RESEARCH Open Access On the EVM computation of arbitrary clipped multi-carrier signals Igal Kotzer * and Simon Litsyn Abstract A common figure of merit in multi-carrier systems is the error vector magnitude (EVM). A method for EVM computation of a multi-carrier signal without any underlying model (e.g., the Gaussianity assumption) was proposed in a previous work of the authors. However, it addressed only the case of identical constellations and power loadings in all tones. In practice, however, the constellation and power loading may vary among the tones (e.g., boosted pilots, waterfilling and zero guard bands). Here the earlier approach is generalized in such a way that it is able to accommodate for an accurate analytical EVM computation in the cases of power loading and different constellations for different tones. Moreover, the derivation is valid for a general magnitude clipping functio n, so that any realistic clipper can be plugged in. 1 Introduction The use of multi-carrier (MC) communication schemes (e.g., OFDM, DMT, etc.) is very common nowadays due to its ability to cope well with channel interference while keeping the receiver complexity low, the ease of spectral mask shapi ng and high spectral ef fici ency. However, on e of MC scheme’s greatest drawbacks is the high peak-to- average power ratio (PAPR) caused by various degre es o f coherent summation in the signal generati on using IFFT [1]. Thus, systems utilizing MC communications must work with a large back-off in the high-power amplifier (HPA), which reduces both the efficiency of the HPA and the average power transmitted, or risk clipping. Based on the understanding that clipping is a nonlinear operation causing both in-band and out-of-band spectral noise and thus is an undesirable operation, methods for reducing the PAPR were devised. For a survey see [1-4]. Most of the power reduction methods are either statistical in nature– that is they do not guaran tee PAPR limits, or iterative–in which required PAPR limits are easier to meet at the expense of computational complexity. Hence, while it is understood that the amount of clipping should be mini- mized, due to practical system limitations clipping cannot be entirely eliminated, but rather be set on a compromise level. Therefore, evaluating the performance of MC sys- tems with clipping becomes relevant. Two prominent criteria for evaluating the perfor- manceofaMCsystemareitscapacity[5-7],andthe system’s error probability [8,9]. However, in engineering practice, the mo st popular measure is the error vector magnitude (EVM). The EVM is a figure of merit for in- band distortion, which does not only quantifies the dis- tortion but in some cases can attribute impairments to various system component s [10]. Due to its popularity and troubleshooting capabilities, the EVM has become a mandatory part of a few communication standards, e.g. [[11], Tables 165, 172]. In [12] the authors express the EVM of an OFDM signal impaired by clipping without relying on the Gaussianity assumption and show that the EVM can be expressed with an arbitrary precision as a power series of the number of tones with constellation-dependent coefficients. It is also shown that for some specific constellations the EVM can be calculated via easy to use expressions without the need for a power series expansion. However, these computa- tions fit the case of MC signals with an identical constella- tion for all tones and no power loading. Yet, real world signal utilize both different constel lations for different tones and power loading. Some o f the to nes are zeroed due to spectral mask considerations, while some tones are boosted (e.g., pilot tones) to allow better channel tracking. A waterfilling solution in high SNR MIMO OFDM or in DMT also requires adjusting power and thus constellation to each tone individually. In this paper we address the issue of various constellations and power loading in the * Correspondence: igalk@eng.tau.ac.il School of Electrical Engineering - Systems, Tel Aviv University, Israel Kotzer and Litsyn EURASIP Journal on Advances in Signal Processing 2011, 2011:36 http://asp.eurasipjournals.com/content/2011/1/36 © 2011 Kotzer and Litsyn; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work i s properly cited. MC signal as well as the effect of an arbitrary magnitude clipping response by giving an analytical expression in the form of a power series for computing the EVM of the gen- eralized cas e. Analysis of clipped signals usually relies on the Gaus- sianity assumption [5]. However, this assumption is not always valid, especially for a mix of BPSK and QAM constellations. Hence, in order to evaluate the perfor- mance of such systems one must resort to numerical evaluations. This work allows to accurately compute the EVM of clipped signals for any constellation mixture and clipping function without the need to redo the numerical evaluation for each desired scenario. The paper is organized as follows. In Section 2 the system model used in this work is introduced. Theorem 3.1 in Section 3 presents the main result of this work. In Section 4 we present simulation results and compare them to the theoretical results about EVM derived in this work. 2 System model The system model discussed in this work is depicted in Figure 1. The vector a =[a 0 , a 1 , , a N-1 ] T denotes the N data symbols vector in the form of constellation po ints, e.g., a Î {+1, -1} N for BPSK. The vector x =[x 0 , x 1 , , x N-1 ] T denotes the time domain discrete time signal and is obtained by applying the inverse discrete Fourier transform on a: x n = 1 √ N N−1  k = 0 a k e i2πkn N ,0≤ n ≤ N − 1 . (1) The vector y denotes the vector x after clipping opera- tion. Two clipping function s we will specifically address are the SSPA clipper [13]: y n = x n  1+  |x n | c  2p  1/2p , (2) and the soft clipper (which is a special case of the SSPA clipper for p ≫ 1): y n =  x n |x n | < c ce ix n |x n |≥c , (3) where c is the clipping level. The noise vector w denot es an AWGN with variance σ 2 w and is independent of a. ˆ x is the noisy clipped discrete time domain signal and ˆ a is the d ata symbols vector reconstructed from t he clipped and noisy signal. For this system we define the EVM as EVM   E{| ˆ a −a| 2 } N · E{|a k | 2 } . (4) Assuming the constellation energy E {| a k | 2 } is known and the noise variance is known, we need to calculate the error power E {| ˆ a −a | 2 } to be able to evaluate the EVM. By virtue of Parseval’s theorem, we have N−1  n=0 |x n | 2 = N−1  k = 0 |a k | 2 . (5) Hence, it immediately follows that E{| ˆ a − a| 2 } = E{| ˆ x − x| 2 } = N−1  n = 0 E{|y n − x n | 2 } + Nσ 2 w . (6) The EVM contribution due to clipping can thus be calculated by computing the quantity E{| y n − x n | 2 } for every 0 ≤ n ≤ N - 1. Obviously, for scenaria with large channel noise we can allow more signal distortion due to clipping as long as it is negligible relative to the chan- nel noise. 3 EVM computation In this section we present the main result. Let f(|x n |) = f ( r) be the energy of the clipped portion of the sample x n , and let us decompose the symbols vector a of leng th N into three groups: • N B groups of BPSK symbols. The symbols of each group, of size N r , are drawn from a constellation with energy E r = b 2 r for 1 ≤ r ≤ N B . • N Q groups of QAM symbols. The symbols of e ach group, of size N s , are drawn from a constellation of size Ms, with constellation coefficients ν s,lm (which are the series expansion coefficients of a function of the constellation -see Appendix A for details.) and energy E s = q 2 s for 1 ≤ s ≤ N Q . For example, for QPSK of the form a k R , a k I ∈ (±1/ √ 2) we have ν s,11 = -1/4, ν s,22 = -1/64, and for 16QAM of the form a k I ∈ (±1, ±3)/ √ 10 , a k I ∈ (±1, ±3)/ √ 10 we have ν s,11 = -1/4 and ν s,22 = -17/1600. • N Z zero tones. Clearly N B + N Q + N Z = N. Then, the follow ing quan- tities are defined: μ 1 = ⎡ ⎣ N Q  s=1 q 2 s N s ν s,11 N − N B  r=1 b 2 r N r 4N ⎤ ⎦ (7) IFFT FFT Clipping Function y ˆ a x a w ˆx Figure 1 Baseband discrete time AWGN channel model. Kotzer and Litsyn EURASIP Journal on Advances in Signal Processing 2011, 2011:36 http://asp.eurasipjournals.com/content/2011/1/36 Page 2 of 7 and μ 2 = ⎡ ⎣ N Q  s=1 q 4 s N s ν s,22 N 2 − N B  r=1 b 4 r N r 32N 2 ⎤ ⎦ . (8) In addition, let ˜ μ 1 = N μ 1 and ˜ μ 2 = N 2 μ 2 . Theorem 3.1. The term  N−1 n = 0 E{|y n − x n | 2 } in (6) can be calculated as follows: N−1  n=0 E{|y n − x n | 2 } = N 2 ∞  q =0 m q (c)N q , (9) where m q (c) depend on the clipping level, the constella- tions, power loading and symbol length. In particular, m 0 (c)and m 1 (c) can be calculated as follows: m 0 (c)=−  ∞ 0 rf (r) exp  r 2 4μ 1  1 2 ˜μ 1 + ˜μ 2 ˜μ 3 1  dr , (10) m 1 (c)=−  ∞ 0 rf (r) exp  r 2 4μ 1  ˜μ 2 2 ˜μ 4 1 r 2 + ˜μ 2 32 ˜μ 5 1 r 4  dr . (11) Proof. See Appendix A. □ 4 Simulation results and discussion 4.1 The Gaussian approximation A common method for analyzing the EVM of an OFDM sig nal uses the central l imit theorem (CLT). By invoking the CLT x n are assumed to be distributed complex nor- mally, i.e. x n ∼ CN ( 0, σ 2 ) ,andthus|x n | ~ Rayleigh(s ). Hence, the EVM can be computed in a straightforward method: N−1  n = 0 E{|y n − x n | 2 } = N  ∞ 0 f (r) r σ 2 exp  − r 2 2σ 2  dr , (12) where f(r)=f(| x n |)is the clipping function in polar coordinates. In this work, when the results are com- pared to the Gaussian approximation it is assumed that s 2 =1. 4.2 Simulation results In the following examples two cases of magnitude clip- ping functions are considered. The SSPA clipper, for which f (|x n |)=f (r)=  r ( 1+ ( r/c ) 2p ) 1/2p − r  2 , and the soft clipper, for which f (|x n |)=f (r)=(r − c) 2 + , where the operation() + denotes taking only the positive part. The soft clipper is a special case of the SSPA clipper for p ® ∞, which can be practically achieved with p > 100. In the following simulations p =200waschosen. Figure 2a demo nstrates the EVM versus clipping level for the mixture of 64 BPSK modulated tones, 320 16QAM modulated tones and 128 zero tones, all randomly spread across the symbol. That is, N B =1,N r =1 =64,N Q =1, N s =1 = 320, ν 1,11 =-1/4andν 1,22 = -17/1600. In this figure all constellation energies are normalized to unity (i.e. b r = q s = 1). Figure 2b demonstrates the EVM versus clipping level for the mixture of 128 B PSK modulated tones with constellation energy bo osted by 3 d B, 128 QPSK modulated tones and 256 16QAM modulated tones (the two latter constellations are with unity constel- lation energy). Namely, f or Figure 2b, the simulation parameters are N B =1,N r =1 = 128, b r = √ 2 , and N Q =2 with N s =1 =128,ν 1,11 = -1/4, ν 1,22 = -1/64, q 1 =1and N s =2 = 256, ν 2,11 = -1/4, ν 2,22 = -17/1600, q 2 =1. It can be clearly seen that as the mixture becomes more diverse in tone constellations and power loading, the mixed signal’s EVM diverges from the Gaussian model. Additionally, as can be expected, the less linear the clipping function, the higher the EVM is. It can be also seen that the analytical computation coincides per- fectly with the simulation. 5 Summary In this paper we present a method for computing the EVM of a MC signal with power loading and various conste llations on various tones that is impaired by clip- ping. This computation does not rely on any underlyi ng model for the signal (such as the Gaussianity assump- tion), making it accurate for any mixture of tone con- stellations and power l oading. A comparison between the simulated and theoretical EVM resu lts shows a per- fect match between the two. The main result of this work can be also used with any arbitra ry magnitude cli pping function for achieving more realistic results for practical uses. Appendix A Proof of the EVM computation equation We define the energy of the clipped portion of the sig- nal as f (x n )=f (x n R , x n I )=|y n − x n | 2 . Any clipping func- tion can be represent ed as a superposition of its effect on the signal’s magnitude (AM/AM) and its effect on the signal’s phase (AM/PM). The AM/AM function can be further represented in terms of |x n |. Thus, f can be defined as f (  x n 2 R + x n 2 I )=f (r ) ,where r =  x n 2 R + x n 2 I is the polar coordinates represe ntation. We wish to calcu- late E{f (x n R + x n I ) } any 0 ≤ n ≤ N -1. We start by repre- senting f ( x R , x I ) by its inverse Fourier transform: f (x R , x I )= 1 2π  ∞ − ∞  f (ω 1 , ω 2 )e i(ω 1 x R +ω 2 x I ) dω 1 dω 2 , (13) Kotzer and Litsyn EURASIP Journal on Advances in Signal Processing 2011, 2011:36 http://asp.eurasipjournals.com/content/2011/1/36 Page 3 of 7 where ˆ f ( ω 1 , ω 2 ) is the Fourier transform of f ( x R , x I ) : ˆ f (ω 1 , ω 2 )= 1 2π  ∞ −∞ f (x R , x I )e −i(ω 1 x R +ω 2 x I ) dx R dx I =  ∞ 0 rf (r) 1 2π  2π 0 e −ir(ω 1 cos(θ)+ω 2 sin(θ)) dθ d r (14) =  ∞ 0 rf (r)J 0  r  ω 2 1 + ω 2 2  dr , (15) where J 0 is the Bessel function of the first kind and zeroth order. Furthermore, x n can be written explicitly as a sum of its real and imaginary parts as follows: x n R = 1 √ N N−1  k=0  a k R cos  2πkn N  − a k I sin  2πkn N  , x n I = 1 √ N N−1  k = 0  a k R sin  2πkn N  + a k I cos  2πkn N  . (16) Thus, we can substitute (16) into(13) and rewrite f ( x R , x I ) as f (x n R , x n I )= 1 2π  ∞ −∞ ˆ f (ω 1 , ω 2 ) exp  i √ N  ω 1 N−1  k=0  a k R cos  2πkn N  − a k I sin  2πkn N   + ω 2 N−1  k=0  a k R sin  2πkn N  + a k I cos  2πkn N   dω 1 dω 2 = 1 2π  ∞ −∞ ˆ f (ω 1 , ω 2 ) exp  i √ N N−1  k=0  a k R  ω 1 cos  2πkn N  + ω 2 sin  2πkn N  + a k I  −ω 1 sin  2πkn N  + ω 2 cos  2πkn N  dω 1 dω 2 . (17) Denoting φ k (α, β)=E  e i(αa k R +βa k I  , (18) and using the fact that a k are independent, we can write: E{f (x n R , x n I )} = 1 2π  (ω 1 ,ω 2 )∈R 2 ˆ f (ω 1 , ω 2 ) · N−1  k = 0 φ k  ω 1 cos( 2πkn N )+ω 2 sin( 2πkn N ) √ N , −ω 1 sin( 2πkn N )+ω 2 cos( 2πkn N ) √ N  dω 1 dω 2 . (19) Therefore, according to(15) E {f (x n R , x n I )} = 1 2π  ∞ 0 rf (r)  (ω 1 ,ω 2 )∈R 2 J 0 (r  ω 2 1 + ω 2 2 ) · N−1  k = 0 φ k  ω 1 cos( 2πkn N )+ω 2 sin( 2πkn N ) √ N , −ω 1 sin( 2πkn N )+ω 2 cos( 2πkn N ) √ N  dω 1 dω 2 dr . (20) We now proceed to calculate the term  N−1 k = 0 φ k of (20) by expanding to a power series the term  N−1 k = 0 ln φ k and then taking the exponent of the series. Unlike [12], if a k are not identically distributed then j k must be computed for every k, or alternatively for every type of constellation and then combined together. We rewrite the arguments of j k as follows: E{f (x n R , x n I )} = 1 2π  ∞ 0 rf (r )  (ω 1 ,ω 2 )∈R 2 J 0 (r  ω 2 1 + ω 2 2 ) N− 1  k=0 · φ k    ζω −kn √ N  ,   ζω −kn √ N  dω 1 dω 2 dr, (21) where ζ = ω 1 + iω 2 , ¯ ζ is the complex conjugate o f ζ and ω = exp(2πi/N). Denoting z = ζω − k n √ N we can write j k as follows: φ k (z)=φ k (z, z)=E exp {i(z · a k R + z ·a k I ) } (22) ( b ) (a) 1 1.5 2 2.5 3 3.5 −60 −50 −40 −30 −20 −10 0 10 c EVM [dB] Gaussian Approx., p=200 Mixed Theory, p=200 Mixed Sim., p=200 Gaussian Approx., p=3 Mixed Sim., p=3 Mixed Theory, p=3 1 1.5 2 2.5 3 3.5 −40 −30 −20 −10 0 10 c EVM [dB] Gaussian Approx., p=200 Mixed Theory, p=200 Mixed Sim., p=200 Gaussian Approx., p=3 Mixed Theory, p=3 Mixed Sim., p=3 Figure 2 Simulated and theoretical EVM versus clipping level for two magnitude clipping functions. (a) Mixture of BPSK, 16QAM and zero tones. (b) Mixture of 3dB Boosted BPSK, QPSK and 16QAM tones. Kotzer and Litsyn EURASIP Journal on Advances in Signal Processing 2011, 2011:36 http://asp.eurasipjournals.com/content/2011/1/36 Page 4 of 7 as in(18). We expand In j k as a power series: ln φ k (z)=lnφ k  ζω −kn √ N  =  l,m ≥ 0 ν (k) lm z l ¯z m , (23) wherewechoosetherepresentation φ k (z)=φ k (z, ¯z)=E exp {i(( z+¯z 2 )a k R +( z−¯z 2 i )a k I ) } .We further assume that the data symbols vector a contains p ≤ N distinct groups of N ι ,1 ≤ ι ≤ p symbols, where each group is drawn from the set of BPSK, QAM or zero constellation points with an average constellation energy of E ι ,1 ≤ ι ≤ p. That is, groups of symbols are distinguished by the constellation type and by t he aver- age constellation energy. Hence, we have N−1  k=0 ln φ k  ζω −kn √ N  = N 1 −1  k=0 ln φ 1  ζω −kn √ N  + N 2 −1  k=0 ln φ 2  ζω −kn √ N  +···+ N p −1  k=0 ln φ p  ζω −kn √ N  . (24) Now we proceed to compute In j k for each type of constellation: • a k = 0:Thisoptionisusuallyusedtogenerate guard bands [11]. For this option j k = 1, and hence in j k =0. • BPSK (a k =±b = b·{±1}): First, it is noted that a k are drawn from a BPSK constellation with energy E bpsk = b 2 . Next, we compute in ln φ k  ζω −kn √ N  for a group of 1 ≤ N bpsk ≤ N bins. Now, using the fact that a k are equi-probable we have φ k (z)= 1 2 e i z+¯z 2 b + 1 2 e −i z+¯z 2 b =cos  b z + ¯z 2  =cos  b  ζω − k n √ N  . (25) By Maclauren’s series expansion we have ln(cos(θ)) = ∞  j =1 ν 2j (2j)! θ 2j , (26) where ν 2 = -1, ν 4 = -2, ν 6 = -16, etc. Now, N bpsk −1  k=0 ln  cos  b  ζω −kn √ N  = N bpsk −1  k=0 ∞  j=1 ν 2j (2j)!  b  1 √ N ζω −kn  2j = ∞  j=1 ν 2j (2j)!  1 2 √ N  2j N bpsk −1  k=0 b 2j (ζω −kn + ¯ ζω kn ) 2j = ∞  j=1 ν 2j (2j)!  b 2 √ N  2j N bpsk −1  k=0 ⎡ ⎣ 2j  m=0  2j m  ( ¯ ζω kn ) 2j−m (ζω −kn ) m ⎤ ⎦ = ∞  j =1 ν 2j (2j)!  b 2 √ N  2j j  s=− j  2j j + s  ζ j+s ¯ ζ j−s N bpsk −1  k=0 ω −2kns . (27) Using N bpsk −1  k = 0 ω −2kns =  N bpsk N|2ns (2ns is a multiple of N ) 0otherwise , (28) (27) becomes N bpsk −1  k=0 ln cos{···}= ∞  j=1 ν 2j (2j)!  b 2 √ N  2j j  s=−j  2j j + s  ζ j+s ¯ ζ j−s N bpsk −1  k=0 ω −2kn s = ∞  j =1 ν 2j (2j)!  b 2 √ N  2j N bpsk j  s=− j  2j j + s  ζ j+s ¯ ζ j−s , (29) where N|2ns,-j ≤ s ≤ j and n Î [0, , N -1]. Next we compute the first two terms of (29), that is for j =1,2, as it is assumed these terms yield sufficient accuracy. The cases of n =0, N 4 , N 2 , 3N 4 require special attention. However, as the impact of the slightly different analyti- cal expression for the above four cases relative to all other n is negligible for practical values of N (e.g., N ≥ 128) these cases will be neglected and treated equally as the rest of the BPSK tones. -j = 1:Ifn ≠ 0, N/2 then the term  j s=−j  2j j + s  ζ j+s ¯ ζ j− s in (29) contains only the term s =0,so 1  s =−1  2 1+0  ζ 1+0 ¯ ζ 1−0 =2|ζ | 2 . (30) -j = 2:Ifn ≠ 0, N/4, N/2,3N/4 then the only possible term in the sum is s = 0, thus the sum is 2  s=−2  4 2+s  ζ 2+s ¯ ζ 2−s =6|ζ | 4 . (31) Going back to (29)and substituting the above expres- sions, we find the following: N bpsk −1  k=0 ln cos{···}= N bpsk  ν 2 2! b 2 2 2 N 2|ζ | 2 + ν 4 4! b 4 2 4 N 2 6|ζ | 4 + ···  = N bpsk  − b 2 |ζ | 2 4N − b 4 |ζ | 4 32N 2 −···  . (32) • M-QAM: The QAM constellation points are drawn from the set a k ∈ q  (±1, ±3, , ±( √ M − 1)) + i · (±1, ±3, , ±( √ M − 1))    2 3 (M 2 − 1)  , (33) Kotzer and Litsyn EURASIP Journal on Advances in Signal Processing 2011, 2011:36 http://asp.eurasipjournals.com/content/2011/1/36 Page 5 of 7 i.e. the QAM constellation is symmetric and the con- stellation energy is E QAM = q 2 . Symmetric QAM con- stellations satisfy ν 00 =0,ν 20 = ν 02 =0,andν 11 <0. In addition, in all the symmetric cases ν lm =0ifl+ m is odd. We proceed by computing the expansion of ln φ k  ζω −kn √ N  for a group of 1 ≤ N QAM ≤ N bins. For the sake of simplicity, the expansion coefficients ν lm here are for the unity energy constellation, hence the expansion coefficients of in j are q l+m ν lm . For example, for QPSK of the form a k R , a k I ∈ (±1/ √ 2) we have ν 11 = -1/4, ν 22 = -1/64, a nd for 16QAM of the form a k R , a k I ∈ (±1, ±3)/ √ 10 we have ν 11 =-1/4and ν 22 = -17/1600. Then, similar to the BPSK case, we have N QAM −1  k=0 ln φ k  ζω −kn √ N  =  l,m≥0 ν lm q l+m N l+m 2 ζ l ¯ ζ m N QAM −1  k=0 ω −kn(l−m) = N QAM  l,m:N|n(l−m) ν lm q l+m N l+m 2 ζ l ¯ ζ m = N QAM  q 2 ν 11 |ζ | 2 N + q 4 N 2 {ν 22 |ζ | 4 + ν 31 ζ 3 ¯ ζ + ν 13 ζ ¯ ζ 3 } + ···  . (34) We next decompose the symbols vector a of length N into three groups: • N B groups of BPSK symbols. The symbols of each group, of size N r , are drawn from a constellation of energy E r = b 2 r for 1 ≤ r ≤ N B . • N Q groups of QAM symbols. The symbols of e ach group, of size N s , are drawn from a constellation of size M s (that is, the coefficients ν lm are constellation dependent and are denoted as ν s,lm )and ener gy E s = q 2 s for 1 ≤ s ≤ N Q . • N Z zero tones. Obviously, N B + N Q + N Z = N. Following(24),the expansions of In j k of all groups are summed: N−1  k=0 ln φ k  ζω −kn √ N  = N B  r=1  − b 2 r N r |ζ | 2 4N − b 4 r N r |ζ | 4 32N 2 −···  + N Q  s=1  q 2 s N s ν s,11 |ζ | 2 N + q 4 s N s N 2 {ν s,22 |ζ | 4 + ν s,31 ζ 3 ¯ ζ + ν s,13 ζ ¯ ζ 3 } + ···  = ⎡ ⎣ N Q  s=1 q 2 s N s ν s,11 N − N B  r=1 b 2 r N r 4N ⎤ ⎦ |ζ | 2 + ⎡ ⎣ N Q  s=1 q 4 s N s ν s,22 N 2 − N B  r=1 b 4 r N r 32N 2 ⎤ ⎦ |ζ | 4 + N Q  s =1 q 4 s N s N 2 [ν s,31 ζ 3 ¯ ζ + ν s,13 ζ ¯ ζ 3 ]+··· (35) Denoting μ 1 =   N Q s=1 q 2 s N s ν s ,11 N −  N B r=1 b 2 r N r 4N  and μ 2 =   N Q s=1 q 4 s N s ν s,22 N 2 −  N B r=1 b 4 r N r 32N 2  we have N−1  k=0 φ k (N −1/2 ζω −kn ) = exp ⎧ ⎨ ⎩ μ 1 |ζ 2 | + μ 2 |ζ 4 | + N Q  s=1 q 4 s N s N 2 [ν s,31 ζ 3 ¯ ζ + ν s,13 ζ ¯ ζ 3 ]+··· ⎫ ⎬ ⎭ = exp{μ 1 |ζ | 2 }exp ⎧ ⎨ ⎩ μ 2 |ζ | 4 + N Q  s=1 q 4 s N s N 2 [ν s,31 ζ 3 ¯ ζ + ν s,13 ζ ¯ ζ 3 ]+··· ⎫ ⎬ ⎭ . (36) Now, using e x =1+x + we have N−1  k=0 φ k (N −1/2 ζω −kn ) = exp{μ 1 |ζ | 2 }· ⎡ ⎣ 1+μ 2 |ζ | 4 + N Q  s=1 q 4 s N s N 2 [ν s,31 ζ 3 ¯ ζ + ν s,13 ζ ¯ ζ 3 ]+··· ⎤ ⎦ . (37) Following (20), we multiply (37) by 1 2 π J 0 (r|ζ | ) and integrate over ℝ 2 . First, we pass to polar coordinates u,θ (i.e. ζ = u exp (iθ)), and observe that all the terms ζ l ¯ ζ m with l ≠ m vanish (since the integral of cos ((l-m)θ)is zero). Therefore, we are left with  ∞ 0 J 0 (ru) exp{μ 1 u 2 }{u + μ 2 u 5 + ···}du . (38) Using [14,(6.631)] we arrive at  ∞ 0 J 0 (ru) exp{μ 1 u 2 }[u + μ 2 u 5 ]du = − 1 2μ 1 1 F 1  1, 1, r 2 4μ 1  − μ 2 μ 3 1 1 F 1  3, 1, r 2 4μ 1  . (39) Using the identities 1 F 1 ( 1, 1, z ) = e z and 1 F 1 ( 3, 1, z ) = e z ( 1+2z + z 2 /2 ) and summing up N times (20), we get N−1  n = 0 E{f (x n R x n1 )} = N  ∞ 0 rf (r) exp  r 2 4μ 1  − 1 2μ 1 − μ 2 μ 3 1 (1 + r 2 2μ 1 + r 4 32μ 2 1 ) −···  dr . (40) Denoting ˜ μ 1 = N μ 1 and ˜ μ 2 = N 2 μ 2 , (40) can be rewritten as N−1  n=0 E{f (x n R x n I )} =N 2  −  ∞ 0 rf (r) exp  r 2 4μ 1  1 2 ˜μ 1 + ˜μ 2 ˜μ 3 1  dr  + N  −  ∞ 0 rf (r) exp  r 2 4μ 1  ˜μ 2 2 ˜μ 4 1 r 2 + ˜μ 2 32 ˜μ 5 1 r 4  dr  + ···  , (41) and following (9) we have m 0 (c)=−  ∞ 0 rf (r) exp  r 2 4μ 1  1 2 ˜μ 1 + ˜μ 2 ˜μ 3 1  d r (42) and m 1 (c)=−  ∞ 0 rf (r) exp  r 2 4μ 1  ˜μ 2 2 ˜μ 4 1 r 2 + ˜μ 2 32 ˜μ 5 1 r 4  dr . (43) Abbreviations CLT: central limit theorem; EVM: error vector magnitude; HPA: high-power amplifier; MC: multi-carrier; PAPR: peak-to-average power ratio. Acknowledgement The authors would like to thank Eyal Verbin for his contribution to this work. Competing interests The authors declare that they have no competing interests. Received: 27 November 2010 Accepted: 8 August 2011 Published: 8 August 2011 References 1. 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Access On the EVM computation of arbitrary clipped multi-carrier signals Igal Kotzer * and Simon Litsyn Abstract A common figure of merit in multi-carrier systems is the error vector magnitude (EVM) magnitude cli pping function for achieving more realistic results for practical uses. Appendix A Proof of the EVM computation equation We define the energy of the clipped portion of the sig- nal as f. expression in the form of a power series for computing the EVM of the gen- eralized cas e. Analysis of clipped signals usually relies on the Gaus- sianity assumption [5]. However, this assumption