Optimal Pilot Sequences for Timing Estimation in Faster-than-Nyquist Systems
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(2) Optimal Pilot Sequences for Timing Estimation in Faster-Than-Nyquist Systems Leila Mounsif, Student Member, IEEE, and Damien Roque , Senior Member, IEEE. Abstract— Traditional synchronization techniques are usually challenged by Faster-than-Nyquist (FTN) systems which drop orthogonal pulses to the advantage of an increased spectral efficiency. In this letter, we studied pilot-aided timing estimation in FTN scenarios, based on the Cramér–Rao bound minimization. We established a closed-form approximation of the bound to explain the relation between optimal pilot sequences and the waveform’s parameters (i.e., signaling density and pulse shape). In particular, we showed that optimal sequences at the Nyquist rate can be non-informative in FTN scenarios. Index Terms— Cramér–Rao bound (CRB), fasterthan-Nyquist (FTN), pilot sequences, synchronization, timing estimation.. we use the Cramér–Rao lower Bound (CRB) as a cost function to design optimal pilot sequences, as already suggested for Nyquist systems in [14]. Our main contributions can be summarized as follows. At an over-critical density and for short-length pilots, we first verify that the CRB remains achievable by a maximum likelihood estimator, thus supporting its relevance as an optimization criterion. Through a closed-form approximation of the bound, we show that optimal timing pilots strongly depend on the signaling density in FTN scenarios and drastically differ from the solutions proposed in [14]. Specifically, optimal sequences at the Nyquist rate can be non-informative in FTN. We exemplify those results in the case of a rootraised-cosine (RRC) pulse to facilitate the comparison with traditional Nyquist systems. We also compare the performance of the proposed CRB-optimal pilots to traditional Zadoff–Chu sequences [15]. Notation: We use Z and R for the set of integer and real numbers, respectively. IN denotes the finite set {0, . . . , N − 1}. ·2 is the Euclidean norm for continuous-time or discrete-time signals, depending on the context. ·F is the Frobenius norm. ·T and ·H refer to transpose and conjugate transpose, respectively. E {·} is the expectation operator.. I. I NTRODUCTION ASTER-THAN-NYQUIST (FTN) linear modulations exploit an over-critical signaling density (i.e., symbol rate greater than the occupied bandwidth) with the aim of increasing spectral efficiency [1], [2]. Even if sophisticated symbol detectors are required to handle interpulse interference (IPI), the FTN strategy shows attractive performance in various power-constrained scenarios [3]–[5]. While many contributions focus on asymptotic or practical throughput of FTN systems, only a few addresses the synchronization problem (i.e., timing, phase or/and frequency recovery at the receiver) [6, Sec. IV]. In a non-data-aided (NDA) approach, traditional synchronization techniques mostly rely on IPI-free observations or on second-order cyclostationarity [7]; both are obviously unsuited to FTN signals. To overcome this issue, a non-linear least squares method has been proposed for carrier-frequency-offset estimation [8]; higher-order cyclostationarity can be used to build timing estimators, even though the observation length quickly becomes prohibitive as the density gets significant [9]. A pilot-aided step is generally required to ease synchronization in most realistic systems (e.g., short/medium bursts, drifting clocks and oscillators) [10]–[12]. A possible solution simply consists in keeping pilots at the Nyquist rate [10]. A full-FTN system may also rely on precoded scattered pilots [13]. Besides those practical solutions, the ultimate performance of pilot-aided synchronization remains largely unstudied in FTN scenarios. In this letter, we investigate the fundamental limits of timing estimation for single-carrier FTN signals. To this extent,. where {ck }k∈IK is the pilot sequence, Ts is the symbol period, ξ is the normalized timing offset (here the parameter of interest), wc (t) is a circularly-symmetric white Gaussian noise with power spectral density 2N0 , g(t) (p ∗ h)(t) is the pulse-channel impulse response with • p(t) a real-valued bandlimited pulse shape, with frequency support (−B/2; B/2); • h(t) a low-pass equivalent channel impulse response, potentially complex-valued.. The work of Leila Mounsif is supported by AID under grant 2018.60.0014. The associate editor coordinating the review of this letter and approving it for publication was E. Bedeer. (Corresponding author: Damien Roque.) The authors are with the ISAE-SUPAERO, Université de Toulouse, 31400 Toulouse, France (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/LCOMM.2020.3045512. The signaling density is defined as ο = 1/(BTs ). The transmission is said FTN iff ο > 1 (i.e., performed at an over-critical signaling density). A discrete-time observation is obtained by sampling (1) at a rate 1/T ≥ B, after a low-pass filter of impulse response v(t). In what follows, such a filter enforces (i) a non-zero gain in the bandwidth of interest (−B/2; B/2) to preserve all the information available for subsequent estimation/detection. F. II. S IGNAL M ODEL : L INEARLY M ODULATED S EQUENCE W ITH S YMBOL T IMING O FFSET We consider a linearly modulated signal transmitted over a frequency-selective channel. The baseband-equivalent received signal is K−1 rc (t) ck g(t − kTs − ξTs ) + wc (t), t ∈ R (1) k=0. Δ.
(3) tasks [16], (ii) a T -orthogonality constraint1 to yield white sampled noise. Even if a realizable analog filter would be used in practice, we choose v(t) = (1/T ) sinc(t/T ) for brevity: r(nT ) (rc ∗ v)(nT ) K−1 ck g(nT − kTs − ξTs ) + w(nT ), n ∈ Z. = k=0. where w(nT ) (wc ∗ v)(nT ). Assuming a reasonable energy decay for g(nT ), there exist integers N1 , N2 such that N N2 − N1 + 1 > 0 and r(nT ) ≈ 0 for n ∈ / {N1 , . . . , N2 }. A finite-length observation is thus appropriate: r = Gξ c + w (2) where • Gξ is the shaping matrix with [Gξ ]n,k g(nT − kTs − ξTs ), n ∈ {N1 , N1 + 1, . . . , N2 }, k ∈ IK ; T • r [r(N1 T ) r(N1 T + T ) · · · r(N2 T )] is the received signal vector; T • c [c0 c1 · · · cK−1 ] is the vectorized pilot sequence; T • w [w(N1 T ) w(N1 T + T ) · · · w(N2 T )] is the noise 2 vector with w ∼ CN 0, σw I N . 2 In the following, we set without loss of generality g2 = 1 and the noise variance is set according to the ratio Es /N0 , where Es represents the per-symbol energy assuming independent and uniformly distributed (i.u.d.) symbols (zero-mean 2 2 and unit-variance): σw v2 (Es /N0 )−1 . III. CRB-O PTIMAL P ILOTS FOR T IMING E STIMATION In the perspective of a deterministic unbiased timing estimation, the pilot sequence could be conveniently chosen to minimize the CRB [14]. Considering an additive white Gaussian noise (AWGN) channel, the expression of the bound is obtained via theSlepian–Bangs formula [17, Eq. (15.52)]: −1 2 H˙H˙ ˙ ξ dGξ . CRB (ξ|c) = (3) c c , G G G ξ ξ 2 σw dξ Using the spectral theorem, we can rewrite ˙ HG ˙ ξ = U H ΛU G (4) ξ. where U is a unitary matrix whose columns (eigenvectors) are denoted uk and Λ is a real-valued diagonal matrix filled with eigenvalues {λk }k∈IK . From (3), the CRB-optimal pilot sequence of bounded energy can be obtained thanks to the maximum principle [18, Th. 6.5]: √ ˙ HG ˙ ξ c s.t. c2 = K (5) cu Kumax = arg max cH G c. ξ. 2. with umax the eigenvector associated with the largest eigenvalue in {λk }k∈IK . In Fig. 1, we exemplify such optimal timing pilots in the case of a frequency-flat channel, with RRC pulses, as described in Appendix A. A frequency-domain representation is obtained by computing ˆ cu = F K cu with F K the unitary discrete Fourier transform (DFT) matrix. In conventional Nyquist scenarios (black markers), optimal sequences are almost unaltered by the signaling density, as already discussed in [14]. However, the opposite comment applies in the FTN case (colored markers). Regardless of the density, optimal pilots are well localized in frequency, thus arguing for an 1 A filter is said T -orthogonal, if the T -spaced replicas of its impulse response form an orthogonal set.. Fig. 1. Amplitude spectra of several optimal timing pilots. General parameters: frequency-flat channel, RRC pulses with roll-off α.. ˙ HG ˙ ξ , as proposed harmonic closed-form approximation of G ξ hereafter. ˙ HG ˙ ξ is banded In addition to its Toeplitz structure, G ξ in many practical cases (i.e., autocorrelation support of dg(nT )/dξ smaller than the K-length pilot sequence). In these ˙ HG ˙ ξ is often circumstances, a circulant approximation for G ξ convenient to exploit a DFT-based eigendecomposition. As a justification, we can invoke an asymptotic convergence in the weak norm sense [19, Lemma 7]: 1 ˙H˙ H ˜ lim √ G (6) ξ Gξ − F K ΛF K = 0 K→∞ F. K ˜ K−1 gathers the eigenvalues: ˜0, . . . , λ ˜ diag λ where Λ √ ˜k = K [F K g˙ ] , k ∈ IK (7) λ c k H ˙ ˙ with g˙ c the “circularized” first column of Gξ Gξ . If the weak norm in (6) is small enough, we can inject the circulant approximation in (5), leading to CRB-approaching pilot sequences: √ 2 ˜ cf Kf max = arg maxc cH F H K ΛF K c s.t. c2 = K (8) ˜ k |, with f max the column of F K associated with the largest |λ k ∈ IK . Besides the low-complexity implementation of (7), the approximate DFT-based eigendecomposition also provides ˙H ˙ a closed-form spectral interpretation of G ξ Gξ , as underlined in the following Theorem (a proof is given in Appendix B). Theorem 1: Considering the signal model in (2), the eigen˙ HG ˙ ξ can be approximated as follows for long pilot values of G ξ sequences: 2 2. K1 (2π)2 Ts k k m m. ˜. λk ≈ −. G KTs − Ts , T KTs Ts m∈Z. k ∈ IK (9) ∞ with G(f ) −∞ g(t)e−j2πf t dt. Theorem 1 connects asymptotically the eigendecomposition ˙H ˙ of G ξ Gξ to the frequency response of the pulse-channel function, characterized by its support (−B/2; B/2). A restriction to ο ≥ 1/2 (i.e., at most 100 % excess bandwidth), limits the.
(4) sum in (9) to its zeroth first terms:
(5) and 2 2 2. k k (2π) Ts ˜ k K1. G. ≈ λ. T KTs KTs. 2 2 . k − K k−K. G. , k ∈ IK . (10) +. KTs KTs. As a consequence, an FTN transmission necessarily introduces zero-approaching eigenvalues. In terms of timing estimation, it means that the corresponding eigenvectors are non˙H ˙ informative. The asymptotic nullity of G ξ Gξ as a function of the signaling density is developed in the following Corollary. Corollary 1: Considering a pulse-shape with frequency support (−B/2; B/2) and a non-zero channel frequency ˙ HG ˙ ξ is response |H(f )| = 0, the nullity of G ξ ⎧ ⎪ ⎪γFTN (K, ο) if ο > 1, ⎨ H K1 ⎪ 1 ˙ G ˙ξ 1 if ≤ ο ≤ 1, dim ker G ≈ ξ 2 ⎪ ⎪ 1 ⎪ ⎩0 if 0 < ο < , 2 (11) with ⎧ K ⎪ + 2 if K/(2ο) ∈ Z+ , ⎨K − ο γFTN (K, ο) (12) K ⎪ ⎩K − 2 if K/(2ο) ∈ R+ \Z+ . 2ο In Nyquist scenarios, there is at most one non-informative pilot sequence, which is necessarily f 0 , the zeroth column of the K-size DFT matrix. This number drops to zero if ο < 1/2. In FTN however, the number of non-informative sequences linearly increases with ο/K which advocates a density-dependent pilot optimization, based on the spectral characteristics of the pulse, as asymptotically emphasized in (10). IV. P ILOT-A IDED T IMING E STIMATION IN FTN S YSTEMS : A CRB-BASED P ERFORMANCE S TUDY In the following, we evaluate the impact of an FTN transmission on the pilot-aided timing estimation performance. To ensure a direct comparison with Nyquist systems (e.g., [14]), we focus on a pure AWGN scenario with RRC pulses (i.e., h(t) = δ(t) and p(t) defined as in Appendix A). A. Relevance of the CRB Criterion for Finite-Length Sequences We first assess the performance of CRB-optimal sequences cu (5) and cf (8) in realistic finite-length scenarios. To this extent, we compare the CRB to the mean-squared-error (MSE) of the maximum likelihood estimate (MLE) ξˆML , characterized by its asymptotic efficiency: for a given pilot sequence c, a a ξˆML (c) ∼ N (0, CRB (ξ|c)) where ∼ denotes an asymptotic distribution (i.e., valid for K large enough). We use a standard ˆ correlation-based to compute ξML and we evaluate approach 2 ˆ ˆ through Monte–Carlo MSE ξML (c) E |ξML (c) − ξ| simulations (20 000 realizations), as described for example in [17, Ch. 7]. In Figure 2, we plot CRB (ξ|cu ) (lines), CRB (ξ|cf ) (non-asterisk markers), MSE ξˆML (cu ) and MSE ξˆML (cf ). Fig. 2. CRB (ξ|cu ) (lines), CRB (ξ|cf ) (non-asterisk markers), MSE ξˆML (cu ) and MSE ξˆML (cf ) (asterisk markers), as a function of the signaling density ρ at Es /N0 = 10 dB. RRC pulses with roll-off α = 0.2 are used. The lowest density depicted (ρ = 25/30) represents the Nyquist reference scenario.. (asterisk markers). We observe a log-like relation between the timing CRB and the density. Even for very short sequences and regardless of the density, both ξˆML (cu ) and ξˆML (cf ) are CRB-approaching, which supports such a pilot optimization criterion in practical cases. Lastly, one can replace cu by cf at almost no penalty in most usual scenarios, except for very-short bursts (here exemplified at K = 10).. B. CRB-Optimal and Zadoff–Chu Sequences For comparison purpose, we consider the so-called Zadoff–Chu (ZC) root sequences which are notably used in Long Term Evolution (LTE) cellular networks [15], [20]. Each T (p) (p) (p) ZC root sequence is denoted czc czc [0] · · · czc [K − 1] with p −jπ K k(k+mod(K,2)) c(p) (13) zc [k] e where p ∈ IK is the root index chosen to ensure gcd(K, p) = 1 (i.e., K and p are relatively prime integers). Among other interesting properties, ZC sequences are constant amplitude zero autocorrelation (CAZAC) waveforms. We notice that ZC and cf sequences share the constant amplitude property and enforce the same energy normalization. However, while ZC sequences enjoy useful circular-crosscorrelation properties [15], their design is irrespective of the pulse shape, in general leading to suboptimal results in terms of CRB. In Figure 3a, we evaluate the impact of the ZC root sequence index p on the CRB performance. Even though its impact is negligible at low transmission densities, slight CRB variations as a function of p can be observed in severe FTN scenarios, ˙H ˙ as hinted by the consequential rank reduction of G ξ Gξ (Corollary 1). From the CRB viewpoint, the superiority of (p) cf with respect to czc also globally increases with ο. This observation is corroborated by Fig. 3b which depicts the CRB as a function of Es /N0 : a performance gap of 4.6 dB at ο = 25/30 and 6 dB at ο = 25/12 is measured between cf and the best ZC sequence identified in the previous FTN scenario (i.e., at p = 31)..
(6) √ Fig. 4. CRB ξ| Kf k (solid lines) and its approximation via Theorem 1 . (31). . (dotted lines) is also depicted for compar(square markers). CRB ξ|czc ison. General parameters: frequency-flat channel, RRC pulses with roll-off α = 0.2, K = 50, Es /N0 = 10 dB.. Fig. 3. Performance comparison between CRB-optimal and ZC sequences. General parameters: frequency-flat channel, RRC pulses with roll-off α = 0.2, K = 50.. Fig. 5. CRB (ξ|cf ) as a function of the RRC’s roll-off α. General parameters: frequency-flat channel, Es /N0 = 10 dB, K = 50. The lowest α is chosen to yield an orthogonal transmission scenario at ρ = 25/30.. ˙ HG ˙ξ C. Asymptotic Spectral Analysis of G ξ In Figure 4, we evaluate the impact of the DFT-based √ ˙ HG ˙ ξ . We first plot CRB ξ| Kf eigendecomposition of G k ξ as a function of k ∈ IK where f k denotes the kth column of the DFT (solid lines). We also depict its approximation obtained by injecting the circulant counter˙ HG ˙ ξ in (3) while using the asymptotic eigenvalpart of G ξ ues derived in Theorem 1 (square markers). At K = 50, the approximation shows its relevance for significant eigenvalues whereas it can be considered as pessimistic for negligible eigenvalues. Interestingly, the best DFT-based pilot sequence at the Nyquist rate with an RRC pulse (here obtained at ο = 25/30) √ is Kf K/2 for K even. In this particular case, we obtain the famous ±1 alternating pattern, as already shown in [14]. This sequence turns out to be non-informative in FTN scenarios, as confirmed by Theorem 1. More generally, Theorem 1 shows thatan optimal sequence at a given density may fall into ˙ HG ˙ ξ at other densities. ker G ξ We can also enumerate the DFT-based pilot sequences that perform better than a given reference, here chosen as the ZC root sequence of index p = 31. As forecasted in Corollary 1, this number decreases with the inverse of the signaling density.. This observation supports the use of CRB-optimal pilots in severe FTN scenarios. D. Impact of the Pulse Shape at Fixed Density In Figure 5, we plot CRB (ξ|cf ) as a function of the RRC’s roll-off factor α. The CRB continuously increases with α and with the density. Under a constant pulse energy constraint, the best performance is observed for the smallest roll-off factor. In this case, the pulse amplitude spectrum |G(f )| is maximized, leading to the CRB minimization, as hinted by (10). This observation suggests a specific pulse design for FTN timing synchronization, with the opportunity to relax the usual orthogonality constraint at a given symbol period. V. C ONCLUSION In this letter, we studied the impact of an FTN transmission on CRB-optimal timing pilot sequences. We verified the relevance of the optimization criterion for short/moderate sequence length via the simulation of the MLE’s meansquared-error. We established a closed-form approximation of the bound, describing its relation with the waveform’s parameters (Theorem 1). We showed that the number of.
(7) non-informative sequences increases with the signaling density (Corollary 1). In particular, a given pilot sequence can be optimal at a given density and non-informative at another density (e.g., ±1 alternating pattern combined with an RRC pulse). In summary, our results support a CRB-based and density-specific pilot design in FTN scenarios. Future work may include a performance evaluation of the proposed sequences in more realistic contexts (e.g., reduced-complexity timing estimators, phase and frequency nuisance parameters). A PPENDIX A ROOT-R AISED -C OSINE (RRC) P ULSE We define the root-raised-cosine impulse response parametrized by a roll-off factor α ∈ [0; 1] and a scale parameter T0 > 0 (see for example (2.2.12) in [7]): pRRC (t) ⎧ α ⎪ 1−α+4 ⎪ ⎪ π ⎪ ⎪ ⎪ for t=0 ⎪ ⎪ ⎪ ⎪ π π 2 α 2 ⎪ ⎪ √ sin cos 1+ + 1− ⎪ ⎪ ⎪ π 4α π 4α 2 ⎪ ⎪ ⎪ T0 ⎨ 1 for t = ± =√ 4α T0 ⎪ t t t ⎪ ⎪ sin π (1 − α) + 4απ cos π (1 + α) ⎪ ⎪ ⎪ T0 T0 T0 ⎪ ⎪ ⎪ 2 ⎪ ⎪ t t ⎪ ⎪ 1 − 4α π ⎪ ⎪ ⎪ T T 0 0 ⎪ ⎪ ⎩ otherwise (14) While the pulse is T0 -orthogonal, it can be used at a symbol period Ts = T0 . Denoting B = (1 + α)/T0 the bilateral bandwidth, the signaling density becomes ο = T0 /(Ts (1+α)). A PPENDIX B P ROOF OF T HEOREM 1 ˙H ˙ We recall the expression of an entry of G ξ Gξ assuming an appropriate H truncation of the time-domain observation: ˙ G ˙ξ G ξ. = Ts2. k,k N2 . g˙ ∗ (nT − kTs − ξTs )g(nT ˙ − k Ts − ξTs ). n=N1. ≈. Ts2. . g˙ ∗ (nT − kTs − ξTs )g(nT ˙ − k Ts − ξTs ). n∈Z. with g(t) ˙ dg(t)/dt. We use Parseval’s identity: H Ts2 ˙ G ˙ξ G ≈ |G˙ 1/T (f )|2 ej2πf (k−k )Ts df (15) ξ T (1/T ) k,k with 1 ˙ k ˙ G1/T (f ) (16) G f− T T k∈Z. ˙ ) j2πf G(f ). Recalling that 1/T > B and and G(f injecting (16) in (15), we obtain B2 H (2πTs )2 ˙ ξG ˙ξ G ≈ f 2 |G(f )|2 ej2πf (k−k )Ts df B T k,k −2 (17). ˜k from (7), while using Finally, we expand the expression of λ H ˙ G ˙ ξ: the banded assumption for G ξ K−1 H kl k ˜k = ˙ ξG ˙ξ G λ e−j2π K ej2π 2 (18) l=0. ≈. (2πTs )2 T. l,K/2. . B 2. −B 2. K/2−1. f 2 |G(f )|2. . k. ej2π(f Ts − K )l df.. (19). l=−K/2. For large K, the Dirichlet kernel in (19) can be approximated by a 1/Ts -periodic Dirac comb which yields (9) and completes the proof. R EFERENCES [1] J. E. Mazo, “Faster than Nyquist signaling,” Bell Syst. Tech. J., vol. 54, pp. 1451–1462, Oct. 1975. [2] F. Rusek and J. B. Anderson, “Constrained capacities for faster-thanNyquist signaling,” IEEE Trans. Inf. Theory, vol. 55, no. 2, pp. 764–775, Feb. 2009. [3] G. 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