Sidelobe Suppression with Orthogonal Projection ... - Semantic Scholar

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Sidelobe Suppression with Orthogonal Projection for Multicarrier Systems Jian (Andrew) Zhang, Senior Member, IEEE, Xiaojing Huang, Senior Member, IEEE, Antonio Cantoni, Fellow, IEEE, Y. Jay Guo, Senior Member, IEEE

Abstract—Sidelobe suppression, or out-of-band emission reduction, in multicarrier systems is conventionally achieved via time-domain windowing which is spectrum inefficient. Although some sidelobe cancellation and signal predistortion techniques have been proposed for spectrum shaping, they are generally not well balanced between complexity and suppression performance. In this paper, an efficient and low-complexity sidelobe suppression with orthogonal projection (SSOP) scheme is proposed. The SSOP scheme uses an orthogonal projection matrix for sidelobe suppression, and adopts as few as one reserved subcarrier for recovering the distorted signal in the receiver. Unlike most known approaches, the SSOP scheme requires multiplications as few as the number of subcarriers in the band, and enables straightforward selection of parameters. Analytical and simulation results show that more than 50dB sidelobe suppression can be readily achieved with only a slight degradation in receiver performance. Index Terms—Multicarrier Systems, sidelobe suppression, orthogonal projection, cognitive radio

I. I NTRODUCTION Thanks to their high spectrum efficiency and simple frequency domain equalization in dense multipath channels, multicarrier systems, such as orthogonal frequency-division multiplexing (OFDM) and precoded-OFDM systems are widely employed in broadband communications. In addition to several design problems in multicarrier systems, such as high peak-toaverage power ratio (PAPR) and sensitivity to carrier frequency offset, sidelobe suppression, which is also known as out-ofband emission reduction, remains as one major challenge. In a multicarrier system, the waveform of each subcarrier is inherently a sinc function, and the power of its sidelobe decays slowly as f −2 , where f is the frequency distance to the mainlobe. This can result in serious interference to systems operating in adjacent bands. The problem is more significant in multicarrier modulation based cognitive radios [1], where instantaneous spare spectrum in primary systems may be used by intelligent secondary systems. Such secondary systems need to ensure that their transmitted signal has very sharp spectrum roll-off to maximize their usable bandwidth and minimize interference to primary systems. Traditionally, windowing techniques, such as the raised cosine windowing, is applied to the time-domain signal waveform for sidelobe suppression [2], [3]. When employing windowing, an extended guard interval is required to avoid signal This is a draft version of the paper accepted by IEEE Trans. on Communications in August 2011. The authors are with the ICT Centre, CSIRO. A. Cantoni is also with the University of Western Australia. Email: {Andrew.Zhang; Xiaojing.Huang; Tony.Cantoni; Jay.Guo}@csiro.au.

distortion, and the spectrum efficiency may be significantly reduced, particularly for systems with large guard intervals. To improve spectrum efficiency, various advanced techniques have been proposed for sidelobe suppression [5]–[15]. Among them, two classes of techniques, Cancellation [7]–[9] and Precoding [10]–[15], have received wide attention recently as they can achieve deep nulls while retaining high spectrum efficiency. Cancellation techniques, such as active interference cancellation (AIC) [7], cancellation carrier (CC) [8] and extended AIC [9], optimize the signal at cancellation subcarriers to cancel out-of-band emission. They can generally achieve good sidelobe suppression, but suffer from signal-to-noise ratio (SNR) degradation in the receiver as either extra power is wasted in the cancellation subcarriers [8], or, intersymbol interference is caused by cancellation signal extended over multiple symbols in the time domain [7], [9]. Precoding techniques optimize a precoding matrix to minimize out-of-band emission. In [10], an optimal compacted window function that does not require extended guard interval is proposed. Although zero-overhead is achieved, its sidelobe suppression and receiver error performance are inferior to those of conventional windowing approach. In [11], [15], new basis sets are developed to replace the rectangularly pulsed multicarrier basis signals, and both sidelobe suppression and error performance are improved. However, a notable increase in complexity is incurred in both the transmitter and receiver. Orthogonal precoders based on subspace extraction are investigated in [13] and [14]. Both approaches have the advantage of maintaining the receiver SNR, but their computation complexity is proportional to the square of the number of subcarriers. In [12], sidelobe suppression is achieved by designing precoders that minimize the discontinuity between consecutive OFDM symbols. To summarize, most sidelobe suppression techniques discussed above are not well balanced between complexity and performance. In addition, the sidelobe power decays faster in OFDM systems with zero padding (ZP) than those with cyclic prefix as explained in [14], and various sidelobe suppression schemes achieve better performance in ZP systems. This suggests that ZP systems are better choices, particularly for cognitive radio systems, in terms of sidelobe suppression. In this paper, we propose a simple and effective out-ofband emission reduction scheme for multicarrier systems. The scheme is primarily designed for ZP systems, but it can also be directly applied to CP systems albeit with degraded performance on sidelobe suppression.. Whilst the scheme is notably

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low in implementation complexity and can be implemented effectively in real hardware, it also offers significant gains in terms of many other performance metrics, such as sidelobe suppression effect, adaptivity and receiver performance. The scheme is based on matrix orthogonal projection and is thus referred to as “sidelobe suppression with orthogonal projection” (SSOP). It uses signal pre-distortion for sidelobe suppression, and employs as few as one reserved subcarrier for recovering the distorted signal in the receiver. The use of orthogonal projection matrix leads to zero emission at some desired frequency points, while simultaneously suppressing emissions at neighboring frequencies. Using the reserved subcarriers, intersymbol interference (ISI) between data symbols introduced by pre-distortion matrix can be completely removed in the absence of noise. The SSOP scheme also has clear physical interpretation, and enables flexible and straightforward parameter configuration. The receiver performance is only slightly degraded due to noise enhancement in the ISI-free receiver. The spectral shaping achievable with the proposed scheme has been validated experimentally in real hardware. The rest of the paper is organized as follows. In Section II, the sidelobe suppression problem is formulated under a general multicarrier system, and the proposed SSOP scheme is presented. In Section III, the sidelobe suppression effect is analyzed, and parameter selection for sidelobe suppression is studied. In Section IV, the ISI-free receiver is investigated. In Section V, both the block and per-symbol SNR analysis is conducted, and the impact of SSOP parameters on SNR is demonstrated with both analytical and numerical results. A simple performance-augmenting approach, motivated by per-symbol SNR analysis, is also presented. In Section VI, numerical and simulation results are presented for OFDM and DFT-OFDM systems. Finally, Section VII concludes the paper. Notation: Lower-case bold face variables (a, b, · · · ) indicate vectors, and upper-case bold face variables (A, B, · · · ) indicate matrices. Frequency domain variables are denoted e · · · ). I denotes the identity matrix, with tilde above (e a, A, E[·] denotes expectation, and (·)T , (·)∗ , (·)−1 and (·)† denote transposition, conjugate transposition, inverse and pseudoinverse, respectively. II. P ROBLEM F ORMULATION AND P ROPOSED S CHEME Consider a general multicarrier system where the total number of subcarriers is N . These N subcarriers may be divided into a few subbands, and each subband includes a block of continuous subcarriers. The number of subcarriers in each subband can be different, and some subbands may not be used for information transmission. This model is applicable to systems such as conventional OFDM, the localized singlecarrier frequency-division multiple access (SC-FDMA) [16], and cognitive radio. To keep complexity low and the scheme general, we concentrate on suppressing the sidelobe for each subband independently. Thus we consider one subband of M subcarriers, where M ≤ N , and denote the M subcarriers as Subcarrier 0, 1, · · · , M − 1. The subcarrier interval is denoted ˜ = as δf . Let the signal assigned to the M subcarriers be x (˜ x0 , x ˜1 , · · · , x ˜M −1 )T . The out-of-band emission or sidelobe

is referred to as the emission generated by the M subcarriers in the frequency range (− inf, 0) and ((M − 1)δf , + inf), and our objective of sidelobe suppression is to reduce the out-ofband emission as much as possible with low implementation complexity. This is achieved by designing a precoding matrix so that the emission at several pre-chosen frequencies become zero. Statistically, the function of sidelobe power corresponds to a segment of the power spectrum density (PSD) of the transmitted signal. Conventionally, PSD analysis for multicarrier systems is based on an analog model with a sinc kernel function [17]. It has been shown that this is an approximation to the actual PSD in a discrete Fourier transform (DFT) based implementation of the multicarrier system where the noninfinite sampling rate in the digital-to-analog converter (DAC) causes periodic expansion of the sinc function [18], [19]. This causes aliasing and the real PSD is thus generally larger than those obtained based on the sinc approximation. The sinc approximation approaches to the real one with the sampling rate increasing, and the PSD converges to the sinc when the sampling period approaches to 0. The sinc approximation is used in this paper in the development and analysis of SSOP. The influence of the sampling rate on PSD will be examined in the simulation results (Section VI). In a multicarrier system with zero padding, the time-domain signal is equivalently windowed by a rectangular function { 1, 0 ≤ t ≤ 1/δf g(t) = (1) 0, elsewhere Note that to represent a causal signal, the rectangular function spans over [0, 1/δf ] instead of [−1/(2δf ), 1/(2δf )], and its Fourier transform is given by sin(πf /δf ) G(f ) = e−jπf /δf . (2) πf /δf Let ω , f /δf be the normalized frequency. With such timedomain windowing, the frequency-domain representation of the m-th subcarrier is ψm (ω) = dm (ω)˜ xm , where dm (ω) = cm (ω)ϕm (ω), and { sign(m − ω) e−jπω sin(πω)/π, ω ̸= m ϕm (ω) , 1, ω = m, { { 1 ω ̸= m 1, α≥0 |m−ω| , cm (ω) , sign(α) , −1, α < 0, 1, ω = m, and |α| denotes the absolute value of α. We note that by using the above expression, we have disregarded the effect of sampling rate in the DAC, or in other words, implicitly assumed that the sampling period approaches to zero. For ω ̸= m, the function cm (ω) represents the envelope of ψm (ω) in the sense that sin(π(m − ω)) 1 ≤ π = cm (ω). π|dm (ω)| = π π(m − ω) π(m − ω) The frequency-domain representation of the signal spectrum due to the superposition of all M subcarriers is b(ω) =

M −1 ∑ m=0

ψm (ω) = d∗ (ω)˜ x=

M −1 ∑ m=0

cm (ω)ϕm (ω)˜ xm , (3)

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where d(ω) = (d0 (ω), d1 (ω), · · · , dM −1 (ω))∗ . For 0 ≤ ω ≤ M − 1, b(ω) is a measure of the in-band frequency components of the transmitted signal; whereas for ω < 0 and ω > (M − 1), b(ω) measures the out-of-band frequency components. In sequel, we refer to b(ω) as the emission of the OFDM signal and will focus on the out-ofband emission in the range of ω < 0 and ω > (M − 1). The goal of sidelobe suppression is to reduce the PSD of the transmitted OFDM signal in designated out-of-band ranges. Our SSOP algorithm reduces the sidelobe power by forcing the out-of-band emission to zero at several carefully chosen frequency points. Based on (3), the emission at p normalized frequencies ςp = {ω0 , ω1 , ..., ωp−1 }, referred to as suppression distances in the SSOP algorithm, can be represented by a p×1 complex vector ˜ = (CΦ)∗ x ˜, b = B∗ x

(4)

where b = (b(ω0 ), b(ω1 ), · · · , b(ωp−1 ))T , B , CΦ where C = {Ci,j }, i = 0, · · · , M − 1, j = 0, · · · , p − 1 is an M × p matrix defined in terms of the envelope functions by Ci,j = ci (ωj ),

(5)

and Φ is a p×p diagonal matrix with the i-th diagonal element being the conjugate of ϕ(ωi ). When ω is not an integer, dm (ω) ̸= 0 for any m. Now, we want to suppress sidelobe by finding a precoding matrix ˜ = P˜ ˜ is an M × 1 vector, so that b = 0 P in x z, where z ˜. Temporarily, we need to limit irrespective of the value of z the suppression distances ωi to be non-integers as b = 0 is naturally true when they are all integers. We will see shortly that this constraint can be removed. One of the solutions to achieving b = 0 is to choose P as P = I − B(B∗ B)−1 B∗

(6)

= I − CΦ(Φ)−1 (CT C)−1 (Φ∗ )−1 Φ∗ CT = I − C(CT C)−1 CT . When C is a matrix of full column-rank, which is usually the case provided that the suppression distances are widely spaced, Q , C(CT C)−1 CT and P = I − Q are actually the orthogonal projection matrixes to the range and null space of CT (and also BT ), respectively. The projection matrixes have some important properties [20], that will be used in subsequent sections Q2 = Q, P2 = P,

(7)

and for any M × 1 complex vector γ 2

2

∥γ − Qγ∥ < ∥γ − Qθ∥ , for any θ ̸= γ, γ ∗ Qγ ≥ 0, γ ∗ Pγ ≥ 0,

(8) (9)

where ∥γ∥ denotes the Euclidean norm of vector γ. It should be noted that B and C share the same orthogonal projection, thus we can directly work on the envelop function without considering the phase and scalar terms. Working with the envelop function also relaxes the requirement of ωi being non-integers. Thus we obtain b = Φ∗ CT P˜ z = 0, for any ω < 0 or ω > M − 1.

(10)

We will show in Section III that forcing emissions to be zeros at the suppression distances leads to significant reduction of emission at all the out-of-band frequencies. To suppress sidelobe symmetrically on two sides of a subband, p should be an even number, and pairs of “symmetric” suppression distances should be used to construct P. For a normalized frequency ωi < 0, its symmetric frequency is ωp−1−i = M − 1 − ωi . For the purpose of sidelobe suppression, the design of the ˜, above pre-distortion matrix is independent of the signal z ˜. However, since and there is no restriction on the values of z rank(P) ≤ M − p, the uncoded signal cannot be recovered in general, unless the distortion introduced by P can be ignored. The distortion due to P can be considered as effectively introducing ISI. For example, with the assumption that each ˜ is a data symbol with zero mean and identical element in z variance, the signal to interference power ratio (SIR) at each symbol ranges from 10.7 to 18.9dB for M = 64, p = 2, ω0 = −32 and ω1 = 95. The ISI will become the performance limiting factor when the noise power is low. To enable ISIfree symbol recovery, we propose to reserve some subcarriers which have zero modulated symbols. The number of reserved subcarriers, q, should not be smaller than p. As will be seen in Section IV, these reserved subcarriers should be spaced as widely as possible in order to minimize noise enhancement at the receiver. At the same time, to minimize out-of-band emission, one reserved subcarrier should be allocated to each edge of the signal band. Thus we propose to allocate reserved subcarriers as those with the following indexes: S = {0, ν − 1, 2ν − 1, · · · , (q − 2)ν − 1, M − 1},

(11)

where ν = floor(M/(q−1)) and floor(α) rounds to the nearest integer less than or equal to α. Let an (M − q) vector ˜s represent the M − q symbols assigned to the unreserved subcarriers before precoding. The vector ˜s represents the data symbols in an OFDM system, or their transformed output in a precoding OFDM system. With ˜ that directly modulates the above considerations the vector x the M subcarriers is given by √ ˜ = λ(I − C(CT C)−1 CT )˜ x z, q ≥ p (12) where the scaling factor λ is introduced so that the mean power ˜ , E(|˜ ˜ = {z0 , z1 , · · · , zM −1 } with of x x|2 ), is 1, and z { 0, k∈S zk = (13) s˜g(k), k ∈ /S and g(k) is a function that maps the indexes of ˜s to the unreserved subcarriers. We will show in Section IV that this arrangement enables symbol recovery without any ISI. In the following subsections we consider three specific arrangements corresponding to different values of p and q. These arrangements are denoted as Case A, Case B, and Case C. Case A and B are very simple for implementation and can achieve significant sidelobe suppression with only slight BER degradation. Case C which uses p = q = 4 provides more flexibility and can achieve more suppression. Unless otherwise noted, M = 64 is used to generate numerical results.

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A. Case A: Single-sided Reduction and p = q = 1 This fulfills single-side sidelobe suppression, and (12) becomes ( ) √ 0 T −1 T ˜ = λ(I − c(c c) c ) x , (14) ˜s where ˜s is now an (M − 1) × 1 vector, and [ 1 ]T 1 1 , ω0 < 0 and c = −ω0 , 1−ω0 , · · · , M −1−ω0 [ ]T 1 1 1 c = M −1−ω0 , M −2−ω0 , · · · , −ω0 , ω0 < 0 for suppressing sidelobe in the region of ω < 0 and ω > M − 1, respectively. Without loss of generality, The region of ω < 0 will be considered for case A hereafter. B. Case B: Double-sided Reduction and p = q = 2 For ω0 < 0, its paired “symmetric” frequency in the other side is (M − 1 − ω0 ). Thus the matrix C in (12) becomes ( )T 1 1 1 · · · M −1−ω −ω0 1−ω0 0 C= , ω0 < 0, 1 1 1 ··· M −1−ω0 M −2−ω0 −ω0 (15)

Consider a set of p − 1 suppression distances ςp−1 obtained from the set ςp by taking out any single suppression distance, ωm , 0 ≤ m ≤ p − 1. The corresponding M × (p − 1) matrix Cp−1 is obtained by removing the m-th column from Cp and can be written as Cp−1 = Cp Ωm ,

where Ωm is a p × (p − 1) matrix with all zeros in the m-th row and only one non-zero element 1 in each column and each of the rest rows. Let θ = Qp−1 d, where Qp−1 = Cp−1 (CTp−1 Cp−1 )−1 CTp−1 is the orthogonal projection matrix of Cp−1 . Then it follows from (8) that ∥ d − Qp d ∥2 < ∥d − Qp θ∥2 =∥ d − Cp Ω(CTp−1 Cp−1 )−1 CTp−1 d ∥2 , =∥ d − Cp−1 (CTp−1 Cp−1 )−1 CTp−1 d ∥2 , = ∥d − Qp−1 d∥2

T

ξp (ω) < ξp−1 (ω).

C. Case C: Double-sided Reduction and p = q = 4 The set of four suppression distances exemplified in Case C in this paper is ς4 = (ω0 − M/2, ω0 , M − 1 − ω0 , 3M/2 − 1 − ω0 ), ω0 < 0. The indexes of the reserved subcarriers are 0, floor(M/3) − 1, 2 floor(M/3) − 1 and M − 1. III. E FFECT OF S IDELOBE S UPPRESSION Although the mainlobe of the signal spectrum varies with different values of p and ωi , the maximum power of the spectrum remains almost unchanged. Thus variation of the power difference between the sidelobe and mainlobe can be well approximated by variation of the sidelobe power. Hence, we use the sidelobe power directly to study the effect of SSOP on sidelobe suppression, in particular, the dependence of the suppression effect on the parameters p and ωi s. For √ convenience, the power coefficient λ, which does not change the power ratio between the sidelobe and mainlobe, will be ignored in this section. We now examine the effect of the suppression distances on the out-of-band emissions. Temporarily, denote the M ×p matrix C with p suppression distances as Cp , and its associated projection matrices as Pp and Qp . At a normalized frequency ω the mean power of the emission after SSOP becomes ∗

∗

˜ | ] = E[|d Pp z ˜| ], ξp (ω) = E[|d x 2

2

(16)

where, for convenience, we simplified d(ω) as d. Assume the elements s˜i , i = 0, · · · , M − q, in ˜s are identically and independently distributed (i.i.d.) random variables with zero mean and variance σs2 . Since Pp is a semi-positive definite matrix, ξp (ω) is maximized when q = 0 and we have ξp (ω) < σs2 d∗ Pp PTp d = σs2 ∥ d − Qp d ∥2 . The difference between ξp (ω) and with p increasing.

σs2

(17)

∥ d − Qp d ∥ decreases 2

(19)

Combining (17) and (19), we get

˜ = [0, s , 0] , where ˜s is an (M − 2) × 1 vector. and z ˜T

(18)

(20)

Thus we have the following theorem: Theorem 1: Given a set of suppression distances ς the use of any subset of ς to define the precoding matrix will result in an increase in the sidelobe power at every frequency in the outof-band range except at the suppression distances where the power is zero; given a set of suppression distances the use of an expanded set obtained by introducing any number of additional distinct suppression distances to define the precoding will result in a decrease in the sidelobe power at every frequency in the out-of-band range except at the suppression distances where the power is zero. Theorem 1 indicates that we can achieve better sidelobe suppression by using more suppression distances in the SSOP scheme. Recalling the requirement of q ≥ p, more reserved subcarriers will be required for successful symbol detection in the receiver without introducing ISI. Repeatedly applying (19) until p = 1, we get ξp (ω) < ξA (ω) ,

(21) −1 T

∥ d − c(c c) c d ∥ ( ) |d∗ c|2 2 ∗ = σs d d − T , c c σs2

T

2

(22)

where ξA (ω) approximates the out-of-band emission in Case A, and it also serves as an upper bound for those using other values of p. Fig. 1 shows some numerical results for normalized values of ∥ d − Qd ∥2 with various suppression distance ωi s. The power is normalized to the maximum one obtained when ω is an integer in the range [0, M − 1]. The result without sidelobe suppression is also plotted for comparison. For clarity of illustration, only envelop of each curve is shown. The figure supports the conclusion in Theorem 1, and shows that significant sidelobe suppression can be achieved by the proposed SSOP scheme.

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IV. DATA S YMBOL R ECOVERY IN R ECEIVER In Section II the use of a specific set of null reserved subcarriers is proposed as means of achieving ISI free recovery of ˜s and then information symbols. A recovery algorithm is described in this section. ISI free symbol recovery is feasible for most scenarios without incurring significant performance degradation. Consider a multipath channel with frequency domain chan˜ = [h ˜0, h ˜ 1, · · · , h ˜ M −1 ]. The received signal nel coefficients h after DFT can be expressed as √ ˜ = λDP˜ ˜, y z+n (23) where D is a diagonal matrix with diagonal elements being ˜ 0, h ˜ 1, · · · , h ˜ M −1 , n ˜ is AWGN noise with zero mean and h 2 variance σn . When a simple zero forcing equalizer is applied, the equalized output becomes 1 1 ˜r , √ D−1 y ˜ = P˜ ˜. z + √ D−1 n (24) λ λ Let ˜rr and ˜rd be sub-vectors constructed from ˜r with indexes corresponding to the reserved subcarriers and data sub˜ r and carriers, respectively, and let Dr and Dd , Cr and Cd , n ˜ d be the matrices and vectors obtained by the corresponding n ˜ respectively. Equation (24) can partitioning of D, C and n then be expressed as √ ˜ d / λ, ˜rd = ˜s − Cd (CT C)−1 CTd ˜s + D−1 (25) d n √ −1 T −1 T ˜ r / λ, ˜rr = −Cr (C C) Cd ˜s + Dr n where Cr is a q×p sub-matrix of C with its k-th row elements being cS(k) (ωi ), i = 0, · · · , p − 1 and S(k) represents the k-th element in S as defined in (11); Cd is an (M − q) × p submatrix of C obtained by removing Cr from C, and Dd , Dr , ˜ r and n ˜ d are similarly defined. n If Cr has full column rank, i.e., rank(Cr ) = p, there exists a pseudo-inverse (Cr )† = (CTr Cr )−1 CTr of Cr such that (Cr )† Cr = I. Applying (Cr )† to (25) yields ⟨˜s⟩ = ˜rd − Cd (Cr )† ˜rr √ √ ˜ d / λ − Cd (Cr )† D−1 ˜ r / λ, = ˜s + D−1 r n d n

level of complexity, particularly for Case A and B, compared to most of other published sidelobe suppression methods. V. SNR A NALYSIS AND PARAMETER C ONFIGURATION In this section, we develop the SNR analysis for the ISI-free SSOP receiver, and investigate the effect of SSOP parameters including p, q and ωi , i = 0, · · · , p − 1 on system performance. Both the block SNR, which is the averaged SNR over a block of symbols in one subband, and the per-symbol SNR, which is the mean SNR at each symbol averaged over the information symbols mapped to this symbol position at different time, are analyzed. Insights on optimizing parameters are provided based on block SNR analysis. Per-symbol SNR analysis also results in simple approaches of improving system performance. A. Mean Block SNR Analysis ˜ in (12) can be computed as The mean signal power of x [ ] E[∥˜ x∥2 ] = λE ˜s∗ (I − Cd (CT C)−1 CTd )˜s ( ) = λσs2 M − q − Tr(Cd (CT C)−1 CTd ) . (27) To achieve unit normalized signal power, one has λ=

1 . − q − Tr(Cd (CT C)−1 CTd ))

(28)

The mean noise power in ⟨˜s⟩ defined in (26) is ˜ r ∥2 ]/λ ˜ d − Cd (Cr )† D−1 (29) η 2 , E[∥D−1 r n d n 2 ( ) σ = n Tr((Dd D∗d )−1 ) + Tr((D∗r Dr )−1 (Cd C†r )T Cd C†r ) , λ and the mean block SNR of ⟨˜s⟩ is given by γSSOP =

(M − q)σs2 . η2

(30)

For AWGN channel, the mean noise power η 2 in (29) can be simplified to

(26)

where ⟨x⟩ denotes the estimate of x. The data symbol vector ˜s can be recovered from ⟨˜s⟩. This receiver achieves ISI-free estimation, and it actually implements zero-forcing equalization to remove the effect of the orthogonal projection matrix P on data symbols. We now consider the complexity of the SSOP transmitter and ISI-free receiver. To minimize the implementation complexity, the projection matrix P is not pre-computed and used directly. Instead, the following matrices are computed: the M × p matrix C, p × (M − q) matrix U , (CT C)−1 CTd and (M − q) × q matrix V , Cd (Cr )† . The pre-computed U and C matrices are stored in the transmitter and the precomputed V matrix is stored in the receiver. The steps and complexity involved in implementing the precoding according to (12) in the transmitter and the recovery of ˜s from ˜r according to (24) are shown in Table I. From the table we can see that the SSOP approach involves only a modest

σs2 (M

η2 =

) σn2 ( M − q + Tr(Cd (CTr Cr )−1 CTd ) . λ

(31)

1) Block SNR Degradation with SSOP: For a system without sidelobe suppression, ˜= x

1 √ ˜, z σs M − q

and the mean block SNR at the receiver is given by γ0 =

1 . Tr((Dd D∗d )−1 )σn2

(32)

As shown in Appendix A, Tr(Cd (CT C)−1 CTd ) < p. When p/(M − q) is small, λ in (28) can be well approximated by 1/(σs2 (M − q)), and γSSOP becomes γSSOP ≈

(

1

). σn2 Tr((Dd D∗d )−1 ) + Tr((D∗r Dr )−1 (Cd C†r )T Cd C†r ) (33)

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Since the diagonal elements of (Cd C†r )T Cd C†r are always positive, Tr((D∗r Dr )−1 (Cd C†r )T Cd C†r ) > 0, which leads to γSSOP < γ0 .

(34)

This indicates that SSOP with ISI-free receiver always degrades the mean block SNR. Fig. 2 shows some numerical results of SNR degradation based on (30) and (32) for Case A, B and C, where M = 64 and ω0 varies on the horizontal-axis in the figure. Both AWGN and Rayleigh channels are tested. The Rayleigh channel has 8 i.i.d. non-zero taps in the time domain. Results for Rayleigh channel are averaged over 2000 realizations. We can see the SNR degradation is in the range of [−0.5, −3]dB for the three cases, larger p generally leads to more SNR degradation, and the degradation is smaller in fading channels compared to AWGN channels. We can show analytically that the SNR degradation for Case A in an AWGN channel is ω0 −10 log10 (1 − M −1−ω )dB. The derivation is omitted due to 0 the page limit. 2) γSSOP as a function of ωi with fixed p = q: From Fig. 2, we can see that γSSOP is a monotonic function of the suppression distance for AWGN channels. γSSOP consistently increases with |ω0 | decreasing in Case A and B, while it decreases in Case C. For Rayleigh fading channels, although the curves are not very smooth, the monotonicity can still be observed. B. Mean Per-Symbol SNR Analysis We have seen in Section V-A1 that SSOP causes block SNR degradation. In this section, we develop the mean per-symbol SNR for two typical systems: DFT-OFDM and OFDM. It is shown that DFT-OFDM has an interesting SNR profile, which can be exploited to improve the performance for both systems. 1) DFT-OFDM systems: In DFT-OFDM systems, ˜s = Fu, where u represents an (M −q)×1 vector of data symbols, and F is the DFT matrix. DFT-OFDM is one type of precoding OFDM systems. It can significantly reduce the peak-to-average power ratio of the signal. This scheme is known as SCFDMA uplink signalling scheme in LTE, and single carrier with frequency domain equalization (SC-FDE) in WiMAX. Following (26), the estimate of u is given by √ √ ˜ d / λ − F∗ Cd (Cr )† D−1 ˜ r / λ. (35) ⟨u⟩ = u + F∗ D−1 r n d n Since each column in Cd represents positive low-frequency signal, each column of F∗ Cd is conjugate symmetric, and gains of the coefficients from the edge to the center decrease. In particular, the first row of F∗ Cd condenses significant portion of the matrix energy as it is obtained via constructive summation of the elements in Cd . Thus the per-symbol SNR distribution over a subband shows the shape of a hump. This can be clearly seen from Fig. 3, which shows the per-symbol SNR variation for Case A, B and C in an AWGN channel. The per-symbol SNR variation is defined as the SNR ratio at each data symbol between systems with and without SSOP applied. As analyzed above, the major degradation appears at the two edges of the data block. For example, there are one symbol for Case A and B, and three symbols for Case

C, which suffer SNR degradation more than 12dB. Most of other symbols see small degradation, and a large amount of symbols in the middle even have higher SNR than those in the system without sidelobe suppression. With M increasing, the maximum degradation will increase as more samples are constructively summed up. Although the results shown in the figure are based on AWGN channels, similar observation can be found in frequency selective channels because fading channel only ˜ r , but has no effect on the coefficients changes the value Dr−1 n in the matrix multiplied to it. To take advantage of this property, we can simply avoid using the symbols with extensive noise enhancement for information transmission. For example, by excluding the first symbol in Case B, about 30% noise power is removed. This symbol can also be used to further reduce the sidelobe power by applying the method proposed in [21]. On the other hand, we may wish to use all the symbols and randomize the noise over all symbols. One approach is to multiply a pseudo-random phase shift term to each subcarrier before doing SSOP. A diagonal phase shfiting matrix is hence multiplied to ˜s, which introduces phase shift to Cd and can distribute noise evenly to the whole subband. 2) OFDM systems: For OFDM systems, ˜s represent the data symbols. From (26), compared to systems without sidelobe suppression, the extra noise term here is √ ˜ r / λ. Since adjoint elements in each row and Cd (Cr )† D−1 r n column of Cd vary slowly, there is no abrupt change of the noise power between adjoint subcarriers. However, motivated by the observation in DFT-OFDM systems, we can still improve the mean per-symbol and block SNRs. The basic idea is to first apply a DFT to ⟨˜s⟩ in (26) to convert the signal to time domain, then set samples which are subject to larger noise power in the DFT output to zeros, finally apply an IDFT to the signal and convert it back to frequency domain for symbol demodulation. Note that this is effective only when the average block SNR is small so that the signal energy is not larger than the noise energy at the point where the noise is to be nulled out. For the configurations used to generate Fig. 3, this SNR threshold is about 14dB for the first symbol in Case B and C. In actual implementation, the above steps can be largely simplified as to be described next. Referring to (35), assume we want to remove significant noise terms at symbols with index set µ. Denote the index set of those retained symbols as ν. Dividing the IDFT matrix into two parts according to the set µ and ν gives F∗ = (Fµ Fν )∗ . Then the process described in the basic idea above can be represented by ⟨˜s⟩1 = Fν F∗ν ⟨˜s⟩ = ⟨˜s⟩ − Fµ (F∗µ ⟨˜s⟩),

(36)

where 0 denotes a zero matrix with the same size to F∗µ . Thus only 2µ(M − q) multiplications are required in this implementation where µ is the size of the index set µ. If only the first noise term, which is always the largest one, is to be removed, the process will be very simple as now Fµ is an all-one row vector, and (36) becomes ⟨˜s⟩1 = ⟨˜s⟩ − mean(⟨˜s⟩).

(37)

7

VI. N UMERICAL AND S IMULATION R ESULTS In this section we present numerical and simulation results to illustrate the effectiveness of the proposed SSOP scheme and to highlight its characteristics. Results for Case B and C are presented and compared to the case without suppression where 0’s are assigned to the two reserved subcarriers at the edges of the subband. The suppression distances are fixed at −64 and 127 for Case B, −64, −32, 95 and 127 for Case C, and the simulated are mainly ZP-systems, unless noted otherwise. To illustrate the sidelobe suppression effect, the PSD as calculated according to the digital signal model [18], [19] is plotted. The PSD is obtained by computing the power of DFT coefficients of each time-domain OFDM symbol over the period of 1/δf + Tg and averaging over thousands of symbols, where Tg is the period of the guard interval. Using a digital model for PSD is to be consistent with the DFTbased implementation of modern multicarrier systems, and to evaluate the effect of the sampling rate of DAC on the proposed SSOP scheme. Signals are over-sampled in both frequency and time domain to generate PSD. Frequencydomain oversampling emulates the interpolation effect in the time domain, and different oversampling rates (OSR) represent different sampling periods in DAC. Time-domain oversampling is used to increase the resolution of the PSD. The OSR is defined as the ratio between the number of samples before and after the particular oversampling operation. A constant timedomain OSR of 4 is used in all the simulations. For clarity, only envelop of the PSD is plotted. To illustrate receiver performance, bit error rate (BER) of the ISI-free receivers is presented for 16QAM and 64QAM modulation. Both uncoded and coded systems are simulated. For coded systems, 1/2-rate convolutional coding and soft Viterbi decoding are used. The channel is assumed to be perfectly known at the receiver, and simple zero forcing equalization is used to remove the effect of the channel.

Approaches smoothing the transition, such as the one proposed in [12], can be combined with the SSOP scheme to improve the sidelobe suppression in CP systems. The effect of different sampling rates on PSD is illustrated in Fig. 5. We can see that sampling period has a notable influence on the sidelobe reduction effect of the SSOP scheme, particularly for the CP systems. With OSR increasing, the sidelobe levels are further reduced, and for ZP systems, they converge to those predicted by the theoretical curves in Fig. 1. Fig. 6 compares the sidelobe suppression effect between the proposed SSOP scheme and the CC [8] and orthogonal precoding (OP) [14] methods in the literature. In the CC method, the power of each cancellation carrier is limited to be no more than that of the mean power of data symbols. The simulated systems are configured to minimize the out-of-band emission close to the mainlobe, to emulate generating deep nulls within a very narrow-band in cognitive radio applications. The SSOP scheme shows slightly better performance than the OP method, and both methods can quickly reduce the sidelobe power. The BER performance of uncoded OFDM systems in AWGN channels is illustrated in Fig. 7. The performance degradation with the application of the SSOP scheme is in the range of 1 to 3dB consistent with the analytical results shown in Fig. 2. The ISI-free receiver with the noise reduction option proposed in Section V-B2 has also been simulated for both AWGN and Rayleigh fading channels with exponential power delay profile. The multipath delay spread is approximately 10 taps and the zero padding is 16-samples long. The coded BER results are shown in Fig. 8 where SSOP Case B is used and the modulation is 16QAM. The noise reduction method improves BER performance for both channels, with a maximum 2dB for the AWGN channel and approximately 1dB for the Rayleigh channel. As pointed out in Section V-B2, the improvement for AWGN channel stops at a SNR close to 14dB.

A. OFDM Systems

B. DFT-OFDM Systems

In the simulated OFDM systems, only one subband is present and the total number of subcarriers is N = M = 64. Fig. 4 illustrates the PSD for both ZP and CP OFDM systems with a guard interval of 16 and frequency-domain OSR of 16. The PSD with a raised cosine window function of rolloff factor 0.11 in the CP system is also shown for comparison. It is noted that the windowing approach may cause significant SNR degradation in ZP systems as the window needs to span within the OFDM symbol period to operate effectively [10]. Although the proposed SSOP scheme is inferior to the windowing approach for the CP systems, it achieves significant sidelobe suppression for the ZP systems. Furthermore, this is achieved with only two or four reserved subcarriers and marginal computation complexity. The good match between the PSD here and the numerical envelop curves in Fig. 1 confirms the suppression analysis and the assumptions made in Section III. We also note that the performance degradation of sidelobe suppression in CP systems is probably due to the nonsmooth transition between two consecutive OFDM symbols.

The DFT-OFDM systems considered in the simulations involve a multi-band scenario, which can be typically found in cognitive radio applications. In the simulations the total number of subcarriers is N = 512 and the number of subcarriers in each band is M = 64. Three subbands, locating on subcarriers [0, 64], [255, 319] and [447, 511], are used for information transmission. SSOP is applied to each subband independently, and M = 64. For a given number of reserved subcarriers q in each subband, a (192 − 3q)-point DFT is applied to (192 − 3q) data symbols, and the outputs are allocated to the data subcarriers in each subbands. In the simulations a Rayleigh fading channel with exponential power decay profile is used. The multipath delay spread is 25 samples and the ZP length is set to be 32 to avoid ISI. Fig. 9 shows the PSD of the transmitted signal for SSOP with Case B, C and without suppression. Similar to the sidelobe suppression effect in the “out-of-band” regions, the emission between subbands is equally reduced. Fig. 10 shows the signal PSD measured with a spectrum analyzer in an

8

actual system under development in CSIRO. The configuration of the system is similar to those used in generating Fig. 9 with the exception of different subbands. As can be seen from the figure, the PSD in-between the subbands of the practical signal, which is slightly increased due to hardware imperfections such as limited word-length and non-linearity of RF components, is quite close to the simulated one. The two edges of the PSD is lower because of the filtering effect in sampling rate conversion and digital to analog conversion. Fig. 11 shows the coded BER for DFT-OFDM systems with 64QAM modulation. The solid and dashed curves are for the cases when the first symbol at the DFT input is not used or used for transmitting information respectively. It is clear that avoiding using the first symbol that is largely contaminated by noise improves performance. VII. C ONCLUSIONS A novel sidelobe suppression with orthogonal projection (SSOP) scheme has been presented. Both analytical and simulation results show that the proposed scheme can achieve significant sidelobe suppression with very high spectrum efficiency and very low computational complexity. It is demonstrated that in typical applications, only 1 or 2 reserved subcarriers are required, and the total computational complexity in a transceiver for SSOP is 3M −2 or 6M −8 multiplications, where M is the number of subcarriers in the subband of interest. The SSOP scheme can be flexibly adapted to a range of different requirements on sidelobe suppression and BER performance. It can also be combined with most frequency domain pre-processing techniques such as power loading where SSOP should be implemented after the processing. The proposed SSOP scheme is very promising for applications in both conventional OFDM systems and cognitive radio systems. It should be noted that BER performance degradation with the simple ISI-free receiver suggests the need to develop advanced receivers such as minimum mean square error and sphere decoding receivers. Improvement to the sensitivity of the SSOP scheme to the frequency-domain oversampling rate is also under investigation. Better extension of the SSOP concept to CP-systems is yet to be investigated to improve the suppression performance. A PPENDIX A: T RACE OF AN ORTHOGONAL PROJECTION MATRIX T

−1

T

For Q = C(C C) C , its trace can be represented by ( ) Tr(Q) = Tr [CTd , CTr ]T (CT C)−1 [CTd , CTr ] ( ) ( ) = Tr Cd (CT C)−1 CTd + Tr Cr (CT C)−1 CTr . (38) Since Q is an orthogonal projection matrix, Tr(Q) equals to its (rank [20], which )is p in the( SSOP scheme. Thus we get ) Tr Cd (CT C)−1 CTd = p − Tr Cr (CT C)−1 CTr . Since (CT C)−1 is a positive definite matrix, we further have ( ) ( ) Tr Cd (CT C)−1 CTd < p and Tr Cr (CT C)−1 CTr < p. (39)

R EFERENCES [1] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Select. Areas Commun., vol. 23(2), pp. 201–220, Feb. 2005. [2] M. S. El-Saadany, A. F. Shalash, and M. Abdallah, “Revisiting active cancellation carriers for shaping the spectrum of OFDM-based cognitive radios,” in SARNOFF’09. Piscataway, NJ, USA: IEEE Press, 2009, pp. 526–530. [3] A. Vahlin and N. Holte, “Optimal finite duration pulses for ofdm,” Communications, IEEE Transactions on, vol. 44, no. 1, pp. 10 –14, Jan. 1996. [4] M. El-Saadany, A. Shalash, and M. Abdallah, “Revisiting active cancellation carriers for shaping the spectrum of ofdm-based cognitive radios,” in Sarnoff Symposium, 2009. SARNOFF ’09. IEEE, 30 2009-april 1 2009, pp. 1 –5. [5] I. Cosovic, S. Brandes, and M. Schnell, “Subcarrier weighting: A method for sidelobe suppression in OFDM systems,” IEEE Commun. Lett., vol. 10 (6), pp. 444–446, Jun. 2006. [6] H. A. Mahmoud and H. Arslan, “Sidelobe suppression in OFDMbased spectrum sharing systems using adaptive symbol transition,” IEEE Commun. Lett., vol. 12 (2), pp. 133–135, Feb. 2008. [7] H. Yamaguchi, “Active interference cancellation technique for MBOFDM cognitive radio,” in Proc. 34th IEEE Eur. Microw. Conf., vol. 2, Oct. 2004, pp. 1105–1108. [8] S. Brandes, I. Cosovic, and M. Schnell, “Reduction of out-of-band radiation in OFDM systems by insertion of cancellation carriers,” IEEE Commun. Lett., vol. 10 (6), pp. 420–422, Jun. 2006. [9] D. Qu and Z. Wang, “Extended active interference cancellation for sidelobe suppression in cognitive radio OFDM systems with cyclic prefix,” IEEE Trans. Veh. Technol., vol. 59 (4), pp. 1689–1695, May 2010. [10] Y.-P. Lin and S.-M. Phoong, “Window designs for DFT-based multicarrier systems,” IEEE Trans. Signal Processing, vol. 53, no. 3, pp. 1015 – 1024, march 2005. [11] C.-D. Chung, “Spectrally precoded OFDM,” IEEE Trans. Commun., vol. 54, no. 12, pp. 2173 –2185, dec. 2006. [12] J. van de Beek and F. Berggren, “N-continuous OFDM,” IEEE Commun. Lett., vol. 13, no. 1, pp. 1 –3, Jan. 2009. [13] R. Xu and M. Chen, “A precoding scheme for DFT-based OFDM to suppress sidelobes,” IEEE Commun. Lett., vol. 13 (10), pp. 776–778, Oct. 2009. [14] M. Ma, X. Huang, B. Jiao, and Y. J. Guo, “Optimal orthogonal precoding for power leakage suppression in DFT-based systems,” IEEE Trans. Commun., vol. 59(3), pp. 844–853, March 2011. [15] H.-M. Chen and C.-D. Chung, “Adaptive spectrally precoded OFDM with cyclic prefix,” in Wireless and Optical Communications Conference (WOCC), 2010 19th Annual, may 2010, pp. 1 –5. [16] H. G. Myung, J. Lim, and D. J. Goodman, “Single carrier FDMA for uplink wireless transmission,” in IEEE Vehicular Technology Magazine, vol. 1(3), Sep. 2006, pp. 30–38. [17] Y.-P. Lin and S.-M. Phoong, “OFDM transmitters: analog representation and DFT-based implementation,” Signal Processing, IEEE Transactions on, vol. 51, no. 9, pp. 2450 – 2453, sep. 2003. [18] G. Cuypers, K. Vanbleu, G. Ysebaert, and M. Moonen, “Intra-symbol windowing for egress reduction in DMT transmitters,” EURASIP J. Appl. Signal Process., vol. Article ID 70387, pp. 1–9, 2006. [19] T. van Waterschoot, V. Le Nir, J. Duplicy, and M. Moonen, “Analytical expressions for the power spectral density of CP-OFDM and ZP-OFDM signals,” IEEE Signal Processing Lett., vol. 17, no. 4, pp. 371 –374, apr. 2010. [20] S.-G. Wang and S.-C. Chow, Advanced Linear Models: Thoery amd Applications. New York: Marcel Dekker, Inc., 1994. [21] M. Ma, X. Huang, and Y. Guo, “An interference self-cancellation technique for SC-FDMA systems,” Communications Letters, IEEE, vol. 14, no. 6, pp. 512 –514, Jun. 2010.

9

2 0 Per−symbol SNR variation (dB)

0

−20

Energy in dB

−40

−60 Case C, ω0= −32 −80

Case A, ω0= −64 Case B, ω0= −64

Case C Case B Case A

−4 −6 −8 −10 −12 −14

Case B, ω0= −32

−100

−2

−16

No suppression

0

10

20

−120 −80

−60

−40

−20 0 20 normalized freq. ω

40

60

80

30 40 Index of data symbol

50

60

100

Fig. 3. Per-symbol SNR variation between DFT-OFDM systems with and without SSOP applied where M = 64.

Fig. 1. Envelop of normalized ∥ d−Qd ∥2 for various suppression distances, where M = 64, and q = p reserved subcarriers are used, and 1 reserved subcarrier in each edge of the subband is used in the case of no suppression. 0 −10 −20 No suppression −30 CP, Case B

Compute U˜ s Compute u , C(U˜ s) ˜−u Compute z Compute V˜ rr Compute ˜ rd − (V˜ rr )

Tx Rx

Numbers of multiplications p(M − q) pM 0 p(M − q) 0

PSD (dB)

−40

Step

Numbers of additions p(M − q) pM M p(M − q) M −q

−50 CP, Case C −60 Windowing −70 ZP, Case C −80

ZP, Case B

−90

TABLE I I MPLEMENTATION OF SSOP WITH ISI- FREE R ECEIVER AND ITS COMPLEXITY: T X FOR T RANSMITTER AND R X FOR R ECEIVER .

−100 −120

−100

−80

−60

−40 −20 0 Normalized Freq. ω

20

40

60

80

Fig. 4. PSD for both CP and ZP OFDM systems with the SSOP scheme and the windowing scheme, where the frequency-domain OSR is 16 and modulation is 64QAM.

0

−0.5

−10

Case C Case A Case B

−20 CP, OSR=4, 8, 16, 64. −30

−1.5

PSD (dB)

SNR Degradation (dB)

−1

−40

−50

−2 −60

ZP, OSR=4, 8, 16, 64

Analytical degradation for Case A −2.5

−70

−80

−3 −140

−120

−100

−80 ω0

−60

−40

−20

Fig. 2. γSSOP /γ0 for Case A, B and C under AWGN (Dashed line) and Rayleigh channels (Solid line).

−90 −150

−100

−50 0 Normalized Freq. ω

50

100

Fig. 5. Influence of OSR on the PSD of OFDM signals with SSOP Case B. Only one period of periodic PSD is displayed for each oversampling rate. Modulation is 64QAM.

10

0

0

10

−10 −1

−20

10

No suppression

CC, Ncc=2 CC, Ncc=4

Rayleigh Channel Normal SSOP Rx, Rayleigh Noise Reduced, Rayleigh Noise Reduced, AWGN Normal SSOP Rx, AWGN No suppression, Rayleigh No suppression, AWGN

−2

Coded BER

PSD (dB)

AWGN −30

OP, R=2

−40

−50 SSOP, p=2 −60

−70 −60

OP, R=4

−50

−40

10

−3

10

SSOP, p=4

−30

−20 −10 Normalized Freq. ω

0

10

20

Fig. 6. PSD of ZP-OFDM signals with different sidelobe suppression techniques where the suppression distances are −15 and 78 for Case B, and −27,−7,70 and 90 for Case C in the SSOP scheme, frequency domain OSR is 8, and modulation is 64QAM. Parameters as shown in the figure are used to ensure same spectrum efficiency for the three schemes.

−4

10

−5

10

6

7

8

9

10 SNR (dB)

11

12

13

14

Fig. 8. BER of coded OFDM systems with 16QAM in AWGN and Rayleigh channels.

−1

10

0 Case B Case C No suppression

−10

−2

10

−20 −30 −3

10

PSD (dB)

Uncoded BER

64QAM

16QAM

−40 −50

−4

10

−5

10

−60

16QAM, Case B 64QAM, Case B 64QAM, No suppression 16QAM, No suppression 16QAM, Case C 64QAM, Case C

−70 −80

−90 −200

14

15

16

17

18 SNR

19

20

21

22

Fig. 7. BER of uncoded OFDM systems with 16QAM and 64QAM modulations in AWGN channels.

−100

0

100

200 300 400 Normalized freq. ω

500

600

700

800

Fig. 9. PSD of DFT-OFDM signals with SSOP Case B, C and without suppression, where the frequency-domain OSR is 1 and the modulation is 64QAM.

11

Fig. 10. Measured PSD of DFT-OFDM signals in the output of the power amplifier in a real hardware system where the SSOP Case B is implemented. The frequency-domain OSR is 8 and the modulation is 64QAM.

No Suppression Case B Case C

−1

Coded BER

10

−2

10

−3

10

14

16

18

20 SNR (dB)

22

24

26

Fig. 11. BER of coded DFT-OFDM systems with 64QAM modulation in Rayleigh fading channels when the first symbol at the DFT input is used (dashed curves) or not used (solid curves).