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Eprints ID: 16212
To cite this version:
Abello, Albert and Roque, Damien and Freixe, Jean-Marie and Mallier,
Sébastien Faster-than-Nyquist Signaling: on Linear and Non-Linear Reduced-Complexity
Turbo Equalization. (2017) Analog Integrated Circuits and Signal Processing, vol. 91 (n° 2).
pp. 267-276. ISSN 1573-1979
Faster-than-Nyquist Signaling: on Linear and Non-Linear
Reduced-Complexity Turbo Equalization
Albert Abell´o · Damien Roque · Jean-Marie Freixe · S´ebastien Mallier
Abstract In the framework of digital video broadcasting by satellite - second generation (DVB-S2), we analyze a faster-than-Nyquist (FTN) system based on turbo equaliza-tion and low-density parity-check (LDPC) codes. Truncated maximum a posteriori(MAP) and minimum mean square error (MMSE) equalizers provide a reduced-complexity im-plementation of the FTN system. On the other hand, LDPC codes allow us to demonstrate attractive performance re-sults over an additive white Gaussian noise (AWGN) chan-nel while increasing spectral efficiency beyond the Nyquist rate and keeping a complexity comparable to that of a cur-rent DVB-S2 modem.
Keywords Faster-than-Nyquist · turbo equalization · low-density parity-check codes · truncated trellis equalizers · linear equalizers
PACS PACS 84.40.Ua
Mathematics Subject Classification (2000) MSC 94A05 · MSC 94A12 · MSC 94A40
1 Introduction
With an increasing use of wireless communications systems, spectral resources become increasingly scarce. In this con-text, new transmission techniques must offer high spectral
A. Abell´o · J.M. Freixe Eutelsat SA, Paris, France
E-mail: [email protected] D. Roque
Institut Sup´erieur de l’A´eronautique et de l’Espace (ISAE-SUPAERO) Universit´e de Toulouse, France
E-mail: [email protected] S. Mallier
Direction G´en´erale de l’Armement, Rennes, France E-mail: [email protected]
efficiency while fulfilling usual constraints in terms of trans-mitted power and bit-error-probability. In the framework of satellite broadcasting applications, new services such as video on demand combined with an increasing need for quality of service confirm the interest for high density trans-mission techniques [8].
Traditional systems usually respect the Nyquist criterion to avoid inter-symbol interference (ISI): the symbol rate is thus bounded by the bilateral bandwidth of the transmitted signal [16]. Consequently, the only way to improve spectral efficiency relies on an extension of the constellation size. However, given a fixed transmission power, the minimum Euclidean distance among symbols is decreased such that the bit-error-rate (BER) necessarily increases as well.
Another strategy referred to as “faster-than-Nyquist” (FTN) signaling was first introduced by J. Mazo in 1975 [14] to improve spectral efficiency without increasing the con-stellation size. This transmission technique yields uncondi-tionally ISI so that non-linear receivers are required in order to reconstruct the sequence of transmitted symbols at the cost of an extra computational load compared to Nyquist systems [9]. Nevertheless, such an algorithmic complexity has prevented the use of FTN techniques for a while.
Recent technological advances combined with iterative equalization and decoding techniques [7, 23] allow FTN sys-tems to be implemented with reasonable complexity. The challenge is therefore to find the best compromise between complexity and spectral efficiency increase. As a conse-quence, practical FTN systems have been presented, demon-strating attractive performance results [12, 15, 19]. FTN combined with low-density parity-check codes (LDPC) has been proposed in the framework of DVB-S2X [2], but finally postponed in order to avoid significant changes in the re-ceiver architecture. More recently, spectral efficiency gains up to 8-20 % have been achieved with the help of linear
equalization structures which suggest an affordable imple-mentation cost in user terminals [18, 13].
As an extension of the work mentioned before, we present a faster-than-Nyquist system achieving attractive performance results over the additive white Gaussian noise (AWGN) channel by means of LDPC codes. Our contribu-tions include the comparison between non-linear and linear equalization criteria, based on maximum a posteriori (MAP) and minimum mean square error (MMSE), respectively. In the former case, the truncation of the discrete-time FTN channel impulse response enables an affordable reconstruc-tion complexity. Furthermore, extrinsic informareconstruc-tion transfer (EXIT) charts are used to evaluate the necessary conditions for system’s convergence. Finally, the convergence analy-sis is exploited to estimate the achievable spectral efficiency for both equalization strategies, allowing us to highlight sys-tem’s capabilites in the framework of a DVB-S2 system.
The paper is organized as follows. Section 2 details the input-output relations of the FTN system over an AWGN channel. This includes a linear stage used for pulse shap-ing/matched filtering and a non-linear stage based on turbo equalization, intended for interference cancellation based on LDPC codes. Two equalization criteria are discussed: trun-cated MAP and MMSE. Section 3 studies BER system per-formance through simulations and also discusses its itera-tive convergence using EXIT charts while emphasizing its low additional complexity compared to actual satellite com-munications systems. To summarize, the achievable spectral efficiency is presented as a bound to system’s performance. Finally, concluding remarks are presented in Section 4.
2 Faster-than-Nyquist signaling
This Section presents the basics of faster-than-Nyquist sig-naling combined with turbo equalization techniques. First of all, a discrete-time model of the system is established con-sidering a linear transmitter/receiver over an AWGN chan-nel. A whitening filter is specified to yield the Forney obser-vation model [9]. Two different equalization techniques are presented next, based on MAP and MMSE criteria, respec-tively. Finally, the turbo equalization principle is introduced as a means to reach the ISI-free system performance with a satisfactory trade-off between performance and complexity.
2.1 Linear FTN system over an AWGN channel
We first consider the linear system depicted in Fig. 1. Let {x[k]}k∈Z⊂ `2(Z) be a sequence of independent and
iden-tically distributed (IID) symbols to be transmitted, where `2(Z) is the set of square summable sequences. More specif-ically, each symbol x[k] is taken in a constellation A (i.e., a finite set of size M). The complex baseband signal at the
g gˇ {x[k]} s(t) n(t) r(t) yc(t) mTs {yc[m]}
Fig. 1 Block diagram of a linear transmission system over an AWGN channel.
output of the transmitter is obtained by associating each x[k] with a pulse shape g(t) ∈L2(R), where L2(R) is the set of
square integrable signals: s(t) =
∑
k∈Z
x[k] g(t − kTs), t∈ R (1)
with Ts the elementary symbol spacing. Over an AWGN
channel, the received signal is denoted r(t) = s(t) + n(t) where n(t) is a circularly-symmetric complex Gaussian noise with zero mean and power spectral density Sn( f ) =
2N0, f ∈ R.
The received signal r(t) is filtered by ˇg(t) ∈L2(R). The resulting signal yc(t) is then sampled synchronously at times
mTs, m ∈ Z. If one denotes ∗ the convolution operator, the
samples of the system equivalent impulse response are given by h[m] = (g ∗ ˇg)(mTs) and the noise samples after reception
filtering are written nc[m] = (n ∗ ˇg)(mTs). Consequently, the
output of the linear system can be split into three terms: yc[m] = yc(mTs) =
∑
k∈Z x[m − k]h[k] + nc[m] (2) = x[m]h[0] | {z } useful term +∑
k∈Z\{0} x[m − k]h[k] | {z } interference term + nc[m] | {z } noise term (3) Within the linear system aforementioned, the transmis-sion density is given by ρ = 1/(TsB) where B is thetransmit-ted signal bandwidth (assumed finite). Based on the frame theory [5, Ch. 7], one remarks that:
– if ρ ≤ 1, interference-free transmission can be per-formed by means of a linear receiver (with Nyquist pulses), in such a case, the system is said orthogonal, namely h[m] ∝ δ0,mwith δ0,mthe Kronecker delta (Fig.
2a) ;
– if ρ > 1, inter-symbol-interference unconditionally ap-pears at the output of the linear receiver, viz. h[m] 6∝ δ0,m
(Fig. 2b) but if one further assumes that symbols are taken from a finite constellation, they can still be recov-ered by means of a non-linear post-processing [9]. In the following, a system will be said faster-than-Nyquist if and only if ρ > 1.
The model presented above can be further simplified thanks to the subsequent assumptions.
−8 −6 −4 −2 0 2 4 6 8 0 0.5 1 Time Amplitude h(t) h(mTs)
(a) Nyquist rate sampling (ρ = 0.87)
−8 −6 −4 −2 0 2 4 6 8 0 0.5 1 Time Amplitude h(t) h(mTs)
(b) Faster-than-Nyquist rate sampling (ρ = 1.8)
Fig. 2 Example of an equivalent impulse response of the linear transmission system, sampled at the Nyquist rate (a) and above the Nyquist rate (b). A raised cosine with roll-off factor α = 0.15 is considered here.
1. The transmitted signal is bandlimited to ensure a suffi-cient isolation in the frequency domain. More precisely, g(t) is a squaroot-raised-cosine (SRRC) impulse re-sponse with roll-off factor α = 0.15 throughout the arti-cle.
2. Since the transmission occurs over an AWGN chan-nel, matched filtering is performed at the receiver side, namely ˇg(t) = g(−t), so that the signal-to-noise ratio is maximized at the output of Fig. 1. Consequently, one obtains a symmetric discrete-time equivalent impulse re-sponse: h[m] = h[−m].
3. Filters g(t) and ˇg(t) are normalized so that h[0] = (g ∗ ˇ
g)(0) = 1, enabling perfect reconstruction provided that Nyquist pulses are used (orthonormal system).
4. By supposing a finite impulse response implementation of the aforementioned filters, the discrete-time equiva-lent impulse response is truncated to 2L + 1 taps: h[m] = 0, |m| > L. In practice, the following energy criterion keeps a reasonable bandlimited assumption:
L
∑
m=−L
|h[m]|2≥ 0.99. (4)
A feasible system can thus be obtained by setting a re-construction delay of L taps, ensuring causality. From (3), the autocorrelation function of the noise sam-ples is given by E{nc[m]n∗c[m + l]} = 2N0v[l] where E{·}
is the expectation operator, (·)∗denotes complex conjugate and v[l] = Z +∞ −∞ ˇ g(τ) ˇg(lTs+ τ) dτ. (5)
Using the matched filter assumption, notice that v[l] = h[l], which implies that a colored noise is obtained at the out-put of the linear receiver if h[l] 6∝ δ0,l. Such scenario arises
for any FTN configuration since ρ > 1 and possibly for non-FTN systems unless Nyquist filters are chosen. For the sake of simplicity and compactness, it is usually desirable
to whiten the noise before applying further interference can-cellation algorithms [6, 20]. In order to derive an equivalent model with white noise, we first express the z transform of v[l]: V(z) = L
∑
l=−L v[l]z−l, z∈ C (6)Since only SRRC filters are considered, g(t) and ˇg(t) are real-valued so that v[l] = v[−l] or equivalently V (z) = V(1/z). This implies that if z0∈ C is a zero of V (z), then
1/z0 is also a zero of V (z). Generalizing this fact to the
2L zeros of the polynomial, assuming (i) V (exp( j2π f )), f ∈ [0; 1) real and non-negative and (ii) any zero of V (z) on the unit circle is of even multiplicity1, one can perform a spectral factorization of the form
V(z) = U (z)U (1/z). (7)
A unique minimum-phase solution can be found by group-ing in U (z) all the roots inside the unit circle, half of the root on the unit circle [21]. As a consequence, U (1/z) represents a maximum-phase system, which implies a stable and anti-causal inverse.
From Fig. 3a, denoting nw[m] the noise samples at the
output of the noise whitening filter, with power spectral den-sity 2N0, one must ensure
U(z)U (1/z)W (z)W (1/z) = 1 (8)
with W (z) the transfer function of the noise whitening filter. The whitening condition (8) can be fulfilled with the triv-ial solution W (z) = 1/U (1/z). The discrete-time equivalent model with a noise whitening filter is depicted in Fig. 3b and characterized with the time-domain input-output relation: yw[m] = (yc∗ w)[m] =
L
∑
k=0
x[m − k]u[k] + nw[m]. (9)
1 Such condition is sometimes not fulfilled and an approximate
h w {x[m]}
{nc[m]}
{yc[m]}
{yw[m]}
(a) Discrete-time equivalent system with colored noise
u {x[m]}
{nw[m]}
{yw[m]}
(b) Discrete-time equivalent system with white noise
Fig. 3 Block diagrams of two discrete-time equivalent linear transmis-sion systems.
Remarkably, both yc[m] and yw[m] constitute sufficient
statistics for MAP detection since both sequences retain all the information in the received signal that is relevant to the detection of the known symbols [4, Sec. 7.3.3].
2.2 MAP equalization
In the context of an FTN transmission, equalization tech-niques should be used in order to mitigate ISI and finally detect the symbol sequence with a sufficiently low error-probability. The first approach developed is based on MAP criterion which is known to minimize bit-error-probability [4, Ch. 7]. Considering a finite-length symbol sequence x = [x[0], . . . , x[N − 1]]T with N > L and using the corresponding
observation sequence y = [yw[0], . . . , yw[N + L − 1]]T, the
de-cided symbol sequence ˆx = [ ˆx[0], . . . , ˆx[N − 1]]T is given by ˆx = argmax x Pr(x|y) (10) = argmax x p(y|x) Pr(x). (11)
In the framework of a turbo equalization scheme, the proba-bility Pr(x) is estimated by the forward error correction de-coder as it will be explained in Sec. 2.4. Otherwise, since IID symbols are considered, Pr(x) can be omitted in the maxi-mization. Assuming uncorrelated noise samples the likeli-hood function in (11) can be factored as:
p(y|x) =
N+L−1
∏
k=0
p(yw[k]|x) (12)
with the symbol likelihood function of the form [6]:
p(yw[k]|x) ∝ exp − 1 2N0 yw[k] − L
∑
l=0 u[l]x[k − l] 2 . (13)The computation of (10) can be performed efficiently by the so-called BCJR algorithm [3]. It consists in the representa-tion of all possible channel states in a trellis diagram and
the recursive computation of symbol likelihoods. The main limitation of this approach is its computational complexity, since the number of states considered in the BCJR trellis grows exponentially with channel’s memory L and with the number of bits per symbol log2M. Within an FTN system, the discrete-time equivalent channel length depends on sys-tem’s density, as justified in (4) under the constraint of ban-dlimited signals.
For example, if we consider the 31-coefficient channel presented in Fig. 2b, a MAP equalizer would require M15 trellis states in order to recursively compute the a posteri-oriprobabilities, with M varying from 4 to 32 within DVB-S2 standard. Facing this impractical scenario, two conse-quences follow:
1. MAP equalization should be restricted to small constel-lation sizes, such as binary phase-shift keying (BPSK) or quadrature phase-shift keying (QPSK) ;
2. the sum in (13) should be truncated to ν < L coeffi-cients enabling a reduced number of trellis states; this approximation is justified by the fact that SRRC filters are used, implying that |v[l]|2 globally decreases as |l| increases (Fig. 2b). Since u[l] results from a minimum-phase spectral factorization of v[l], it also tends to zero as l increases [17, Sec. 5.6.3].
2.3 MMSE equalization
Compared to the MAP equalization approach, the MMSE criterion leads to sub-optimal performance in terms of bit-error-probability but benefits from a linear computational complexity as a function of channel’s memory L (for any constellation size). The structure of a such equalizer is de-picted in Fig. 4. − f U y[k] ˜x[k] ˜y[k] z[k]
Fig. 4 Block diagram of an MMSE equalizer with a priori informa-tion.
If a turbo equalizer is used (Sec. 2.4), a priori symbols ˜
x[k], k ∈ {0, . . . , N − 1} are estimated thanks to the forward error correction decoder. Otherwise, they are set to zero. The observation sequence after partial interference cancellation is given by [22]:
˜y[k] = y[k] − U˜x[k], k∈ {0, . . . , N − 1} (14) where y[k] = [yw[k], yw[k − 1], . . . , yw[k − F + 1]]T is a set of
channel observations of length F > L with (·)Tthe transpose
Encoder Interleaver mappingSymbol u Turbo equalizer {nw[m]}
{a[n]} {b[i]} {c[i]} {x[k]} {yw[m]} { ˆa[n]}
Fig. 5 Block diagram of a complete faster-than-Nyquist system with turbo equalization.
of symbol estimates using a priori information where ˜x[k − d] is set to zero in order not to cancel the useful term, with d a reconstruction delay. U is the Toeplitz matrix of size (F × F + L) formed by the discrete-time equivalent impulse response u[l]. After interference cancellation, linear MMSE filtering is performed:
z[k] = fT˜y[k] (15)
where f = [ f [0], . . . , f [F − 1]]T represents the feedforward filter impulse response. Such filter is computed in order to minimize the mean square error between the estimates z[k] and the transmitted symbols x[k − d] at a time k − d: f = argmin ˜f E ˜f T ˜y[k] − x[k − d] 2 (16)
where ˜f = [ ˜f[0], . . . , ˜f[F − 1]]T represents the filter variable to be optimized.
2.4 FTN-induced ISI mitigation using turbo equalization The non-linear system using turbo equalization is depicted in Fig. 5. Let {a[n]} be a sequence of IID bits. This se-quence is LDPC encoded to produce {b[i]}. To decorrelate the sequence of coded bits, an interleaver is used, yielding the sequence {c[i]}. After a bit-to-symbol mapping opera-tion, the sequence of transmitted symbols {x[k]}k∈{0,...,N−1} is linearly filtered by the discrete-time equivalent channel presented in Sec. 2.1. Within FTN transmission conditions, we choose to mitigate the resulting interference thanks to a turbo equalizer [23]. This technique improves the perfor-mance of the equalizers described above by using a priori information provided by the forward error correction de-coder.
Turbo equalization (Fig. 6) consists in an iterative pro-cess involving two main blocks: the equalizer performs in-terference cancellation while the decoder estimates the se-quence of coded bits that will be sent back to the equalizer as an a priori information.
The iterative exchange between constituent blocks of the turbo equalizer may converge to the orthogonal (ISI-free) system performance. One notes two necessary conditions for system convergence [11]:
1. the sequence between equalizer and decoder must be in-terleaved and conversely de-inin-terleaved in order to sta-tistically decorrelate the bits;
2. extrinsic information must be exchanged between con-stituent blocks of the turbo equalizer in order to avoid local convergence. This implies subtracting output and input sequences for each constituent block before send-ing back the information.
In addition, log-likelihood ratios (LLRs) are used at each iterative step instead of binary values. The LLR of a bit a is given by
L(a) = lnPr(a = 1)
Pr(a = 0) (17)
where the sign of L(a) provides the hard (binary) decision on aand the magnitude |L(a)| is the reliability of this decision.
MMSE or MAP equalizer − Deinterleaver LDPC decoder − Interleaver y L( ˆc[k]|y) Lext( ˆc[k]|y) Lext(ˆb[k]|y) L(ˆb[k]|p) Lext(ˆb[k]|p) Lext( ˆc[k]|p) ˆ a[k]
Fig. 6 Block diagram of a turbo equalizer with MMSE or MAP equal-ization and LDPC decoding.
By using (17), we define L( ˆc[k]|y) the LLRs of equal-ized interleaved bits ˆc[k] conditionally to y. If the MAP equalizer is used, L( ˆc[k]|y) is computed from the a pos-teriori probability in (10). Otherwise, in the case of MMSE equalization, L( ˆc[k]|y) results from a input soft-output conversion of (15), as detailed in [23]. Similarly, Lext( ˆc[k]|y) and Lext(ˆb[k]|y) are the extrinsic LLRs of the
estimated bits ˆc[k] and ˆb[k] knowing y. We denote p = [Lext(ˆb[0]|y), . . . , Lext(ˆb[N − 1]|y)]T the sequence of
extrin-sic LLRs after de-interleaving, knowing y. Therefore we can define L(ˆb[k]|p) the LLRs of the estimated values ˆb[k] condi-tionally to p, Lext(ˆb[k]|p) and Lext( ˆc[k]|p) the extrinsic LLRs
of the estimated values ˆb[k] and ˆc[k] conditionally to p. The estimated sequence { ˆa[n]} results from performing Iiterations within the turbo equalizer. Finally, it is important to note that the non-linear system presented in this Section would require, with respect to the DVB-S2 standard system,
an additional computational load brought by the equalizer and by its iterative use along with the channel decoder.
3 Simulations
3.1 Introduction to the simulation framework
In this Section, simulations are performed in order to evalu-ate system’s BER performance as a function of Eb/N0where
Ebrepresents the per-uncoded-bit energy and N0refers to the
noise power spectral density.
The system performs LDPC encoding with code rate Rc = 1/2 unless otherwise stated. The parity check
ma-trix from the DVB-S2 standard is considered [1] along with the use of a random interleaver. Symbols are IID and fol-low a BPSK mapping. At the receiver side, iterative MMSE or MAP equalization and LDPC decoding are performed. For each turbo equalizer iteration, denoted “turbo iter”, the LDPC decoder performs a specified number of inner itera-tions.
The turbo equalizer block contains N = 64800 sym-bols as specified in the DVB-S2 standard. In addition, pulse shapes g(t) and ˇg(t) are SRRC with roll-off α = 0.15. The impulse response is truncated to 2L + 1 coefficients so that the whitened equivalent response has L + 1 coefficients. In order to bound complexity, a model with a reduced mem-ory ν is used in the MAP equalizer thus allowing a reduced trellis processing of 2νstates, as described in Section 2.2.
In addition, extrinsic information transfer (EXIT) charts are also computed [10] . They represent the average mu-tual information between the LLR at the output of a block (equalizer or decoder) and the transmitted symbol x, denoted I(LE; x), as a function of the average mutual information be-tween the input LLR and the transmitted symbol, denoted I(LA; x). These curves allow to accurately predict the
itera-tive behavior of the turbo equalizer by representing the mu-tual information both for the equalizer and the decoder and by considering that the output information of one of them becomes the input information for the other one and so forth.
3.2 System BER performance analysis
BER performance according to a given number of turbo it-erations is shown in Fig. 7 for ρ = 1.4 and a response with 2L + 1 = 9 taps. For each turbo iteration, the LDPC decoder performs 10 iterations. A Nyquist rate coded system is taken as a reference, implying ρ = 0.87 (dashed lines). Notice that in this configuration, the turbo equalizer has a convergence threshold of approximately 2 dB (i.e., the lowest Eb/N0
value from which BER can be decreased by iterating within the turbo equalizer). Since we want the FTN system to con-verge to Nyquist performance after a given number of
itera-0 2 4 6 8 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0(dB) BER FTN turbo iter 1 FTN turbo iter 2 FTN turbo iter 3 FTN turbo iter 4 Nyquist
Fig. 7 FTN MMSE-LDPC system performance for ρ = 1.4 and α = 0.15 with 10 LDPC iterations. 0 2 4 6 8 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0(dB) BER FTN turbo iter 1 FTN turbo iter 2 FTN turbo iter 3 FTN turbo iter 4 FTN turbo iter 5 Nyquist
Fig. 8 FTN MMSE-LDPC system performance for ρ = 1.4, α = 0.15 and 5 LDPC iterations.
tions, we can say that this will only be possible beyond this working point. In the next section, EXIT charts show that the FTN convergence threshold remains almost unchanged as the number of LDPC iterations (and thus Nyquist perfor-mance) increases, for a given equalizer. Therefore, we say that the equalizer becomes the limiting factor with respect to the LDPC decoder in the FTN turbo scheme. For example, if we set the number of LDPC iterations to 5 instead of 10 (Fig. 8), we observe that the convergence threshold remains the same whereas Nyquist performance has decreased. This allows us to reach convergence while reducing LDPC com-plexity. This result underlines the need for a joint configu-ration of the equalizer and the LDPC decoder. This will de-pend on system requirements specification in terms of target BER and decoding complexity.
Performance for full-complexity MAP equalization (ν = L= 4) is presented in Fig. 9. We can observe that the con-vergence threshold is lower than that of MMSE equaliza-tion so that convergence is reached after 3 turbo iteraequaliza-tions at Eb/N0= 4 dB. The MAP performance increase comes
0 2 4 6 8 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0(dB) BER FTN turbo iter 1 FTN turbo iter 2 FTN turbo iter 3 FTN turbo iter 4 Nyquist
Fig. 9 FTN MAP-LDPC system performance for ρ = 1.4 and α = 0.15 with 5 LDPC iterations. 0 2 4 6 8 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0(dB) BER FTN MMSE (ν = 4) FTN MAP (ν = 4) FTN MAP (ν = 3) FTN MAP (ν = 2) Nyquist
Fig. 10 FTN MAP-LDPC and MMSE-LDPC system performance for ρ = 1.4 and α = 0.15 with 5 LDPC iterations and 2 turbo iterations.
at the cost of complexity growing exponentially with L. A possible strategy to reduce complexity while maintaining good performance results is therefore to consider a truncated response to be considered by the MAP equalizer. To com-pare this approach with MMSE equalization, we present in Fig. 10 system’s performance for ρ = 1.4 (60% spectral ef-ficiency increase) for different values of memory ν of the truncated response. We observe that our system outperforms MMSE equalization while keeping reasonable complexity (the recursive algorithm computes 22to 23states).
In light of this results and for the sake of example, if we take typical DVB-S2 parameters, that is, satellite transpon-der bandwidth of 36 MHz, QPSK modulation and code rate Rc= 1/2, the given bit rate of 30 Mbit/s could be increased
up to 48 Mbit/s while keeping a constant bandwidth and equivalent BER performance for Eb/N0> 4 dB.
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I(LA; x) I( LE ;x ) MMSE eq. (2 dB) MMSE eq. (3 dB) MMSE eq. (4 dB) MAP eq. (2 dB) MAP eq. (3 dB) MAP eq. (4 dB) LDPC dec.
Fig. 11 EXIT chart for MAP-LDPC and MMSE-LDPC turbo equal-ization with ρ = 1.4, Eb/N0∈ {2, 3, 4} dB and 5, 10, 50 LDPC
itera-tions. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I(LA; x) I( LE ;x ) MMSE eq. (1.4) MMSE eq. (1.6) MMSE eq. (1.8) MAP eq. (1.4) MAP eq. (1.6) MAP eq. (1.8) LDPC dec.
Fig. 12 EXIT chart for MAP-LDPC and MMSE-LDPC turbo equal-ization with Eb/N0= 5 dB, ρ ∈ {1.4, 1.6, 1.8} and 5, 10, 50 LDPC
iterations. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 I(LA; x) I( LE ;x ) MMSE eq. (ν = 5) MAP eq. (ν = 4) MAP eq. (ν = 3) MAP eq. (ν = 2) LDPC dec.
Fig. 13 EXIT chart for MAP-LDPC and MMSE-LDPC turbo equal-ization with Eb/N0= 4 dB, ρ = 1.6 and 5, 10, 50 LDPC iterations.
3.3 Convergence analysis with EXIT charts
EXIT charts allow us to predict system’s convergence by representing the exchange of extrinsic information between the two main blocks within the turbo equalizer (equalizer and decoder). This way, the convergence threshold can be determined as the lowest Eb/N0 for which there is no
in-tersection point between the equalizer and decoder curves. Additionally, the more the latter are separated, the less it-erations are needed to reach convergence. As we observed through BER performance in Fig. 8 and 9, we confirm through EXIT charts in Fig. 11 a convergence threshold of Eb/N0= 2 dB for MMSE equalization and a slightly smaller
one for the full-complexity MAP approach. On the other hand, an increase in convergence speed can be obtained when LDPC iterations are increased (solid lines).
We observe in Fig. 12 the impact of a density increase in system’s convergence threshold and speed. It is also in-teresting to note that as the full convergence approaches (I(LA; x) = I(LE; x) = 1) the convergence speed is similar for
all densities. We also note that the threshold increases with density as it does with a decrease in Eb/N0. This behavior
limits the achievable spectral efficiency of the faster-than-Nyquist system as it will be presented below.
Finally, to evaluate the truncated MAP approach, we in-troduce in Fig. 13 the EXIT charts for ρ = 1.6 (yielding 2L + 1 = 11) and for different values of truncated memory ν and we compare them to the full-complexity MMSE equal-izer. We observe that this solution can outperform MMSE while keeping a memory as low as ν = 3 and therefore al-lowing a reasonable equalization complexity.
3.4 Evaluation of the achievable spectral efficiency
We have presented BER results as well as EXIT charts both allowing to analyze and compare system’s performance for MMSE and MAP turbo equalization in faster-than-Nyquist signaling. We have seen that truncated MAP approaches of-fer better performance than MMSE and that complexity can be limited by properly bounding the channel’s response to be considered by the equalizer.
In order to present system’s capabilities in a more gen-eral framework, the achievable spectral efficiency (Fig. 14) is computed here by using the convergence analysis pro-vided by EXIT charts. We define the achievable spectral effiency for the specified FTN system as the maximum spec-tral efficiency for which the decoding error probability can be made arbitrarily small for a given number of iterations. To do this, the convergence threshold has been determined for code rates Rc= {1/2, 2/3, 3/4} and for system’s densities
ρ = {1.4, 1.6, 1.8}. In all cases, the standardized LDPC par-ity check matrix in DVB-S2 is used (the decoder performs up to 50 inner iterations).
First of all, one notes that the efficiency difference be-tween MAP and MMSE equalization increases with the code rate. For instance, the combination (ρ = 1.4, Rc= 1/2)
shows a 0.3 dB gap between MAP and MMSE whereas the difference is increased to 0.7 dB for (ρ = 1.4, Rc= 2/3) and
up to 1.3 dB for (ρ = 1.4, Rc= 3/4).
Another interesting result is the efficiency difference in-crease that we find between MMSE and MAP equalization as system’s density increases. In this way, the difference of 0.3 dB for (ρ = 1.4, Rc= 1/2) is increased to 0.5 dB
for (ρ = 1.6, Rc= 1/2) and reaches 0.7 dB for (ρ = 1.8,
Rc= 1/2).
Finally, we note that the achievable spectral efficiency is constrained by system’s convergence threshold, which is to be computed for a given combination of equalizer/decoder and for a given density ρ and code rate Rc. This
representa-tion allows to choose the most efficient configurarepresenta-tion in each case.
4 Conclusion
In this work, we have presented a faster-than-Nyquist sys-tem that could be considered as an evolution of DVB-S2X standard for direct-to-home satellite broadcast. We have shown a spectral efficiency gain with respect to the ISI-free system at the cost of a computational load increase.
The comparison of MMSE and MAP equalization tech-niques allows us to find a compromise between system per-formance and complexity. In particular, MAP equalization using truncated discrete-time FTN channels can outperform MMSE solutions by ensuring BER performance conver-gence at a lower signal-to-noise ratio threshold. However, in virtue of its exponential complexity, the MAP approach is to be considered only for small constellation sizes, in par-ticular for BPSK and QPSK.
On the other hand, upper bounds for the achievable spec-tral efficiency as a function of code rate and system’s density have been found by means of a joint BER and EXIT speci-fication for the FTN system presented here.
Further study on faster-than-Nyquist systems may in-clude the optimization of the reduced complexity non-linear approach and the exploration of alternative pulse shapes other than SRRC.
Acknowledgements This work has been supported by the Direction G´en´erale de l’Armement(DGA) under the CIFRE grant 10/2015/DGA.
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Albert Abell´o received his M.Sc. double degree in electrical engineering from the Universitat Polit`ecnica de Catalunya, Barcelona and in aerospace engineering and satellite telecommu-nications from the Institut Sup´erieur de l’A´eronautique et de l’Espace (ISAE-SUPAERO), Toulouse in 2015. Since november 2015 he is engaged in a Ph.D. program at Eutelsat S.A. and ISAE-SUPAERO under a grant from the French armament procurement agency. His research interests include satellite communications, faster-than-Nyquist, turbo equalization and interference cancellation.
Damien Roque was
born in Chambery, France, in July 1986. He received the Ph.D. degree from the University of Grenoble in 2012 and the Engineer degree in telecommuni-cations from the Ecole´ Nationale Sup´erieure des T´el´ecommunications (ENST) de Bretagne in 2009. From 2009 to 2013, he worked at the Grenoble Images Paroles Signal Automatique (GIPSA-lab), Grenoble, France. Since 2013, he is an associate professor at the Institut Sup´erieur de l’A´eronautique et de l’Espace (ISAE-SUPAERO), University of Toulouse, France. His research interests are in the area of interference mitigation techniques for high throughput satellites, faster than Nyquist signaling, multicarrier modulations.
Jean-Marie Freixe
is the head of the in-terference group within the deployment and in-novation department at Eutelsat S.A., France. He received the M.Sc. degree in electrical engineering and space telecommuni-cations from the Ecole´ Nationale Sup´erieure des T´el´ecommunications (ENST) de Bretagne in 1982. His research interests include
space communications, interference cancellation, transmis-sion, turbo equalization, MIMO techniques, modulations and forward error correction.
S´ebastien Mallier was born in Rennes, France, in July 1973. He received the Ph.D. degree from INSA Rennes. In 2002, he joined the French armament pro-curement agency in Infor-mation Warfare Technology Center as a technical system engineer in telecommunica-tions and electronic warfare. His research interests are in the area of waveform design and signal analysis.
