A Simple ICI Suppression Method
Utilizing Cyclic Prefix for OFDM Systems in the
Presence of Phase Noise
Chun-Ying Ma, Student Member, IEEE, Chun-Yen Wu, and Chia-Chi Huang
Abstract—An extremely low-complexity inter-carrier
interfer-ence (ICI) suppression method for OFDM systems in the presinterfer-ence of phase noise is proposed in this paper. The core idea is to utilize the ISI-free part of cyclic prefix (CP), which is simply ignored in traditional methods. We linearly combine the ISI-free samples in CP with the corresponding samples in the OFDM symbol to suppress ICI, and this scheme is called ‘CP combining’ through-out this paper. In addition, the optimum combining coefficients, in the sense of ICI power minimization, are derived, and a set of near-optimum coefficients are proposed to reduce the complexity. Simulation results show that the proposed CP combining method improves the system performance by 0.5 ´ 1.5dB. Furthermore, the proposed method can be incorporated with other phase noise mitigation methods to further improve the system performance.
Index Terms—Phase noise, ICI suppression, CP combining,
low-complexity, windowing, 60GHz.
I. INTRODUCTION
O
RTHOGONAL Frequency Division Multiplexing(OFDM) is the key modulation technique widely used in modern and next-generation wireless communication systems, such as WiMAX, LTE-Advanced, and wireless local area networks, mainly for three reasons. First, it can effectively combat the inter-symbol interference (ISI) caused by a multipath channel by dividing a wideband frequency-selective channel into many narrowband frequency-flat subchannels. Second, the receiver is simple since it requires only one Fast Fourier Transform (FFT) chip and a one-tap equalizer. Last but not least, power and bit loading algorithms can be employed to further increase the power and spectral efficiency.
However, phase noise, which is caused by the imperfection of the local oscillator, is detrimental to OFDM systems. Phase noise causes not only common phase error (CPE), which rotates the received signal, but also inter-carrier interference (ICI), which destroys the orthogonality among subcarriers. The effect of phase noise has been analyzed in [1]–[7], and it is shown from these works that phase noise is a critical
Manuscript received March 13, 2013; revised July 16 and September 5, 2013. The editor coordinating the review of this paper and approving it for publication was R. Dinis.
The authors are with the Inst. of Communications Engineering, National Chiao Tung University. No. 1001 Ta Hsueh Rd., Hsinchu, Taiwan 300, R.O.C. (e-mail: ma chun ying.cm93@g2.nctu.edu.tw, andy1988.tw@yahoo.com.tw, huangcc@cc.nctu.edu.tw).
This work was supported by National Science Council of the Republic of China (Taiwan) under grant NSC 100-2220-E-009-025.
Digital Object Identifier 10.1109/TCOMM.2013.091513.130197
issue on system performance. Consequently, many phase noise mitigation methods have been proposed in literature e.g., [8]– [22]. In [8], [9], simple methods for CPE estimation and phase noise suppression were proposed. The authors in [10] proposed a low-complexity ICI mitigation method by dividing the entire OFDM symbol into a number of subblocks. In [11]–[14], [17], [18], joint estimation algorithms for channel impulse response, frequency offset, and phase noise were proposed. In [12], the authors reduced the complexity of ICI mitigation algorithms by treating the time-domain phase noise as static within K samples. In [13]–[16], the authors reduced the overall complexity by interpolating the time-domain phase noise process. Through approximating phase noise by a band-limited process, Petrovic at el. proposed a method to reduce the complexity of phase noise parameter estimation in [19]. In [20], the phase noise process was modeled as a power series to reduce the complexity in parameter estimation. A low-complexity method incorporating error correcting codes and adaptive algorithms to track phase noise was proposed in [21]. Finally, a blind compensation method was proposed in [22]. However, since it only works for constant modulus modulation, this method is limited in its applications.
In this paper, we concentrate on a different scheme: the cyclic prefix (CP) recycling scheme. The CP recycling scheme is motivated by the fact that in some situations the specified CP length is much longer than the delay spread and there are a considerable number of ISI-free samples in CP. Con-ventionally, these ISI-free samples are simply discarded, and apparently it is a waste of resources. Therefore, the core idea of this scheme is to design a method to recycle the ISI-free samples in CP to improve the system performance. Originally, the CP recycling scheme was used to maximize signal-to-noise ratio (SNR) in the presence of frequency offset in [23]– [26], and it was also applied to suppress the ICI caused by time-varying channels in [27]–[32]. Recently, in [33], [34], Tchamov et al. directly applied the combining weights of [28] to suppress the phase noise effect.
In this paper, the CP recycling scheme is applied to suppress the ICI power incurred by phase noise. Although this idea has already been used in [33], [34], nevertheless, the heuristic combining weights adopted in [33], [34] are apparently not optimum in the sense of ICI minimization. To the best of our knowledge, the optimum combining weights have not been investigated. In this paper, we derive the optimum combining weights and provide a set of near-optimum combining weights.
Fig. 1. Block diagram of the OFDM transceiver system model in the presence of phase noise.
Moreover, it should be noted that the proposed method can be incorporated with other phase noise mitigation methods since the output of the proposed method can be treated as a
cleaner (or less ICI-polluted) input to the traditional phase
noise mitigation methods, e.g., [8]–[14], [17]–[22].
The contributions of this paper are organized as follows: (1) we derive the optimum combining weights in the sense of ICI power minimization; (2) we propose a set of near-optimum combining weights to reduce complexity; (3) through simula-tions, we show that the proposed method, whose complexity is extremely low, provides about 0.5 to 1.5 dB gain in BER performance; (4) we systematically analyze the relationship between ICI power level and the length of CP.
Throughout this paper, we use the following notations. Boldface uppercase and lowercase letters denote matrices and vectors, respectively. The superscript p¨qT and p¨qH denote
transpose and Hermitian of a matrix or vector, respectively. The superscriptp¨q˚is the conjugate of a complex variable. We usex „ CN pμ, Σq to represent that x is a complex Gaussian distributed random vector with meanμ and covariance matrix Σ. FN is the N ˆ N discrete Fourier transform (DFT) matrix
whose pm, nqth entry is given by ?1
Nexpp´j 2πmn
N q, with
m, n P t0, 1, ..., N ´ 1u. We define diagtxu as a diagonal
matrix with vectorx on its diagonal. 1 is a column vector with appropriate dimension containing all ones. We use OMˆN to
represent an all-zero matrix with dimension M ˆ N . Finally, trp¨q denotes trace operation.
The rest of the paper is organized as follows. In Section II, the OFDM signal model and the phase noise model are presented. The proposed method is described in Section III. Subsequently, the derivations of the optimum and the near-optimum combining coefficients are presented in Section IV. Finally, simulation results and conclusion are given in Section V and VI, respectively.
II. SYSTEMMODEL
A. OFDM Signal Model
The baseband system model of an OFDM system in the presence of phase noise is illustrated in Fig. 1. We use
s ““sp0q sp1q ¨ ¨ ¨ spN ´ 1q‰T (1) to represent the transmit information symbol vector, where
N is the OFDM symbol length. Each entry of s is
in-dependently drawn from a finite alphabet A, i.e., spkq P
A, @k P r0, N ´ 1s. Without loss of generality, we
normalize the average transmit power to unity; hence, E“ssH‰ “ I
N. To generate an ordinary OFDM symbol,
the information symbol vector s is transformed into time
domain via an N -point Inverse Fast Fourier Transform (IFFT) operation, and then a CP with size Ng is inserted to
prevent ISI. The time-domain baseband transmitter output x fi “xp´Ngq ¨ ¨ ¨ xp´1q xp0q ¨ ¨ ¨ xpN ´ 1q
‰T
can be mathematically described as
x “ TcpFHNs, (2)
whereTcp is the CP-inserting matrix defined as
Tcp“ “ IT cp ITN ‰T (3) andIcpis the matrix collecting the last Ngrows of the identity
matrixIN. The transmitted signalx is then passed through the
multipath channel whose channel impulse responseh is math-ematically defined as“hp0q ¨ ¨ ¨ hpL ´ 1q‰T, where L ´ 1 is the maximum path delay. We assume that the CP length is longer than the maximum multipath delay, that is, Ngą L´1.
We define ˘y fi“˘yp´Ngq ¨ ¨ ¨ ˘ypN ´ 1q
‰T
as the convolution of the transmitted signal and channel impulse response. Then, the received signal˘y is corrupted by phase noise and Additive White Gaussian Noise (AWGN). The time-domain noise-corrupted received signal y fi “yp´Ngq ¨ ¨ ¨ ypN ´ 1q
‰T is given by y “ ˘D˘y ` ˘ε, (4) where ˘ D fi diagt“epjϕp´Ngqq ¨ ¨ ¨ epjϕpN´1qq‰Tu (5) is the matrix corresponding to the phase noise effect, ϕpnq is the phase noise at sample time n, and ˘ε „ CN p0, σ2IpN`Ngqq is the time-domain AWGN.
At the receiver side, the first L ´ 1 samples of the received signal, the so-called ISI-polluted samples, are dropped, and the remaining signal r fi “rp´Ng` L ´ 1q ¨ ¨ ¨ rpN ´ 1q
‰T
, whose dimension ispN ` Ng´ L ` 1q ˆ 1, is given by
r fi“OpN`Ng´L`1qˆpL´1q IpN`Ng´L`1q ‰
y. (6)
Therefore, the received signalr can be represented as
r “ DHx ` ε, (7) whereD fi diagt“epjϕp´Ng`L´1qq ¨ ¨ ¨ epjϕpN´1qq‰Tu, H “ ¨ ˚ ˚ ˚ ˚ ˝ hpL ´ 1q ¨ ¨ ¨ hp0q 0 ¨ ¨ ¨ 0 0 hpL ´ 1q ¨ ¨ ¨ hp0q . .. ... .. . . .. . .. . .. . .. 0 0 ¨ ¨ ¨ 0 hpL ´ 1q ¨ ¨ ¨ hp0q ˛ ‹ ‹ ‹ ‹ ‚, (8) and ε is the vector consisting of the last N ` Ng´ L ` 1
elements of˘ε.
B. Phase Noise Model
Phase noise is the random perturbation caused by an imper-fect local oscillator. For a local oscillator with center frequency
fc, the oscillator output at time instant t can be written as
exptjp2πfct ` ϕptqqu, where ϕptq is the phase noise at time
instant t. In this paper, the phase noise model recommended by IEEE 802.11ad task group [35] is used. The power spectrum
density (PSD) of the phase noise is a one-pole/one-zero model given as Sϕpfq “ K01 ` pf{f zq2 1 ` pf{fpq2 , (9) where fzand fpare the zero and pole frequency, respectively.
In [35], the default value of each parameter is: fz“ 100MHz,
fp“ 1MHz, and K0“ ´90dBc/Hz.
By definition, the auto-correlation function of the phase noise is the inverse Fourier transform of Sϕpfq, i.e.,
ρpτ q fi Erϕptqϕpt ` τ qs “ F´1tSϕpfqu, (10) and it is derived in [21] as ρpτ q “ K0f 2 p fz2 δpτ q ` K0πfp ˜ 1 ´fp2 fz2 ¸ e´2πfp|τ| . (11) III. THEPROPOSEDMETHOD
A. Cyclic Prefix Combining
For convenience, we define q “ Ng´ L ` 1. Then, the last
q received samples in CP, namely rp´qq, ¨ ¨ ¨ , rp´1q, are the
so-called ISI-free samples. Since these samples are originated from the last q samples of the OFDM symbol, they can be linearly combined with their corresponding OFDM samples to improve the performance. The combined signal vtpnq can
be described as vtpnq “ " rpnq , if 0 ď n ď N ´ q ´ 1 μnrpnq ` ϑnrpn ´ N q , if N ´ q ď n ď N ´ 1 (12) where tμnuNn“N´q´1 andtϑnuNn“N´q´1 are the combining
coef-ficients which have the following relationship
μn “ 1 ´ ϑn, @n P rN ´ q, N ´ 1s. (13)
The constraint (13), the so-called Nyquist constraint [24], is made to ensure the orthogonality when there is no phase noise. After the CP combining, the combined signal vt fi
rvtp0q, ¨ ¨ ¨ , vtpN ´1qsT is transformed into frequency domain
via an FFT operation. As a result, the frequency-domain combined signal v fi rvp0q, ¨ ¨ ¨ , vpN ´ 1qsT can be written
as
v “ FNvt. (14)
The CP combining operation is illustrated in Fig. 2. It should be noted that since only q samples are utilized to combat phase noise, the proposed method can have a significant gain only when q is a significant fraction of N .
B. The Design Goal and The Solutions
The combined signalv can be expressed as
v “ ˜Ip0qΛs ` b ` z, (15) where ˜Ip0q “1 N ˜N´q´1 ÿ m“0 ejϕpmq` Nÿ´1 n“N´q pμnejϕpnq` ϑnejϕpn´Nqq ¸ , (16)
Fig. 2. Implementation of the CP combining.
is the effective common phase error (CPE) which
rotates the frequency-domain received signal,
Λ “ diagtrΛp0q, ¨ ¨ ¨ , ΛpN ´ 1qsTu represents the channel
frequency response,b denotes the ICI term, and z represents the noise term. One of the major contributions of this paper is to find the optimum combining coefficients tμnuNn“N´q´1 and
tϑnuNn“N´q´1 that suppress the ICI power; the design goal can
be mathematically described by the following optimization problem: min tμnu,tϑnu ErbHbs s.t. μn“ 1 ´ ϑn, @n P rN ´ q, N ´ 1s. (17) The optimum solution of (17) will be derived in Section IV. Furthermore, in order to reduce the computational complexity, a set of near-optimum combining coefficients are also derived in Section IV. The set of near-optimum combining coefficients can be explicitly written as
" μ‹n “ α ` pN ´ 1 ´ nqβ ϑ‹n “ 1 ´ μ‹n , @n P rN ´ q, N ´ 1s, (18) where β “ 2πfpTs´ 2πNf 2 pTs2 1 ´ e´2πfpN Ts´ 2π 3NfpTs, (19) α “ 12 ´12pq ´ 1qβ, (20)
and Tsis the sampling period.
It is noteworthy that in (19) the calculation of β involves only N, Ts and fp. In practice, the parameter fp can be
obtained either by measurement or from the specifications of the VCO [11]. Since all these parameters are irrelevant to channel realizations, β can be calculated off-line. Furthermore, the calculation of α, tμnu, and tϑnu is extremely simple and
depends only on the the size of ISI-free region q.
C. Comparison with Other Windows
For convenience of comparison, we use the terminology
window as which is used in the literature [23]–[26]. We define
wnfi $ ’ ’ & ’ ’ % ϑN´n, if n P r´q, ´1s 1, if n P r0, N ´ q ´ 1s μn, if n P rN ´ q, N ´ 1s 0, otherwise (21)
Fig. 3. Illustration of various windows withq “ 100.
as the window coefficient for sample n. As introduced in Sec-tion I, there have been several windows, designed for different kinds of problems, in literature. Here, we briefly compare the proposed window with other windows in literature. In the following, we assume that N “ 512 and Ts“ 37.9ns. Fig. 3
shows each window with q “ 100. Rectangular window rep-resents the conventional OFDM receiver that simply removes the CP. Constant window is proven to be the maximizer of the received SNR in [24], and it was used in [28] and in [33] as a heuristic window function to mitigate Doppler-induced ICI and phase-noise-induced ICI, respectively. The Franks window, which minimizes the ICI incurred by Doppler effect, is coincidentally similar to the proposed window. In the ISI-free region, the slope of the Franks window, which minimizes the time variations of the channels, is 1{N « 0.002, whereas the slope of the proposed window, which mitigates the ICI caused by the phase noise, is β « 0.0013.
We can also compare these windows in frequency domain. For this purpose, we introduce the normalized frequency response defined as [24] W pf q “ 1 N 8 ÿ n“´8 wne´ j2πnpf{Δfsubq N , (22)
where Δfsub is the subcarrier spacing. Fig. 4 shows the
normalized frequency response of each window with q “ 100. It is evident that the other three windows suppress the side lobes when compared with the rectangular window; therefore, they can be used to suppress the ICI effect. Nevertheless, they have different shapes because they are specifically designed for different kinds of ICI sources.
IV. DERIVATION OF THEOPTIMUMCOMBINING
COEFFICIENTS AND THENEAR-OPTIMUMCOMBINING
COEFFICIENTS(18)
The optimization problem (17) is difficult to solve directly; therefore, we introduce the technique originally developed for the high-mobility ICI self-cancellation technique [31] and transform the problem (17) into a tractable equivalent optimization problem.
Fig. 4. Normalized frequency response of four different receiver window shapes withq “ 100.
Fig. 5. Illustration of the time-domain segments.
A. Preliminaries
For the purpose of analysis, we define received time-domain
segment and frequency-domain segment as follows.
Definition 1. Time-domain segment d, denoted as ypdq, is defined as
ypdq““rp´dq ¨ ¨ ¨ rpN ´ 1 ´ dq‰T
. (23) Definition 2. Frequency-domain segment d, or ypdqF , is defined
as
ypdqF “ FNypdq. (24)
Fig. 5 illustrates the relationship of each time-domain segment. Equivalently, we can rewrite (23) as
ypdq“ Rpdqr, (25) where Rpdq““O NˆpNg´L´dq IN ONˆd ‰ (26) is the matrix that takes the desired components ofr. Moreover, from (7), we can express (25) as
ypdq“ DpdqHpdqFH Ns ` εpdq, (27) where Hpdq“ RpdqHT cp, (28a) εpdq“ Rpdqε, (28b) Dpdq“ diagt”ejϕp´dq ¨ ¨ ¨ ejϕpN´1´dqıTu (28c)
are the corresponding equivalent time-domain channel matrix, AWGN vector, and the phase noise matrix of time-domain seg-ment d, respectively. Therefore, the corresponding frequency-domain segment d can be written as
ypdqF “ FNypdq“ ΦpdqΛpdqs ` epdq, (29) where epdq“ F Nεpdq, (30a) Φpdq“ FNDpdqFHN, (30b) Λpdq“ F NHpdqF H N (30c)
are the frequency-domain AWGN, phase noise matrix, and channel frequency response matrix of segment d, respectively. To be more specific, the channel frequency response matrix is a diagonal matrix given by
Λpdq“ diagt“Λpdqp0q ¨ ¨ ¨ ΛpdqpN ´ 1q‰T
u, (31)
whereΛpdqpkq is the channel frequency response on subcarrier
k of segment d, and the frequency-domain phase noise matrix
Φpdqis a circulant matrix which can be explicitly written as
Φpdq“ ¨ ˚ ˚ ˚ ˚ ˝ I0pdq I1pdq ¨ ¨ ¨ INpdq´1 INpdq´1 I pdq 0 ¨ ¨ ¨ INpdq´2 .. . . .. ... ... I1pdq I2pdq ¨ ¨ ¨ I0pdq ˛ ‹ ‹ ‹ ‹ ‚, (32) where Ikpmq“ 1 N Nÿ´1 n“0 exppjϕpn ´ mqqej2πnk N . (33)
The diagonal part of Φpdq will rotate the received signal constellation and cause CPE. On the other hand, the off-diagonal part ofΦpdqwill destroy the orthogonality of OFDM and incur ICI.
B. Problem Reformulation, Optimum Combining Weights, and Near-Optimum Combining Weights
In this subsection, we will reformulate the ICI minimization problem (17) as a tractable equivalent optimization problem.
The CP combining operation illustrated in Fig. 2 is equiv-alent to making the q ` 1 frequency-domain segments be weighted and combined. To show this equivalence, we first let
ud denote the combining weight corresponding to segment d.
Then, it can be easily verified that the weighted and combined frequency-domain signal v in (14) can be represented as
v “ q ÿ d“0 Gpdqypdq F ud (34) if we let μn“ N´1´nÿ d“0 ud, ϑn “ q ÿ d“N´n ud, (35) and q ÿ d“0 ud“ 1, (36) where Gpdqfi diagt”ej2π0dN ej2π1dN ¨ ¨ ¨ ej2πpN´1qdN ıT u (37) represents the phase-shift compensation matrix that compen-sates the linear phase shift of each frequency-domain segment. Consequently, for the following mathematical analysis, the optimum solution is derived with respect to tudu instead of
tμnu and tϑnu.
We define two matrices ΦpdqCPE and ΦpdqICI corresponding to the diagonal and off-diagonal parts of the matrix Φpdq, respectively, i.e., ΦpdqCPEfi diagt “ Ipdqp0q ¨ ¨ ¨ Ipdqp0q‰Tu “ Ipdqp0qIN (38) ΦpdqICI fi Φpdq´ Φpdq CPE. (39)
We also define the combining weight vector u fi ru0 u1 ¨ ¨ ¨ uqsT. Then the combined signal v can be
expressed in a more compact form:
v “ m ` b ` z, (40) where m “”Gp0qΦp0q CPEΛp0qs ¨ ¨ ¨ GpqqΦ pqq CPEΛpqqs ı u, b “”Gp0qΦp0q ICIΛp0qs ¨ ¨ ¨ GpqqΦ pqq ICIΛpqqs ı u, z ““Gp0qep0q ¨ ¨ ¨ Gpqqepqq‰u (41)
represent the combined signal term, ICI term, and noise term, respectively. Thus, the average ICI power is given by
1 NE “ bHb‰“ 1 Nu HE“CHC‰u “ 1 Nu HΩu, (42) where C fi”Gp0qΦp0q ICIΛp0qs ¨ ¨ ¨ GpqqΦ pqq ICIΛpqqs ı , (43) andΩ fi ErCHCs. The pi
1, i2qth entry ofΩ can be derived
as ¯γ“N Rφp0q ´ |i2´ i1|pRφp0q ´ RφpNqq ‰ ´ ¯γ Nÿ´1 n“´N`1 Rφpn ` |i2´ i1|q ˆ 1 ´|n| N ˙ , (44) where ¯γ “ 1 N Nÿ´1 k“0 |Λp0qpkq|2, (45)
and Rφpnq is the discrete-time auto-correlation function of
exppjϕpnqq, which is defined as
Rφpnq fi Erejϕpmqpejϕpm`nqq˚s “ Erejpϕpmq´ϕpm`nqqs.
(46) The derivation of (44) is given in Appendix A.
Now, the ICI power minimization problem (17) can be re-formulated as the following equivalent optimization problem:
min uPRq`1 u
TΩu
s.t. 1Tu “ 1.
(47) Note that Ω is well-defined (each entry is given in (44)). Since (47) is a convex optimization problem, any solution that
satisfies the KKT condition is the global optimum solution [36]. The KKT condition of (47) is given as
Ωu “ λ1, and 1Tu “ 1, (48)
where λ is a Lagrange multiplier. The optimum solution u is given by
u“ Ω´11
1TΩ´11 (49)
if Ω is invertible.
However, as is suggested by (49), the calculation of the optimum solution u is quite complicated since a matrix inversion is involved. From the numerical results of (49), we approximate the optimum combining weights u by the following explicit expression1
u‹““α β β ¨ ¨ ¨ β α‰T
, (50) and later we will show that
α “ 12 ´q ´ 12 β (51) and β “ 2πfpTs´ 2πNf 2 pTs2 1 ´ e´2πfpN Ts´ 2π 3NfpTs. (52) It should be noted that by using the relationship (35), we can transform u‹ in (50) back to the corresponding combining coefficients tμ‹nu and tϑ‹nu in (18). That is to say, the
near-optimality of (50) infers the near-near-optimality of (18). In the following subsection, we will show that (50) is a near-optimum solution.
C. Validation of the Near-Optimality ofu‹ (50)
By (11) and assuming that ϕpnq « 0, the discrete-time auto-correlation function Rφpnq can be derived and approximated
as Rφpnq « 1 ` κ0e´2πfp|n|Ts, (53) where κ0 fi K0πfp ` 1 ´ f2 p{fz2 ˘ . By (44) and (53), the pi1, i2qth entry ofΩ is approximated as ¯γ ˆ N p1 ` κ0q ´ κ0pp1 ´ e´2πfpN Tsq´ Nÿ´1 n“´N`1 p1 ` κ0e´2πfp|n`p|Tsq ˆ 1 ´ |n| N ˙ ˙ , (54)
where p fi |i1´ i2|. By Taylor Series Expansion, (54) can be
approximated as ¯γNp1 ` κ0q ´ ¯γκ0pp1 ´ e´2πfpN Tsq ´ Nÿ´1 n“´N`1 ¯γ ˆ 1 ` κ0 ´ 2κ0πfpTs|n ` p| ` 2κ0π2fp2Ts2pn ` pq2 ˙ ˆ 1 ´ |n| N ˙ . (55)
1This approximation is only valid for the IEEE 802.11ad phase noise model [35] and Wiener phase noise model, whereas the optimum solutionugiven by (49) is valid for all kinds of phase noise sources. Therefore, for a general phase noise process, the optimum combining weights can be calculated via (49) and (44).
For brevity, (55) can be expressed as
C1´ κ1p ` κ2 Nÿ´1 n“´N`1 |n ` p| ˆ 1 ´|n| N ˙ loooooooooooooooomoooooooooooooooon paq ´ κ3 Nÿ´1 n“´N`1 pn ` pq2ˆ1 ´|n| N ˙ looooooooooooooooomooooooooooooooooon pbq , (56) where C1“ ¯γp1 ` κ0q ˜ N ´ Nÿ´1 n“´N`1 ˆ 1 ´ |n| N ˙¸ , (57) κ1“ p1 ´ e´2πfpN Tsq¯γκ 0, (58) κ2“ 2πfpTs¯γκ0, (59) and κ3“ 2π2fp2Ts2¯γκ0. (60)
We remove the absolute value operations of paq in (56), and thenpaq can be derived as
paq “ ´p ÿ n“´N`1 |n`p| ˆ 1´|n| N ˙ ` ÿ ´p`1ďnď0 |n`p| ˆ 1´|n| N ˙ ` Nÿ´1 n“1 |n ` p| ˆ 1 ´|n| N ˙ “ Nÿ´1 n“p pn ´ pq´1 ´ n N ¯ ´ ÿ 0ďnďp´1 pn ´ pq´1 ´ n N ¯ ` Nÿ´1 n“1 pn ` pq´1 ´ n N ¯ . (61) Furthermore, paq in (56) can be rearranged as a polynomial function of p, given as paq “2 Nÿ´1 n“1 n ´ 1 ´ n N ¯ ´ 2 ÿ 1ďnďp´1 pn ´ pq´1 ´ n N ¯ ` p “C2´3N1 p3` p2`3N1 p, (62) where C2 is a constant. Similarly, we remove the absolute value operations of pbq in (56) and derive pbq as
pbq “ Nÿ´1 n“´N`1 pn ` pq2ˆ1 ´ |n| N ˙ “ ÿ´1 n“´N`1 pn`pq2´1 `n N ¯ ` p2`Nÿ´1 n“1 pn`pq2´1 ´n N ¯ “C3` Np2, (63) where C3is a constant. By (62) and (63), we can rewrite (56) as a polynomial function of p as
where
b “ 3N1 κ2´ κ1, (65)
c “ κ2´ Nκ3, (66)
and C4 is a constant. In practice, the third-order term of (64), namely, κ2{p3Nqp3, can be neglected due to the following two reasons: (1) For most practical wireless communication systems, κ2{p3Nq ! c, e.g., for IEEE 802.11ad specification
κ2{p3Nq « 1.5 ˆ 10´6 and c « 9.3 ˆ 10´4. (2) In practice,
the ISI-free region q is quite small such that the gap between
p3and p2is not large enough to compensate for the difference produced by their corresponding coefficients. Hence, (64) can be approximated as
C4` bp ` cp2. (67)
In summary, we have shown that Ω can be approximated by
Ω « » — — — — – f p0q f p1q ¨ ¨ ¨ f pqq f p1q f p0q . .. fpq ´ 1q .. . . .. . .. ... f pqq f pq ´ 1q ¨ ¨ ¨ f p0q fi ffi ffi ffi ffi fl, (68) where f ppq “ C4` bp ` cp2, @p P r0, qs. (69)
By (68) and (50), the ith element ofΩu‹ equals to
α`f piq ` f pq ´ iq˘` β ˜q´i´1 ÿ m“1´i f p|m|q ¸ . (70) Eq. (70) can be further derived as
α“bi ` ci2` bpq ´ iq ` cpq ´ iq2‰ ` β q´i´1ÿ m“1´i ` b|m| ` cm2˘` C5 “αcp2i2´ 2qiq ` βb q´i´1ÿ m“1´i |m| ` βc q´i´1ÿ m“1´i m2` C6 “ipi ´ qq r2αc ` βb ´ βc ` βcqs ` C7, (71)
where C5, C6, and C7 are all constants. It is implied by the conditionΩu‹“ λ1 that
2αc ` βb ´ βc ` βcq “ 0. (72)
Similarly, from the other condition 1Tu‹ “ 1, we can
conclude that
2α ` pq ´ 1qβ “ 1. (73)
From (72) and (73), we can derive that
α “12`2bc pq ´ 1q “ 12´q ´ 12 β β “ ´c b “ 2πfpTs´ 2πNfp2Ts2 1 ´ e´2πfpN Ts´ 2π 3NfpTs . (74)
As a result, we have shown that (50) is approximately the optimum solution since the KKT condition (48) approximately holds. Equivalently, the near-optimality of (18) is validated as well.
Fig. 6. The PMF ofq.
TABLE I SIMULATION PARAMETERS.
Sample Rate 2.64GHz CP Length 128
Carrier Frequency 60 GHz # of Data Subcarriers 336 Subcarrier Spacing 5.15625MHz # of Pilot Subcarriers 16
FFT Length 512 samples # of DC Subcarriers 3
V. SIMULATIONRESULTS
A. Simulation Parameters
Our simulation parameters are mainly based on the spec-ification proposed by IEEE 802.11ad task group [37]. The simulation parameters are listed in Table I. We define P as the pilot index set, and the details ofP are defined in [37].
The channel is simulated according to the IEEE 802.11ad 60GHz channel model [38]; specifically, conference room STA-STA sub-scenario is applied, and we use the default setting of [38] throughout our simulations. In this channel model [38], the channel realizations are generated based on a ray-tracing algorithm which takes the 60GHz electromagnetic waves propagation properties into account. In the conference room STA-STA sub-scenario, a 4.5mˆ 3mˆ 3m conference room is considered, and both the transmitter and the receiver are located on a table in the center of the room. For each channel realization, the positions of the transmitter and the receiver are both uniformly distributed on the top of the table. Additionally, each transceiver equips a steerable directional antenna, whose half-power antenna beamwidth is 300. For more details, please refer to [38]. We execute a Monte Carlo experiment to get the probability mass function (PMF) of the number of ISI-free samples, namely q, for this channel. The PMF of q is shown in Fig. 6. The expected value of q is 84.9, and the standard deviation of q is 13.44.
The phase noise model used is based on [35], as described in section II, and each parameter is set to its default value in [35].
B. Assumptions in Simulations
We assume that channel is static within a period containing 40 OFDM symbols. Each simulation result is averaged over 5000 independent channel realizations. We assume that there
Fig. 7. BER performance of 16QAM OFDM in the presence of phase noise.
is a preamble in each OFDM data frame, and by utilizing the preamble, the channel frequency responseΛ can be accurately estimated via a channel estimation method robust against phase noise effect, e.g., [11]–[14], [17], [18]. For simplicity, we assume that the channel frequency response is perfectly
estimated.
We apply the method proposed in [8] to estimate the CPE. According to [8], the CPE is estimated as
ˆIp0q “ ř kPP vpkqs˚pkq˜Λ˚pkq ř kPP |spkq˜Λpkq|2 . (75)
Note that in [8] the CPE can be iteratively updated via the help of the detected data. For simplicity, we do not use the iterative estimator in this paper. After CPE estimation, the combined signal v is fed into a CPE compensator (CPEC), a one-tap equalizer, and a data detector; the aforementioned process can be mathematically described as ˆs “ Π ˜ ˜ Λ´1v ˆIp0q ¸ , (76) where Πp¨q is an entry-wise quantization function which quantizes each entry to its nearest constellation point, and ˆs is the detected data. It is worth mentioning that almost all ICI cancellation methods (e.g., [8]–[22], [33], [34]) can be applied to further mitigate the residual ICI.
C. Simulation Results
The BER performance for OFDM transmissions employing 16QAM modulation in the presence of phase noise is plotted in Fig. 7. For brevity, the method without CP combining is called “conventional method” in the following. We compare the BER performance of the conventional method and the proposed method with and without CPEC in this figure. The bottom curve is the phase-noise-free case, which serves as the ideal case. As shown from the figure, we observe that the performance of the proposed method is indistinguishable from the ideal case for Eb{N0 less than 17dB. And the
performance of the proposed method is about 0.5dB better than the conventional method at the BER levels being equal to10´2 and10´3.
Fig. 8. BER performance of 64QAM OFDM in the presence of phase noise.
Similarly, Fig. 8 shows the BER performance of 64QAM OFDM transmissions in the presence of phase noise. Without CPEC, both the proposed method and the conventional method exhibit an error floor. Nevertheless, the proposed method outperforms the conventional method and has about1dB gain at the BER level10´2. When CPEC is included, the error floor effect is extensively improved. In this situation, it is observed that the proposed method still outperforms the conventional method. For example, with BER being equal to 10´3, the proposed method is about1.5dB better than the conventional method. However, in this case, the performance gap between the ideal case and the proposed method is evident because the ICI effect is more severe for 64QAM. As mentioned earlier, the proposed method can be in conjunction with other ICI mitigation methods to further improve BER performance. Here we use the ICI mitigation method described in [15] as an example. As shown in Fig. 8, the performance is greatly improved via using the method proposed in [15]. In this case, the proposed method still provides an observable performance improvement, e.g., it has a0.3dB gain at the BER level 10´3. It is worth mentioning that the aforementioned performance improvements resulted from the proposed method, although not large, is remarkable because the proposed method is extremely simple.
The coded BER performance of 16QAM and 64QAM are shown in Fig. 9(a) and Fig. 9(b), respectively. The LDPC code with code rate 1{2 defined by IEEE 802.11ad specification [37] is used in simulations. For simplicity, the traditional Bit Flipping (BF) algorithm is used to decode the LDPC code. In Fig. 9(a) and Fig. 9(b), a specific channel realization, which is a typical channel realization drawn from the ran-dom channel generator [38]. The considered specific channel realization is given as r´0.02527 ´ j0.12424, ´0.00878 `
j0.04851, 0.246 ` j0.08669, ´0.87908 ´ j0.3457, 0.00258 ´ j0.00365, ´0.01489´ j0.0798, ´0.05142 ` j0.0956s, and the
corresponding lags arer0, 2, 9, 20, 22, 37, 39s samples, respec-tively. From Fig. 9(a) and Fig. 9(b), it is observed that the proposed method clearly outperforms the conventional method when error control coding is included.
Fig. 10 shows the BER performance as a function of phase noise severity, and here we apply the Wiener phase noise
(a) Coded BER performance of 16QAM (b) Coded BER performance of 64QAM Fig. 9. The coded-BER performance in the presence of phase noise.
model instead of the phase noise model described in Section II. Besides the phase noise model used in this paper, Wiener phase noise is another commonly used model for a free-running oscillator [1]–[5], [7]–[10], [14]–[20]. In general, the Wiener phase noise process ϕpnq can be described as
ϕpnq “ ϕpn ´ 1q ` ξpnq, (77) where ξpnq „ N p4πΔf3dBTsq and Δf3dB is the one-sided
3dB linewidth of the Wiener process [1]. It can be easily proven that the proposed method can be directly applied to the Wiener phase noise except that the pole frequency fp
should be substituted byΔf3dB. In Fig. 10, the relative phase noise linewidth δP N is defined as Δf3dB{Δfsub [19]. It is
evident that the BER of the proposed method is always lower than or indistinguishable from the conventional method. For
δP N being small enough, 10´6 for example, the BER of
the conventional method almost merges with the ideal case. Surprisingly, the BER of the proposed method is lower than the phase-noise-free case when δP N ă 3 ˆ 10´5. This is
because the proposed method utilizes the CP information, which is simply discarded in the phase-noise-free case. The CP compensation method suppresses not only the ICI power caused by phase noise but also the thermal noise power. That is to say,
Er}z}2s ă σ2. (78)
The proof of (78) is given in Appendix B.
Fig. 11 shows the percentage of the reduction in ICI power, which serves as a metric to evaluate the ICI suppression capability, as a function of q. The formal definition of the percentage of the reduction in ICI power is
„
1 ´ uiTTΩuΩi j
p%q, (79)
where i fi “1 0 ¨ ¨ ¨ 0‰T. In [33], [34], the combining weights of [28] were applied to suppress the ICI caused by phase noise. The ICI suppression capability of the combining weights [28], denoted as Svensson’s weights, is compared with that of the proposed optimum and near-optimum weights in Fig. 11. The phase noise model introduced in Section II is used here. Since this metric is not dependent on channel
Fig. 10. OFDM BER performance for a free-running oscillator as a function of relative phase noise linewidthδP N (QPSK,Eb{N0“ 10dB).
realizations for a fixed q value, we do not assume any specific channel model. From Fig. 11, we observe the following facts. First, it is obvious that we can reduce the ICI power to an acceptable level by extending the length of CP. In other words, we can make a trade-off between spectral efficiency and the ICI power induced by phase noise. As shown in Fig. 11(a), the ICI power can be effectively suppressed at the expense of the loss in spectral efficiency, e.g., for q “ N {4 the reduction in ICI power is about 20%. Taking q “ N for another example, the ICI power is reduced more than half in this situation, which means that a repetition-code-like transmission scheme2 can effectively reduce the ICI power incurred by
phase noise. Second, from Fig. 11(a), we can observe that the proposed optimum and near-optimum combining weights clearly outperform Svensson’s weights, especially for q ě 200. Third, the proposed near-optimum weights have a evident performance gap while q approaches N . The reason is that we have made an approximation in (64) that the third order term is negligibly small, which is only valid when q is much
2Traditional transmission schemes such as OFDM [27], single carrier block transmission (SCBT), and code-division multiple access (CDMA) can be transmitted in such a manner.
(a) q “ 1, ¨ ¨ ¨ , 512 (b) A zoom-in section Fig. 11. The percentage of the reduction in ICI power as a function ofq.
smaller than N . However, for a reasonable value of q, the performance of the proposed near-optimum weights is almost identical to that of the optimum weights. Last, as shown in Fig. 6, for the simulation channels used in this paper, q usually ranges from 40 to 60. Hence, we provide a zoom-in section in Fig. 11(b) so that we can focus on the ICI suppression capability of each method for the range of q in which we are interested. As shown in Fig. 11(b), the performance of the proposed near-optimum weights is indistinguishable from that of the optimum weights, and the proposed weights clearly outperform Svensson’s weights. Even though the performance improvement is not tremendous, we still recommend to use the proposed near-optimum weights rather than Svensson’s weights, for they have essentially the same complexity.
VI. CONCLUSION
In this paper, a simple and effective ICI suppression method in the presence of phase noise for OFDM transmission is proposed. The method is based on the utilization of the ISI-free samples in CP, which are traditionally abandoned. We derive the optimum combining coefficients for ICI power minimization and propose a set of near-optimum combining coefficients for complexity reduction. Simulation results show that the proposed method is about 0.5 to 1.5dB better than the conventional method with negligible amount of additional computation complexity. This paper validates the fact that an additional performance gain can be achieved by properly exploiting the unused resources in CP.
APPENDIXA THEDERIVATIONS OF(44)
Through (42) and (43), we can rewrite the pi1, i2qth entry
ofΩ as E”sHΛpi1qHΦpi1qH ICI Gpi1qHGpi2qΦpiICI2qΛpi2qs ı (80a) “tr´E”sHΛpi1qHΦpi1qH ICI Gpi1qHGpi2qΦ pi2q ICI Λpi2qs ı¯ (80b) “tr´E”Λpi1qHΦpi1qH ICI Gpi1qHGpi2qΦ pi2q ICI Λpi2qss Hı¯ (80c) “tr´E”Φpi1qH ICI Gpi1qHGpi2qΦpiICI2q ı Λpi2qΛpi1qH¯. (80d)
From (80a) to (80b), we use the fact that (80a) is a scalar, hence it is equal to its trace. To derive (80c) from (80b), we use the equalitytrpABq “ trpBAq. Since we assume ErssHs “
IN, together with the equality trpABq “ trpBAq, we get
(80d). For convenience, we define
K fi Λp0qΛp0qH. (81)
Hence,ΛpvqΛprqH “ KGpr´vq. So, (80d) becomes
tr´ErΦpi1qH ICI Gpi2´i1qΦ pi2q ICI sKGpi1´i2q ¯ . (82) Since ΦpiICI1q “ Φpi1q´ Φpi1q
CPE, (82) can be divided into the
following four terms
tr´E”Φpi1qHGpi2´i1qΦpi2qKGpi1´i2qı¯ (83a) ´tr´E”Φpi1qH
CPE Gpi2´i1qΦpi2qKGpi1´i2q
ı¯ (83b) ´tr´E”Φpi1qHGpi2´i1qΦpi2q CPEKGpi1´i2q ı¯ (83c) `tr´E”Φpi1qH CPE Gpi2´i1qΦ pi2q CPEKGpi1´i2q ı¯ . (83d)
By (30b), the first term (83a) can be expressed as
p83aq
“tr´Gpi1´i2qFErDpi1qHFHGpi2´i1qFDpi2qsFHK¯
“1 N Nÿ´1 k“0 |Λp0qpkq|2rpN ´|i 1´ i2|qRφp0q`|i1´i2|RφpNqs . (84)
The second term (83b) equals to tr´ErGpi1´i2qΦpi1qH CPE Gpi2´i1qΦpi2qsK ¯ “ tr´ErΦpi1qH CPE Φpi2qsK ¯ “ tr ¨ ˚ ˚ ˝E » — — – ¨ ˚ ˚ ˝ I0pi1q O . .. O I0pi1q ˛ ‹ ‹ ‚ H¨ ˚ ˚ ˝ I0pi2q ˆ ˆ ˆ . .. ˆ ˆ ˆ Ipi2q 0 ˛ ‹ ‹ ‚ fi ffi ffi fl K ˛ ‹ ‹ ‚ “ 1 N2 Nÿ´1 m“0 Nÿ´1 n“0 Rφpm ´ n ` |i2´ i1|qtrpKq “ ¯γ N ¨ ”Nÿ´1 n“1 ˆ Rφpn ` |i2´ i1|q ` Rφp´n ` |i2´ i1|q ˙ pN ´ nq ` NRφp|i2´ i1|q ı , (85) where ‘ˆ’ stands for ‘don’t care’. The first equality holds because ΦpiCPE2q, Gpi2´i1q, andGpi1´i2q are all diagonal matrix so that they are interchangeable in matrix multiplications. Hence Gpi2´i1q andGpi1´i2q cancels with each other.
The third term (83c) equals to tr´ErΦpi1qHΦpi2q CPEsK ¯ “tr ¨ ˚ ˚ ˝E » — — – ¨ ˚ ˚ ˝ I0pi1q ˆ ˆ ˆ . .. ˆ ˆ ˆ Ipi1q 0 ˛ ‹ ‹ ‚ H¨ ˚ ˚ ˝ I0pi2q O . .. O I0pi2q ˛ ‹ ‹ ‚K fi ffi ffi fl ˛ ‹ ‹ ‚. (86) The last term (83d) equals to
tr´ErΦpi1qH CPE Φ pi2q CPEsK ¯ “tr ¨ ˚ ˚ ˝E » — — – ¨ ˚ ˚ ˝ I0pi1q O . .. O I0pi1q ˛ ‹ ‹ ‚ H¨ ˚ ˚ ˝ I0pi2q O . .. O I0pi2q ˛ ‹ ‹ ‚K fi ffi ffi fl ˛ ‹ ‹ ‚. (87) Apparently, both (86) and (87) are identical to (85). Hence, the pi1, i2qth entry of Ω equals to
¯γ „ pN ´ |i2´ i1|q ¨ Rφp0q ` |i2´ i1| ¨ RφpNq ´ Rφp|i2´ i1|q j ´ ¯γ N Nÿ´1 n“1 „ Rφpn ` |i2´ i1|q ` Rφp´n ` |i2´ i1|q j ¨ pN ´ nq . (88) By simple mathematics, (88) can be rewritten as
¯γ“N Rφp0q ´ |i2´ i1|pRφp0q ´ RφpNqq ‰ ´ ¯γ Nÿ´1 n“´N`1 Rφpn ` |i2´ i1|q ˆ 1 ´|n| N ˙ . (89) APPENDIXB PROOF OFEr}z}2s ă σ2
By [31], for a given combining weight vector u, the combined noise power is given as
Er}z}2s “ uTΨu, (90) where Ψ “ σ2 N ¨ ˚ ˚ ˚ ˝ N N ´ 1 . . . N ´ q N ´ 1 N . . . N ´ q ` 1 .. . ... . .. ... N ´ q N ´ q ` 1 . . . N ˛ ‹ ‹ ‹ ‚. (91)
By applying (50) into (91), we have
N σ2Eru ‹TΨu‹s “2α2` 2αβNpq ´ 1q ` αβp2N ´ qqpq ´ 1q ` β2N pq ´ 1q2 ´ β2 ˜ qN ` 2 q ÿ i“1 pq ´ iqpN ´ iq ¸ ´ αβqpq ´ 1q ă2α2` 2αβNpq ´ 1q ` αβp2N ´ qqpq ´ 1q ` β2N pq ´ 1q2 “N ´12q r1 ´ pq ´ 1qβqs (92) In practice,1 ´ pq ´ 1qβ ą 0; therefore, we can conclude that u‹TΨu‹ă σ2.
ACKNOWLEDGMENT
The authors would like to thank anonymous reviewers and the Editor for their constructive advices and comments that help improve this paper.
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Chun-Ying Ma was born in 1985, Tainan, Taiwan.
He received his B. Eng in communications from National Chiao Tung University, Taiwan, in 2008. He then joined the Wireless Communication Lab of National Chiao Tung University pursuing a Ph.D. degree in the area of wireless communications. His current research interests include high-mobility OFDM communication systems, green radio, wire-less resource allocation, and phase noise suppression algorithms.
Chun-Yen Wu was born in Taiwan, R.O.C. He
received his B.S. degree in Electrical Engineering from Nation Taipei University of Technology in 2010 and M.S. degree in Communications Engineer-ing from National Chiao Tung University in 2012. He is currently a software engineer with MediaTek Corp. in Hsinchu City, Taiwan.
Chia-Chi Huang was born in Taiwan, R.O.C. He
received the B.S. degree in electrical engineering from National Taiwan University in 1977 and the M.S. and Ph.D. degrees in electrical engineering from the University of California, Berkeley, in 1980 and 1984, respectively. From 1984 to 1988, he was an RF and communication system engineer with the Corporate Research and Development Center, General Electric Company, Schenectady, NY, where he worked on mobile radio communication system design. From 1989 to 1992, he was with the IBM T.J. Watson Research Center, Yorktown Heights, NY, as a Research Staff Member, working on indoor radio communication system design. Since 1992, he has been with National Chiao Tung University, Hsinchu, Taiwan, and currently as a Professor in the Department of Electrical and Computer Engineering.
