Overlapped frequency-time division multiplexing

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1 April 29, 16(2): he Journal of China Universities of Posts and elecounications Overlapped frequency-tie division ultiplexing JIANG Hui ( ), LI Dao-ben chool of Inforation Engineering, Beijing University of Posts and elecounications, Beijing 1876, China Abstract A technique naed overlapped frequency-tie division ultiplexing (OVDM)) is proposed in this article. he technique is derived fro Nyquist syste and frequency-tie division ultiplexing syste. When the signals are copactly overlapped without the orthogonality in tie doain, the technique is naed overlapped tie division ultiplexing (OVDM), whereas when signals are copactly overlapped without the orthogonality in frequency doain, the technique is called overlapped frequency division ultiplexing (OVDM). o further iprove spectral efficiency, the OVDM in which signals are overlapped both in frequency doain and in tie doain is explored. OVDM does not depend on orthogonality whatever in tie doain or in frequency doain like Nyquist syste or ODM syste, but on the convolutional constraint relationship aong signals. herefore, not only the spectral efficiency but also the reliability is iproved. he siulations verify the validity of this theory. Keywords overlap, OVDM, spectral efficiency 1 Introduction It has been known since the 192 s that Nyquist pulses can be used to send data at the Nyquist rate without intersybol interference (II) over band-liited channels. his fact has played a ajor role in the design and ipleentation of data transission over the telephone network [1]. It is natural to concern about the degradation caused by the presence of intersybol interference when Nyquist pulses are used to send data at a rate faster than the Nyquist rate. he OVDM presented in this article is the solution. Orthogonal frequency division ultiplexing (ODM) [2] is a well-known odulation ethod with high spectral efficiency and particularly its high tolerance of ultipath interference in a wireless ediu. In ODM, plural subcarriers are transitted with orthogonal frequency spacing. Usually, the frequency spacing is chosen as the inverse of sybol duration s. In this case, no interference between subcarriers appears even when ultilevel or ultiphase techniques such as M-PK and M-QAM are used to odulate the subcarriers. o solve the high copaction of frequency spacing in order to iprove the spectral efficiency, the Received date: Corresponding author: JIANG Hui, E-ail: jianghuiphd@gail.co DOI: 1.116/ (8) technique OVDM is presented in this article is the technique. he above two new techniques not only increase syste s spectral efficiency but generate coding constraint relationship naturally. Consequently, the perforance of reliable transission can be iproved. According to ourier transfor [3 4], when tie is liited, the interference in frequency doain is dooed to occur, vice versa. he detection in the two techniques is that guard interval is introduced in tie or frequency doain at the expense of spectral efficiency or that the axiu likelihood detection (MLD) is applied to hold spectral efficiency. However, the MLD only increases the coplexity of detection, and it is no use at all to iprove spectral efficiency. herefore, it would be better for increasing the interference aong signals to further proote counication syste s spectral efficiency. One realization of above analysis is that signals are overlapped in both tie doain and frequency doain, and the technique is called OVDM. his article is organized as follows: ect. 2 outlines the principle of overlapped ultiplexing syste, as well as OVDM and OVDM. OVDM is presented in ect. 3. And the perforance analysis and the siulations are introduced in ect. 4. inally, ect. 5 concludes the article.

2 Issue 2 JIANG Hui, et al. / Overlapped frequency-tie division ultiplexing 9 2 OVDM and OVDM principle of overlapped ultiplexing It is well known that signals should be overlapped to iprove spectral efficiency. However, it is generally believed that serious interference can be iported except orthogonality. Now, let us see able 1 and ig. 1, where there are three signals (A, B, C) that will be overlapped. Because of the interference, it is ipossible that the signals are deodulated inerrably in the conventional way. able 1 Overlapped ultiplexing of three signals equence Data cobination nuber A B C Overlapped D E G H I J K In ig. 1, let the Y denote pulse tie duration, and x is tie unit t (of course Y can also denote spectru, and x is frequency unit f ). he overlapped interval is Y /3, which eans that there are 3 sybols in one duration. If each signal is processed individually, they cannot be deodulated accurately. However, it can be seen fro able 1 and ig. 1 that if the three signals are seen as a sequence of a whole, the sequence shape is only one of eight shapes (D, E,, G, H, I, J, K) in any one duration. herefore, we know that there is one correspondence (bijective relation) between the overlapped signals and the sequence fro the chart. 2.1 OVDM OVDM is also a tie division ultiplexing schee. he ain difference between conventional DM and OVDM is that OVDM allows overlapping aong serial sybol signals. herefore, in the sae odulation schee, the spectral efficiency of OVDM is higher than that of conventional DM. In essence, the overlapping aong serial sybol signals can be seen as a natural convolutional coding constraint relationship rather than interference. he ore the degree signals overlap, the longer constraint length is generated, and subsequently, the higher coding gain and spectral efficiency are obtained. Under the sae signal-to-noise ratio threshold, the bit error rate (BER) perforance of OVDM is better than that of conventional DM with the sae spectral efficiency. he scheatic diagra of OVDM is shown in ig. 2. As is presented in ig. 2, the atheatical odel of transitted signals in equivalent lowpass wavefor is as follows: L 1 l = ig. 1 Principle of overlapped ultiplexing s() t = u() l a( t lδ) ; t [, ) (1)

3 a 1 he Journal of China Universities of Posts and elecounications 29 where the pulse shaping wavefor at () = t [, ); is the sybol period; is the tie shift of the transitted signals and Δ = / K( K Z + ) ; K is the nuber of overlapped sybols (obviously, if K = 1, OVDM is the sae as interference free syste; K = 2, OVDM is the Nyquist syste); L is the total nuber of sybol in a frae; u(l) is the transitted sequence; is the one frae duration. the etric M is easily obtained: L K M = vl () hkul ( ) ( k) l= k= 2 (5) ig. 2 cheatic representation of OVDM hrough AWGN channel, the received signal can be expressed as follows: L 1 rt () = ulat () ( lδ ) + nt (); t [, ) (2) l = where nt () is white Gaussian noise with power spectru density N. A better way than aking sybol-by-sybol decision is that the decision based on the entire received signal axiu-likelihood sequence detection (MLD) [5] is utilized to iniize the following Euclidean distance: 2 L u = arg in r() t u() l a( t lδ) dt (3) u l = he physical eaning of Eq. (3) is to search the axiu likelihood data sequence u whose corresponding wave is s() t for approaching rt () ost closely. ro ig. 3, the received signal is the convolution between transitted sybols and syste ipulse response in any counication channel. he equivalent convolutional coding odel is illustrated in ig. 4, where the tap coefficient hk ( ) ( k=,1,..., K 1) is the entire result of channel gain and pulse shaping wavefor. ig. 4 Equivalent convolutional coding odel of OVDM It is shown that OVDM syste has alost the sae atheatical odel as intersybol interference and the detection algorith is also siilar [1,6]. hrough constructing interference, that is, convolutional coding constraint on its own initiative, OVDM obtains coding gain and high spectral efficiency. 2.2 OVDM In the structure of ODM, the space between subcarriers Δ f = 1/, ( is sybol duration), that is, Δ f = 1, which guarantees orthogonality aong subcarriers. If the space between subcarriers becoes saller, that is, Δ f = λ 1, it is naed OVDM. Here, fi = i ( K) = f+ iλδ f, where K = 1 λ, K 1. he scheatic representation of OVDM is shown in ig. 5. L is the total nuber of subcarriers in a frae, and the OVDM signals can be expressed as: L j2π ti [ ( K )] s() t = direct t e ; t i= 2 (6) ig. 5 cheatic representation of OVDM ig. 3 yste odel of OVDM Based on convolutional coding odel, i.e., K 1 vl () = hkul ( ) ( k) + nl () (4) k = ig. 6 is the odel of OVDM syste (Obviously,when K = 1, it is an ODM syste). Generally, when K 1, it is called OVDM syste. Because orthogonality does not always occur, OVDM signals cannot be exained by orthogonality but by the MLD. When N is large in the OVDM syste, Eq. (6) can be obtained by the ethod of inverse discrete ourier transfor (ID). Rectangle function can be oitted for concise description [6]: () L j2πtik ( ) i e ; i= s t = d t (7) If s(t) is sapled by t = k/ N( k =,1,..., N 1), we can

4 Issue 2 JIANG Hui, et al. / Overlapped frequency-tie division ultiplexing 11 obtain that L s j2π( ik KL) sk() t = s = die ; k L (8) N i= syste bandwidth is B. Each subcarrier bandwidth in the ulticarriers syste is not strictly restricted. o iprove the spectral efficiency further, the transitted data ight be overlapped both in frequency doain (i.e. applying OVDM in frequency doain, and the frequency shift Δ B = 1( K ) ) and in tie doain (i.e. applying OVDM in tie doain, and the tie shift Δ = K ). herefore, according to igs. 6 and 3, the cheatic representation of OVDM is shown in ig. 8. ig. 6 Model of OVDM syste based MLD It is equivalent to adding (K 1)L zeros at the end of di( i=,1,..., L 1) before ID transfor and the anterior N data after ID are transitted. In the siulation, the inverse fast ourier transfor (I) can be applied, and the odel of OVDM based on I and MLD is shown in ig. 7. ig. 8 cheatic representation of OVDM ig. 7 Model of OVDM based on I and MLD If the prior probabilities of transitted sybols are equal, the MLD will be optiu. However, the coplexity of MLD algorith grows exponentially with the increase of subcarriers. 3 OVDM he product of tie-bandwidth of pulse shaping filter puzzled the OVDM and OVDM. When tie is liited, the interference in frequency doain is dooed to occur, vice versa. In addition, there is tradeoff between the coplexity of detection and the spectral efficiency in the two techniques. he detection in the is that guard interval is introduced in tie or frequency doain at the expense of spectral efficiency or that the MLD is applied to hold spectral efficiency; however, the MLD only increases the coplexity of detection, and it is no use at all to iprove the spectral efficiency. herefore, it would be better to increase the interference aong signals to further proote counication syste s spectral efficiency. One realization of above analysis is that signals are overlapped in both tie doain and frequency doain and this technique is called OVDM. Assue that sybol period equals (after the odulation, filter, and channel transission) and the total In this syste, there are L subcarriers and L tie shifts in one frae; the spectru of each carrier holds the total syste bandwidth, and they are overlapped by each other; K is the overlapped nuber within one data sybol spectru in frequency doain; the data is odulated at M levels and K is the overlapped nuber within one data sybol duration in tie doain. he frae length = [ + ( L ) Δ ] in tie doain, and the total syste bandwidth (zero-bandwidth) B = ( 1 ) [2 + ( L 1)/ K]. When t [, + ( L ) Δ ], the transitted data in one frae is as follows: LL j2π flδb( t tδ ) 2 E u f, f ( )e l t a t t l Δ (9) t= fl = where a f l () t is the unitary shaping wavefor of duration tie and its spectru is A f ( f f ) l lδb ; u fl, t is the data that is odulated on the fl( fl f) subcarrier; E is the sybol energy. Let f u 1 L 1, ul 1,1 u L L, L f u L 2 L 2, ul 2,1 u L 2, L U (1) f u, u,1 u, L U denote the total transitted data in one frae ( LL sybols). ro Eq. (1), we can see that the sybols can be considered as being located at points in a lattice separated by

5 12 he Journal of China Universities of Posts and elecounications 29 Δ and Δ B. In addition, the U can also be expressed as follows: U, t, t1, tl U U U (11) where U, t [ u f, t, u 1,,..., 1, ] ; f t u fl t U, t is the whole data in frequency doain when t ( tδ, tδ + ]. Let a, t ( t) [ a, ( ), 1, ( ),..., f t t t Δ a f t t t Δ a f 1, ( )] L t t t Δ (12) where a t, () t is the frequency shaping wavefor and its spetru is: j2πftδ [ Af( f), Af1( f ΔB),..., AfL ( f ( L ) Δ B)]e (13) where a f l () t is wavefor of subcarrier fl( fl f), and its spectru is Afl ( f flδ B). herefore, fro Eq. (12), the transitted data can be described as follows: V() t = 2 E U, t a () t (14) 1, t t t And fro Eq. (13), the spectru of the transitted data can be presented as follows: V( f) = 2 E U, t A, t ( f) (15) 1 t t he received signals are as follows: 1 Vt () = 2 E U, t a, () () t t+ nt 2 t t (16) where E = αe is received signal energy, and α is the channel decline; and a, t ( t) [ a ( ), ( 1 ),..., 1( )] f t tδ a f t tδ a f t t L Δ (17) where a () t is the received wavefor of subcarrier f l fl( fl f) ; nt () is white Gaussian noise with power spectru density N. In addition, the spectru of the received signals is 1 V( f) = 2 E U, t A, t ( f) + n ( f) (18) 2 t t where A, t ( f) [ Af( f), Af1( f ΔB),..., j2πftδ Af ( f ( L ) ΔB)] e (19) L 1 the spectral efficiency of OVDM is as follows: LLQ LLQ L K, L K η = B L L K K KKQ= 2KKη (2) OVDM adopts low order odulation, and the overlapping used in tie and frequency doains not only prootes spectral efficiency but also generates coding constraint. Meanwhile, these advantages also increase the decoding coplexity. he decoder coplexity of OVDM ( 1) is O( 2 QL K ). We can see that the decoder coplexity of OVDM is exponentially increasing with the growth of Q, L, and K. According to Eq. (2), there is a linear relationship between Q, L, K, and η. herefore, the coplexity of the syste grows exponentially with the increase of spectral efficiency. When Q, L, and K are large enough, the decoder becoes too coplex. In this case, soe other fast decoding algoriths need to be considered such as sei-definite prograing (DP), iterative algorith to reduce the coplexity. Whereas, any ethod to reduce the coplexity is at the expense of signal-to- noise ratio threshold. ake QPK as an exaple, OVDM syste utilizes rectangle pulse shaping; L = 3; L = 1 ; K = K = 2; therefore η = = (bit/(s Hz)) (21) he siulation result with above syste paraeters is shown in ig. 9, where OVDM has soe power gain over 16 QAM (no cyclic prefix ODM syste in AWGN channel). 4 Perforance analysis and siulation results Let M be the odulation order, and M = 2 Q, i.e. each sybol carries Q bits; s denotes sybol duration tie; the referenced spectral efficiency is η = Q /2 (zero-bandwidth is 2/ ), and subsequently the syste bandwidth is B = [2 + ( L 1)/ K]/ ; the frae length is = [1 + ( L 1) / K] ; ig. 9 iulation on perforance of OVDM syste in AWGN channel With the increase in the nuber of overlapped signals, OVDM would obtain larger and larger power gain over

6 Issue 2 JIANG Hui, et al. / Overlapped frequency-tie division ultiplexing 13 QAM-ODM fro the above analysis. he paraeters of the IU-VA channel are listed in able 2, and the channel is assued to be stationary in each frae, and the channel estiation and synchronization are perfect. able 2 Delay/ μs Paraeters of IU-VA Channel Average path gains/db In ig. 1, the ODM and OVDM hold the alost sae spectral efficiency. shown in ig. 1. In fact, the perforance of real ODM is usually worse by about 1 db than that of ideal ODM because of the power loss caused by cyclic prefix (CP). hus, the perforance of OVDM is better than ODM but this is at the expense of increasing detection coplexity. 5 Conclusions he OVDM technique in which signals are overlapped both in frequency doain and in tie doain to further iprove spectral efficiency is presented in this article. he technique does not depend on orthogonality whatever in tie doain or in frequency doain like Nyquist syste or ODM syste but on the convolutional constraint relationship aong signals. hus, not only the spectral efficiency but also the reliability can be prooted. Although the perforance of OVDM is better than others, the obvious advantage is at the expense of increasing detection coplexity. herefore, a fast detection ethod is still the key question of OVDM. Acknowledgeents his work was supported by the National Natural cience oundation of China (96435). References ig. 1 iulation on perforance coparison between OVDM and ODM in IU-VA channel he first siulation was perfored without channel coding, and the second siulation was perfored with the rate-1/2 convolution code and bit interleaving. he convolution code is [ ] whose constraint length is 7. Besides, the perforance of ODM systes with different CPs will differ slightly despite the sae odulation. herefore, the ideal ODM syste without power loss caused by CP is taken into account for an easy and fair coparison, and the results are 1. Proakis J G. Digital counications. 4th ed. New York, NY, UA: McGraw Hill, Van N R, Prasad R. ODM for wireless ultiedia Counications. Norwood MA, UA: Artech House, 2 3. Zheng J L, Ying Q X, Yang W L. ignals and systes. Beijing, China: Higher Education Press, 2: (in Chinese) 4. Bringha E O. he fast ourier transfor and its applications, Englewood Cliffs, NJ, UA: Prentice Hall, Li D B. he statistical theory of signal detection and estiation. 2nd ed. Beijing, China: cience Press, 25: (in Chinese) 6. Bai Z Q, Li D B. A transission in II channel. Journal of Beijing University of Posts and elecounications, 25, 28(6): (in Chinese) (Editor: ZHANG Ying)