobility constraints

The assumption used when modelling a fading channel by time-multiplexed AWGN channels, is that the channel is slowly varying in time [4]. Then the channel can be modelled as having blockwise (almost) constant fading, and the CSNR can be considered as almost constant for the time period used to transmit an given amount of information (such as a codeword). This is the case when the coherence time of the channel is much larger than the time slot used to transmit a block of coded information [1]. Coherence time is a relatively vague term, being defined as a period of time where the correlation of two samples of the amplitude of the noise is “relatively high”. Increased mobility, in terms of faster relative motion of receiver and transmitter, decreases the coherence time of a wireless channel. The period used to transmit a single channel symbol is denoted TS = 1/B [s] (Nyquist transmission), and with a block length of M symbols, the total transmission of one codeword takes T = M · TS [s]. The channel variation due to the relative movement of transmitter and receiver manifests itself as Doppler shift [13, Sec. 4.1]. The Maximum Doppler shift is given as fm = v/λc

[Hz]

(14)

where v [m/s] is the relative velocity of the receiver-transmitter movement, and λc [m] is the wavelength of the arriving plane wave. Using that carrier frequency can be written as fc = c/λc where c [m/s] is the speed of light, this results in the following relationship between maximum Doppler shift and carrier frequency vfc . (15) fm = c Clearly, mobility in a transmission environment changes the state of a wireless channel both with and without obstructing objects in the transmission path. The criterium for both time-flat and frequency-flat fading channels is that the correlation between received samples is “high enough”. To find the correlation between two components of a signal we use the definition in [14, Eq. (2.93)] Cov(a1 , a2 )

ρ(∆f, ∆t) = p

Var(a1 )Var(a2 )

,

(16)

where Cov(·, ·) is the covariance and Var(·) is the variance. The parameter ai represents the signal component at frequency fi and at time ti . We define ∆f = f2 − f1 and ∆t = t2 − t1 . If we assume isotropic scattering, the correlation coefficient for two signal components separated by ∆f Hz and ∆t seconds is equal to [14, Eq. (2.98)] ρ(∆f, ∆t) =

J02 (2πfm ∆t) , 1 + (2π∆f )2 σ 2

(17)

where J0 () is the zero-order Bessel function of the first kind [3, Eq. (8.441.1)]. In our case we may set ∆f = 0 since we assume flat fading. The channel can be viewed as approximately constant over each codeword if ρ(0, T ) > ρ0 where the value of ρ0 is chosen sufficiently high. From the assumptions, the only phenomenon that has to be restricted to make sure that the channel does not vary over each codeword, is the velocity of relative movement between the transmitter and the receiver. Also note that our assumption of a slowly varying channel not only ensures that the channel stays within a single fading region during transmission of a sequence or block of channel symbols, but also that the predicted channel quality will be sufficiently accurate. Using Eq. (15) and Eq. (17), the maximum velocity for an almost constant channel quality over a codeword is given as √ c · J0−1 ( ρ0 ) vmax = , (18) T 2πfc were J0−1 () is the inverted zero-order Bessel function of the first kind. The function is not oneto-one, but from Fig. 5 we see that we want the smallest positive solution of J0−1 (). In table 1 we have calculated the value of the inverse function for some appropriate values of ρ0 .

5 5.1

Simulations Component codecs

In our simulations a random information bit stream was coded by N = 8 different LDPC codes and subsequently modulated using QAM and PSK constellations with Sn = 2n+1 different symbols, n ∈ {1, 2, . . . , 8}. Each parity check matrix was generated by a random number

Figure 5: Envelope correlation versus normalized time lag.

ρ0 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.90

√ J0−1 ( ρ0 )

0.141 0.201 0.246 0.285 0.319 0.350 0.379 0.406 0.432 0.456

√ Table 1: Calculated values for J0−1 ( ρ0 .)

generator that generates rows sequentially and checks whether or not the weight criteria holds for each new row. The codes are indexed with the integer n. kn is the number of information bits, and un is the number of parity bits in a block of M channel symbols from a constellation of size Sn . The codes were designed such that the rate of LDPC code n is rn = n/(n + 1) information bits per coded bit, and such that the number of parity bits un per binary codeword for code n is constant and equal to the block length M for all n. Thus, when code n is employed each QAM symbol carries Rn = log2 (Sn )−1 = n information bits. Table 2 shows the parameters for the LDPC codes simulated. The coded bits were modulated to channel symbols using Gray n 1 2 3 4 5 6 7 8

rn 1/2 2/3 3/4 4/5 5/6 6/7 7/8 8/9

kn 200 400 600 800 1000 1200 1400 1800

Sn 4 8 16 32 64 128 256 512

Rn 1 2 3 4 5 6 7 8

Table 2: Simulated LDPC codes with block length M = 200 channel symbols.

mapping [1, Sec. 5.3.4]. The iterative decoder was set up to terminate if the decoded information was equal to the transmitted information (thus, no errors detected) or if the number of iterations reached 100 (resulting in detected bit errors). Note that there exists more than one LDPC code for a given rate and block length, since the parity check matrix is generated in a stochastic manner, using only a set of Hamming weights for the rows and columns of the binary parity check matrix. We do not claim that the codes presented here are the best codes; rather we have constructed a set of good codes that are shown to be useful in a rate-adaptive scheme. The BER versus CSNR performance for the codes is shown in Fig. 6. We used Monte Carlo simulations to find each point on the curves. To give good consistency of the simulated BERSNR relationship we simulated 100 errors per point found on the curve, where one error was defined as a block of M = 200 channel symbols decoded in error (not just one single information bit decoded in error). The parameters a, b, c, and d in Eq. (7) were found using nonlinear curve fitting, employing the least squares method (we used the lsqcurvefit routine in the Optimization Toolbox of Matlab). Then, by setting BER0 = 10−3 and 10−4 the thresholds {γn } were obtained. Table 3 shows both the approximation parameters and the thresholds for all of the simulated component codecs.

5.2

Comparisons to previous schemes

In [5] Hole, Holm, and Øien presented a rate adaptive system employing trellis coded modulation (hereafter referred to as the TCM scheme). We have included their results to be able to benchmark the quality of our system by comparing results. The TCM scheme used 4-dimensional channel symbols, i.e., sequences of two channel symbols collected constellations of the same sizes

Figure 6: Simulated LDPC codes.

γn n 1 2 3 4 5 6 7 8

an 14.0168 13.0823 13.4064 13.4105 0.0272 0.0513 0.3947 0.0621

bn 1.0620 6.1563 9.7041 13.6210 18.0941 21.0407 24.3590 27.2904

cn 4.2335 4.0004 4.3240 4.5389 6.5885 4.5350 2.5404 3.0309

dn 1.4013 1.3176 1.3151 1.5632 2.0968 2.8911 7.3215 5.4137

BER0 = 10−3

BER0 = 10−4

2.7 8.0 11.4 14.9 18.3 21.3 24.4 27.3

3.1 8.4 11.8 15.3 18.5 21.5 24.7 27.6

Table 3: Values of the parameters used to estimate the BER-CSNR relationship, and the threshold values for the fading regions.

used in our system. In [5], Hole et. al. reported that their TCM scheme performed 1.9 bits/s/Hz better than an uncoded rate adaptive system.

5.3

Spectral efficiency of the rate adaptive scheme

The ASE (see Eq. (9)) for our rate adaptive system is plotted versus the average CSNR in Figs. 7 through 9 for BER0 = 10−3 , and for three different values of the Nakagami parameter

Figure 7: MASE and ASE for rate adaptive systems employing LDPC codes and TCM, with m = 1. (m = 1, 2, 4), along with the MASE (see Eq. (6)) of the channel and the ASE of the TCM scheme. The difference between ASE of the rate adaptive system and the MASE of the channel is plotted in Figs. 10 through 12. Observe that the ASE of our scheme is always between 1 and 2 bits/s/Hz away from the MASE. Figs. 7 through 12 show that our LDPC scheme outperforms the TCM scheme for low values of CSNR (¯ γ < 27 [dB]). This is in accordance with the results from [12], the main difference being the significantly increased confidence in our results. In fact, our results are also slightly better then the results in [12]. The small improvement could be caused by LDPC codes with a slightly better BER performance, or it could be the result of improved confidence in simulated data. The highest rate component codec in the TCM scheme had a spectral efficiency of 8.5 bits/s/Hz, while our highest rate component codec has a spectral efficiency of the slightly lower 8 bits/s/Hz. This means that our scheme cannot perform with the same ASE for high values of the average CSNR. This can be observed in Figs. 7 through 12. If we want a higher capacity LDPC-based system also at higher CSNRs, we must find a ninth component codec with a higher spectral efficiency that performs well at higher CSNRs.

Figure 8: MASE and ASE for rate adaptive systems employing LDPC codes and TCM, with m = 2.

Figure 9: MASE and ASE for rate adaptive systems employing LDPC codes and TCM, with m = 4.

Figure 10: Difference between MASE and ASE for rate adaptive systems employing LDPC codes and TCM, with m = 1.

Figure 11: Difference between MASE and ASE for rate adaptive systems employing LDPC codes and TCM, with m = 2.

Figure 12: Difference between MASE and ASE for rate adaptive systems employing LDPC codes and TCM, with m = 4.

5.4

Mobility

A realistic example will show how allowed mobility is reduced with increased block lengths. The upcoming ETSI standard HiperLAN/2 [15] is specified to use carrier frequencies around fc = 5.4 GHz, and a symbol period of 4 µs. In the case of our scheme which has 200 symbols per block, maximum velocity according to Eq. (18) becomes 1.56 m/s (or 5.62 km/hr) which is about walking speed, when the correlation is chosen to be ρ0 = 0.99 during transmission. If ρ0 is set to 0.90 the maximum velocity increases to 5.04 m/s. It is however not entirely clear whether codeword length or the return path delay is the most significant factor in limiting the mobility. In Fig. 13 maximum speed of the LDPC-based scheme is plotted against different values of the envelope correlation bound ρ0 ∈ [0.90, 0.99] and of block length M ∈ [1, 450]. The figure shows how increased block lengths and increased correlation demand reduce the upper bound on the velocity of the relative movement of transmitter and receiver. The bound falls according to Eq. (7) (which for the simulated data is approximately exponential) for increasing block lengths. Reducing the correlation coefficient ρ0 increases this bound slightly.

6

Concluding remarks

The LDPC encoded rate-adaptive system presented shows a very promising behavior. Although it only performs better than previous schemes for low CSNRs, it is a good indication on how to create schemes with high average spectral efficiency using block codes.

Figure 13: Upper bound on velocity of the receiver as a function of envelope correlation and block length, when carrier frequency fc = 5.4GHz, and transmission time of one channel symbol is Ts = 4µs. It is also shown, by an example using parameters specified by the HiperLAN/2 standard, how increased code block length may decrease the allowed mobility in a rate-adaptive system. The results strongly indicate the trade-off between good BER performance (i.e., longer codewords) and mobility.

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