Vector orthogonal frequency division multiplexing ... - IEEE Xplore

www.ietdl.org Published in IET Communications Received on 4th December 2013 Revised on 14th February 2014 Accepted on 15th April 2014 doi: 10.1049/iet-com.2013.1064

ISSN 1751-8628

Vector orthogonal frequency division multiplexing system over fast fading channels Wen Zhou1,2, Lisheng Fan1,2, Hongbin Chen3 1

Department of Electronic Engineering, Shantou University, Shantau, People’s Republic of China The National Mobile Communications Research Laboratory, Southeast University, Nanjing, People’s Republic of China 3 School of Information and Communication, Guilin University of Electronic Technology, Guilin, People’s Republic of China E-mail: [email protected] 2

Abstract: The performance of the vector orthogonal frequency division multiplexing (V-OFDM) system over fast fading channels is investigated. The channel is time varying within one V-OFDM data-block period, which causes the inter-carrier interference (ICI). With the help of an equivalent V-OFDM system model and an auxiliary transform matrix, a novel mathematical expression for the received signal with the ICI signal is derived, which clearly expresses how the signal of a target vector block (VB) is interfered by other VBs. The ICI signal is analysed and two theorems about its property are proposed. The first theorem presents the expression of the power of the ICI signal, showing that the ICI signal power increases with the VB number. The other one indicates that ICI signals at different subcarriers are not correlated. With the two theorems, a novel detection method utilising the correlation of the ICI signal is therefore proposed. Numerical results verify the validity of the derived theorems and simulation results demonstrate that the proposed minimum mean square error (MMSE) detection can suppress the ICI effect and it is superior to the conventional MMSE detection in terms of both detection mean square error and bit error rate performance.

1

Introduction

Orthogonal frequency division multiplexing (OFDM) is a transmission technique with high data rate as it combines many data streams with low rates. A lot of aspects about the OFDM system such as channel estimation, the peak-to-average ratio (PAR) suppressing technique were studied over past years. Currently, OFDM, together with multi-input multi-output (MIMO) has been adopted as the key technique of the physical layer in 3GPP long term evolution advanced [1, 2]. Recently, a few derivatives of the conventional OFDM system appear, such as flash-OFDM [3, 4], orthogonal frequency and code division multiplexing (OFCDM) [5, 6], vector OFDM (V-OFDM) [7–15] and constellation-rotated V-OFDM [16, 17]. The derivatives are proposed to overcome the shortcomings of the conventional OFDM system that include high PAR and sensitivity to carrier frequency offset and so on. Such systems can be used in some special scenarios. For instance, flash-OFDM, combining OFDM with frequency hopping technique, is often used in the scenario with real-time services. V-OFDM, first proposed in [7], is a generalised version of OFDM and it has some advantages over the conventional one, such as lower PAR and better system performance in terms of bit error rate (BER). Therefore it receives a lot of attention recently [7–15]. In [8], a structure of layered fast Fourier transform (FFT) was introduced. Based on such a structure, a novel asymmetric OFDM system that bridges the 2322 & The Institution of Engineering and Technology 2014

conventional OFDM system and single carrier system was proposed. In fact, the asymmetric OFDM system in [8] is equivalent to the V-OFDM system, and the corresponding mathematical equivalence between them was later shown in [14]. Zhang and Xia [10] studied iterative decoding methods for V-OFDM systems with convolutional codes. Particularly, a few low-complexity demodulation schemes using the linear cancellation technique were proposed. Cheng et al. [13] presented a comprehensive study of V-OFDM over multipath channels, thoroughly investigating the diversity gain and coding gain of each vector block (VB). Li et al. [9] investigated the performance of V-OFDM with linear receivers including the zero-forcing (ZF) and minimum mean square error (MMSE) receivers. The detection signal-to-noise ratio (SNR) gap between MMSE and ZF receivers was shown, and the diversity order for V-OFDM with the two kinds of receivers was also analysed. Currently, the papers investigating the V-OFDM system performance under fast fading channels are few. Most of them focus on the system with quasi-static fading channels. In the case that the channel is time varying even within one V-OFDM data block, the ICI effect will arise. Corresponding performance of V-OFDM systems with existing detection methods in such a scenario is worthy of investigation. It is known that the V-OFDM system becomes a conventional OFDM system for the VB size is one. The researches on analysis of the ICI and how to mitigate it in OFDM systems were comprehensively IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

www.ietdl.org investigated in the past [18–21]. For instance, the authors in [19] proposed a general ICI self-cancellation method which can be implemented through the windowing technique. However, the ICI effect in V-OFDM systems may exhibit different traits from that in OFDM systems, and the corresponding suppressing methods are also worthy of exploiting. Therefore we conduct a thorough study of ICI analysis and investigate the performance of V-OFDM systems with various detectors. The main contributions of this paper can be summarised as follows.

matrix X, the notations X*, X T, X H denote the conjugate, transpose, Hermitian transpose, respectively; Tr(X) denotes the trace of X; [X]i, j is its (i, j)th entry, while [X ]i1 :i2 ,j1 :j2 , [X ]:,j1 :j2 and [X ]i1 :i2 ,: are a submatrix by selecting rows i1, …, i2 and columns j1, …, j2, a submatrix by selecting columns j1, …, j2, and a submatrix by selecting rows i1, …, i2, respectively. I is an identity matrix with appropriate dimensions. Notation (k)L stands for k modulo L, and int(.) stands for extraction of the integer part of a number.

† First, a novel and equivalent V-OFDM system model is introduced to facilitate the derivation of the received signal in frequency domain in the V-OFDM system with fast fading channels. Based on such a model and with the aid of a transform matrix, the expression for the received signal in frequency domain is provided. It clearly shows how the signal of a target VB is disturbed by other VBs. † Second, two theorems together with a corollary about the property of the ICI signal are presented. The first theorem provides the expression for the power of the ICI signal on each subcarrier and it shows that the power of the ICI signal at a subcarrier does not depend on its index. With the first theorem, we then prove that larger VB size M results in lower ICI signal power in a corollary. The second theorem shows that ICI signals at different subcarriers are not correlated. † Third, based on the analysis of the ICI signal, a novel MMSE detection method that utilises the correlation of the ICI signal is proposed. Simulation results demonstrate the superiority of the proposed method over the conventional MMSE detection, in terms of both the detection mean square error (MSE) and BER performance.

2

The rest of the paper is organised as follows. Section 2 briefly reviews the basic V-OFDM system model and Section 3 describes the V-OFDM system with fast fading channels. A detailed analysis of the ICI signal is conducted in Section 4, and then a novel MMSE detection method is proposed in Section 5. The system performance with various settings is presented in Section 6, followed by conclusions in Section 7. Notations: Scalars are denoted by normal letters, bold-italic-face lower-case letters are used for vectors, and bold-italic-face upper-case letters for matrices. For a vector x, diag(x) denotes the diagonal operation, [x]i denotes its ith element and ||x|| represents its Frobenius norm. For a

Brief review of the basic V-OFDM system

The V-OFDM system is a generalised version of the conventional OFDM system. The basic V-OFDM system model is depicted in Fig. 1, where the modulated symbols are processed block by block. At the transmitter, N modulated symbols, denoted by xfi , i = 0, . . . , N − 1 , are to be sent out, where N is the number of subcarriers in one data block. Denote the originally transmitted signal T vector by x = x0 x1 · · · xN −1 , and it is transformed to an M × L matrix A with its (i, j)th entry being [A]i,j = x jM +i ,

i [ {0, . . . , M − 1},

(1) j [ {0, . . . , L − 1} T The VB l, denoted by xl = xlM · · · xlM +M −1 , is the lth column of A. The transmitter performs L-point IFFT on each ˜ Then, the relationship row of A and obtains a new matrix A. ˜ denoted by x˜ i , and xl can be between the column i of A, expressed as x˜ i =

L−1 1 x e j(2pl/L)i L l=0 l

(2)

˜ are chosen as the cyclic prefix The last P column vectors of A (CP) vectors, such that the CP length PM is larger than the maximum delay of the multipath channel. After adding CP, the signal in time domain is sent out serially by parallel-to-serial (P/S) transformation. At the receiver, a serial of reverse operations are conducted. They include removing CP, column-wise blocking the ˜ received signal in time domain, ytl , into an M × L matrix B, ˜ and taking L-point FFT over each row of B. The received VB l, denoted by yl, is the lth column of B, and it can be

Fig. 1 Basic V-OFDM system model IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

2323

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www.ietdl.org expressed as [7, 10, 11] yl = H l xl + wl ,

l [ {0, . . . , L − 1}

(3)

where the additive white Gaussian noise (AWGN) vector T wl = wl0 wl1 · · · wlM −1 , with its each entry following the Gaussian distribution with zero mean and s2w variance. The equivalent channel matrix Hl is given by H l Ul Hl = Ul H

1 2p m(l + sL) U l s,m = √ exp −j N M

(5)

where T y f = y0f y1f · · · yNf −1 , x f = x0f xNf −1 ]T ; the ( p, q)th entry of G is given by

(7)

x1f

···

[G] p,q = G p,q

and

l = diag Hl , Hl+L , . . . , Hl+(M −1)L H

(6)

In (6), the channel frequency response, {Hi, i = 0, …, N − 1}, is obtained from N-point FFT of the channel impulse response T h = h t0 h t1 · · · h tv , where v is the channel order, tv is the delay of the vth path.

3 The V-OFDM system with fast fading channels Previous section has illustrated the basic V-OFDM system with quasi-static fading channel. This section considers the V-OFDM system with fast fading channels, that is, the channel impulse response is time varying within a V-OFDM data block. 3.1

y f = Gx f +w′

(4)

where Ul is an M × M unitary matrix with its entry being

It is clear that the vector x is unchanged. The aim of adding those modules is to facilitate the derivation of the ICI expression. Then, the last PM elements (or subcarriers) of x are selected as the CP. Obviously, the transmitter in Fig. 2 is equivalent to that in Fig. 1. At the receiver, we add three counterparts including MR0, MR1 and MR4. According to [18], the received signal after N-point FFT is given by

An equivalent V-OFDM system model

We consider an equivalent V-OFDM system model depicted in Fig. 2. At the transmitter, three auxiliary modules that ˜ is include MT2, MT3, and MT4 are added. The matrix A transformed to a length-N vector x = xi , i = 0, . . . , N − 1}, followed by the N-point FFT and IFFT operations.

=

−1 v 1 N h(n, tl )e−j(2p/N )n(p−q) e−j(2p/N )qtl N n=0 l=0

(8)

in which w′ is an AWGN vector with its each element having zero mean and M s2w variance. h(n, tl) is the channel impulse response of path l at sampling point n within one data-block, tl is the channel delay of path l. We assume that the channel paths are statistically independent and the power of path l is s2l . The channel correlation function is given by

2 p fd T E h n1 , tl1 h∗ n2 , tl2 = d l1 − l2 s2l J0 n1 − n2 N (9) where J0(·) is the zero-order Bessel function of the first kind, δ(·) is the Kronecker delta function, fd is the maximum Doppler shift and T is the time period of an V-OFDM data block excluding the CP. On the other front, the relationship between xif and xi can be expressed as (Appendix 1) x f = Dx

(10)

Fig. 2 Equivalent V-OFDM system model 2324 & The Institution of Engineering and Technology 2014

IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

www.ietdl.org

T

T

where x = x0 · · · xN −1 , x f = x0f x1f · · · xNf −1 , and the transform matrix D is defined in Appendix 1. Note that the matrix D represents the relationship between the originally modulated signal xi and its corresponding frequency-domain signal xif , which shows that the transmitted signal spectrum after a series of operations on the original signal. If L = N, that is the V-OFDM becomes the conventional OFDM, D is an identity matrix and the originally modulated signal is indeed the frequency-domain signal, as already known to us. The introduction of D is to facilitate the expression of received signal as well as the analysis of the ICI signal afterwards. If we remove the part on the right side of dashed line in Fig. 2, that is xif is directly input to MR1, the received signal yi will equal to the originally transmitted signal xi. Therefore by this observation, the relationship between signals yif and yi is given by y = D−1 y f T where y = y0 · · · yN −1 , y f = y0f With (10), (11), and (7), we have

···

(11) T yNf −1 .

Fig. 3 Structure of D, with M = 2, L = 4 and N = 8

3.2 −1

−1

y = D GDx + D w

′

(12)

From (12), the received VB l, yl, can be expressed in the following form yl = S user x + S ICI xICI,l +wˆ l l l l user signal

(13)

ICI term

(14)

(15)

For better understanding of (13) and (15), the structure of and S ICI D −1GD, together with S user l l , is plotted in Fig. 3. Finally, for the sake of performance evaluation, the system SNR is defined as the SNR of the received signal. Without loss of generality, we assume that the sum of all paths’ 2 power is one, that is l sl = 1, and the power of the originally transmitted signal is also one, that is E|xi|2 = 1. Therefore the system SNR is expressed as SNR =

1/sw2

(16)

Note that the definition above is suitable for the V-OFDM system with either quasi-stationary fading channels or fast fading channels. IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

Lemma 2: The non-zero element’s index q, in row p of D in (10), satisfies q [ Y(p) = (p)L M + m, m = 0, 1, . . . , M − 1

ˆ l is an AWGN vector, having the same statistics properties w with wl in (3). wˆ l , S user and S ICI are an M × 1 column l l vector, an M × M matrix and an M × (N − M) matrix, respectively. They are given by wˆ l = D−1 w′ lM :lM +M −1 , S user = D−1 GD lM:lM +M −1,lM :lM +M −1 l S ICI = D−1 GD lM:lM +M −1,{0,...,lM −1} 1. The following theorem illustrates the crosscorrelation between two ICI signals. □ Theorem 2: In the case of m1 ≠ m2 or l1 ≠ l2, theICI signal at the m1th subcarrier of VB l1, denoted by yICI,l1 , is m1

n − n2 −j(2p/N )[(n1 −n2 )(l−l)] e J0 2pfd T 1 N n − n2 −j(2p/L)[(n′1 −n′2 )(l−l)] e J0 2pfd T 1 N

(36)

n − n2 L−1 −j(2p/L)[(n′1 −n′2 )(l−l)] × J0 2pfd T 1 l=0 e N =m+n′ M 2


www.ietdl.org uncorrelated with that at the m2th subcarrier of VB l2, that is

∗ yICI,l2 E yICI,l1 = 0. m1

l=l,i,j′ ,k,n ,n 1 2

′

with (5), one obtains (see (45)) Noting that M −1

yICI,l

= m

UH l

The correlation between expressed as

L−1 l=l

yICI,l

˜ G l,l Ul xl

+

=

and yICI,l

M −1

can be

m2

∗ Ex,h yICI,l m yICI,l m 1 2 ⎧ ⎛ ⎞∗ ⎫ ⎨ ⎬ L−1 L−1 ˜ l,l Ul xl ˜ l,l Ul xl ⎝ UH ⎠ G G = Ex,h U H l l ⎩ ⎭ l=l

= Ex,h =

UH l

UH l L−1 l=l

L−1 l=l

˜ l,l Ul xl U H G l

Ul

l=l

m1

L−1 l=l

H ˜ l,l Ul xl G

m1 ,m2

m1 ,m2

m1

M −1

=

l=l,i,j′ ,k

U ∗l i,m Eh 1

∗ ˜ ˜ G l,l i,j′ G l,l ′ U l k,m k,j

M, 0,

if n1 − n2 = n′ M else

(47)

ej(2p/M )[(m1 −n1 )i−(m2 −n2 )k ] ×

M −1

ej(2p/M )[n1 −n2 ]j = 0 ′

j′ =0

Therefore for l = l = l and m ≠ m , E 1 2 1 2 x,h yICI,l m1 ∗ yICI,l m } = 0. 2 Second, we consider the case where two subcarriers do not belong to the same VB, that is l1 ≠ l2. The correlation between and yICI,l2 can be expressed as yICI,l1 m1

Ex,h

∗ m2

in which n′1 , n′2 , n′ [ Z, 1 − L ≤ n′ ≤ L − 1, it can be deduced that one cannot find a pair (n1, n2) such that

yICI,l

′

j′ =0

with (31), one has Ex,h yICI,l

else

i=0,k=0

˜ l,l G ˜H Eh G l,l

0,

(46)

n1 = m1 + n′1 M , n2 = m2 + n′2 M

ej(2p/M )[n1 −n2 ]j =

m2

(43)

if

and

m1

M 2,

(42)

m

ej(2p/M )[(m1 −n1 )i−(m2 −n2 )k ]

i=0,k=0

First, consider the case where two subcarriers belong to the same VB. From (27), the ICI signal at the subcarrier m in VB l, [yICI,l]m, is given by

′

(44)

m2

∗ 1 n − n2 U l i,m U l k,m × 2 J0 2pfd T 1 1 2 N N

× e−j(2p/M )[n1 (i−j )−n2 (k−j )] e−j(2p/N )[(n1 −n2 )(l−l)]

m2

Proof: This theorem states that the ICI signals at different subcarriers are not correlated. The problem can be divided into two cases according to whether two subcarriers belong to the same VB. Our proof consists of two parts. The first part is to consider the case where two subcarriers belong to the same VB, that is l1 = l2 = l, m1 ≠ m2, and to prove that ∗ Ex,h yICI,l m yICI,l m = 0. The second part is to consider 1 2 the case where two subcarriers do not belong to the same VB, that is l1 ≠ l2, and to show that Ex,h yICI,l1 ∗ m1 yICI,l2 } = 0, no matter whether m1 = m2 holds or not.

=

m2

∗ yICI,l1 yICI,l2 m1

m2

⎧⎡ ⎤ ⎛⎡ ⎤ ⎞∗ ⎫ ⎪ ⎪ ⎬ ⎨ L−1 L−1 ⎜⎣ H ⎟ H ˜ ˜ ⎣ ⎦ ⎦ = Ex,h U l1 G l1 ,l Ul xl G l2 ,l Ul xl ⎝ U l2 ⎠ ⎪ ⎪ ⎭ ⎩ l=l l=l 1 2

2

m1

m2

∗ Ex,h yICI,l m yICI,l m 1

=

=

e

2

j(2p/N )(m1 −m2 )l

MN 2

l=l,i,j′ ,k,n ,n 1 2

n − n2 −j(2p/M )[n1 (i−j′ )−n2 (k−j′ )] −j(2p/N )[(n1 −n2 )(l−l)] e e e j(2p/M )(m1 i−m2 k ) · J0 2pfd T 1 N

e j(2p/N )(m1 −m2 )l n1 − n2 −j(2p/N )[(n1 −n2 )(l−l)] e J 2 p f T 0 d MN 2 N l=l,n1 ,n2

×

M −1

e

j(2p/M )[(m1 −n1 )i−(m2 −n2 )k ]

×

M −1

i=0,k=0


(45)

3

e

j(2p/M )[n1 −n2 ]j′

j′ =0

2329


www.ietdl.org ⎡

⎛

L−1 H

= ⎣Ex,h ⎝U l1 ⎡ =

⎣U H l1

L−1 l=l ,l 1 2

l=l 1

⎛

⎞H ⎞⎤

L−1 H

˜ l ,l Ul xl ⎝U l G 1 2

respectively. The first term T1 can be written as

˜ l ,l Ul xl ⎠ ⎠⎦ G 2

l=l 2

m1 ,m2

⎤

˜ l ,l G ˜H ⎦ Eh G l2 ,l U l2 1

l,i,k,n ,n ,n −n =n′ M 1 2 1 2

ej(2p/N )(m1 l1 −m2 l2 ) = MN 2

where 1 − L ≤ n’ ≤ L − 1. Since

L, if n′ = 0 , we have 0, else ej (2p/M ) m1 i − m2 k T1 =

j(2p/M )(m1 i−m2 k )

e

def

i,k,n1 =n2 = n

l=l1 ,l=l2 ,i,k,n1 ,n2

M −1

=

∗ yICI,l1 yICI,l2 m2

ej(2p/M )[(m1 −n)i−(m2 −n)k ]

ej(2p/M )(m1 i−m2 k )

n − n2 −j(2p/M )[n1 i−n2 k ] −j(2p/N )[n1 l1 −n2 l2 ] e e × J0 2pfd T 1 N × e j(2p/N )[n1 −n2 ]l × M ⎧ j(2p/N )(m1 l1 −m2 l2 ) ⎨ e = ej(2p/M )(m1 i−m2 k ) ⎩ N2 ′ l,i,k,n1 ,n2 ,n1 −n2 =n M

n − n2 −j(2p/M )[n1 i−n2 k ] e × J0 2pfd T 1 N × e−j(2p/N )[n1 l1 −n2 l2 ] ej(2p/N )[n1 −n2 ]l l=l ,i,k,n ,n ,n −n =n′ M 1 1 2 1 2

n − n2 ej(2p/M )(m1 i−m2 k ) J0 2pfd T 1 N

× e−j(2p/M )[n1 i−n2 k ] e−j(2p/N )[n1 l1 −n2 l2 ] ej(2p/N )[n1 −n2 ]l n − n2 ×− ej(2p/M )(m1 i−m2 k ) J0 2pfd T 1 N ′

l=l2 ,i,k,n1 ,n2 ,n1 −n2 =n M

× e−j(2p/M )[n1 i−n2 k ] e−j(2p/N )[n1 l1 −n2 l2 ] ej(2p/N )[n1 −n2 ]l

j(2p/N )(m1 l1 −m2 l2 ) def e = T1 − T2 − T3 2 N

(49) In the following, we derive T1, T2, and T3 in above equation, 2330 & The Institution of Engineering and Technology 2014

M 2, 0,

if n = m1 + n′1 M , n = m2 + n′2 M else

(52)

in which n′1 , n′2 [ Z, it follows that for m1 ≠ m2, M −1 j(2p/M )[(m1 −n)i−(m2 −n)k ] = 0 and T1 = 0; Otherwise, i,k=0 e def let m1 = m2 = m, and T1 can be expressed as

l=l1 ,l=l2 ,i,k,n1 ,n2 ,n1 −n2 =n′ M

×1

i,k=0

with (47), (48) is derived as

′

ej(2p/L)n l =

Noting

+

ej(2p/N )(m1 l1 −m2 l2 ) MN 2

l=0

(51)

(48)

m1

L−1

i,k

j(2p/M )(m1 i−m2 k )

j′

−

l=0

′

ej(2p/L)n l

× e−j(2p/M )n[i−k ] e−j(2p/N )n[l1 −l2 ] N −1 j(2p/M ) (m −n)i−(m −n)k ] [ 1 2 × L = L n=0 e−j(2p/N )n[l1 −l2 ] · e

n − n2 × e−j(2p/M )[n1 i−n2 k ] × e J0 2pfd T 1 N 3 −j(2p/M ) n −n j′ −j(2p/N )[n1 (l1 −l)−n2 (l2 −l)] ] [ 1 2 × e ×e

=

L−1

(50)

n − n2 −j(2p/M )[n1 (i−j′ )−n2 (k−j′ )] e × J0 2pfd T 1 N × e−j(2p/N )[n1 (l1 −l)−n2 (l2 −l)]

Ex,h

n − n2 ej(2p/M )(m1 i−m2 k ) J0 2pfd T 1 N

× e−j(2p/M )[n1 i−n2 k ] e−j(2p/N )[n1 l1 −n2 l2 ] ×

l=l1 ,l=l2 ,i,j′ ,k,n1 ,n2

m1 ,m2

ej(2p/N )(m1 l1 −m2 l2 ) = MN 2

T1 =

T1 =

ej(2p/M )(m1 i−m2 k ) e−j(2p/M )n[i−k ] def

i,k,n1 =n2 = n

e−j(2p/N )n[l1 −l2 ] × L N −1 −j(2p/N )n l −l [ 1 2] × M 2 =L× e n=0

(53)

−1 −j(2p/N )n[l1 −l2 ] = 0, and T1 is e Since l1 ≠ l2, it follows Nn=0 also zero. To sum up, T1 = 0 for l1 ≠ l2, whether m1 = m2 or not. Using the similar logic in deriving T1, it is easy to verify T 2 = T3 = 0. Therefore with (49), for the second case, Ex,h yICI, l1 m 1 ∗ yICI, l2 m 2 Finally, combining the results of the two cases, it is concluded that the ICI signals at different subcarriers are uncorrelated and Theorem 2 has been proved. □

5

The proposed MMSE detection method

In [9], two linear detection methods that include ZF and MMSE detection methods are proposed and studied. The MMSE detection utilises the SNR information and therefore has a better BER performance than the ZF detection. However, when the ICI effect cannot be ignored, only considering the AWGN results in unmatched SNR. In fact, it is found that at a given normalised Doppler shift, a BER floor will occur in the simulation, which is because of the fact that the power of the ICI signal is not incorporated in IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

www.ietdl.org MMSE detection. This motivates us to propose a novel MMSE detection method for V-OFDM systems over fast fading channels. 5.1

Proposed MMSE detction

To start with, recall that the MMSE detection method is characterised by an MMSE weight matrix of the received signal in frequency domain and the matrix is given by [9] −1 −1 H C MMSE = HH Hl , l l H l + SNR I l = 0, . . . , L − 1

(54)

where Hl and SNR are defined in (4) and (16), respectively. Note that the MMSE detection only uses the SNR information while not taking the ICI effect into consideration. Then, with (13), we define an equivalent noise as wS =

S ICI l xICI,l

+ wˆ l

(55)

The correlation matrix of the noise above, RwS wH , is utilised in S

the proposed MMSE detection method, and the MMSE corresponding MMSE weight matrix, C˜ l , is designed as

−1 MMSE = HH HH C˜ l l H l + RwS wH l

h n, t1 · · · h n, tv ]T , depends on each sample time n. In this scenario, we use the time average of h(n) within one data block to calculate the channel frequency response {Hk}, that is {Hk, k = 0, …, N − 1} = FFTN{(1/N) ∑nh(n, tl), l = 0, …, v}. Furthermore, in the simulation, we assume that the channel estimation is ideal, which means that the average of h(n) is perfectly known whereas h(n) in each time n is not required.

5.2

Detection MSE evaluation

MSE plays an important role in the system design, which can show how accurately a signal can be recovered at the receiver. Given a detection method, that is the conventional method in (54) or the proposed one in (58), the corresponding data detection MSE for VB l can be defined as 2 MSEl = Ex,h C l yl − xl where C l = C˜ l in (27), we have

MMSE

(59)

C MMSE . With the received signal l

or

MSEl

H H˜ ˜ = Eh Tr C l U H l G l,l U l − I C l U l G l,l U l − I

(56)

+

S

L−1 l=l

˜ ClU H l G l,l U l

H ˜ ClU H l G l,l U l

+

s2w I

H H˜ ˜ = Eh Tr C l U H l G l,l U l − I C l U l G l,l U l − I 3 L−1 H H H 2 H ˜ ˜ G l,l G ll,ll U l C l + sw C l C l +C l U l

Clearly, the proposed MMSE detection method incorporates the correlation of the ICI signal. As in Theorem 2, the ICI signals at different subcarriers are uncorrelated. Meanwhile it is easy to derive H ˆ l SICI ˆl RwS wˆ H = Ex,wˆ S ICI l xICI,l w l xICI,l + w

l=l

S

= R ICI ICI H RxICI,l xH sl

=

sl

ICI,l

+ PICI (l, m)I = 2 = sw + PICI (0, 0) I

s2w I

(60)

+ Rwˆ l wˆ H l

s2w

+ PICI (l, m) I, ∀l, m

Therefore the average detection MSE over all VBs is given by MSE =

(57) where Ex,wˆ (.) denotes the statistical expectation with respect to the originally modulated signal x and AWGN noise; R ICI ICI H , RxICI,l xH and Rwˆ l wˆ H denote the autocorrelation sl

sl

ICI,l

l

ˆ l , respectively. Therefore the matrices of S ICI l , xICI,l and w weight matrix for the proposed MMSE detection method is finally written as 2 −1 H MMSE C˜ l = HH Hl l H l + sw + PICI (0, 0) I

(58)

in which PICI(0, 0) is given in Theorem 1. Note that in the case that the channel is quasi-static, we have PICI(0, 0) = 0 and (58) becomes (54). Remark: In the case of V-OFDM with quasi-static fading channels, Hl in (54) or (58) is obtained from the channel frequency response, {Hk}, which FFT is the N-point of channel impulse response h = h t0 h t1 · · · h tv ]T . However, for V-OFDM over fast fading channels, the channel impulse response, denoted by h(n) = h n, t0 IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

·

3

ClCH l

L−1 1 MSEl L l=0

(61)

Since an analytical expression for detection MSE is hard to solve, we use numerical method to evaluate it. For one V-OFDM data block, its MSE evaluation process can be summarised as the following steps. Step 1: Given a normalised Doppler shift, generate the channel impulse response h(n, tl). ˜ l,l with its entry Step 2: Obtain the matrix G by (8), and then G ˜ l,l = Gm1 L+l,m2 L+l . G m ,m 1

2

Step 3: Calculate the detection matrix Cl from (54) or (58), which represent the conventional MMSE detection and proposed one, respectively. Step 4: Calculate the average MSE by (60) and (61).

6

Numerical and simulation results

Both numerical method and computer simulation are used to investigate the performance of V-OFDM systems over fast fading channels. The bandwidth of the system is 2.56 MHz. We employ a multipath Rayleigh fading channel in the 2331


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Fig. 5 ICI signal power at each subcarrier with a normalised Doppler shift and b number of VBs.

simulation. The channel has six paths with an equal power profile and a delay profile of [0 1 2 3 4 5] in unit of sampling point. The Doppler spectrum of each path is the classical U-shaped spectrum of Clarke [22]. One V-OFDM data block has 128 subcarriers with its CP length being 10 sampling points. Thus, one V-OFDM data block excluding the CP, denoted by T, is 0.05 ms. In addition, the transmitted signal is QPSK modulated. 6.1 The power of ICI signal with normalised Doppler shift fdT and number of subcarriers N Fig. 5 shows the ICI signal power at each subcarrier with various normalised Doppler shifts and Ls . The ICI signal power at each subcarrier is calculated from Theorem 1, by numerical method. In Fig. 5a, four cases with the number of VBs L = 2, 4, 8 and 128 are plotted for comparison. Note that the V-OFDM system with N = L = 128 is actually an OFDM system. We observe that the ICI signal power increases with fdT; given fdT, larger L yields larger ICI signal power. For example, given fdT = 0.05, the ICI power is −25.1 dB at L = 2 and it increases to −23.9 dB at L = 8. Thus, the results imply that smaller L or larger M can

Fig. 6 BER performance of the V-OFDM system with ZF and MMSE detection methods, over quasi-static fading channels 2332 & The Institution of Engineering and Technology 2014

reduce the ICI signal power, which is consistent with Corollary 1. In Fig. 5b, we set fdT = 0.07. It can be seen that the ICI power reaches an asymptotic level of about −20.9 dB at L ≥ 8. Since computing PICI(l, m) with large L is time consuming, the result indicates that PICI(l, m) can be calculated accurately enough by smaller L. 6.2 The detection MSE and BER performance for the conventional MMSE and proposed MMSE detection methods Fig. 6 shows the BER performance of the V-OFDM system with ZF and MMSE detectors, under quasi-static fading channels. We also plot the BER performance of the conventional OFDM system for comparison. Observe that the curves with ZF detectors almost overlap with that of the conventional OFDM, except that the BERs of ZF detectors are a little worse than that of the conventional OFDM at low SNR. The above results are also consistent with that in Fig. 10 [9]. On the other hand, the BER performance of the V-OFDM system with MMSE detection is superior to the conventional OFDM. The trend is clearer with increasing

Fig. 7 BER performance of the V-OFDM system with ZF and MMSE detection, over fast fading channels IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

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Fig. 8 Comparisons between the conventional MMSE and proposed MMSE a Detection MSE; b BER

SNR and VB size M, which is because of the fact that the diversity order increases with the VB size M. However, for M ≥ 32, further increasing M cannot bring more performance gain, which is because of the fact that the maximum diversity order is min(M/2−2, 5) + 1 = 6 according to [9], and M = 32 or 64 has the same diversity order 6. Fig. 7 depicts the BER performance of the V-OFDM system with various detectors, under fast fading channels. The Doppler shift is 1200 Hz so that the normalised Doppler shift is 0.06. The following results can be observed from this figure. First, the V-OFDM with ZF detection is always worse than the conventional OFDM. Increasing M just makes the system even worse, especially at low SNR. Second, the V-OFDM with MMSE detection is superior to the conventional OFDM. Since larger M or smaller L can reduce the ICI, we find that, up to M = 32, increasing M can bring performance gain. Also because of the ICI, the BER cannot decrease significantly in the case of SNR > 26 dB. Fig. 8 compares the detection MSE and BER performance of the conventional MMSE and proposed MMSE with different Ms and normalised Doppler shifts. The detection MSE is calculated by the four steps in the previous section. The following results can be observed from this figure. First, the detection MSE of the proposed detection is less than that of the conventional one, especially at high SNR; at a given SNR and M, the MSE increases with the normalised Doppler shift. Second, for the conventional method, at about SNR = 26 dB, an MSE floor occurs, which is consistent with its BER performance in Fig. 7. Third, from the Sub Figure 8b, the proposed MMSE with different Ms outperforms the conventional MMSE, especially at high SNR. With larger M, the performance advantage is more obvious. Fourth, we also see the BER performance is basically consistent with the corresponding MSE performance in Fig. 8a. Therefore we conclude that the proposed method always outperforms the conventional MMSE detection within a wide normalised Doppler shift range, in terms of both BER and MSE.

7

Conclusions

In this paper, the effect of ICI resulting from fast fading channels on the V-OFDM system was thoroughly studied. We derived the expression of the received signal IET Commun., 2014, Vol. 8, Iss. 13, pp. 2322–2335 doi: 10.1049/iet-com.2013.1064

consisting of ICI by introducing an equivalent V-OFDM system model. A transform matrix was also introduced and its main traits were characterised by two lemmas. With the aid of it, the properties of the ICI signal consisting of its power and correlation were analysed, and the corresponding results were presented by two theorems and a corollary. Based on the two theorems, a novel MMSE detection incorporating the ICI correlation matrix was therefore proposed. Both numerical method and simulation were adopted to verify the ICI analysis and investigate the performance of V-OFDM with various system configurations. The results showed the followings. First, it was demonstrated that the ICI power increases with VB number L by numerical method and hence Corollary 1 was verified. Second, for the ZF detection, its BER performance with different Ms is marginally the same with that of the conventional OFDM. Third, for the conventional MMSE detection, at high SNR, increasing SNR cannot improves the system performance because of the ICI. Fourth, the proposed MMSE detection can eliminate the ICI effect to some degree and it always outperforms the conventional MMSE detection in terms of both BER and detection MSE.

8

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61162008 and 61372129), Guangdong Province Natural Science Fund (No. S2012040006342, S2012010010062, and S2013040016857), the Open Research Fund of the National Mobile Communications Research Laboratory, Southeast University (No. 2013D04), and the Training Program of the Outstanding Young Teachers in Higher Education Institutions of Guangdong Province (No. Yq2013070). The author would also like to thank the anonymous reviewers for their valuable efforts in improving the quality of this paper.

9

References

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www.ietdl.org 2 Chen, L., Chen, W., Wang, B., et al.: ‘System-level simulation methodology and platform for mobile cellular systems’, IEEE Commun. Mag., 2011, 49, (7), pp. 148–155 3 Oguma, H., Kameda, S., Takagi, T., et al.: ‘Uplink throughput performance of FH-OFDMA improved by 16 QAM: effect estimation and validation in MBWA system field trial’. Proc. IEEE PIMRC, 2009, pp. 2050–2054 4 Riihimäki, V., Värääki, T., Vartiainen, J., et al.: ‘Techno-economical inspection of high-speed internet connection for trains’, IET Intell. Transp. Syst., 2008, 2, (1), pp. 27–37 5 Miridakis, N.I., Vergados, D.D., Papadakis, E.: ‘A receiver-centric OFCDM approach with subcarrier grouping’, IEEE Commun. Lett., 2012, 16, (6), pp. 761–764 6 Zhou, Y., Ng, T.S., Wang, J., et al.: ‘OFCDM: a promising broadband wireless access technique’, IEEE Commun. Mag., 2008, 46, (3), pp. 38–49 7 Xia, X.-G.: ‘Precoded and vector OFDM robust to channel spectral nulls and with reduced cyclic prefix length in single transmit antenna systems’, IEEE Trans. Commun., 2001, 49, (8), pp. 1363–1374 8 Zhang, J., Luo, L., Shi, Z.: ‘Quadtrature OFDMA systems based on layered FFT structure’, IEEE Trans. Commun., 2009, 57, (3), pp. 850–860 9 Li, Y., Ngebani, I., Xia, X.-G., et al.: ‘On performance of vector OFDM with linear receivers’, IEEE Trans. Signal Process., 2012, 60, (10), pp. 5268–5280 10 Zhang, H., Xia, X.-G.: ‘Iterative decoding and demodulation for single-antenna vector OFDM systems’, IEEE Trans. Veh. Technol., 2006, 55, (4), pp. 1447–1454 11 Zhang, H., Xia, X.-G., Cimini, L.J., et al.: ‘Synchronization techniques and guard-band configuration scheme for single-antenna vector-OFDM systems’, IEEE Trans. Wirel. Commun., 2005, 4, (5), pp. 2454–2464 12 Zhang, H., Xia, X.-G.: ‘A guard band configuration scheme for single-antenna vector OFDM systems’. Proc. ACSSC, 2004, pp. 827–831 13 Cheng, P., Tao, M., Xiao, Y., et al. : ‘V-OFDM: On performance limits over multi-path Rayleigh fading channels’, IEEE Trans. Commun., 2011, 59, (7), pp. 1878–1892 14 Li, Y.: ‘On mathematical equivalence between vector OFDM and quadrature OFDMA’, IEEE Trans. Commun., 2013, 61, (2), pp. 813–814 15 Zhou, G., Pingzhi, F., Hao, L.: ‘Frequency-domain scrambling differential detection and equalization for DFT scrambling vector OFDM system’. Proc. IEEE VTC, 2012, pp. 1–5 16 Han, C., Hashimoto, T.: ‘Performance analysis of constellation rotated vector OFDM over fast fading channel’. Proc. IEEE WCNC, 2012, pp. 97–102 17 Han, C., Hashimoto, T., Suehiro, N.: ‘Constellation-rotated vector OFDM and its performance analysis over rayleigh fading channels’, IEEE Trans. Commun., 2010, 58, (3), pp. 828–838 18 Song, W.-G., Lim, J.-T.: ‘Channel estimation and signal detection for MIMO-OFDM with time varying channels’, IEEE Commun. Lett., 2006, 10, (7), pp. 540–542 19 Seyedi, A., Saulnier, G.: ‘General ICI self-cancellation scheme for OFDM systems’, IEEE Trans. Veh. Technol., 2005, 54, (1), pp. 198–210 20 Mostofi, Y., Cox, D.C.: ‘ICI mitigation for pilot-aided OFDM mobile systems’, IEEE Trans. Wirel. Commun., 2005, 4, (2), pp. 765–774 21 Wu, H.-C., Huang, X., Xu, D.: ‘Novel semi-blind ICI equalization algorithm for wireless OFDM systems’, IEEE Trans. Broadcasting, 2006, 52, (2), pp. 211–218 22 Rappaport, T.S.: ‘Wireless communications principles and practice’ (Publishing House of Electronics Industry, 2002)

The signal xi is the IFFT of xi and it can be written as

xkM +m =

9.1

Appendix

(63) Substituting (63) into (62), we have

xfk =

xkf =

xi e−j2pki/N ,

k = 0, . . . , N − 1

i=0

2334 & The Institution of Engineering and Technology 2014

(64)

1

2

Observe that i1 ej2pi1 (i2 −k)/L is L only if (i2 − k) is an integer multiple of L; otherwise it is zero. Moreover, since i2 ∈ [0, M − 1], we have i2 = (k)L, in which (k)L denotes k modulo L. Therefore with such an observation, (64) can be further derived as

xfk = =

−1 1M x e−j2pkm/N × L L m=0 (k)L M +m M −1

xf(k)L M +m e−j2pkm/N

(65)

m=0

Equation (65) can be written in the vector form, that is, x f = Dx

(66)

T T where x = x0 · · · xN −1 , xf = xf0 xf1 · · · xfN −1 , and D is an N by N matrix whose ( p, q)th entry is e−j2pmp/N , for m = q − (p)L M , m [ {0, . . . , M − 1} 0, otherwise (67)

Or equivalently,

D p,q =

(62)

i1 =0 m=0

xi1 M +m e−j2pk(i1 M +m)/N

L−1 M −1 L−1 1 x ej2pi1 i2 /L e−j2pk(i1 M +m)/N L i =0 m=0 i =0 i2 M +m 1 2 M −1 L −1 j2pi (i −k)/L 1 −j2pkm/N 1 2 = x e × e L m=0 i =0 i2 M +m i

Appendix 1

N −1

L−1 M −1

=

D p,q =

This appendix presents the relationship between the signals T xi and xif in Fig. 2. First, let x W x0 . . . xN −1 , and the relationship between xif and xi is

k [ {0, . . . , L − 1},

m [ {0, . . . , M − 1}

9

L−1 1 x ej(2p/L)ik , L i=0 iM +m

e−j2p(q)M p/N , for m = q − (p)L M , m [ {0, ..., M − 1} 0, otherwise (68)


www.ietdl.org 9.2

Appendix 2

This appendix gives a proof of Lemma 1. Let dj be the jth column of D. From the definition of D in (68), we have d Hj1 d j2 =

N −1

D∗k,j1 Dk,j2

k=0

=

N −1

9.3 ej(2p/N )m1 k e−j(2p/N )m2 k

k=0

=

⎧ N −1 ⎨ ⎩ k=0 0,

ej(2p/N )[ j1 −j2 ]k , if m1 and m2 [ {0, . . . , M − 1} else (69)

where m1 = j1 − (k)LM, m2 = j2 − (k)LM. Then, we need to find all k’s satisfying the condition: m1 and m2 ∈ {0, …, M − 1}. For one thing, when j1 and j2 belong to the same VB, that is there exists some integer i0 for which j1, j2 ∈ {i0M, …, i0M + M − 1}, choosing k = i0 + i’ L can make the condition hold, where i’ = 0,…, M − 1. With such k’s, (69) can be derived as d Hj1 d j2 =

M −1

ej(2p/N )( j1 −j2 )(i0 +i L) ′

(70)

i′ =0

Observe that in the case of j1 = j2, it follows d Hj1 d j2 = M ; ′ otherwise, d H d = ej(2p/N )[ j1 −j2 ]i0 × M′ −1 ej(2p/M )[ j1 −j2 ]i = 0. j1

On the other hand, when j1 and j2 do not belong to the same VB, that is there does not exist an integer i0 for which j1, j2 ∈ {i0M, …, i0M + M − 1}, we cannot find a k such that m1 and m2 ∈ {0, …, M − 1}. In this case, d Hj1 d j2 = 0. √ To sum up, if j1 ≠ j2, d Hj1 d j2 = 0; otherwise ||d j ||F = M . Hence Lemma 1 has been proved.

j2

i =0


Appendix 3

This appendix proves Lemma 2. First, we study the property of non-zero elements in a row of D. With (67), for row p, Dp, q ≠ 0 only if q = m + ( p)LM, m ∈ {0, …, M − 1}. Therefore it is straightforward that condition (17) holds. Second, for column q of D, according to (68), a non-zero element’s index should satisfy ( p)L = (q − m)/M

(71)

Generally, the integer q can be written as q = int(q/M )M + m0

(72)

where m0 ∈ {0, …, M − 1}. Furthermore, as (q − m)/M should be an integer, it follows m = m0. Hence (71) becomes ( p)L = int(q/M), or equivalently, p = int(q/M ) + nL,

n = 0, . . . , M − 1

(73)

Therefore condition (18) has been proved.

2335
