FTN multicarrier transmission based on tight Gabor frames
- Alexandre Marquet^{1, 2}Email author,
- Damien Roque^{2},
- Cyrille Siclet^{1} and
- Pierre Siohan^{3}
https://doi.org/10.1186/s13638-017-0878-3
© The Author(s) 2017
Received: 28 October 2016
Accepted: 5 May 2017
Published: 30 May 2017
Abstract
A multicarrier signal can be synthesized thanks to a symbol sequence and a Gabor family (i.e., a regularly time–frequency shifted version of a generator pulse). In this article, we consider the case where the signaling density is increased such that inter-pulse interference is unavoidable.
Over an additive white Gaussian noise channel, we show that the signal-to-interference-plus-noise ratio is maximized when the transmitter and the receiver use the same tight Gabor frame. What is more, we give practical efficient realization schemes and show how to build tight frames based on usual generators. Theoretical and simulated bit-error probability are given for a non-coded system using quadrature amplitude modulations. Such a characterization is then used to predict the convergence of a coded system using low-density parity-check codes. We also study the robustness of such a system to errors on the received bits in an interference cancellation context.
Keywords
1 Introduction
In most of current communication systems, information symbols can be transmitted and reconstructed thanks to linear operations. More precisely, the synthesis and analysis families used in the transmitter and the receiver form biorthogonal frames (also known as Riesz bases). In a single-carrier band-limited scenario, this requires the Nyquist criterion to be respected [1]. In other words, the transmission rate must be lower than the bilateral bandwidth of the transmitted signal.
With an increasing need of spectral efficiency driven by overcrowded frequency bands, the main strategy relies on an increase of constellation size while keeping a constant transmission power, bandwidth, and symbol rate (below the Nyquist limit). This choice induces a decrease of the minimum Euclidean distance between symbols, and the transmitted signal becomes more sensitive to noise, thus increasing bit-error probability [2].
A more unusual way to improve spectral efficiency is to increase the symbol rate until the Nyquist criterion is overridden, leading to unavoidable inter-pulse interference (IPI). This idea has been proposed by J. Mazo under the denomination “faster-than-Nyquist” (FTN) [3]. His work showed that up to a certain point, an increase of the Nyquist symbol rate keeps the minimal distance between symbols unchanged. As a consequence, considering the work of G.D. Forney on the optimal detection in presence of inter-symbol interference, one can achieve identical bit-error probability using an optimal receiver [2].
FTN transmission techniques can be extended to multicarrier modulations [4]. In this case, denoting F _{0} the inter-carrier spacing and T _{0} the multicarrier symbol duration, it can be shown that if ρ=1/(F _{0} T _{0})>1, then the synthesis and analysis families, respectively used for transmission and reception, can no longer be biorthogonal but can still form overcomplete frames [5–7]. This leads to IPI both in time and/or frequency. Numerous studies focus on the realization of coded multicarrier FTN systems using, in particular, series or parallel concatenations [8] as well as turbo equalization techniques [9]. Studies of these latter systems over additive white Gaussian noise (AWGN) channels show great performance, confirming their relevance, even if their intrinsic complexity makes their design and performance comparison particularly demanding in terms of simulation time.
In this article, we study a linear multicarrier system operating with overcomplete Gabor frames (i.e., a generalization of an FTN system), as it plays a fundamental role in more complex systems (e.g., decision feedback equalizers, forward error correction, turbo equalizers). Our work includes guidelines for the design of optimal transmission and reception Gabor frames with respect to the signal-to-interference-plus-noise ratio (SINR) criterion over an AWGN channel. An efficient realization scheme is proposed and assessed with several classical pulse shapes (i.e., square-root-raised-cosine, rectangular...) built using the Wexler–Raz theorem.
This article is constructed as follows. Section 2 establishes input–output relations of the system in presence of noise, based on frame theory. This theoretical framework enables a closed-form expression of the SINR and the theoretical bit-error probability (BEP), assuming a circularly symmetric Gaussian distribution of the interference term. We then give in Section 3 practical efficient realization schemes. Next, Section 4 shows how we can easily find pulse shapes generating dual and tight frames. Section 5 first focuses on the statistical properties of the interference in an empirical way, as to confirm the relevance of its Gaussian approximation. We then present bit-error rate (BER) simulations to verify our theoretical results. In particular, we show how our closed-form BEP expression can predict the performance of a coded system. The last simulation scenario analyzes the relevance of interference cancellation techniques in this communication context. Finally, conclusions and insights are presented in Section 6.
and the \(\mathcal {L}_{2}\)-norm of x _{1}(t) is given by \({\left \|x_{1}\right \|}=\sqrt {{\left \langle x_{1},x_{1} \right \rangle }}\). The ℓ _{2}-inner product and its induced norm is defined similarly in the case of discrete-time signals.
2 System model
2.1 Input–output relations in presence of white Gaussian noise
On the contrary, in order to increase the spectral efficiency of the system (for a fixed number of bits per symbol), this article focuses on the case where ρ>1. Thus, this increase in spectral efficiency is counterbalanced by an induced interference. Indeed, when ρ>1, g is necessarily a linearly dependent Gabor family, but it may be an overcomplete frame of \({\mathcal {L}_{2}({\mathbf {R}})}\), i.e., a family for which (5) is valid not only for \(x \in {\mathcal {H}}_{{\boldsymbol {g}}}\) but also for every \(x \in \mathcal {L}_{2}({\mathbf {R}})\). In this case, (2) is always stable and \({\mathcal {H}}_{{\boldsymbol {g}}} = \mathcal {L}_{2}({\mathbf {R}})\). However, g cannot be a basis of \({\mathcal {L}_{2}({\mathbf {R}})}\).
where \({\boldsymbol {\check {g}}}~=~\{\check {g}_{m,n}\}_{(m,n) \in {\boldsymbol {\Lambda }}}\) is an analysis Gabor family, r(t) = s(t) + n(t) is the signal seen by the receiver where n(t) is a zero-mean circularly symmetric Gaussian noise, independent from the symbols, and whose bilateral power spectral density is γ _{ n }(f) = 2N _{0}, f∈R.
2.2 Interference and noise terms analysis
All the conditions for applying the central limit theorem are thus not fulfilled, but as shown by our simulations in Section 5.1, the Gaussian approximation is accurate for the sake of error–probability estimation. That is why in the following, we will assume \(i_{p,q} \sim \mathcal {CN}(0,\sigma _{i}^{2})\) independent from the noise. This is analogous to a case where the symbols would have been transmitted through an AWGN channel characterized by a signal-to-noise ratio (SNR) given by (18). It is interesting to note that the noise term n _{ p,q } is a colored zero-mean random variable following a Gaussian distribution.
2.3 Theoretical error probability with a linear M-ary system
We now restrict our analysis to the case where the symbols c are taken from an P-ary constellation, such as quadrature amplitude modulation (P-QAM) or phase-shift keying (P-PSK) with P the size of the constellation.
Approximating the interference distribution with a normal distribution, let us use classical formulas for BER in presence of AWGN [12, 13]. Given that these classical formulas usually give BER as a function of E _{ b }/N _{0} (with E _{ b } = E _{ s }/ log2(P) the per-bit energy), the only adaptation to be made in order to take into account the intereference is to change E _{ b }/N _{0} to SINR/ log2(P).
3 Discrete-time implementation of the linear system
In this section, we derive a discrete-time efficient implementation of the linear multicarrier system. Surprinsingly, such a process does not seem well documented in the literature. In the following, we first derive the input-output relation of a causal discrete-time system. Then, we develop an efficient time-domain implementation using the fast Fourier transform algorithm and finite impulse response filtering.
3.1 Discrete-time equivalent linear system
Let us recall the multicarrier transceiver defined by (2) and (7) with a finite number of subcarriers M and a finite number of multicarrier symbols K, such that Λ={0,…,M−1}×{0,…,K−1}. The transmission generator g(t) is supposed to have a bandwidth W _{ g }. It results an overall system bandwidth W=(M−1)F _{0} + W _{ g } that can be approximated by M F _{0} hereafter assuming |W _{ g }−F _{0}|/(M F _{0})≪1. In practice, it is generally the case if we consider a large number of subcarriers. As a consequence, the signal can be sampled at critical rate 1/T _{ s }=M F _{0} and we denote N the number of samples per multicarrier symbol such that T _{0}=N T _{ s }. Note that the density can be rewritten as ρ = M/N, and considering a unique multicarrier symbol, the FTN case is illustrated in the discrete-time domain by a number of samples per multicarrier symbol N less than the number of subcarriers M.
Given (22) and (23), it is usually desirable to use short-length generators in order to keep a low latency transceiver. In the following, for the sake of clarity, causality of the transceiver will be implicit.
3.2 Time domain efficient implementation of the linear system
Symbol rotation is performed at the transmitter and at the receiver side in order to account for the delay D required to yield a causal transceiver. Since such operations are dual and have no consequence on the performance of the system, they can be omitted at both ends for simulation purpose.
The discrete Fourier transform step should be implemented, thanks to the FFT algorithm, in order to ensure a computational complexity O(M logM). In Figs. 2 and 3, operations denoted FFT and its inverse (IFFT) refer to a column-wise implementation of [15]. Interestingly, but without computational complexity gain, rotations and discrete Fourier transform operations can be merged together with the help of the chirp-z transform, as proposed in [16].
4 Selection of the generators
We have seen in Section 2.1 that a Gabor family g may be a Riesz basis of \(\overline {\text {Span}}({\boldsymbol {g}})\) if ρ≤1 or a frame of \(\mathcal {L}_{2}({\mathbf {R}})\) if ρ>1. The case ρ ≤ 1 corresponds to “slower-than-Nyquist” (STN) or Nyquist rate (ρ = 1) systems, and it is the only possibility to obtain a perfect reconstruction (PR) system. On the contrary, Gabor families with ρ > 1 constitute FTN multicarrier systems. The PR case (ρ ≤ 1) has been well studied and is also often referred to as filtered multitone (FMT) systems [20], or oversampled orthogonal (or biorthogonal) frequency-division multiplexing (OFDM and BFDM) systems [21]. Such systems are also said to be orthogonal when the same generator is used at transmission and reception and biorthogonal if otherwise.
Duality relations between STN and FTN multicarrier communication
System | STN | FTN |
---|---|---|
Frequency spacing | \(\tilde {F}_{0} ~=~1/T_{0}\) | \(F_{0} = 1/\tilde {T_{0}}\) |
Time spacing | \(\tilde {T}_{0} ~=~1/F_{0}\) | \(T_{0}~=~1/\tilde {F_{0}}\) |
Density | \(\tilde {\rho } ~=~1/(\tilde {F}_{0} \tilde {T}_{0})~=~F_{0} T_{0}~<~1\) | \(\rho ~=~1/(F_{0} T_{0})~=~ \tilde {F}_{0} \tilde {T}_{0} ~>~1\) |
Interference mitigation | PR with biorthogonal frames | Dual frames |
Noise reduction | PR with orthogonal frames | Tight frames with A _{ g } = 1 |
Transmission generator | \(\tilde {g} ~=~ {g/\sqrt {\tilde {\rho }}} ~=~ \sqrt {\rho }\ g\) | \(g ~=~ {\tilde {{g}}/\sqrt {\rho }} ~=~ \sqrt {\tilde {\rho }}\ \tilde {{g}}\) |
Reception generator | \(\tilde {\check {g}} ~=~ {\check {g}/\sqrt {\tilde {\rho }}} ~=~ \sqrt {\rho }\ \check {g}\) | \(\check {g} ~=~ {\tilde {{\check {g}}}/\sqrt {\rho }} ~=~ \sqrt {\tilde {\rho }}\ \tilde {{\check {g}}}\) |
There are various ways to get PR-FMT filters. The more obvious one is to simply use at transmission and reception the same rectangular filter \(\tilde {{g}}(t)\) = \( \tilde {{\check {g}}}(t) ~=~ \sqrt {\tilde {F}_{0}} \Pi (\tilde {F}_{0} t) ~=~ \tilde {F}_{0}\) if \(|t|~<~1/(2 \tilde {F}_{0})\) and \(\tilde {g}(t)\) = \( \tilde {\check {g}}(t)~=~0\) if \(|t|~>~1/(2 \tilde {F}_{0})\). This leads to an orthogonal PR-FMT system. By duality, choosing \({g}(t)\,=\, {\check {g}}(t)\) = \(1/\sqrt {\rho T_{0}} \Pi (t/T_{0})\) leads to tight frames. This will be referred as the \(\text {RECT}_{T_{0}}\) generator in the rest of this text.
Setting \(\tilde {g}(t)~=~ \sqrt {\tilde {F}_{0}} \Pi (\tilde {F}_{0} t)\) and \(\tilde {\check {g}}(t)~=~ \sqrt {\tilde {F}_{0}} \Pi (t/\tilde {T}_{0})\) leads to a PR biorthogonal STN system, but it is not orthogonal since \(\tilde {{g}} ~\neq ~ \tilde {{\check {g}}}\). By duality, this allows us to obtain dual frames which are not tight: \({g}(t) ~=~1/\sqrt {\rho T_{0}} \Pi (t/T_{0})~=~ \text {RECT}_{T_{0}}(t)\) and \({\check {g}}(t) ~=~1/\sqrt {\rho T_{0}} \Pi (t/\rho T_{0}))~=~\text {RECT}_{\rho T_{0}}(t)\).
Another classical way to obtain orthogonal FMT systems is to use square-root-raised-cosine (SRRC) generators. Indeed, they meet the Nyquist criterion for a \(\tilde {T}_{0}\) time spacing on each subband. What is more, since their frequency occupancy is \([-(1~+~\alpha)/(2\tilde {T}_{0}); (1~+~\alpha)/(2\tilde {T}_{0})]\), where α designates the roll-off factor, there is no inter-carrier interference granted that \((1~+~\alpha)/\tilde {T}_{0} \leq \tilde {F}_{0}\) which is also equivalent to \(\alpha ~\leq ~1/\tilde {\rho }~-~1\). Finally, we thus obtain tight frames with SRRC generators with roll-off factor α ≤ ρ − 1. In here, we will use α = ρ − 1 and a truncation length equal to 32T _{0}.
Finally, we have chosen to also use two types of discrete-time-optimized orthogonal filters published in [20]. The first one is designed in order to minimize out-band energy (OBE), and the second one maximizes time–frequency localization (TFL). They also have the property of having a minimal time duration. With a proper adaption as described previously, they lead to tight frames.
5 Simulations
5.1 Empirical study of the interference term
In this sub-section, we discuss the relevance of the Gaussian approximation of the interference. To this extent, we measure 3.6 × 10^{6} realizations of the interference term i _{ p,q } by performing a transmission of M = 64 subcarriers over K = 50000 multicarrier symbols for different values of ρ, using a QPSK constellation and tight frames. The variance of the obtained samples is then normalized thus giving standardized versions of i _{ p,q } depending on ρ, whose empirical probability density functions and cumulative distribution functions (CDF) are comparable. Thus, we will simply denote by i|ρ the random variable whose realizations are i _{ p,q }, with the knowledge of ρ. The behavior described here has been observed to be similar for various generators forming tight frames.
χ ^{2} and Kolmogorov–Smirnov (KS) statistical tests results for the null hypothesis that the real (\(\mathcal {R}\)) and imaginary part (\(\mathcal {I}\)) of the interference come from a normal distribution
Pulse shape | ρ | KS rejected | KS p value | χ ^{2} rejected? | χ ^{2} p value | ||||
---|---|---|---|---|---|---|---|---|---|
\(\mathcal {R}\) | \(\mathcal {I}\) | \(\mathcal {R}\) | \(\mathcal {I}\) | \(\mathcal {R}\) | \(\mathcal {I}\) | \(\mathcal {R}\) | \(\mathcal {I}\) | ||
TFL | 16/15 | Yes | Yes | 2.1 × 10^{−6} | 2.1 × 10^{−6} | Yes | Yes | <10^{−6} | <10^{−6} |
8/5 | Yes | Yes | <10^{−6} | <10^{−6} | Yes | Yes | <10^{−6} | <10^{−6} | |
OBE | 16/15 | Yes | Yes | 3.4 × 10^{−5} | 3.4 × 10^{−5} | Yes | Yes | <10^{−6} | <10^{−6} |
8/5 | Yes | Yes | <10^{−6} | <10^{−6} | Yes | Yes | <10^{−6} | <10^{−6} | |
SRRC | 16/15 | Yes | Yes | <10^{−6} | <10^{−6} | Yes | Yes | <10^{−6} | <10^{−6} |
8/5 | Yes | Yes | <10^{−6} | <10^{−6} | Yes | Yes | <10^{−6} | <10^{−6} | |
\(\text {RECT}_{T_{0}}\) | 16/15 | Yes | Yes | <10^{−6} | <10^{−6} | Yes | Yes | <10^{−6} | <10^{−6} |
8/5 | Yes | Yes | <10^{−6} | <10^{−6} | Yes | Yes | <10^{−6} | <10^{−6} |
5.2 Linear system performance
The simulations presented in this sub-section consist in the transmission of K = 5000 multicarrier symbols over M = 128 subcarriers with a QPSK constellation. They were run for various generators, as presented in Section 4.
In terms of performance, for this kind of non-coded multicarrier FTN system, Fig. 10 shows that the bit-error rate (BER) rapidly rises with the density. We can also see in Fig. 11 that a lower-bound of the BER appears as noise becomes negligible compared to interference. In addition, and in accordance with the expression of the SINR, the performance gets worse if the frames used are not tight nor canonical dual. These results confirm the needs to develop non-linear detectors allowing for a more efficient IPI mitigation.
5.3 Use in a coded system with iterative decoding
As highlighted in the previous sub-section, a linear system is not sufficient to combine practical BER performance with attractive density gains. To overcome this issue, one can add forward error correction (FEC). In this work, we focus on FEC schemes enabling iterative decoding (e.g., LDPC, turbo codes) due to their good performance at an affordable computational complexity.
5.4 Performance with interference cancellation
From the expressions of the bit-error probabilities (20) (21), it is obvious that the FTN linear system shows worse performance compared to the orthogonal case. Besides, from the expression of the received signal (8), one can notice that the performance of the orthogonal system can be retrieved by removing the interference induced by the FTN system, allowing for an improvement of the spectral efficiency of the transmission while keeping the same BER.
6 Conclusions
Through this article, we specified a linear multicarrier system based on the use of overcomplete Gabor frames, enabling an increase in signaling density in the time and/or the frequency domain and leading to a bidimensional FTN system. Consequently, an increase of the spectral efficiency beyond (bi)-orthogonal systems (for a given constellation size) yields interference between pulse shapes.
The results presented in this article compare the performance of FTN multicarrier systems based on the parameters of their linear stage (e.g., time–frequency lattice density, transmission/reception generators...) in the presence of additive white Gaussian noise. Additionally, guidelines for efficiently choosing transmission/reception generators and implementing the discrete-time equivalent linear system are provided.
Finally, we have shown that the performance of linear systems should be studied before designing more complex receiver structures (e.g., LDPC/turbo decoders, turbo equalizers). Besides, such a separate analysis helps to lower simulations’ computational complexity.
Future work includes the assessment of the complete FTN multicarrier system over more realistic channel models (e.g., fading...). More precisely, it would be interesting to evaluate the robustness of such a system in the presence of an imperfect channel estimation.
Declarations
Acknowledgements
The authors would like to address a particular thanks to Dr. Laurent Ros for his valuable advices and relevant remarks concerning this work.
Authors’ contributions
This article is an extended version of AM, CS, DR, and PS’ “Analysis of a Multicarrier Communication System Based on Overcomplete Gabor Frames,” Cognitive Radio Oriented Wireless Networks, Springer International Publishing, p.387, 2016, Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, 978-3-319-40352-6. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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