Practical Synchronization Waveform for Massive Machine-Type-Communications

Abstract — This paper proposes a practical synchronization waveform that is resilient t o frequency error for m achine-type- communications (MTC) with applications in massive Internet - of - Things (IoT). Mathematical properties of the waveform are derived, which are keys to addressing the practical issues. In particular, it is shown that t his type of w aveform is a symptotically optimal in the presenc e of a frequency error, in th e sense that its asymptotic performance is the same as the optimal matched-filter detector that is free of frequency error. This asymptotic property enables optimization o f the w avefo rm under the constraints imposed by an application. It is also sho wn that such optimized waveform comes in pairs, which facilitates the formation of a new waveform capable of frequency error estimation and timing refinement at the receiver in addition to the resilience to receiver frequency errors. Index Terms — Massive m achine-type-communications (mMTC), mass ive Internet- of -Things (mIoT), m MTC synchronization waveform, sy stem acquisition, frequency offset/error estimation. I. I NTRODUCTI ON Machine-type -communications (MT C) is a ty pe of wireless communications that support fully automatic data generation, exchange, processing, and actuation among intelligent machines, without or w ith low intervention of humans including utilities, sensing , health care, m anufacturin g, and transportation [1], [2] . Wh ereas massive MT C ( mMTC) characterized by simultaneous support of a massive number of MTC devices is becoming the prominent communicat ion paradigm f or a wide range of emerging s mart service s with a typical application in the massive Internet of Things (m IoT ) market where devices sendi ng bits of infor mation to other machines, servers, clouds, or humans account for a mu ch larger proportion in wireless communication applicatio ns. Although cellular networks such as GSM and L TE have long been us ed for MTC in various IoT applications, its capability to support mMTC is rather limited. Nevert heless, this is not surprising since cellular technolo gies were d eveloped for “human - type” communicatio ns (HTC) in the f irst place . T o provide a s olution th at are built on top of the traditional cellular network for mMTC, the Third Generatio n Partnership Project (3GPP) d edicated an immense effort during LTE Release 13 to develop a new radio access technology k nown as Narro w-Band Internet o f Things (NB-IoT ) [3] as p art of t he long ter m evolution proce ss towards a more versatile uni versal communication technology . As such, NB -IoT inherits most functionality from the legacy LTE system, such a s the transmission frame structure and the data trans m issio n waveform (i.e., OFDM) . T he most noticeable changes are probably the reduced m inimum system bandwidth from 1.4 MHz to 180 kHz, m ainly f or exploiting t he refar med GSM spectrum that are channelized 200 kHz per carrier, and th e redesigned synchronization waveform for b etter serving mIo T use cases that are commonly c haracterized by low -cost, extended co verage, and the unique short burst, long sleep transmission pattern [4]. Synchronization is t he first and arguably the most challenging step of MTC in mIoT applications d ue to the largest time and frequenc y uncertainties present in transcei vers. Because of th e low-cost nature of an mIoT device, a local oscillator of the device may suffer from a large frequency erro r that creates an offset in carrier f requency between the i ncoming signal and the recei ver . For instance, t he initial frequenc y error of the local oscillator can be as large as 20 pp m for an mMTC device [5] . Moreover, the short-burst nature o f an mMT C transmission makes synchro nization of each tra nsmission a more do minant factor in the o verall transmission efficiency as compared to that in HTC; and the p rolong ed sleep duration (to save b attery) causes the local oscillator to drift away fro m the de fault frequency. T hese unique characteristics of mMT C further burden the synchro nization process. Ye t, data transmissions can p roceed only a fter time and frequenc y synchronization i s established as req uired by data transmission waveforms. Since the purpo ses o f the wav efor m used fo r synchronization and the ones for data transmission ( i.e., the multiple access wavefor ms) are different, the design requirements for these two types of waveforms are v ery different as well. Compared to the data transmission or multiple access waveform [6]-[13 ], the synchronization waveform for m MTC so far has received much less attention in 5 G technology development. Rece ntly in [14] , a general synchronizatio n waveform resilie nt to frequency er ror is derived. It is shown that the effect o f a freq uency error on the well-known m a tched filter-based detec tor of such ty pe of waveform is simply a time shift of the d etection p eak p osition without incurring a significant loss in detection energy. However, for u se as a practical s ynch ronization waveform, issues like (1) w aveform parameter selection to m eet application -specific requirements; and (2) the capability f or frequenc y error estimation and timing refinement th at is essential to a sy nchronization signal still remain to be resolved. This pap er proposes a practical Practical Synchronization W aveform for Massi ve Machine-T ype-Communications M. Wang, and JJ Zhang This work has been submitted to the IEEE for p ossibl e p ublicatio n. Copy right may be transferre d without notice, a fter w hich this version may n o longer b e accessible. synchronization wavefor m that solves these important issues while retaining the freq uency error resilience p roperty . In Section II , w e briefly review the derivation of the synchronization waveform proposed in [14]. In Sectio n III, we deal with the first issue, i.e., the selection of wavefor m parameters , detailing the optimization of the waveform under the co nstraints imposed b y a practical app lication, which is exemplified by NB-IoT . Key mathematical properties of the waveform are derived along t he way as needed . In Section IV , we shift our f ocus to the second issue, presenting the means for frequency err or esti mation a nd ti ming refinement exploiti ng the unique properties of this type of waveforms derived . Fin ally, Section V concludes t his paper. II. F REQUENCY O FFSET R ESILI ENT W AVEFORM Assuming the b aseband sync hronization wavefor m is represented as   xt , the local co py of this synchronization waveform of a matched-filter based detector (also k nown as th e cross-correlation based detector) in effect becomes   2 j ft x t e   , in the presence of a freq uency offset f  . The cross-correlation funct ion bet w een the inco ming synchronization si gnal   xt and the local copy can be w ritten as       2 2 2 1 , T j ft T f x t e x t d t           E , (1) where   2 2 2 T T x t dt    E . In the absence of a frequency offset, i.e. , 0 f  , the maximum output of the co rrelator happens when 0   ,   0 , 0 1 f      , (2) providing the op timal detection per formance as asserted by the well-known matched-filter detection theory. While in the presence of a frequency o ffset, 0 f  (unknown to the receiver), the phase ra mping component , 2 j ft e   , introduced by the frequency offset ef fectively creates a mismatch between the incoming synchronizatio n signal and t he local cop y of the waveform , breaking th e ver y premise of the op timality o f a matched-filter detector , thereby inevitably resulting in a lo ss in detection energy, i.e.,   0 , 0 1 f      , (3) and ergo a lo ss in detection p erformance. This phe nomenon is well-documented in the liter ature [14] -[19], and b ecomes prominent in mIoT wh en the frequency offset is likely large – typically to an extent t hat the resulting mismatch totally fails a matched filter-based detector. In NB-IoT , this issue is dealt with by repeating a waveform multiple time s consecutivel y i n time, a nd a differential correlator betw een rep eated signals (also called an autocorrelator) is emplo y ed at the receiver to suppress the phase ramping eff ect . Spec ifically, the w aveform is repeated 11 times to form a synchronization signal with a total length o f ~780 μ s . Although very effective (in rem ovin g phase ramping) , this ty pe of autocorr elator-based detector is not optimal, causin g at least 3 dB degradation at its best ( as SNR   ) w it h r espect to the optimal matched-filter detecto r (without frequency offset). T he degradation quickly increas es as S NR deteriorates , e.g., close to 5 dB degradatio n at an SNR o f -5 d B, and ~10 dB at -10 dB . This b ehavior , known a s the “noise amplification” pheno m enon, could be problematic in a low SNR scenario which is not uncommon in mIoT deployments. In search for a solution such that the optimality of a matched filter detect or can be m aximally retained under frequency offset , i n [14] , matched-filter (i.e ., cross-correlato r) based detection with freque ncy offset i s re-exami ned using the Ca uchy- Schwartz inequalit y, which indicates that       22 22 2 22 1 , TT j ft TT f x t e d t x t d t            E .( 4 ) That is,     , 1 , and f f x t      , (5) with equality if and onl y if     2 j ft x t e C x t       , (6) where C  is a non- ze ro constant. The significance o f this result is obvious in that it claims the mathematical existence of s uch a w aveform t hat atta ins the optimality of a m atched filter even in the presence of a frequency offset bet ween the received signal a nd the local wa ve form, as long as condition (6) is met. It can be s hown that t he w aveform that sat isfies condition (6) is of the following general form     2 , , j t t x t e        . (7) The matched filter output energ y based on this type of waveform transforms the frequenc y error of the detector , f  , into a time shift away fro m the o riginal po sition by a n amount of 1 ˆ f      . (8) III. P ROTOT YPE W AVEF ORM AND O PTI MALITY In the p revious section, we have briefly revie wed the derivation of a general wavefor m in [14] th at has the capabilit y of conve rt ing a frequency offset between the transmitted signal and the detector into a time offset . Ho wever, the question regarding how to take advantage of this property of this type of waveform to build a p ractical synchronization wa veform remains to b e ans wered since to be a prac tical s y nchronization waveform, it must allo w us to select w avefor m para meters to satisfy the applicatio n requirements; and more importantly, estimate th e frequency and time offset/error at the receiver, which is, after all, one of the essential functions of a synchronization signal, nevertheles s un available in [14]. In this section, we f irst form a prototype w avefor m, and s how its asymptotic opti m ality. W e then utilize thi s p roperty for optim iz ing the w avefor m subject to practical constraints . The analytical results fro m this section pave the way to Sectio n IV . A. Waveform Constraints In the mathe m atical treatment of Sectio n II , it is implicitly assumed that   x t extends be yond the correlation interval,   2 , 2 TT  . In practice, a synchronization waveform i s ti me- bounded w ithin length T , i.e.,   , ( ), 2 2 ˆ 0, oth erwise x t T t T xt        . (9) This pr actical form of   x t defined in (7) is the prototype for the follo wing a nalysis, and serves as the buildi ng block for creating the ultimate synchro nization waveform in Sectio n IV . As s uch, when a frequency error ca uses a s hift o f the correlation peak from 0   to ˆ  , it crea tes a time offset or misalignment, ˆ  , b etween the r eceived w a veform and the local one, causing that only partial signal energ y can be detecte d, which consequently results in a lo ss in detection peak energy. Therefore, the effect of this time misalignment on the detection energy (and ergo t he detectio n p erformance) needs to be taken into account in prac tical design s. It is ap parent that the misalign m ent, ˆ  , between the incoming signal and the local wavefor m must be less tha n the waveform length, T , in order for the co rrelator to o utput nonzero detection ener gy, i.e., ˆ T   . ( 10 ) It is not difficult to find that t he loss in detection energ y due to a time offset ˆ  is     2 1 ˆ ˆ 1 T      . ( 11 ) From ( 10 ) it is clear that   ˆ 0 1    ( 12 ) in linear, or   ˆ 0   ( 13 ) in dB, for ˆ 0   (i.e., 0 f  ) . Hence, the practical form of the mathematical waveform   x t , i.e., the pro toty pe   , ˆ xt  given in (9) , no longer attains the maximum detection energy o f an actual matched filter. I n fact, it is   ˆ   dB aw ay from the optimal. The selection of the wavefor m parameters, i.e. ,  and  , therefore needs to minimize th is defic it , or maximize   ˆ  , i.e.,   , ˆ , a r g m ax       . ( 14 ) From (8 ) this d eficit is found to be a sole function of parameter  (not a function o f  ). Substituting (8) into ( 11 ) follows that       2 1 2 1 1, Tf         ( 15 ) where 1 0 Tf      . ( 16 ) Clearly,     1 ( or 0 dB) , i.e., no loss with respect to the optimal, when 0   or 0 f  . Equation ( 14 ) then beco mes   a r g m ax      , ( 17 ) which is equivalent to a r g m ax .     ( 18 ) From (8) , ( 10 ) and ( 16 ), it is clear that   . ( 19 ) It is not difficult to see from ( 15 ) that, for nonzero  , i.e., f  0,   1 o r 0 d B    , ( 20 ) as     . We thus have the following proposition: Proposition 1 : Waveform   ˆ xt defined in (9) asymptotically attains t he optimality of a matched-filter detector, in the presence of a f requency error . The rate of convergence is determined by  defined in ( 16 ), which is irrelevant of SNR . This asymptotic behavior of   , ˆ xt  is graphicall y shown in Fig. 1, wh ere the d otted line is the asymptote, i.e.,     0 (dB) for   0 (i.e., 0 f  ), as promised by an actual matched-filter detector that has the full knowledge of t he i nput signal frequency, i.e., ze ro freq uency error , and the cases with   0 ( f  0):   0.0256 and 0 .0512   kH z μs , corresponding to T  780 μ s and 2 T  390 μ s , respectively, given f  1 GHz  20 ppm = 20 kHz. Here we have borrowed the NB -IoT parameters, the pri m ary synchronization sign al length ( T  780 μ s ), and maximum frequency error ( f  20 p pm), as example. It is now evident that the desi gn of the proto type w avefor m   , ˆ xt  optimized for a particular application becomes maximization of the magnitude of the w avefor m para m eter  , subject to the spec ific constraint imposed b y the application. Fig. 1. Illustration of the asymptotic property of the prototype waveform ˆ () xt in the presence of a non-zero f requency error , where the c ases for 1 7 80 2 0    =0.0256 and 1 3 90 2 0    =0.0512 (kHz/ μ s ) are shown. B. Frequency E rror constraint From (8) and ( 10 ), it is clear that 1 fT    . ( 21 ) This i m plies that for a m aximum s upported frequency er ror requirement, m a x f  , the selection of  must satisfy 1 m a x : fT   S . ( 22 ) That is to say, for a given wavefor m length, the larger the maximum frequency error that a p ractical sy stem is designed to tolerate, the greater the magnitude of  is required. For NB -IoT, the maximum supported frequenc y er ror is ma x f  20 ppm, co rresponding to ~ 20 kHz at 1 GHz carrier frequency. Hence the solution to ( 22 ) is † 1 1 1 1 : ,   SO ( 23 ) where   † 1 , 0.0256     O ( 24 ) with T  780 μ s . For the same frequency error toler ance but a shorter waveform length , e.g., 2 390 T  μ s , the corresp onding solution is   † 1 , 0.0512     O . ( 25 ) C. Spectral Co nstraints Both ( 18 ) and ( 22 ) d emand a large  . Nevert heless, in practice, the selection of  cannot be arbitrar ily large and is often li mited by the applicati on- sp ecific constraints. A typical example of s uch constraints is the sp ectral req uirements , e.g ., the maximum occ upied band width restriction, and the adj acent channel leakage r atio ( ACLR) req uirement [20][21 ]. The occupied bandwidth is the width of a frequency band such that, below the lower and above the upper frequency limits, the powers emitted are each equal to a specified perce ntage 2  (e.g., 1%   ) of the total transmitted po w er, while ACLR i s the ratio of the power centred on the assigned cha nnel frequenc y to th at centred on an adj acent channel frequency . Therefore, the optimization of the waveform in ( 18 ) can be reformulated as 1 2 3 a r g m ax    S S S , ( 26 ) where 1 S is the constraint from the supported maximum frequency error requirement given i n ( 22 ), 2 S is the o ccupied bandwidth require ment, and 3 S the ACLR require ment. In detail, 2 S requires that a specified percent of the waveform energy be co nfined w ithin a given bandwidth W   , i.e.,   /2 2, /2 1 ˆ : 1 W W f df T      S X , ( 27 ) where     2 2 2 , , 2 ˆ ˆ , T j ft T f x t e dt f          X , ( 28 ) is the p ower spectr um of waveform   ˆ xt , and noting that   2 2 , 2 ˆ T T x t dt T     E . Using NB-IoT as example, 200 W  kHz, 1%   , and T  780 μ s [22] . To see the im plication of 2 S on the w aveform para meters  and  , let us first look at ( 27 ) with equality, fro m which t he (a) (b) Fig. 2 Graphical illustration of occupied bandwidth constraint, 2 S , on the parameters  and  : (a) the occupied bandwidth su rfa ce  W for 780 T  μ s ; and (b) the contour † C formed by ,  ’s whose † W  200 kHz ( 53 dBHz) for 780 T  μ s . 2 T  390 μ s is also plotted . occupied bandwidth , W , of w aveform   , ˆ xt  can be plotted as a function of  and  , which pr oduces a surface as illustrated in Fig. 2 (a). Every point on this sur face is a pair of  and  values, ,  , associated with a W   that corresponds to a p articular bandwidth, within which 1   99% of the energ y of   , ˆ xt  is confined. T his surface, henceforth re ferred to as the occu pied bandwidth surface , i s denoted as   ,   W , and mathematical ly represented as     /2 , /2 1 ˆ , 1 . W W W f df T                 W X ( 29 ) Among these -  pairs on  W , we are particularly interested in the one s that correspond to a given W , † W , i.e.,     † † , , W       CW , ( 30 ) which forms a closed symmetric contour † C on the -  plane as illustrated in Fig. 2 (b) w here † 200 W  kHz (or 53 d BHz) and 99%   , which can be viewed a s the intersection between the occupied bandwidth surface   ,   W and the † 200 W  kHz plane [see Fig. 2 (a)]. In fact, it can be further sho wn that any -  pair that falls within the area enclosed by † C satisfies ( 27 ). If w e denote this enclosure (including † C ) as † 2 O , w e can conc lude t hat 2 S in ( 27 ) is satisfied by † 2 ,   O . Consequently,   , ˆ f  X with parameters  and  selected from any point in † 2 O meets the maxi mum occupied bandwidth require ment , 2 S . The occupied bandwidth requirement i n ( 27 ) thus implies † 2 2 2 2 : ,   SO , ( 31 ) as graphically sho wn in F ig. 2 (b) . 3 S , the ACLR restrictio n requires that   3, ˆ : ( ), f f f     S XM . ( 32 ) where () f M is the spectral mask. The ACLR from NB -IoT indicates that for ad jacent channel with 300 kHz offset is 40  dB c and w it h 500 kHz is 50  dB c , which can be rep resented as 40 (dBc) 300 (kHz) 500 (kHz) () 50 (dBc) 50 0 (kHz) f f f         M ( 33 ) as shown in Fig. 3, w here f is th e frequency offset with respect to the carrier f requency . It is not surprising to see the relaxed ACLR require ment s ince t he adj acent cha nnel interference from NB -IoT is suppressed by the 1/12 frequenc y reuse plan for GSM, recalling that NB-IoT m ainly targets the refarmed GSM bands. Nevert heless, the solution to ( 32 ) can be found to be † 3 3 3 3 : ,   SO , ( 34 ) which is graphicall y shown in Fig. 4. Now we are r eady to find the so lution to ( 26 ) by combinin g ( 23 ), ( 31 ), and ( 34 ) , 1 2 3 † † † 1 2 3 , , a r g m ax arg max 0.251 , 0 .             S S S OOO ( 35 ) for 780 T  μ s . The solution in ( 35 ) gives the o ptimal p arameters in the sense that the synchronizatio n waveform,   , ˆ xt  , parameterized by ,   has the least per form ance lo ss (with respect to the optimal matched-filter) a mong all the wavefor ms in     , ˆ , xt    , constrained by the maxi mum frequenc y error tolerance, maximum occupied band w idth, and ACLR spectrum mask. Waveform   , ˆ xt   is hence referred to as the optimal wavefo rm in the same se nse. The p ower spectrum   , ˆ f   X of the optimal wavefor m   , ˆ xt   is plotted in Fig. 3. From ( 11 ) or Fig. 1 , the co rresponding d egradations of the optimal w a veform parameterized by ( 35 ) at various magnitudes Fig. 4. Graphical illustration of the ACLR restrictio n, 3 S , on wave form parameters  and  , for 780 T  μ s as well as 2 T  390 μ s . Fig. 3. The ACLR requirement: the NB -IoT spectrum mask () f M and the   , ˆ f   X of the optimal w aveform. of f requenc y errors is plotted in Fig. 5. It is seen th at the degradation is less than 1 dB at a maximum frequency erro r of 20 kHz, and d w indles to 0 dB as the frequency error becomes zero, which is significa ntly less th an a differential detector which suffers much greater than 3 d B degradation , even without frequency error s. It is noted that t he opti mal value of  is a function of the ti me d uration T of the waveform. Fo r instance, followin g the same opti mization proced ure, the opti m al  for 2 39 0 T  μ s can be found to be   1 2 3 a r g m ax 0.481 kHz/ μ s      S S S . ( 36 ) IV. P RACTICAL W AVEFORM So far w e have sho wn that the prototype waveform d efined in (9) possesses certai n useful mathematical p roperties that not only pro vide rob ust detection per formance agai nst frequency errors but also facilitate optimizatio n to meet the applicatio n requirements. Ho wev er, i n ad dition to the primary role of detecting the presence and timing o f a s ystem, a nother essential function of a synchronization signal is to pro vide frequency synchronization. To this end, the frequency error f  needs to be o btained a fter signal   , ˆ xt  is de tected. From (8 ) , f  is linearly related to ˆ  , the ti me shift of the d etection peak from the actual timing po sition, however also un known to the receiver. It is thus clear that, in its origi nal for m, the prototype waveform cannot be used as a practical synchronization signal. To solve this dile mma, we need the following pro perty, which is alrea dy seen fro m the solutions to ( 26 ), and is generally true. Proposition 2 : The o ptimal solutions to ( 26 ) co m e in symmetric pairs, i.e ., if ,   is t he sol ution to ( 26 ), the n ,    is also a solution. This is because the constraint , i S ( i  1, 2,3), is symmetric in terms of  and  . Consequently, † i O is symmetric, i.e., (1) symmetric about 0   , i.e.,     † † ,, ii      OO ; ( 37 ) (2) symmetric about 0   , i.e.,     † † ,, ii      OO ; ( 38 ) and (3) symmetric about 0 , 0   , i.e.,     † † ,, ii        OO . ( 39 ) This s ymm etr y property is straight forward for 1 i  , while for i  2, and 3 it is a direc t result fro m the fact that   , ˆ f  X is symmetric, i.e.,             22 2 /2 2 ,, /2 / 2 / 2 22 / 2 / 2 ˆ ˆ . T j ft T TT j t t j t t j ft j ft TT f x t e dt e e dt e e dt                           X ( 40 ) Substituting t  with  results in             22 , / 2 / 2 22 / 2 / 2 , ˆ ˆ , TT jj j f j f TT f e e d e e d f                       X X ( 41 ) and ( 37 ) follows. Similarly, it can be sho wn that     ,, ˆ ˆ ff       XX , ( 42 ) which leads to ( 38 ) since       † † † † † † / 2 / 2 ,, / 2 / 2 /2 , /2 1 1 ˆ ˆ 1 ˆ , f WW WW W W f df d TT f df T                XX X ( 43 ) wherea s ( 39 ) d irectly follows from ( 37 ) and ( 38 ). The follo w ing pr oposition is a direct extension of P roposition 2. Proposition 3 : T he optimal waveforms come in pairs, meaning that i f ,   parameterizes an optimal waveform,     2 , , 2 2 ˆ 0 , otherw ise j t t e T t T xt                , ( 44 ) then ,    also yields an op timal waveform,     2 , 22 , ˆ 0 , ot herwis e j t t T t T e xt                  . ( 45 ) Together, ( 44 ) and ( 45 ) constitute the o ptimal waveform pair . Following the same example from the previou s section, waveforms   ,0 ˆ xt  , ( 46 ) and     , 0 , 0 ˆ ˆ x t x t      , ( 47 ) where =0 . 2 5 1  (kH z μ s) with 780 T  μ s , form an optimal pair that are co -conjugate . Fig. 5 . Plot of the performance degr adation (re lative to the optim al matche d- filter , 0d B) of the optimal wavefor m   ,0 ˆ xt  against t he frequen cy error   f  ranging from 20  kHz to +20 kHz . It is evident that the opti mal waveform pair not only provide the same per formance but al so produce the sa me ti me shift, on ly in o pposite directions. T hat is to say, a frequency erro r f  causes the output of t he matched filter to   , ˆ xt    to shift by 1 ˆ f     ( 48 ) in response to an inco ming signal   , ˆ xt    , as opposed to 1 ˆ f      ( 49 ) in response to   , ˆ xt   fr om its corresponding matched filter [cf. (8) ]. This property lead s to a simple means to obtain the ti me shift (and then t he actual timing and the frequenc y erro r) by constructing a composite wavefor m co mposed of   ,0 ˆ xt  followed by its cou nterpart   ,0 ˆ xt   , i.e., ,0 ,0 ˆ ( ), 4 4 () ˆ ( 2 ), 4 3 4 x t T t T xt x t T T t T             ( 50 ) as d epicted in Fi g. 6 (a) , where the length of   xt is T , P is the transmission per iod, and   ,0 ˆ xt  has length 2 39 0 T  μs with 0 . 4 8 1   (kH z/ μ s) . For a giv en f  , this gives rise to tw o correlation p eaks , o ne of which is loca ted at 11 ˆ t   ( 51 ) in response to   ,0 ˆ xt  at the output o f a matched filter to   ,0 ˆ xt  , where 1 1 ˆ f      ( 52 ) according to ( 8) , and the other at 22 ˆ 2 tT   ( 53 ) in resp onse to   ,0 ˆ xt   at the output of another matched filter to   ,0 ˆ xt   , where 1 2 ˆ f     ( 54 ) They are separated by a distance of 21 tt  or     1 21 ˆ ˆ ˆ 2 2 2 d T T f            . ( 55 ) Fig. 7 is the sample outputs of the paired-correlato r d etector in response to the composite synchronizat ion waveform, where (a) is the case with a frequency error o f 20 kHz, (b) 0, and (c) 20  kHz. Solving ( 55 ) for f  immediately yields ˆ 22 T fd        . ( 56 ) Since ˆ d can be directly measured after detection, f  can then be calculated. Apparently, the distance between the two detection peaks is ˆ 2 T d  , in the absence of a freque ncy error. The composite waveform e nables the e stimation o f a frequency error, and yet has an efficient detection implementation by taking advantage of the fact that most of the computations ca n be shared bet w een the pair ed corr elators due to the co-conjugate nat ure. It is also possible to transmit th e pair ed opti m al waveforms P sec apart, as illustrated in Fig. 6 (b ), where they are transmitted alternatively in diff erent transmission opportunities. In this case, the detector output peaks are sep arated further apart, and the correspo nding frequency error is calculated as   ˆ 2 f d P     . ( 57 ) With the knowledge of the frequency error, the timing error due to th e ti me shift attributable to f  is read ily a vailable from (8). A cor rection can then be made to the detected ti ming to remove the effect of t he frequency er ror on the detected timing. Fig. 8 plots the cumulated error d istribution function (CDF) of the esti mated receiver freq uency error via numerical simulations, where waveform in F ig. 6 (a) with parameter pair, , 0.4 81 , 0      [see ( 36 )], is constructed and transmitted by a base station with power p  43 dBm. Bet w een two transmissions, rando m data are also tran smitted to mimic r eal life situations. The maximum coupli ng loss between the transmitter and the receiver of an mIoT device is 144 dB plus 2 0 dB additional penetration loss (for deployment deep inside a building, e.g., basement). T he total propagation loss is thus  164 dB. T he receiver sensitivit y is then P P P P P T T (a) (b) A paired optimal waveform Fig. 6 . Il lustration of the paire d optimal w aveforms based on Pro position 3 that e nables fre qu ency error estimation, w here red denotes   ,0 ˆ xt  and blue   ,0 ˆ xt   . 43 d Bm 164 dB 121 dBm p       . ( 58 ) The received signal is filtered with a 20 0 kHz band width filter operating at a sam ple rate of 4 20 0  kHz. The noise figure o f the mIoT rec eiver is 5   dB . The noise power at the receiver is thus (a) 20 f  kHz (b) 0 f  (c) 20 f    kHz Fig. 7 Sample o utputs of the p aired correlator s in response to a n incoming s equence of   xt in ( 50 ) with = 0 . 4 8 1 ( k H z/ μ s)  and data stre am s, resulting in paired peaks separate d by ˆ d : (a) 20 f  kHz, (b) 0 f  , and (c) 20 f    kHz. 0 53 dB Hz 174 dB m /Hz +5 dB = 116 dB m , N W N       ( 59 ) where 0 N is the noi se po w er spectral density (i.e., 174  dBm/Hz). The corr esponding signal SNR at t he receiver is 121 dBm ( 116 dB m) 5 dB N        , ( 60 ) which is the minimu m SNR that an mIoT rec eiver oper ates at . A frequency error f  , uniformly d istributed i n the ra nge from 20  kHz to +20 kHz, is intro duced into the incoming signal, w hich then passes through a filter of bandwidth 200 kHz at a sampling rate o f 1600 kHz. T he detector employs a p air of matched filters to the paired waveforms, a nd jointly detects the incoming paired signals . T he frequency error is estimated using ( 55 ) after the paired signals are successf ully detected. The evaluation i s performed at SNR leve ls of 5, 0 , and -5 dB. It is observed from Fig. 8 that about 95 percent of f  estimation errors fall with in 400 Hz, corresponding to ˆ  estimation error of less than 2 μs . V. C ONCLUSION In this pap er, we propose a practical design of the frequency-error -resilient synchronization waveform for massive MT C. W e derive and exp loit the key m athematical properties o f the prototype waveform for wavefo rm optimization, frequency erro r estimation, a s well as ti ming refinement. The design is ex emplified by a specific pr actical application , i.e. , LTE NB -IoT . W e show that the practical for m of this waveform is asymptoti cally optimal, in the sense that its asymptotic detection energy in the pr esence of freque ncy er ror is the same as an op timal matched filter which has full knowledge of the input signal frequency (i.e., free of frequenc y error). Based o n this p roperty, the practical design proble m boils down to maximization o f the waveform parameter, i.e.,  , under the constraints present in the applicatio n. We further show that the optimal parameter of this type of waveform comes in symmetric pairs, which facilitates the crea tion of a unique synchro nization waveform consisti ng of paired optimal waveforms for determ ining the frequency error and refined timing at the recei ver. R EFERENCES [1] E. Dutkie wicz, X. C osta-Per ez, and I . Z . Kov acs, “Massive machine-type c ommunications ,” IEEE Network , vo l. 31, no. 6, pp. 6-7, Nov. 2017. [2] M. W ang, W. Yan g, J . Z ou, B. Ren, M. Hua, J. Zhang, X. You, “Cel lular machine- ty pe communications: Physical challenge s and sol utions,” IEEE Wireless C ommun ., vol. 23, no. 2, pp. 108-117, A pril 2016. [3] 3rd G eneration Partne rship Pr oject T echnical Specification G roup Radio Access Networ k Evolved Universal Te rrestrial Rad io A ccess (E-UTRA ) Physical channels and modulation (Release 13), TS 36.211 ver. 13.2.0, June 2016. [4] From G SM to LTE-Advanced Pro and 5 G: An I ntroduction t o M o bile Networks and Mobile Broadba nd, Martin Sauter, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA . [5] Y. -P. E. Wang, X. Lin, A. Adhikary , A. Gro vlen, Y. Sui, Y. Blankenship, J. Ber gman, an d H. S. Razaghi , “A primer on 3GPP narro wband I nt erne t of Things,” IEE E Commun. Mag . , vol. 55, n o. 3, pp. 117-123, M ar. 2017. [6] G. Berardinell i, K. I. Pedersen, T. B. Sørensen, and P. Mogensen , “Gene ralized DFT -Spre ad-OFDM as 5 G waveform, ” IEEE Co mm un. Mag. , vol. 54, no. 11 , pp. 99 -1 05 , Nov . 2016. [7] X . Zhang, L. C hen, J. Qiu, and J. Abdoli , “ On the waveform for 5G, ” IEEE Commun. M ag. , vol. 54, no. 11, pp. 74 - 80 , Nov. 2016. [8] A. A. Zaidi, R. Baldemair, H. Tullber g, H. Björkegre n, L. Sundström, J. Medbo, C. Kilinc, an d I. D. Silva , “ W aveform and numerol ogy to support 5G service s and requ irements, ” IEEE Commun. Mag. , vol . 54, no. 11, pp. 90 - 98, Nov. 201 6. [9] J. Yli-Kaakinen , T. Levanen , S. Valko nen , K. Pajukosk i , J. Pirskanen , M. Renfors, and M. Valkama , “ Efficient Fast-Convolution-Base d wavef orm processing fo r 5G physical lay er, ” IEEE J. Sel. Are as Commun. , vol. 35, no. 6, pp. 1309-13 26, Jun. 2017. [10] P . Guan , D. Wu , T. Tian, J. Zhou, X . Zhang, L. Gu, A. Benjebbour, M. Iwabuchi, and Y. K ishiyama , “ 5G fie ld trials: OF DM-Based wavefor ms and mixed numero logies, ” IEEE J. Sel. Areas Commun. , vol. 35, no. 6 , pp. 1234-1243, Jun. 2017. [11] Y . L iu, X. Chen, Z. Zhong, B. Ai, D . Miao, Z. Zhao, J. Sun, Y. Teng, and H. Guan , “ Wave form design for 5G networks: Analysis and comparison, ” IEEE A ccess , vol. 5, pp. 19282-19292, Ja n. 2017. [12] B. Farhang-Boroujeny and H. Moradi , “ OFDM inspired wavef orms for 5G , ” IEEE Commun. Su rve ys Tuts. , vol. 18, no. 4, pp. 2474 -24 92, 4th Quart., 2016. [13] J. M. Hamamr eh and H. Arslan, "Secure orthogonal transform division multiplexing (OTDM) w aveform for 5G and beyond," IEEE Commun. Lett. , vol. 21, no. 5 , pp. 1191-1194, May 2017. [14] J. Zhang, M. Wang, M . Hua, W. Yang, and X. You, “Robust synchronization waveform design for massive I oT , ” IEEE Trans. on Wireless Commu n. , vol. 16, no. 11, pp. 7551-7559, Nov . 2017. [15] H . Kroll, M. Kor b, B. Weber , S. Willi, an d Q. Huang, “Maximum - likelihood detection for energy-ef ficient t imi ng acquisit ion in NB- IoT,” in Proc. IEEE Wireless Commu nications an d Networkin g Conference Workshops (W CNCW), San F rancisco, USA , Mar. 2017, pp. 1- 5. [16] M. Hua, M. Wang, W. Yang, X. You, F. Shu , J. Wang, W. Sheng, and Q. Chen , “Analy sis of th e frequency offset effe ct on random access signals,” IEEE Trans . Commun. , vol. 61, no. 11, pp. 4728-4740, Nov. 2013. [17] M. H ua, M. Wang, K . W. Yang, and K . J. Zou, “A nalysis of the frequency offset effect on Zadoff –Chu sequence timing p erf ormance,” Fig. 8 Frequency error   f  estimation error CDF with rece iver signal SNR levels of 5, 0, and -5 dB. The freque ncy error is uniformly distributed in the range from - 20 kHz to +20 kHz . IEEE Trans. Comm un. , vol. 62, no. 11, pp. 4024-4039, Nov . 2014. [18] H . Puska and H. Saarnisaar i, “Matc hed filter time and fre quency synchronization method for OFDM systems u sing PN -seq uence preambles,” i n IEEE 18t h Internati onal Symposium on Personal, Indoor and Mobile Radi o Communicat ions , Athens, 2007, pp. 1- 5. [19] F . Tufvsson, O. Ed fors, and M. Faulkner, “Time and fre quency synchronization for OFDM using P N- sequence preambles,” in Proc. IEEE Vehicular T echnology Conf. , vo l. 4, Sep. 1999, pp. 2203-2207. [20] I TU- R reco mmendation SM. 328: “ Spe ctra and b andw idth of emissions” . [21] I TU-R recomme ndation SM.329 : “ Unwanted emissio ns in the spurious domain ” . [22] T hird Generation Partnershi p Program (3GPP) : Evolved Universal Terre strial Ra dio Access Base Station (BS) radio transmission and reception, Te ch. Specification 3 6.104 V 14. 0.0, Jun. 2016.