232 Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243

Frontiers of Information Technology & Electronic Engineering www.jzus.zju.edu.cn; engineering.cae.cn; www.springerlink.com ISSN 2095-9184 (print); ISSN 2095-9230 (online) E-mail: [email protected]

Data recovery with sub-Nyquist sampling: fundamental limit and a detection algorithm∗ #

Xiqian LUO, Zhaoyang ZHANG†‡ College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China †E-mail: [email protected] Received June 28, 2019; Revision accepted Nov. 26, 2019; Crosschecked Oct. 20, 2020

Abstract: While the Nyquist rate serves as a lower bound to sample a general bandlimited signal with no information loss, the sub-Nyquist rate may also be sufficient for sampling and recovering signals under certain circumstances. Previous works on sub-Nyquist sampling achieved dimensionality reduction mainly by transforming the signal in certain ways. However, the underlying structure of the sub-Nyquist sampled signal has not yet been fully exploited. In this paper, we study the fundamental limit and the method for recovering data from the sub-Nyquist sample sequence of a linearly modulated baseband signal. In this context, the signal is not eligible for dimension reduction, which makes the information loss in sub-Nyquist sampling inevitable and turns the recovery into an under-determined linear problem. The performance limits and data recovery algorithms of two different sub-Nyquist sampling schemes are studied. First, the minimum normalized Euclidean distances for the two sampling schemes are calculated which indicate the performance upper bounds of each sampling scheme. Then, with the constraint of a finite alphabet set of the transmitted symbols, a modified time-variant Viterbi algorithm is presented for efficient data recovery from the sub-Nyquist samples. The simulated bit error rates (BERs) with different sub-Nyquist sampling schemes are compared with both their theoretical limits and their Nyquist sampling counterparts, which validates the excellent performance of the proposed data recovery algorithm.

Key words: Nyquist-Shannon sampling theorem; Sub-Nyquist sampling; Minimum Euclidean distance; Under-determined linear problem; Time-variant Viterbi algorithm https://doi.org/10.1631/FITEE.1900320 CLC number: TN911.72

1 Introduction maximum frequency of the bandlimited signal, the samples capture all the information and the original The Nyquist-Shannon sampling theorem estab- analog signal can be perfectly reconstructed from lishes a sufficient condition for a sample rate that the samples. The lower bound of the sampling permits a discrete sample sequence to capture all the rate, which equals twice the bandwidth, is called information of a bandlimited continuous-time sig- the Nyquist rate. However, it is also possible to nal. With a sampling rate of no less than twice the sample and recover an analog signal with a certain time and/or frequency structure at a sub-Nyquist ‡ Corresponding author rate perfectly with no information loss. It is called * Project supported by the National Natural Science Founda- tion of China (Nos. 61725104 and 61631003) and Huawei Tech- sub-Nyquist sampling (SNS), and it is an efficient nologies Co., Ltd. (Nos. HF2017010003, YB2015040053, and way to reduce the hardware cost on sampling, signal YB2013120029) # A preliminary version was presented at the 11th International processing, and storage. Conference on Wireless Communications and Signal Processing, October 23–25, 2019, China Sub-Nyquist sampling has been widely studied ORCID: Xiqian LUO, https://orcid.org/0000-0002-6219-5240; Zhaoyang ZHANG, https://orcid.org/0000-0003-2346-6228 as one of the fundamental problems in signal pro- c Zhejiang University Press 2021 cessing. Early works mainly focused on nonuniform Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 233 sampling of the multiband signal with known fre- lem, i.e., sub-Nyquist sampling on the receiver side, quency support (Scoular and Fitzgerald, 1992; has a similar property in information conservation. Venkataramani and Bresler, 2001). Landau (1967) The above problem of sub-Nyquist sampling of showed that the average sampling rate of a multi- a general signal is a fundamental problem in signal band signal can be reduced to twice the occupied processing. Unlike sub-Nyquist sampling in com- bandwidth; that is, the “Landau rate,” which is typ- pressive sensing, sub-Nyquist sampling of a general ically far smaller than the corresponding Nyquist signal without a special structure usually results in rate, serves as the lower bound for sampling multi- an under-determined linear problem, which is not band signals. sparse. Furthermore, it brings about energy loss as Later, many researchers attempted to general- a result of the reduction of the number of available ize the problem to multiband signals with unknown samples, which may in turn degrade the performance spectral support (Feng and Bresler, 1996; Herley of data recovery. In this study, we take a closer look and Wong, 1999). Multicoset sampling (Domínguez- at the fundamental aspects of this problem. Specif- Jiménez et al., 2012) and the modulated wideband ically, we study the performance limit and the data converter (Mishali and Eldar, 2010) are two widely recovery algorithm for data recovery from the sub- used sub-Nyquist sampling methods for multiband Nyquist sample sequence of a general baseband sig- signals with unknown spectral support. Such a prob- nal, which is linearly modulated with symbols from lem could be addressed well under a compressive a finite alphabet. In the studied system, the trans- sensing framework in some specific cases such as cog- mitter transmits symbols at the Nyquist rate, while nitive radio, where the spectral occupation of the the receiver samples the received signal uniformly signal is unknown but sparse (Mishali and Eldar, at a sub-Nyquist rate. Unlike sub-Nyquist sampling 2009; Tropp et al., 2010; Sun et al., 2013). In such a in compressive sensing, the general baseband signal context, compressive sensing is introduced to sample is sparse neither in the time domain nor in the fre- signals with a rate much lower than the Nyquist rate. quency domain. As a result, its degree of freedom The sparsity of the signal in the frequency domain is not reducible. We consider two practical sub- guarantees adequate detection accuracy. Of course, Nyquist sampling schemes, i.e., direct sub-Nyquist the multiband signal is not the only kind of signals sampling (DSNS) and filtered sub-Nyquist sampling that can be perfectly sampled and reconstructed with (FSNS). The former directly samples the modulated sub-Nyquist sampling. In the recent works from Lu signal at some sub-Nyquist rate, while the latter and Do (2008) and Mishali et al. (2011), a model for conducts the sampling after a perfect anti-aliasing the union of subspaces was proposed to reveal the low-pass filtering. For both sampling schemes, the underlying structure of the signals for which sub- impacts of sub-Nyquist sampling on the error per- Nyquist sampling works. Chen et al. (2013) stud- formance in data recovery are investigated, and a ied the impact of sub-Nyquist sampling on channel practical algorithm to resolve the very challenging capacity. data recovery problem is proposed. On the other hand, the early studies and re- The main contributions of our work can be sum- cent revisits to faster-than-Nyquist signaling (Mazo, marized as follows: 1975; Mazo and Landau, 1988; Liveris and Georghi- 1. The fundamental performance limit of data ades, 2003; Fan et al., 2017) revealed that there exists recovery of a sub-Nyquist sampling system for gen- some threshold of the signaling rate on the transmit- eral linearly modulated baseband signals is investi- ter side, which is beyond the Nyquist rate and is gated. The minimum Euclidean distances between called the “Mazo limit.” Within the limit, it is found two different sample sequences for both DSNS and that the minimum Euclidean distance between the FSNS are derived as the indicators of the upper transmitted signals does not decrease as the symbol bounds of data recovery performances. The derived rate increases beyond the Nyquist rate. This finding minimum Euclidean distances are proportional to supports a potentially efficient signaling scheme to the sampling rates within certain thresholds, and approach the constrained channel capacity for future thus provide a bound of the performance loss re- communications (Anderson et al., 2013). What is sulting from sub-Nyquist sampling. more, it inspires us to consider whether its dual prob- 2. A low-complexity but efficient time-variant 234 Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243

Viterbi detection algorithm is proposed to recover The signal is bandlimited to [−π/T, π/T ] and its data from the sub-Nyquist sampled data sequence. spectrum is  As the original signal is a general linearly modu-   −jωmT π π lated signal which lacks any inherent sparsity, data X(ω)= EsT um e R − , , recovery is an under-determined problem. Using the m T T constraint of a finite alphabet of transmitted sym- (3) bols and the priori knowledgement of the intrinsic where R(−π/T, π/T ) is the box function that equals interference structure of the sample sequence, the al- 1 when ω ∈ (−π/T, π/T ) and 0 otherwise. To gorithm can reliably recover the data. Its bit error demonstrate the spectrum change in the sampling rate (BER) performance is simulated and compared process, we show the spectrum in Fig. 2a with a tri- with the theoretical limits and its Nyquist sampling angle, which is not the actual shape of the spectrum counterpart, which validates the algorithm’s excel- but can show the spectral overlapping clearly. lent performance. The signal is transmitted over an additive white Gaussian noise (AWGN) channel. So, the received ( )= ( )+ ( ) ( ) 2Systemmodel signal is y t x t η0 t ,whereη0 t is AWGN with power spectral density N0/2. In this study, we focus on the sub-Nyquist sam- At the receiver, the received signal is first pling problem for a linearly modulated baseband sig- matched filtered with the sinc filter as    nal with the constraint of a finite symbol set. 1 t yM(t)=y(t) ∗ sinc = x(t)+η(t), (4) As shown in Fig. 1, at the transmitter, the trans- T T mitted symbols {um} are selected from a finite al- where η(t) denotes the bandlimited Gaussian with phabet set and sent with period T . The transmitted spectral density N0/2 within its bandwidth. The symbol sequence m umσ(t − mT ) is then filtered ( ) with a pulse shaping filter g(t) to limit the band- bandlimited transmitted signal x t has no change width. The transmitted signal can be denoted as but the noise becomes bandlimited.   When the filtered signal is sampled with the x(t)= Es umg(t − mT ), (1) Nyquist rate 1/T , the samples are orthogonal as m  s where Es is the average symbol energy. E yn,O = yM(t)|t=nT = un + ηn,O, (5) Here, we assume that binary phase shift keying T

(BPSK) modulation and the sinc pulse shaping filter where ηn,O for all n’s are independent and identically ∈{+1 −1} ∼ (0 2) are used. Then, um , and the signal is distributed as ηn,O N ,N0/ . The spectrum of    δ( − )  the Nyquist sampled waveform n yn,O t nT Es t x(t)= um sinc − m . (2) can be demonstrated as Fig. 2b, where δ(t) is the T T m unit impulse function. Spectrum X(ω) is replicated Pulse shaping AWGN Matched filtering with interval 2π/T and there exists no spectrum g(t) η(t) h(t) overlapping. If the filtered signal is directly sampled with the u x(t) y(t) y (t) m M sub-Nyquist rate τ/T,whereτ<1,thesamplesare

Nyquist sampling the linear combinations of the transmitted symbols as yn, O nT yn, =[x(t)+η(t)]| DSNS D t=nT /τ y  n, D  nT/τ Es n (6) y (t) M = um sinc − m + ηn,D. FSNS T m τ yn, F nT/τ Sub-Nyquist sampling without pre-filtering is called Low-pass filter direct sub-Nyquist sampling in this study. The Fig. 1 System model. Reprinted from Luo and Zhang spectrum of the direct sub-Nyquist sampled wave- (2019), Copyright 2019, with permission from IEEE form n yn,D δ(t − nT/τ) can be demonstrated as Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 235

(a) Amplitude (b) Amplitude

τ 0 τ ω 2 0 2 ω TT TT TTT T

(c) Amplitude (d) Amplitude

2τ 0 2τ ω 2τ τ 0 τ 2τ ω T T T T TT TT

Fig. 2 Comparison of the spectra of the original signal (a), Nyquist sampled signal (b), DSNS signal (c), and FSNS signal (d). Reprinted from Luo and Zhang (2019), Copyright 2019, with permission from IEEE shown in Fig. 2c. Spectra X(ω) overlap on the high- 3 Minimum Euclidean distance frequency end. To avoid spectral overlapping, a low-pass filter is Pairwise error probability (PEP) is the error u often added before sampling. The sub-Nyquist sam- probability that when signal 1 is transmitted, the u pling with pre-filtering is called filtered sub-Nyquist receiver detects the distorted version 2 instead of u y y sampling in this study. Suppose the ideal low-pass 1. Suppose 1 and 2 are the noiseless sample filter τ/T · sinc(tτ/T ) is used. Then the low-pass sequences corresponding to the transmitted symbol u u y filtered signal is sequences 1 and 2 respectively, and is the sam-    ple sequence with noise when u1 is transmitted. A τ tτ detection error occurs when y is closer to y2 than to yF(t)=[x(t)+η(t)] ∗ · sinc T T y  1,thatis,  Es τ (u → u |y)= ( 2(y y ) 2(y y )) = τ um sinc t − mτ + ηF(t), P 1 2 P d , 1 >d , 2 . (10) T m T (7) As a result, the error probability above depends on where ηF(t) is the bandlimited Gaussian noise the noise power normalized Euclidean distance be- y y with a plain power density N0/2 within its band tween 1 and 2. The PEP of the system depends [−πτ/T, πτ/T]. The spectrum of the signal without on the minimum normalized Euclidean distance for u u noise is the low-pass filtered X(ω) as all possible 1’s and 2’s. In this section, the minimum noise power nor- πτ πτ XF(ω)=X(ω)R − , , (8) malized Euclidean distances of DSNS and FSNS un- T T der different sampling rates are derived. Compared where R(−πτ/T, πτ/T) is the box function that to the minimum distance with Nyquist sampling, equals 1 when ω ∈ (−πτ/T, πτ/T) and 0 otherwise. the derived minimum Euclidean distances of DSNS When the filtered signal yF(t) is uniformly sam- and FSNS show how the detection performances de- pled with intervals of T/τ,thesamplesare grade with the decreasing sampling rates beyond the Nyquist rate. = ( )| yn,F yF t t=nT /τ In Nyquist sampling, DSNS, and FSNS, the   (9) samples are linear combinations of the transmitted = Es sinc( − )+ τ um n mτ ηn,F, symbols. For a linear system, the Euclidean distance T m between y1 and y2 is the function of the error pattern where ηn,F for all n’s are independent and identically e = u1 − u2: distributed as N(0,N0/2). The spectrum of the 2 2 2 2  d = |y1 − y2| = |Hx1 − Hx2| = |He| . (11) sampled signal m yn δ(t − nT/τ) is the repetition of XF(ω) with intervals 2πτ/T (Fig. 2d). There is The difference between the sample sequences, . no overlapping as a result of low-pass filtering. whichisdefinedasye = y1 − y2 = He,is 236 Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 equivalent to the sample sequence corresponding to (orthogonal transmission) case is given by the transmitted symbol sequence e. Calculating the  2 τ1  2 dD −jπωm minimum Euclidean distance is to search over all pos- min =min bme dω 2 2 m sible nonzero e’s to find min d2 =min|He| .With b dO,min bm∈{±1,0} 0 e e  1 τ    2 this fact, we can calculate the minimum normalized −jπωm j2πτm + bme 1+e dω . 2 m Euclidean distances for DSNS and FSNS with BPSK τ1 modulation and sinc pulse shaping. (15) As for BPSK modulation, the elements in the Proof Obviously, the sample sequence and the error pattern e = u1 −u2 have three possible values: original transmitted data sequence constitute a lin- 2, −2,and0. Define half of the error pattern as b = ear system, with the transmitted data sequence as 1 input and the sample sequence as output. As a re- e, the elements of which satisfy bm ∈{1, −1, 0}. 2 sult, for two distinct original data sequences, the To calculate the Euclidean distance between two difference between the two corresponding sample se- distinct sample sequences, the following lemma is quences is equal to the sample sequence correspond- first introduced: ing to the original data sequence of the error pattern ( ) Lemma 1 Suppose that x t is a continuous-time 2b. Given the original transmitted symbol sequence signal which is sampled uniformly with interval T , {2bm}, the total energy of the corresponding sam- { } and the corresponding sample sequence is xn .De- ples is calculated as the Euclidean distance between ( ) (j ) note the spectrum of x t as XC ω . The total en- sample sequences. ergy of the sample sequence can be calculated by After matched filtering at the receiver, the integrating its power spectrum as continuous-time signal before sampling is band- 2 [−π π ]      limited to /T, /T as 2 1 2πk |xn| = X j ω− dω.  √  2π C −jωmT n T 2π/T k T 2 EsT bme , |ω| < π/T , ( )= m (12) XM ω 0, otherwise, Proof Denote the discrete Fourier transform of (16) { } (ejω) the sample sequence xn as XD . According where the noise is also bandlimited AWGN with a to Parseval’s theorem for discrete-time signals, the two-sided spectral density of N0/2. total energy of all the samples {xn} is equal to the In DSNS, the signal is directly under-sampled integration of its power spectrum over one period: uniformly with time interval T/τ, which is larger ∞ π  1   2 than its symbol interval T . Define the partition point 2 = ejω =2 − 1 xn 2π XD dω. (13) as τ1 τ . As shown in Fig. 2c, the higher n=−∞ −π frequency bands beyond the partition point τ1π/T <   jω |ω| < π/T of the spectrum of the sampled signal The relationship between XD e and XC(ω) is X(ω) will overlap, while the lower band |ω| <τ1π/T     +∞ stays the same as the spectrum of the original signal. jω 1 ω 2πn XD e = XC − . (14) The spectrum of the discrete sample sequence can be T T T n=−∞ represented by the spectrum of the continuous-time Combining Eqs. (13) and (14), Eq. (12) is proved. signal according to Eq. (14) as For Nyquist sampling in Eq. (5), obviously the   +∞ jω τ τ minimum Euclidean distance occurs when there is XD e = XM (ω − 2πn) . (17) T T only one nonzero element in b, and the minimum n=−∞ 2 normalized distance is d =8Es/N0.Inthefol- O,min The noise after sampling is no longer white. Its lowing, we present two basic results for the minimum power density is N0/2 in the lower band and N0 Euclidean distances of the DSNS and FSNS systems, in the higher band. Thus, a noise whitening filter respectively. should be used as Theorem 1 The minimum normalized Euclidean    1 | | π distance between two distinct sample sequences in jω , ω < τ1/τ, WD e = (18) DSNS with respect to that in the Nyquist sampling 1/2, πτ1/τ < |ω| < π. Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 237

After whitening filtering, the whitened noise has the Besides, the sampled noise remains to be AWGN (2 ) 2 power ofN0τ/ T and the spectrum of the signal is with a power spectrum density of N0/ . jω jω XD e · WD e . Normalized by the noise power N0τ/(2T ),the With whitened noise, the energy of the sam- total energy of the sample sequence can be calculated ple sequence ye normalized by the noise power can according to Eq. (12) as be calculated according to Parseval’s theorem in  2 |yn, | Eq. (13) as n e N0τ/(2T )  | |2        2 n yn,e T jω jω 2 1  2π = XD e · WD e dω τk (2 ) π = XF j ω − dω (23) N0τ/ T N0τ 2π πN0 T  2πτ/T k 8 τ1  2 Es −jπωm τ  2 = bme dω 8Es −jπmω N0 0 m = bme dω.  N0 m τ    2 0 1 −jπωm j2πτm + bme 1+e dω . 2 m Compared to the minimum Euclidean distance τ1 8 (19) Es/N0 of Nyquist sampling, the normalized dis- tance of FSNS is Compared to the minimum Euclidean distance 2 τ  2 8Es/N0 of orthogonal transmission, the normalized dF −jπmω = bme dω. (24) 2 m Euclidean distance of DSNS is given by dO,min 0

2 τ1  2 dD −jπωm The minimum of the normalized Euclidean distance 2 = bme dω d 0 m of FSNS in Eq. (24) is obtained by searching all O,min τ    2 nonzero b’s. 1 −jπωm j2πτm + bme 1+e dω. b 2 m We search the sequence for the minimum Eu- τ1 2 2 2 clidean distances dD,min  min dD and dF,min  (20) b 2 In Eq. (20), the first term comes from the lower- min dF under different sampling rates τ/T.The b frequency band where there is no overlapping, and upper bounds and minimum normalized Euclidean 2 2 2 2 the second term comes from the overlapping higher- distances dD,min/dO,min and dF,min/dO,min versus τ frequency band. The minimum of the normalized are shown in Fig. 3. We can see that at the same Euclidean distance of DSNS in Eq. (20) is obtained sampling rate, the minimum normalized Euclidean by searching all nonzero b’s. distance of DSNS is much lower than that of FSNS, Similarly, we have the following result for the indicating the negative effect of frequency overlap- FSNS system: ping in DSNS on the performance. Theorem 2 The minimum normalized Euclidean distance of FSNS with respect to that of the Nyquist 1.0 b=[−1] b=[−1 1 −1 1 −1 1 −1 1] sampling (orthogonal transmission) case is given by 0.9 Filtered SNS b=[−1 1 −1 1 −1 1 −1] b=[−1 1 −1 1 −1 1] b=[−1 1 −1 1 −1] 2 τ  2 0.8 b=[−1 1 −1 1] dF −jπmω b=[−1 1 0 −1 1] min =min me d b=[−1 1 −1 0 1 −1 1] 2 b ω. b b bm∈{±1,0} m 0.7 =[−1 1 −1] dO,min 0 b=[−1 1 0 −1 1 0 −1 1] b=[−1 1 −1 1 0 −1 1 −1 1] (21) 0.6 b=[0 −1 1 −1 0 1 −1 1] Direct SNS Proof In FSNS, the received signal is first matched 0.5 filtered and then low-pass filtered with an ideal low- 0.4 pass filter τ/T · sinc (tτ/T ). The spectrum of the Normalized distance signal after low-pass filtering is 0.3  √  0.2 −jωmT 2 EsT bme , |ω| <τπ/T , 0.1 XF(ω)= m 1.0 0.9 0.8 0.7 0.6 0.5 0, otherwise. τ (22) Fig. 3 Minimum of the normalized distance of two dis- Then the low-pass filtered signal is sampled with in- tinct sample sequences in DSNS and FSNS. Reprinted terval T/τ. As shown in Fig. 2d, there is no overlap- from Luo and Zhang (2019), Copyright 2019, with ping in the frequency domain of the sampled signal. permission from IEEE 238 Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243

Based on the above two theorems and also as in- band and the second term from the overlapping fre- τ1 e−jπωm 2d dicated by the curves in Fig. 3, we have the following quency band. The first part 0 m bm ω two important observations: is the same as the Euclidean distance of FSNS, ex- Theorem 3 For FSNS, the minimum normalized cept that the integral limit is τ1 =2τ − 1 instead of 2 2 Euclidean distance satisfies dF,min/dO,min = τ when τ. According to Theorem 3, the minimum value of the sampling rate decreases below the Nyquist rate the first term is no more than 2τ − 1. within the range τ ∈ [0.802, 1], and it is less than τ Next, if we choose the error pattern b = when τ<0.802. [... 0 ... 0 − 10... 0 ...] with the mth el- Proof The equation for the minimum normalized ement as nonzero, the non-overlapping part of the EuclideandistanceofFSNSinTheorem2 is similar Euclidean distance in Eq. (20) just equals 2τ − 1.By to the minimum normalized Euclidean distance of choosing the position m to satisfy 2τm =2k +1, faster-than-Nyquist (FTN) signaling in Mazo (1975). where k is an arbitrary integer, the overlapping part For BPSK modulation and sinc pulse shaping, the turns out to be zero. Thus, the minimum Euclidean minimum Euclidean distance of FTN signaling with distance for DSNS is 2τ − 1. symbol rate 1/(τT) is When the sampling rate τ begins to decrease

τ  2 from 1, the minimum distance remains as 2τ − 1, 2 1 −jπmω min d =min bme dω. FTN m corresponding to some error pattern b with only one b bm∈{±1,0} τ 0 (25) nonzero element at some position. The minimum Comparing the Euclidean distance of the FTN sig- sampling rate τ at which the minimum distance re- naling above with the minimum Euclidean distance mains as 2τ − 1 can also be obtained by search. The of FSNS with a sampling rate τ/T in Eq. (21) yields combinatorial complexity for search can be greatly reduced by considering only those b that make the d2 F,min second part no more than 2τ − 1 since the minimum 2 = τ. (26) dFTN,min distance never exceeds 2τ − 1. The minimum sam- 2 − 1 The minimum Euclidean distance of FTN sig- pling rate for keeping the linear distance τ is =0855 naling in Eq. (25) was calculated in Mazo and Lan- τ . . dau (1988) and Hajela (1990). It is obtained by In conclusion, the upper bound of the minimum searching the possible error patterns b. To reduce Euclidean distance decreases proportionally to the the search complexity, several constraints such as sampling rate within some thresholds for both FSNS patterns and the number of nonzero blocks, are made and DSNS. to b. FTN signaling has a limit on the signaling rate called the Mazo limit. When increasing the signaling 4 Time-variant Viterbi algorithm for rate within the Mazo limit, the minimum normal- data recovery ized Euclidean distance between two different sig- nals will not decrease as the signaling rate increases. 4.1 Challenges in data recovery For BPSK and sinc pulse shaping, the Mazo limit is τ =0.802. As shown in Eqs. (6) and (9), the sub-Nyquist BasedontherelationshipbetweenFTNand sampling forms an under-determined linear system FSNS in Eq. (26), the minimum normalized Eu- which has a time-variant system impulse response. clidean distance of FSNS is equal to τ within the With BPSK modulation, the transmitted symbols {+1 −1} Mazo limit τ ∈ (0.802, 1], and will drop below τ as are constrained to be chosen from , .So,the the sampling rate decreases beyond the Mazo limit. data recovery problem from the sub-Nyquist sampled Theorem 4 For DSNS, the minimum normalized sequences in Eqs. (6) and (9) can be formulated as 2 2 Euclidean distance satisfies dD,min/dO,min =2τ − 1 min y − Hu2, (27) when τ ∈ [0.855, 1], and it is less than 2τ − 1 when u∈{+1,−1}M τ<0.855. 2 Proof As shown in Eq. (20), the normalized Eu- where ·2 denotes the  norm of a vector, and H clidean distance for DSNS is composed of two terms, is the equivalent channel response matrix which is of the first term from the non-overlapping frequency full row rank with size N × M. Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 239

As an under-determined problem that has more the neighboring symbols after being weighted by the inputs than outputs, linear equalization methods are equivalent channel impulse responses. As a result of incapable for data recovery in this case. The conven- non-integer (fractional) sub-Nyquist sampling, the tional sparsity detection algorithms, such as Bases equivalent channel impulse responses change from Pursuit and the greedy algorithm used in compressed sample to sample, as shown in Fig. 4, resulting in a sensing, are also unable to work since there is no time-variant interference structure and a relatively sparsity in the sampled signal. Although in the- large and dynamic constraint length as well as state ory, the complicated maximum likelihood sequence space. Due to the above reasons, the conventional detection approach (Goldsmith, 2005) may be ap- VA does not apply to this case. plicable, the computational complexity becomes too high in practice because the search space becomes 4.2 Algorithm description extremely large as the data length increases and, to Here, we propose an efficient time-variant VA to make matters worse, a much severer inter-symbol in- recover data from the sub-Nyquist sampled sequence. terference (ISI) arises when the sampling is not at First, the overall equivalent channel impulse re- the Nyquist rate. sponse is truncated to reduce the number of states. The Viterbi algorithm (VA) (Forney, 1973) is Let L be a tunable parameter denoting the number a simplified realization of maximum likelihood se- of neighboring interfering symbols at each side of the quence estimation. It can recover data from the out- dominating symbol(s) associated with a sample af- puts of an ISI channel with low computational com- ter the truncation. The number of dominating sym- plexity. However, in the case of SNS and as far as VA bol(s) could be one or two as shown in Fig. 4; thus, is concerned, the ISI structure varies from sample to the length of the truncated impulse response is either sample, so does the state space. Let us elaborate 2L +1or 2L +2. Obviously, the truncation length more on these. Note that each output sample is L provides a tradeoff between the performance and associated with either one or two dominating input the computational complexity. symbols (i.e., they constitute the principal compo- Next, the dynamic trellis graph is constructed, nents of that sample), as indicated by the red dotted over which the new VA is conducted. For truncated lines in Fig. 4. As a matter of fact, the number of impulse response with length 2L +1, the constraint dominating symbols changes every several samples, length is normally 2L +1(or equivalently the mem- which also makes the state transitions of the trel- ory is 2L); thus, the resultant number of states is 22L. lis graph varying (see Fig. 4 for illustration). The When a new sample is obtained, the Euclidean mea- interferences contained in each sample come from sure is calculated along the paths which start from the current states and go to the next states. Note u u u u u Symbols m m+1 m+2 m+3 m+4 that for a certain sample that is associated with two dominating symbols, the constraint length is 2L +2; Samples thus, there will be four branches (corresponding to y y y y n n+1 n+2 n+3 two unknown single-bit symbols) starting from each H 0.24 1 0.24 n of the current 22L states and ending to four of the next 22L states (Fig. 5a). H n+1 0 0.94 0.50 Without loss of generality, let us take FSNS with sampling rate τ =0.8 for illustration. With the sam- pling rate τ =0.8=4/5, four samples are generated Hn+2 −0.16 0.76 0.76 −0.16 within the duration of five symbols (Fig. 4). As de- scribed above, each sample is associated to one or H n+3 0.50 0.94 0 two dominating symbols. As depicted in the up- per dotted-line block in Fig. 4, each of the 1st,2nd, th and 4 samples yn,yn+1,andyn+3 has one dom- Fig. 4 Structure of the time-variant inter-symbol in- rd terference with τ =0.8 and 2L =2. Reprinted from inating symbol, while the 3 one yn+2 has two. Luo and Zhang (2019), Copyright 2019, with permis- Alongside the dominating symbols of each sample, sion from IEEE there are two interfering symbols, one for each side. 240 Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243

States Stage 1 Stage 2 Stage 3 Stage 4 ing two symbols, each state transition corresponds S0: [−1, −1] to one or two specific input symbols. All the above S1: [−1, +1] symbols are summed up after being weighted by the

S2: [+1, −1] truncated channel coefficients to obtain the predicted sample value. Then the square error between the S3: [+1, +1] predicated sample value and the real sample value (a) serves as the branch metric. Finally, the path metric

In Current state can be achieved by accumulating the branch metrics (out) S0 S1 S2 S3 along that path. The calculation of the metrics for −1 +1 −1 +1 each state of each stage is illustrated in Fig. 5b. Note

S0 (−1.48) (−1) (−1.44) (−0.44) that at stage 3 the states are driven by two input −1 +1 −1 +1 S1 (0.52) (1) (0.44) (1.44) symbols and there are four next states for each cur- −1 +1 −1 +1 rent state; thus, four symbols are used to predict the Next state S2 (−1) (−0.52) (−1.44) (−0.44) sample value. At the other three stages, only three −1 +1 −1 +1

S3 (1) (1.48) (0.44) (1.44) symbols are used to obtain the prediction. Finally,

Stage 1 Stage 2 with the trellis dynamically growing, the path met- ric calculation and decision-making are conducted in −1, −1 −1, +1 +1, −1 +1, +1 −1 +1 (−1.2) (−1.52)(−0.32) (0) (−1.44) (−1.44) the same way as in the conventional VA. The path −1, −1 −1, +1 +1, −1 +1, +1 −1 +1 with the minimum accumulated metric survives and (0.32) (0) (1.84) (1.52) (0.44) (0.44) the corresponding input symbols along that path will −1, −1 −1, +1 +1, −1 +1, +1 −1 +1 (−1.52) (−1.84) (−0) (−0.32) (−0.44) (−0.44) be the final detection output. −1, −1 −1, +1 +1, −1 +1, +1 −1 +1 Note that the computational complexity of the (0) (−0.32) (1.52) (1.2) (1.44) (1.44) time-variant VA is of the same order as that of the Stage 3 Stage 4 conventional VA, with only a slight difference in the (b) treatment of the non-regular stages where there are Fig. 5 The trellis graph (a) (reprinted from Luo and two dominating input symbols. Zhang (2019), Copyright 2019, with permission from IEEE) and state transition (b) of a time-variant VA 4.3 Simulation results with τ =0.8 and 2L =2. In (b), each term in the form represents a trellis branch connecting the cur- In this subsection, the bit error rate (BER) per- rent state and the next state, the inputs and outputs formances of the above time-variant VA detection denote the input symbol values and the predicted sample values respectively, and the “×”intheform algorithm for both the DSNS and FSNS systems are stands for an impossible state transition simulated. The simulation results are then compared with both their theoretical limits and the bench- Therefore, here 2L =2and the number of states is marks of their Nyquist sampling counterparts. four. The truncated time-variant impulse responses Suppose the block length of the transmitted Hn,Hn+1,Hn+2,andHn+3, for the four successive symbols is M = 200. At a sampling rate τ,the samples yn,yn+1,yn+2,andyn+3, respectively, are number of samples is N = τ · M ( represents shown in Fig. 4. As a new sample comes, the VA de- the rounding up operation). The equivalent chan- tector enters a new stage. At each stage, the first two nel impulse response is truncated to length 2L +1or symbols constitute the current state space, shown as 2L+2,whereL is first set to four. Figs. 6 and 7 show theblackdisks.Theotheroneortwowhitedisksat the performance curves of BER versus the sampling each stage correspond to the unknown input symbols rate in DSNS and FSNS with sinc pulse shaping, which lead to state transitions. Note that the state respectively. Compared to the case of Nyquist sam- transition of the third stage is different, at which pling, the BER performance for both SNS systems there are four branches starting from each of the degrades as the sampling rate decreases. FSNS al- current four states and ending at the next four states ways has a lower BER than DSNS probably at the (Fig. 5a). same sampling rate because FSNS has anti-aliasing At each stage, the path metric is calculated as filtering. follows.Giventhecurrentstateandthecorrespond- For DSNS with τ =0.9, 0.8,and0.7, the square Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 241 normalized Euclidean distances are d2 =0.8, 0.5, equivalent channel impulse response degrades the and 0.1, respectively, while for FSNS with τ =0.9, performance, especially when the signal-to-noise ra- 0.8,and0.7, the square normalized Euclidean dis- tio (SNR) becomes high. On the other hand, the per- tances are d2 =0.9, 0.8,and0.5, respectively. Com- formance loss is due to the non-optimality of time- paring the theoretical performance and the simulated variant VA detection. BERs in Figs. 6 and 7, there is a trend that the BER Next we show the impact of the truncation of performance approaches the theoretical limit as the the channel impulse response, which causes addi- sampling rate increases. However, there are gaps tional interference and degrades the performance. between the theoretical performances and the sim- Fig. 8 presents the BER performances of FSNS with ulated ones. On one hand, the truncation of the truncation length 2L +1 = 5, 7, 9, with sinc pulse

100 100 DSNS, τ=0.7 FSNS, τ=0.7 DSNS, τ=0.8 FSNS, τ=0.8 10−1 DSNS, τ=0.9 10−1 Nyquist FSNS, τ=0.9 d2=0.1 Nyquist −2 2 −2 10 d =0.5 10 d2=0.5 d2=0.8 d2=0.8 −3 −3 10 10 d2=0.9 BER BER

10−4 10−4

10−5 10−5

10−6 10−6 0 5 10 15 20 0 5 10 15 20 SNR (dB) SNR (dB) Fig. 6 BER of DSNS with sinc pulse shaping. Fig. 7 BER of FSNS with sinc pulse shaping. Reprinted from Luo and Zhang (2019), Copyright Reprinted from Luo and Zhang (2019), Copyright 2019, with permission from IEEE. The solid lines 2019, with permission from IEEE. The solid lines show the simulated BERs, and the dotted lines are show the simulated BERs, and the dotted lines are the theoretical BERs calculated based on the mini- the theoretical BERs calculated based on the mini- mum normalized Euclidean distances in Fig. 3 mum normalized Euclidean distances in Fig. 3

(a) (b) 100 100 τ=0.9, 2L+1=5 τ=0.8, 2L+1=5 τ=0.9, 2L+1=7 τ=0.8, 2L+1=7 τ=0.9, 2L+1=9 τ=0.8, 2L+1=9 −1 −1 10 Limit d2=0.9 10 Limit d2=0.78

10−2 10−2

10−3 10−3 BER BER

10−4 10−4

10−5 10−5

10−6 10−6 0 5 10 15 0 5 10 15 SNR (dB) SNR (dB)

Fig. 8 BER of FSNS with different truncation lengths under sinc pulse shaping with τ =0.9 (a) and τ =0.8 (b) 242 Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 shaping and sampling rate τ =0.9 and 0.8.We have been proposed. The performance loss has been can see that the effect of truncation is more obvi- estimated by the minimum Euclidean distances of ous when the SNR becomes higher. As the trun- the sample sequences for DSNS and FSNS. While cation length increases, the residual interference is DSNS suffers from frequency overlapping, FSNS has reduced and the error floor decreases as well, and better performance due to its anti-aliasing filtering. more importantly, the BER performance gradually For FSNS with BPSK modulation and sinc pulse approaches the theoretical limit given by the mini- shaping, the minimum distance remains proportional mum Euclidean distance. to the sampling rate within the τ =0.802 limit, To alleviate the negative effect of truncation, and it drops more rapidly beyond this limit. The we test the performance when the root raised co- Viterbi algorithm is adapted for detection in this sine (RRC) filter, instead of the sinc pulse filter, under-determined time-variant linear system. The is used for shaping. In the case of RRC shaping, proposed time-variant Viterbi algorithm is able to the ISI decays more rapidly, so that the truncation recover the transmitted signal from the sub-Nyquist length can be much smaller than that of sinc pulse samples, and its performance approaches the theo- shaping, which greatly reduces the computational retical limit as its truncation length grows. complexity. Fig. 9 shows the BER of DSNS and FSNS with RRC shaping with roll-off factor α =0.3 Contributors and single-side truncation length L =3.Thereis Xiqian LUO and Zhaoyang ZHANG designed the no error floor in the observation window caused by research. Xiqian LUO did the simulation and drafted the the truncation. With alleviated effect of truncation, manuscript. Zhangyang ZHANG revised the manuscript. it shows the good performance of the time-variant Xiqian LUO revised and finalized the paper. Viterbi algorithm. Compliance with ethics guidelines Xiqian LUO and Zhaoyang ZHANG declare that they 100 DSNS, τ=0.7 have no conflict of interest. DSNS, τ=0.8 10−1 DSNS, τ=0.9 FSNS, τ=0.7 References FSNS, τ=0.8 FSNS, τ=0.9 Anderson JB, Rusek F, Öwall V, 2013. Faster-than-Nyquist 10−2 Nyquist sampling signaling. Proc IEEE, 101(8):1817-1830. https://doi.org/10.1109/JPROC.2012.2233451 10−3 Chen YX, Eldar YC, Goldsmith AJ, 2013. Shannon meets BER Nyquist: capacity of sampled Gaussian channels. IEEE 10−4 Trans Inform Theory, 59(8):4889-4914. https://doi.org/10.1109/TIT.2013.2254171 Domínguez-Jiménez ME, González-Prelcic N, Vazquez-Vilar −5 10 G, et al., 2012. Design of universal multicoset sampling patterns for compressed sensing of multiband sparse 10−6 signals. Proc IEEE Int Conf on Acoustics, Speech and 0246810121416 18 20 SNR (dB) Signal Processing, p.3337-3340. https://doi.org/10.1109/ICASSP.2012.6288630 Fig. 9 BER of DSNS and FSNS with RRC pulse shap- Fan JC, Guo SJ, Zhou XW, et al., 2017. Faster-than-Nyquist ing. Reprinted from Luo and Zhang (2019), Copy- signaling: an overview. IEEE Access, 5:1925-1940. right 2019, with permission from IEEE https://doi.org/10.1109/ACCESS.2017.2657599 Feng P, Bresler Y, 1996. Spectrum-blind minimum-rate sam- pling and reconstruction of multiband signals. Proc IEEE Int Conf on Acoustics, Speech, and Signal Pro- 5 Conclusions cessing, 3:1688-1691. https://doi.org/10.1109/ICASSP.1996.544131 Forney GD, 1973. The Viterbi algorithm. Proc IEEE, 61(3): In this paper, the sub-Nyquist sampling of a 268-278. https://doi.org/10.1109/PROC.1973.9030 linearly modulated baseband signal has been inves- Goldsmith A, 2005. Wireless Communications. Cambridge tigated. While FTN signaling increases the symbol University Press, Cambridge, USA, p.327-340. rate beyond the Nyquist rate at the transmitter, SNS https://doi.org/10.1017/CBO9780511841224 Hajela D, 1990. On computing the minimum distance for reduces the sampling rate below the Nyquist rate at faster than Nyquist signaling. IEEE Trans Inform The- the receiver. Direct SNS and low-pass filtered SNS ory, 36(2):289-295. https://doi.org/10.1109/18.52475 Luo and Zhang / Front Inform Technol Electron Eng 2021 22(2):232-243 243

Herley C, Wong PW, 1999. Minimum rate sampling and Trans Signal Process, 57(3):993-1009. reconstruction of signals with arbitrary frequency sup- https://doi.org/10.1109/TSP.2009.2012791 port. IEEE Trans Inform Theory, 45(5):1555-1564. Mishali M, Eldar YC, 2010. From theory to practice: sub- https://doi.org/10.1109/18.771158 Nyquist sampling of sparse wideband analog signals. Landau HJ, 1967. Necessary density conditions for sampling IEEE J Sel Top Signal Process, 4(2):375-391. and interpolation of certain entire functions. Acta Math, https://doi.org/10.1109/JSTSP.2010.2042414 117(1):37-52. https://doi.org/10.1007/BF02395039 Mishali M, Eldar YC, Elron AJ, 2011. Xampling: signal Liveris AD, Georghiades CN, 2003. Exploiting faster-than- acquisition and processing in union of subspaces. IEEE Nyquist signaling. IEEE Trans Commun, 51(9):1502- Trans Signal Process, 59(10):4719-4734. 1511. https://doi.org/10.1109/TCOMM.2003.816943 https://doi.org/10.1109/TSP.2011.2161472 Lu YM, Do MN, 2008. A theory for sampling signals from a Scoular SC, Fitzgerald WJ, 1992. Periodic nonuniform sam- union of subspaces. IEEE Trans Signal Process, 56(6): pling of multiband signals. Signal Process, 28(2):195- 2334-2345. https://doi.org/10.1109/TSP.2007.914346 200. https://doi.org/10.1016/0165-1684(92)90035-U Luo X, Zhang Z, 2019. Data recovery from sub-Nyquist Sun HJ, Nallanathan A, Wang CX, et al., 2013. Wideband sampled signals: fundamental limit and detection algo- th spectrum sensing for cognitive radio networks: a survey. rithm. Proc 11 Int Conf on Wireless Communications IEEE Wirel Commun, 20(2):74-81. and Signal Processing, p.1-6. https://doi.org/10.1109/MWC.2013.6507397 https://doi.org/10.1109/WCSP.2019.8927914 Tropp JA, Laska JN, Duarte MF, et al., 2010. Beyond Mazo JE, 1975. Faster-than-Nyquist signaling. Bell Syst Nyquist: efficient sampling of sparse bandlimited sig- Techn J, 54(8):1451-1462. nals. IEEE Trans Inform Theory, 56(1):520-544. https://doi.org/10.1002/j.1538-7305.1975.tb02043.x https://doi.org/10.1109/TIT.2009.2034811 Mazo JE, Landau HJ, 1988. On the minimum distance Venkataramani R, Bresler Y, 2001. Optimal sub-Nyquist problem for faster-than-Nyquist signaling. IEEE Trans nonuniform sampling and reconstruction for multiband Inform Theory, 34(6):1420-1427. signals. IEEE Trans Signal Process, 49(10):2301-2313. https://doi.org/10.1109/18.21281 Mishali M, Eldar YC, 2009. Blind multiband signal recon- https://doi.org/10.1109/78.950786 struction: compressed sensing for analog signals. IEEE