Symbol Error Rate of OFDM Systems with Carrier Frequency Offset ...

Symbol Error Rate of OFDM Systems with Carrier Frequency Offset and Channel Estimation Error in Frequency Selective Fading Channels Marco Krondorf, Ting-Jung Liang and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Technische Universit¨at Dresden, D-01062 Dresden, Germany {krondorf,liang,fettweis}@ifn.et.tu-dresden.de, http://www.ifn.et.tu-dresden.de/MNS Abstract— In this paper we present an analytical approach to evaluate the symbol error rate (SER) of OFDM systems subject to carrier frequency offset (CFO) and channel estimation error in Rayleigh flat fading and frequency selective fading channels. Based on correct modeling of the correlation between channel estimates and received signals with carrier frequency offset, the symbol error rate can be numerically evaluated by averaging symbol error rate on different subcarriers using an analytical expression of double integrals. The results illustrate that our analysis can approximate the simulative performance very accurately if the power delay profile of fading channels and carrier frequency offset are known.

I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) is a widely applied technique for wireless communications, which enables simple one-tap equalization by cyclic prefix, but the sensitivity of OFDM systems to carrier frequency offset is higher than that of single-carrier systems. In present OFDM standards, such as IEEE802.11a/g or DVB-T, preamble (or pilots) are used to estimate and compensate the carrier frequency offset (CFO) and channel impulse response but after the CFO estimation and compensation, the residual carrier frequency offset still destroys the orthogonality of the received OFDM signals and the channel estimates, which worsen further the symbol error rate (SER) of OFDM systems by equalization. In the literature, the effects of carrier frequency offset on symbol error rate are mostly investigated under the assumption of perfect channel knowledge. The papers [4] and [2] consider only the effects of carrier frequency offset (without channel estimation and equalization) and give exact analytical expressions in terms of SNR-loss or bit error rates for AWGN channel. The authors of [5] extend their analysis to frequencyselective fading channels and derive the correspondent uncoded bit error rates for OFDM systems assuming also perfect channel knowledge. Furthermore, the authors of [1] has tried to analyze the joint effect of carrier frequency offset and channel estimation error on uncoded bit error rate for OFDM systems, but the assumption in [1] (channel estimates are uncorrelated to channel estimation error or uncorrelated to received signals) does not hold in real OFDM systems, especially when carrier frequency offset is large.

In this paper, after introducing the OFDM system model (Sec.II) and the probability density function analysis [6] (Sec.III), we explain how to model the correlation between channel estimates and received signals for deriving the symbol error rates of OFDM systems with carrier frequency offset in Rayleigh flat fading channels (Sec.IV) and Rayleigh frequency selective fading channels (Sec.V) respectively. Finally, Sec. VI presents some numerical examples before the conclusions in Sec.VII. II. OFDM S YSTEM M ODEL We consider an OFDM system with N-point FFT. The data is modulated by a QAM modulator to different OFDM data subcarriers, then transformed to a time domain signal by IFFT operation and prepended by a cyclic prefix, which is chosen to be longer than the maximal channel impulse response (CIR) length L taps. In this paper, we assume that the residual carrier frequency offset is a given deterministic value (in practice, the residual carrier frequency offset is a Gaussian-like distributed random variable.) and a one-OFDM-symbol-long preamble is used for channel estimation. Furthermore, we assume static channel characteristics in one OFDM burst. The sampled signal for the lth subcarrier after the receiver fast Fourier transform processing can be written as N 2

Yl = Xl Hl I(0) +

−1 X

k=− N 2 ,k6=l

Xk Hk I(k − l) + Wl

(1)

where Xl represents the transmitted complex QAM modulated symbol on subcarrier l and Wl is a complex Gaussian noise sample. The coefficient Hk denotes the frequency domain channel transfer function on subcarrier k, which is Fourier transform of the CIR h(τ ) with maximal L taps Hk =

L−1 X

h(τ )e−j2πkτ /N

(2)

τ =0

the coefficient I(k − l) represents the impact of the received signal at subcarrier k on the received signal at subcarrier l due to the residual carrier frequency offset [2] I(k − l) = ejπ((k−l)+∆f )(1−1/N )

sin(π((k − l) + ∆f )) N sin(π((k − l) + ∆f )/N )

where ∆f is the residual carrier frequency offset normalized to the subcarrier spacing. In equation (1), we do not model the time variant common phase shift introduced by the residual frequency offset ∆f on different OFDM data symbols [5], because this time variant common phase term is considered to be robustly estimated and compensated by continuous pilots inserted in the OFDM data symbols. In addition, later in this paper the summation P N2 −1 P will be abbreviated as k6=l . k=− N 2 ,k6=l Subsequently, we consider the preamble-based Least Square (LS) frequency domain channel estimation to obtain the channel state information on subcarrier l. P ′ YP,l k6=l XP,k Hk I(k − l) + Wl ˆ Hl = = I(0)Hl + (3) XP,l XP,l followed by the frequency domain zero-forcing equalization before data detection Yl (4) Zl = ˆl H where XP,l and YP,l denote the transmitted and received preamble on subcarrier l. The Gaussian noise of preamble part 2 2 Wl′ has the same variance as Wl of data part (σW ′ = σW ). l l The power of preamble signals and the average power of transmitted data signals on all subcarriers is normalized to 2 one (|XP |2 = σX = 1). III. P ROBABILITY D ENSITY F UNCTIONS A NALYSIS The author of [6] has suggested a correlation model regarding channel estimation for single-carrier systems and derived the correspondent symbol error rate (SER) or bit error rate (BER) of QAM-modulated signals transmitted in flat Rayleigh and Ricean channels. In this section, we will give a short review of the contribution of [6] and extend the results to OFDM systems for frequency selective fading channels with both carrier frequency offset and channel estimation error in the next sections. The single-carrier transmission model without carrier frequency offset for Rayleigh flat fading channels can be written as y = hx + w (5) where y, h, x and w denote the complex baseband representation of the received signal, the channel coefficient, the 2 transmitted data and the Gaussian noise with variance σw respectively. ˆ is assumed to be biased and In [6], the channel estimate h used for zero forcing equalization as follows y ˆ = αh + ν with h (6) z= ˆ h where α denotes the bias in the channel estimates and ν is a zero-mean complex Gaussian noise with variance σν2 . The channel coefficient h and Gaussian noise ν are assumed to be ˆ are correlated, we can write uncorrelated. Since y and h ryhˆ σy ˆ y= h+η (7) σhˆ

where the zero-mean Gaussian random variable η with variance ση2 = E{|η|2 } = σy2 (1 − |ryhˆ |2 ) ˆ and is uncorrelated to h ryhˆ =

ˆ ∗} E{y h σhˆ σy

ˆ 2 } and σ 2 = E{|y|2 } σhˆ2 = E{|h| y

with

Substituting equation (7) into equation (6), the equalized received signal takes the form z= r

ryhˆ σy σhˆ

+

η ˆ h

(8)

ˆ σy

where b = yσhˆ represents the bias of equalized received h signal z and hˆη denotes the zero mean noise part of the equalized received signal z. It should be noted that the bias b is highly correlated to the transmitted data symbol x and can be formulated by [6] b(x)

= ℜ{b} + jℑ{b} = br + jbi   σν2 1 − = x α α(|α|2 σh2 + σν2 )

(9)

where we define the channel power σh2 = E{|h|2 } in one tap. Furthermore, according to [6] [3], the joint pdf of z = zr + jzi with given transmitted symbol x in cartesian coordinates [6] can be expressed by a2 (x) fz|x (z|x) = (10) 2 2 2 π(|z − b(x)| + a (x)) z=zr +jzi where

a2 (x) =

ση2 σ 2 |x|2 σ 2 σ2 = ν 4 h + w2 2 σhˆ σhˆ σhˆ

(11)

The result of (10) can be used to calculate the symbol error rate (SER) of a given QAM constellation [6]. As an example, consider a QPSK constellation with four points x1 = √12 (+1, +1), x2 = √12 (−1, +1), x3 = √12 (+1, −1) and x4 = √12 (−1, −1). For each constellation point xm , where m denotes the constellation point index, the parameters b(xm ) ≡ bm,r + jbm,i and parameter a2 (xm ) ≡ a2m has to be calculated seperately for evaluating the symbol error rate using Eq.(12), where for each constellation point xm , the parameters zr,min , zr,max , zi,min and zi,max represent the lower and upper boundaries of the m-th symbols cartesian decision region [6]. For the QPSK constellation point x1 , we have (zr,min , zr,max , zi,min and zi,max ) = (0, ∞, 0, ∞) and the symbol error probability Pe can be expressed by averaging over all QPSK constellation points as follows Pe =

4 1 X Pe (xm ) 4 m=1

(13)

Additionally, the symbol error rate of different modulation schemes, such as 16-QAM, can be found in [6].





zr −bm,r

√ 2  (zi − bm,i )arctan am +(zi −bm,i )2  p Pe (xm ) = 1 −   2π a2m + (zi − bm,i )2



IV. S YMBOL E RROR R ATE A NALYSIS FOR R AYLEIGH F LAT FADING C HANNELS In this section we will derive the symbol error rate of OFDM systems with carrier frequency offset and channel estimation error in Rayleigh flat fading channels (Hl = Hk = H, ∀ k, l ∈ [1, ..., N ]) based on Eq. (12) in Sec. II. Firstly, we can rewrite the channel estimates of subcarrier l in Eq.(3) to be P   Wl′ k6=l XP,k I(k − l) ˆ + (14) Hl = I(0)H 1 + I(0)XP,l XP,l and from Eq.(14), an Eq.(6)-like expression can be given ˜+ ˆl = α H ˜lH

Wl′ XP,l

˜ + ν˜l =α ˜l H

˜l X

(16)

˜ l , that is no longer a where we define the effective symbol X deterministic value but a stochastic quantity due to i.i.d. data symbols on subcarriers k 6= l. Giving a certain transmit symbol Xl and assuming randomly transmitted data signals Xk with k 6= l, we can decompose ˜ l as follows the effective symbol X P X I(k − l) ˜ l = Xl + k6=l k = Xl + ICIl X I(0) ˜ l due to the random which shows the stochastic nature of X ICI part. Applying the central limit theorem (although, strictly ˜ l has discrete distribution), we assume that the speaking, X inter-carrier interference term is a complex zero-mean Gaussian random variable ICIl = p + jq. The mutual uncorrelated

 zr,max zi,max   

zr,min

  

(12)

zi,min

real and imaginary parts p and q have the same variance for all constellation points 2 σICI = l

1 − |I(0)|2 2|I(0)|2

Secondly, according to Eq.(9) and Eq.(11), we calculate the ˜ l ) = bl,r + jbl,i and a2 (X ˜l ) for M-QAM parameters bl (X l ˜ l on subcarrier l in Rayleigh flat effective data symbols X fading channels. ! 1 σν2˜ ˜ ˜ (17) bl (Xl ) = Xl − 2 + σ2 ) α ˜l α ˜ l (|˜ αl |2 σH ˜ ν ˜ ˜ l |2 σ 2 σν2˜ |X ˜ H 2 )2 (σH ˆl

(15)

where α ˜ l is a deterministic quantity with given subcarrier index l, a set of preamble symbols XP,k and a fixed frequency offset. For σν2˜l , which represents the AWGN variance of the 2 channel estimates, we have σν2˜l = σW for all subcarriers, if 2 |XP,l | = 1 for all subcarrier indexes l. Applying the same method above for Eq.(1), we can use the ˜ to obtain an Eq.(5)-like same definition of effective channel H expression as follows P   X I(k − l) ˜ Xl + k6=l k +Wl Yl = H I(0) | {z } ˜X ˜ l + Wl H

zi −bm,i

(zr − bm,r )arctan √ 2 am +(zr −bm,r )2 p 2π a2m + (zr − bm,r )2

˜l ) = a2l (X

˜ and effective bias α by defining effective channel H ˜l P   k6=l XP,k I(k − l) ˜ = I(0)H, α H ˜l = 1 + I(0)XP,l

=

+



+

2 σW 2 σHˆ

(18)

l

2 2 2 2 2 2 where σH αl |2 |I(0)|2 σH + σW for ˜ = |I(0)| σH and σH ˆ l = |˜ subcarrier l. As an example, for one QPSK constellation point with the index m = 1, X1,l = √12 (1, 1) = √12 (1 + j) on subcarrier l, ˜ 1,l ) = b1,l,r + jb1,l,i and parameter we need to calculate bl (X 2 ˜ al (X1,l ) separately for each effective symbol realization

˜ 1,l = X1,l + p + jq = √1 (1 + j) + p + jq X 2 in Eq. (12). Subsequently, the symbol error rate on subcarrier l for the m-th constellation point can be expressed by the following double integral P¯e (Xm,l ) =

Z∞ Z∞

−∞ −∞

p2 +q2

2 Pe (Xm,l + p + jq) − 2σICI l dp dq e 2 2πσICI l

(19) Finally, to obtain the general symbol error rate we have to average Eq.(19) over all subcarriers with index l and M-QAM constellation points with index m as follows P˜e =

1 MN

N/2−1

X

M X

P¯e (Xm,l )

(20)

l=−N/2 m=1

V. S YMBOL E RROR R ATE A NALYSIS FOR R AYLEIGH F REQUENCY S ELECTIVE FADING C HANNELS In this section we will extend the symbol error rate of OFDM systems with carrier frequency offset and channel estimation error in frequency selective fading channels. Firstly, we define the cross-correlation coefficient of channel transfer functions between subcarrier k and l in frequency selective fading channels rk,l =

E{Hk Hl∗ } 6= 1 ∀ k 6= l 2 σH

(21)

2 where the channel gains σH are the same for all subcarriers. Assuming mutual uncorrelated channel taps of the CIR and applying Eq. (2), we can derive ) (L−1 L−1 XX ∗ −j2πkτ /N j2πlγ/N rk,l = E h(τ )h (γ)e e τ =0 γ=0

=

L−1 X τ =0

 E |h(τ )|2 e−j2π(k−l)τ /N

(22)

where the values E{|h(τ )|2 } = στ2 denotes the tap power of the fading channels with different power delay profile (PDP) and the cross correlation properties of the channel coefficients is written as follows Hk = rk,l Hl + Vk where Vk a complex zero-mean Gaussian with variance 2 σV2 k =σH (1 − |rk,l |2 ) and E{Vk Hl∗ }=0. Substituting the cross-correlation properties of the channel coefficients into Eq.(3), the channel estimate can be written as P   X I(k − l)rk,l ˆ l = I(0)Hl 1 + k6=l P,k H + νˇl I(0)XP,l ˇlα =H ˇ l + νˇl (23) ˇ l = I(0)Hl and where we define the effective channel as H the effective noise term of the channel estimates with variance σν2ˇl is given by   1 X XP,k I(k − l)Vk + Wl′  (24) νˇl = XP,l k6=l

and the bias factor of channel estimates α ˇ l can be expressed by P   k6=l XP,k I(k − l)rk,l α ˇl = 1 + I(0)XP,l Secondly, substituting the cross-correlation properties of the channel coefficients into Eq.(1), we obtain P   k6=l Xk I(k − l)rk,l Yl = Hl I(0) Xl + I(0) X + Xk I(k − l)Vk + Wl k6=l

ˇl + W ˇl ˇlX = H

ˇ l with variance where the effective noise term W written as X ˇl = W Xk I(k − l)Vk + Wl

(25)

2 σW ˇl

can be (26)

k6=l

ˇ l is and the stochastic effective transmit symbol X P X I(k − l)rk,l ˇ l = Xl + k6=l k X I(0)

and the uncorrelated real part p and imaginary part q of the inter-carrier interference have the same variance for all MQAM constellation points. P 2 2 k6=l |I(k − l)| |rk,l | 2 σ ˇICIl = 2|I(0)|2 According to Eq.(9) and Eq.(11), we calculate the parameters ˇ l ) = bl,r + jbl,i and a2 (X ˇ l ) for M-QAM effective data bl (X l ˇ symbols Xl on subcarrier l in frequency selective fading channels. ! σν2ˇl 1 ˇ ˇ (29) bl (Xl ) = Xl − 2 + σ2 ) α ˇl α ˇ l (|ˇ αl |2 σH ˇl ν ˇl 2 ˇ l |2 σ 2ˇ σW σν2ˇl |X ˇl Hl ˇl ) = + (30) a2l (X 2 )2 2 (σH σ ˆ ˆ H l

k6=l

where data power is assumed to be 1 and data symbol Xk is uncorrelated to data symbol Xl (Vk is correlated to Vl ). Similarly, replacing Vk = Hk − rk,l Hl in Eq.(24), we get the variance of the effective AWGN term of the channel estimate XX 2 ∗ σν2ˇl = σW + XP,k XP,m I(k − l)I ∗ (m − l) k6=l m6=l

2 ∗ ×(rk,m − rk,l rm,l σH )

ˇ l can Applying the central limit theorem, the ICI part of X be express ed by P k6=l Xk I(k − l)rk,l ICIl = = p + jq (28) I(0)

(32)

Furthermore, same as in Sec.IV, given a certain effective ˇ m,l = Xm,l + ICIl = Xm,l + p + jq together with symbol X an appropriate decision region of the related M-QAM data symbol Xm,l on subcarrier l, where m is the constellation point index, we get the symbol error probability as follows Z∞ Z∞ p2 +q2 2 Pe (Xm,l + p + jq) − 2ˇσICI ¯ l dp dq Pe (Xm,l ) = e 2 2πˇ σICI l −∞ −∞

(33) Finally, to obtain the general symbol error rate we have to average Eq.(33) over all subcarriers l and M-QAM constellation points with index m as Pˇe =

(27)

l

2 2 2 2 2 where σH αl |2 |I(0)|2 σH + σν2ˇl for ˇ l = |I(0)| σH and σH ˆ l = |ˇ subcarrier l. From Eq.(29) and Eq.(30) we observe, that the parameters 2 2 σW ˇ l and σνˇl have to be calculated exactly to obtain reliable 2 results. The exact expression of σW ˇ l is obtained by taking 2 2 2 Eq.(26), σVk = σH (1 − |rk,l | ) as follows   X
2  2 2 σW |I(k − l)|2  1 − |rk,l |2 (31) ˇ l = σW + σH

1 MN

N/2−1

X

M X

P¯e (Xm,l )

(34)

l=−N/2 m=1

VI. N UMERICAL E XAMPLES In numerical examples, we consider a OFDM system with 64-point FFT. The data is QPSK-modulated to different subcarriers, then transformed to a time domain signal by IFFT operation and prepended by a 16-tap long cyclic prefix. The

data is randomly generated and one OFDM symbol preamble was used for channel estimation. The used BPSK training symbols of the preamble in frequency domain is given by

0

10

XP,l = (−1)l for subcarrier index l = [−32 : 1 : 31]

1 −Dτ /L e , τ = 0, 1, . . . , L − 1 C PL−1 where the factor C = τ =0 e−Dτ /L is chosen to normalize PL−1 2 2 the τ =0 στ = 1 (or σH = 1 on all subcarriers) through all realizations. Additionally, the maximal channel impulse response is shorter than cyclic prefix (16 taps) in our numerical examples. For the numerical integration of Eq.(19) and Eq.(33) we used the Matlab 7.0.1 build-in numerical integration function of tolerance 1e-6 and upper/lower bounds of ±1e4. Fig.1 and Fig. 2 present the calculated and simulated symbol error rates vs. SNR with given carrier frequency offset ∆f for Rayleigh flat fading channels and frequency selective channels with 8 tap exponential power delay profile (D=7) respectively. The results illustrate that our analysis can approximate the simulative performance very accurately, if the power delay profile of fading channels and carrier frequency offset are known.

SER

In addition to a Rayleigh flat fading channels, we consider a Rayleigh frequency selective fading channel having an exponential power delay profile described by

−1

10

∆f 1% calc ∆f 1% Sim ∆f 5% calc ∆f 5% Sim ∆f 10% calc ∆f 10% Sim ∆f 15% calc ∆f 15% Sim ∆f 30% calc ∆f 30% Sim

−2

10

στ2 =

−1

SER

10

∆f 1% calc ∆f 1% Sim ∆f 15% calc ∆f 15% Sim ∆f 25% calc ∆f 25% Sim ∆f 30% calc ∆f 30% Sim

−2

−3

10

0

5

10

−4

10

0

5

10

15

20 SNR in dB

25

30

35

40

Fig. 2. Comparison of calculated and simulated SER vs. SNR under different ∆f in frequency selective channels with 8 taps exponential power delay profile

results is, the system engineers might use our analytical tools to check the performance (uncoded symbol error rates) of OFDM systems using different frequency synchronization algorithms. R EFERENCES [1] H. Cheon and D. Hong. Effect of channel estimation error in OFDMbased WLAN. In Proc. IEEE Communication Letters, volume 6, pages 190–192, May 2002. [2] K.Sathananthan and C. Tellambura. Probability of error calculation of OFDM with frequency offset. In Proc. IEEE Transactions on Communications, volume 49, pages 1884–1888, November 2001. [3] A. Papoulis. Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1984. [4] T. Pollet, M. van Bladel, and M.Moeneclaey. BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise. In Proc. IEEE Transactions on Communications, volume 42, pages 191–193, 1995. [5] L Rugini, P. Banelli, and S. Cacopardi. BER of OFDM systems impaired by carrier frequency offset in multipath fading channels. In IEEE Transactions on Wireless Communications, volume 4, pages 2279– 2288, September 2005. [6] S. K. Wilson and J. M. Cioffi. Probability density functions for analyzing multi-amplitude constellations in Rayleigh and Ricean channels. In IEEE Transactions on Communications, volume 47, March 1999.

0

10

10

−3

10

15 SNR in dB

20

25

30

Fig. 1. Comparison of calculated and simulated SER vs. SNR under different ∆f in Rayleigh flat fading channel

VII. C ONCLUSIONS In this paper we present how to analytically evaluate the symbol error rate of OFDM systems subject to carrier frequency offset and channel estimation error in Rayleigh flat fading and frequency selective fading channels. The results show that the symbol error rate can be calculated exactly when the power delay profile of fading channels and carrier frequency offset are known. One possible application of our