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EURASIP Journal on Wireless Communications and NetworkingVolume 2010, Article ID 176083, 14 pages doi:10.1155/2010/176083 Research Article Bit Error Rate Approximation of MIMO-OFDM Syste

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EURASIP Journal on Wireless Communications and Networking

Volume 2010, Article ID 176083, 14 pages

doi:10.1155/2010/176083

Research Article

Bit Error Rate Approximation of MIMO-OFDM Systems with

Carrier Frequency Offset and Channel Estimation Errors

Zhongshan Zhang,1Lu Zhang,2Mingli You,2and Ming Lei1

1 Department of Wireless Communications, NEC Laboratories China (NLC), 11th Floor Building A, Innovation Plaza TusPark, Beijing 100084, China

2 Research & Innovation Center (R&I), Alcatel-Lucent Shanghai Bell, No 388 Ningqiao Road, Pudong, Shanghai 201206, China

Correspondence should be addressed to Zhongshan Zhang,zhang zhongshan@nec.cn

Received 23 February 2010; Revised 10 August 2010; Accepted 16 September 2010

Academic Editor: Stefan Kaiser

Copyright © 2010 Zhongshan Zhang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

The bit error rate (BER) of multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems with carrier frequency oﬀset and channel estimation errors is analyzed in this paper Intercarrier interference (ICI) and interantenna interference (IAI) due to the residual frequency oﬀsets are analyzed, and the average signal-to-interference-and-noise ratio (SINR) is derived The BER of equal gain combining (EGC) and maximal ratio combining (MRC) with MIMO-OFDM is also derived The simulation results demonstrate the accuracy of the theoretical analysis

1 Introduction

Spatial multiplexing multiple-input multiple-output

(MI-MO) technology significantly increases the wireless system

capacity [1 4] These systems are primarily designed for

flat-fading MIMO channels A broader band can be used

to support a higher data rate, but a frequency-selective

fading MIMO channel is met, and this channel experiences

intersymbol interference (ISI) A popular solution is

MIMO-orthogonal frequency-division multiplexing (OFDM), which

achieves a high data rate at a low cost of equalization and

demodulation However, just as single-input

single-output-(SISO-) OFDM systems are highly sensitive to frequency

oﬀset, so are MIMO-OFDM systems Although one can

use frequency oﬀset correction algorithms [5 10], residual

frequency oﬀsets can still increase the bit error rate (BER)

The BER of SISO-OFDM systems impaired by frequency

oﬀset is analyzed in [11], in which the frequency oﬀset is

assumed to be perfectly known at the receiver, and, based on

the intercarrier interference (ICI) analysis, the BER is

eval-uated for multipath fading channels Many frequency oﬀset

estimators have been proposed [8,12–14] A synchronization

algorithm for MIMO-OFDM systems is proposed in [15],

which considers an identical timing oﬀset and frequency

offset with respect to each transmit-receive antenna pair In [10], where frequency offsets for different transmit-receive antennas are assumed to be different, the Cramer-Rao lower bound (CRLB) for either the frequency offsets or channel estimation variance errors for MIMO-OFDM is derived More documents on MIMO-OFDM channel estimation by considering the frequency offset are available at [16,17] However, in real systems, neither the frequency offset nor the channel can be perfectly estimated Therefore, the residual frequency offset and channel estimation errors impact the BER performance The BER performance of MIMO systems, without considering the effect of both the frequency offset and channel estimation errors, is studied in [18,19]

This paper provides a generalized BER analysis of MIMO-OFDM, taking into consideration both the frequency oﬀset and channel estimation errors The analysis exploits the fact that for unbiased estimators, both channel and frequency oﬀset estimation errors are zero-mean random variables (RVs) Note that the exact channel estimation algorithm design is not the focus of this paper, and the main parameter of interest is the channel estimation error Many channel estimation algorithms developed for either SISO or MIMO-OFDM systems, for example, [20–22], can be used to

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perform channel estimation The statistics of these RVs are

used to derive the degradation in the receive SINR and the

BER Following [10], the frequency oﬀset of each

transmit-receive antenna pair is assumed to be an independent and

identically distributed (i.i.d.) RV

This paper is organized as follows The MIMO-OFDM

system model is described in Section 2, and the SINR

degradation due to the frequency oﬀset and channel

esti-mation errors is analyzed in Section 3 The BER, taking

into consideration both the frequency oﬀset and channel

estimation errors, is derived in Section 4 The numerical

results are given in Section 5, and the conclusions are

presented inSection 6

conjugate transpose The imaginary unit is j = √ −1.R{ x }

and I{ x } are the real and imaginary parts of x,

respec-tively arg{ x } represents the angle of x, that is, arg { x } =

arctan(I{ x } /R{ x }) A circularly symmetric complex

Gaus-sian RV with meanm and variance σ2 is denoted byw ∼

CN (m, σ2) IN is theN × N identity matrix, and ON is the

is theith entry of vector a, and [B]mnis themnth entry of

matrix B.E{ x }and Var{ x }are the mean and variance ofx.

2 MIMO-OFDM Signal Model

Input data bits are mapped to a set ofN complex symbols

drawn from a typical signal constellation such as phase-shift

keying (PSK) or quadrature amplitude modulation (QAM)

The inverse discrete fourier transform (IDFT) of these N

symbols generates an OFDM symbol Each OFDM symbol

has a useful part of durationTsseconds and a cyclic prefix of

lengthTg seconds to mitigate ISI, where Tg is longer than

the channel-response length For a MIMO-OFDM system

withNttransmit antennas andNrreceive antennas, anN ×1

vector xn trepresents the block of frequency-domain symbols

sent by thentth transmit antenna, wherent ∈ {1, 2, , Nt }

The time-domain vector for the ntth transmit antenna is

given by mn t = Es/NtFxn t, whereEsis the total transmit

power and F is theN × N IDFT matrix with entries [F]nk =

(1/ √

N)e j2πnk/N for 0 ≤ n, k ≤ N −1 Each entry of xn t is

assumed to be i.i.d RV with mean zero and unit variance;

that is, σ2 = E{| xn t[n] |2} = 1 for 1 ≤ nt ≤ Nt and

0≤ n ≤ N −1

The discrete channel response between thenrth receive

antenna and ntth transmit antenna is hn r,n t = [hn r,n t(0),

hn r,n t(1), , hn r,n t(Ln r,n t −1), 0T

max− L nr ,nt]T, whereLn r,n t is the maximum delay between the ntth transmit and the nrth

receive antennas, and Lmax = max{ Ln r,n t : 1 ≤ nt ≤ Nt,

1 ≤ nr ≤ Nr } Uncorrelated channel taps are

assumed for each antenna pair (nr,nt); that is,

E{ h ∗ n r,n t(m)hn r,n t(n) } = 0 when n / = m The corresponding

frequency-domain channel response matrix is given by

Hn r,n t = diag{ H n(0)r,n t,H n(1)r,n t, , H n(N r,n − t1)} with H n(n) r,n t =

L nr ,nt −1

attenuation at thenth subcarrier In the sequel, the channel

power profiles are normalized asL nr ,nt −1

d =0 E{| hn r,n t(d) |2} =1 for all (nr,nt) The covariance of channel frequency response

is given by

C H(n)

nr ,nt H(p,q l) =

Lmax−1

d =0

Eh ∗ n r,n t(d)hp,q(d)

e − j2πd(l − n)/N,

0≤ d ≤ Lmax, 0≤ l, n ≤ N −1.

(1)

Note that ifnr = / p and nt = / q are satisfied simultaneously, we

assume that there is no correlation betweenhn r,n t andhp,q Otherwise the correlation betweenhn r,n tandhp,qis nonzero

In this paper, ψn r,n t andεn r,n t are used to represent the initial phase and normalized frequency oﬀset (normalized

to the OFDM subcarrier spacing) between the oscillators

of the nt-th transmit and the nrth receive antennas The frequency offsets εn r,n t for all (nr,nt) are modeled as zero-mean i.i.d RVs (Multiple rather than one frequency offset are assumed in this paper, with each transmit-antenna pair being impaired by an independent frequency offset This case happens when the distance between different transmit

or receive antenna elements is large enough, and this big distance results in a different angle-of-arrive (AOA) of the signal received by each receive antenna element In this scenario, once the moving speed of the mobile node is high, the Doppler Shift related to different transmit-receive antenna pair will be different.)

By considering the channel gains and frequency oﬀsets, the received signal vector can be represented as

y=yT

1, yT

2, , y T

N r

T

where yn r = Es/NtN t

n t =1En r,n tFHn r,n txn t + wn r, En r,n t =

diag{ e jψ nr ,nt, , e j(2πε nr ,nt(N −1)/N+ψ nr ,nt)} and wn r is a vector

of additive white Gaussian noise (AWGN) with wn r[n] ∼

CN (0, σ2

w) Note that the channel state information is available at the receiver, but not at the transmitter Conse-quently, the transmit power is equally allocated among all the transmit antennas

3 SINR Analysis in MIMO-OFDM Systems

This paper treats spatial multiplexing MIMO, where inde-pendent data streams are mapped to distinct OFDM symbols and are transmitted simultaneously from transmit antennas

The received vector yn r at the nrth receive antenna is thus

a superposition of the transmit signals from all the Nt

transmit antennas When demodulating xn t, the signals from the transmit antennas other than thentth transmit antenna constitute interantenna interference (IAI) The structure of MIMO-OFDM systems is illustrated inFigure 1, whereΔ f

represents the subcarrier spacing

Here, we first assume thatεn r, and Hn r, for each (1 ≤

i ≤ Nt,i / = nt) have been estimated imperfectly; that is,

εn r, = εn r, +Δε n r, andH n r, = Hn r, +ΔHn r,, whereΔε n r,

andΔHn r, = diag{ ΔH(0)

n r,, , ΔH(N −1)

n r, }are the estimation errors of εn r, and Hn r, (ΔH(n)

n r, = H n(n) r, − H n(n) r, represents the estimation error of H n(n) r,), respectively We also assume

that each xi / = n is demodulated with a negligible error After

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· · ·

H1N t

N r

Nt

x1

H11

1

xN t

r11

.

1 2

e j2π( f c+ε11·Δ f )t

e j2π( f c+ε Nt1·Δ f )t

wN r

r1

yN r

xΛ 1

IFFT

CP

CP P/S

P/S Transmit antenna 1

Transmit antennaN t

y

IAI cancellation

CFO estimation

Channel estimation

Demodulation Combining, e.g.,

EGC, MRC

1 2

w1

rN r1

Figure 1: Structure of MIMO-OFDM transceiver

estimatingεn r,n t, that is,εn r,n t = εn r,n t +Δε n r,n t,εn r,n t can be

compensated for and xn t can be demodulated as

rn r,n t =FHE H

n r,n t

⎛

⎝yn r −

Es Nt

N t

i =1,i / = n t

En r,F H n r,xi

⎞

⎠

=

Es

NtF

HE H

n r,n tEn r,n tFHn r,n txn t

snr ,nt

+

Es

Nt

N t

i =1,i / = n t

FHE H

n r,n t

En r,FHn r, − En r,F H n r,xi

Υnr ,nt + FHE H

n r,n twn r

wnr ,nt

,

(3)

where E n r, is derived from En r, by replacing εn r, with

εn r, andΥn r,n t and w n r,n t are the residual IAI and AWGN

components of rn r,n t, respectively (WhenNt is large enough

and the frequency oﬀset is not too big (e.g.,1), from the

Central-Limit Theorem (CLT) [23, Page 59], the IAI can be

approximated as Gaussian noise.)

3.1 SINR Analysis without Combining at Receive Antennas.

The SINR is derived for the ntth transmit signal at the

nrth receive antenna The signals transmitted by antennas

other than thentth antenna are interference, which should

be eliminated before demodulating the desired signal of

thentth transmit antenna Existing interference cancelation algorithms [24–27] can be applied here

Let us first define the parameters m(n n,l) r,n t = (sin[π(l −

n − Δε n r,n t)]/N sin[π(l − n − Δε n r,n t)/N])e jπ(N −1)(l − n)/N,

m(n n,l) r,i / = n t =(sin[π(l − n + εn r, − εn r,n t)]/N sin[π(l − n + εn r, −

εn r,n t)/N])e jπ(N −1)(l − n)/N, andm (n n,l) r,i / = n t =(sin[π(l − n + εn r, −

εn r,n t)]/N sin[π(l − n + εn r,− εn r,n t)/N])e jπ(N −1)(l − n)/N, 0≤ l ≤

N −1 Based on (3), thenth subcarrier (0 ≤ n ≤ N −1) of thentth transmit antenna can be demodulated as

rn r,n t[n] =

Es

Ntsn r,n t[n] +Υn r,n t[n] +wn r,n t[n]

=

Es

(n,n)

n r,n t H(n)

n r,n txn t[n]

+

Es Nt

l / = n

m(n n,l) r,n t H n(l) r,n txn t[l]

η nr ,nt(n) = H nr ,nt(n) α(nr ,nt n) +β(nr ,nt n)

+

Es Nt

N t

i =1,i / = n t

m(n n,n) r, H n(n) r,xi[n]

λ(nr ,nt n)

−

Es Nt

N t

i =1,i / = n t

m(n n,n) r, H (n)

n r,xi[n]

λ(n)

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Es Nt

l / = n

N t

i =1,i / = n t

m(n n,l) r, H n(l) r,xi[l]

ξ nr ,nt(n)

−

Es Nt

l / = n

N t

i =1,i / = n t

m(n n,l) r, H (l)

n r,xi[l]

ξ nr ,nt(n)

+wn r,n t[n]

=

Es

(n,n)

n r,n t H(n)

n r,n txn t[n] + H(n)

n r,n t α(n)

n r,n t

+β(n)

n r,n t+Δλ(n)

n r,n t+Δξ(n)

n r,n t+wn r,n t[n],

(4) where η n(n) r,n t is decomposed as η n(n) r,n t = H n(n) r,n t α(n n) r,n t +β(n n) r,n t,

which is the ICI contributed by subcarriers other than the

of ICI into the format ofHα + β is referred to [11].) We can

easily prove thatα(n n) r,n t andβ n(n) r,n t are zero-mean RVs subject

to the following assumptions

(1)εn r,n t is an i.i.d RV with mean zero and varianceσ2

for all (nr,nt)

(2)Δε n r,n tis an i.i.d RV with mean zero and varianceσ2

res

for each (nr,nt)

(3)H n(n) r,n t ∼ CN (0, 1) for each (n r,nt,n).

(4)ΔH(n)

n r,n t is an i.i.d RV with mean zero and variance

σ ΔH2 for each (nr,nt,n).

(5)εn r,n t,Δε n r,n t,H n(n) r,n t, and ΔH(n)

n r,n t are independent of each other for each (nr,nt)

Given these assumptions, let us first defineΔλ(n)

n r,n t = λ(n n) r,n t −

λ(n n) r,n t as the interference contributed by thenth subcarrier of

the interfering transmit antennas, that is, the co-subcarrier

inter-antenna-interference (CSIAI), and define Δξ(n)

n r,n t =

ξ n(n) r,n t − ξ n(n) r,n t as the ICI contributed by the subcarriers other

than thenth subcarrier of the interfering transmit antennas,

that is, the intercarrier-interantenna interference (ICIAI)

Then we derive Var{ α(n n) r,n t }and Var{ β(n n) r,n t }as

Var

α(n n) r,n t

= Es

⎧

⎨

⎩

C −1

H(nr ,nt n) H nr ,nt(n)

2

l / = n

m(n,l)

n r,n t C H(l)

nr ,nt H nr ,nt(n) 2

⎫

⎬

⎭

∼ Es

⎧

⎨

⎩

l / = n

sin(π Δε n r,n t)

2

·

Lmax−1

d =0

E

hn r,n t(d)2

e − j2πd(l − n)/N

2⎫

⎪

= π2σres2 Es

Nt

l / = n

C H(n)

nr ,nt H nr ,nt(l) 2

N2sin2[π(l − n)/N],

(5)

Var

β(n)

n r,n t

= Es

Nt

· E

⎧

⎨

⎩

l / = n

m(n,l)

n r,n t

2

×

C H(l)

nr ,nt H(nr ,nt l) − C H −1(n)

nr ,nt H nr ,nt(n)

C H(l)

nr ,nt H nr ,nt(n) 2

∼ π2σres2 Es

3Nt −Var

α(n)

n r,n t

,

(6)

whereC H(l)

nr ,nt H nr ,nt(n) is given by (1) The demodulation of xn t[n]

is degraded by either η n(n) r,n t or IAI (CSIAI plus ICIAI) In this paper, we assume that the integer part of the frequency offset has been estimated and corrected, and only the fractional part frequency offset is considered Considering small frequency offsets, the following requirements are assumed to be satisfied:

(1)| εn r,| 1 for all (nr,i),

(2)| εn r,n t |+| εn r,| < 1 for all (nr,nt,i),

(3)| εn r,n t |+| εn r,| < 1 for all (nr,nt,i).

Condition 1 requires that each frequency offset should be much smaller than 1, and conditions 2 and 3 require that the sum of any two frequency offsets (and the frequency offset estimation results) should not exceed 1 The last two conditions are satisfied only if the estimation error does not exceed 0.5 If all these three conditions are satisfied simultaneously, we can representλ(n n) r,n t, λ(n)

n r,n t,ξ n(n) r,n t, andξ (n)

n r,n t

as

λ(n)

n r,n t =

Es Nt

N t

i =1,i / = n t

m(n n,n) r, H n(n) r,xi[n]

=

Es Nt

N t

i =1,i / = n t

sin!

π"

εn r, − εn r,n t

#$

N sin!

π"

εn r, − εn r,n t

#

/N$H n(n) r,xi[n],

(7)

λ(n n) r,n t =

Es Nt

N t

i =1,i / = n t

m(n n,n) r, H (n)

n r,xi[n]

=

Es Nt

N t

i =1,i / = n t

sin!

π"

εn r, − εn r,n t

#$

N sin!

π"

εn r, − εn r,n t

#

/N$ H n(n) r,xi[n],

(8)

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ξ n(n) r,n t =

Es

Nt

l / = n

N t

i =1,i / = n t

m(n n,l) r, H n(l) r,xi[l]

∼

Es

Nt

l / = n

N t

i =1,i / = n t

(−1)(l − n)sin!

π"

εn r, − εn r,n t

#$

× e jπ(N −1)(l − n)/N H n(l) r,xi[l],

(9)

ξ(n)

n r,n t =

Es

Nt

l / = n

N t

i =1,i / = n t

m(n n,l) r, H (l)

n r,xi[l]

∼

Es

Nt

l / = n

N t

i =1,i / = n t

(−1)(l − n)sin!

π"

εn r, − εn r,n t

#$

× e jπ(N −1)(l − n)/N H (l)

n r,xi[l].

(10) Therefore, the interference due to the nth subcarrier of

transmit antennas (other than thentth transmit antenna, i.e.,

the interfering antennas) is

Δλ(n)

n r,n t = λ(n)

n r,n t − λ(n)

n r,n t

=

Es

Nt

·

N t

i =1,i / = n t

⎡

⎣π2

"

εn r, − εn r,nt+"

Δε n r,/2##

H n(n) r,Δε n r,

3

−

'

1− π2

"

εn r, − εn r,n t

#2 6

(

ΔH(n)

n r,

⎤

⎦xi[n]

+o"

Δε n r,,ΔH n r,

# ,

(11)

Δξ(n)

n r,n t = ξ(n)

n r,n t − ξ(n)

n r,n t

=

Es

Nt

l / = n

N t

i =1,i / = n t

(−1)l − n+1 e jπ(N −1)(l − n)/N

·

+

π cos

π

εn r, − εn r,n t+Δε n r,

2

H n(l) r,Δε n r,

+ sin"

π"

εn r, − εn r,n t

##

ΔH(l)

n r,

,

xi[l]

+o"

Δε n,,ΔH n,

#

(12)

with o( Δε n r,,ΔH n r,) representing the higher-order item of

Δε n r, andΔH n r, It is easy to show thatΔλ(n)

n r,n t andΔξ(n)

n r,n t

are zero-mean RVs and that their variances are given by

E-

Δλ(n)

n r,n t

2

= Es

Nt

N t

i =1,i / = n t

× E

⎧

⎪

⎡

⎣π2

"

εn r, − εn r,n t+"

Δε n r,/2##

H n(n) r,Δε n r,

3

⎤

⎦

2⎫

⎪

+ Es Nt

N t

i =1,i / = n t

E

⎧

⎨

⎩

.'

1− π2

"

εn r, − εn r,n t

#2

6

(

ΔH(n)

n r,

/2⎫

⎬

⎭

∼ (Nt −1)π4Es

9Nt

⎛

⎝2σ2

σ2 res+σ4 res+EΔε4

n r,

4

⎞

⎠+(Nt −1)Es

Nt

· σ2

ΔH ·

⎡

⎣1 +π4

Eε4

n r,

+ 8σ2

σ2 res+ 2σ4

+ 2σ4 res

18

−2π2

"

σ2

+σ2 res

# 3

⎤

⎦,

(13)

E-

Δξ(n)

n r,n t

2

= Es

Nt

l / = n

N t

i =1,i / = n t

1

N2sin2[π(l − n)/N]

· E

-+

π cos

π

εn r, − εn r,n t+Δε n r,

2

H n(l) r,Δε n r,

+ sin"

π"

εn r, − εn r,n t

##

ΔH(l)

n r, ,20

∼ (Nt −1)Es

3Nt

⎡

⎣π2σ2 res− π4

⎛

⎝2σ2

σ2 res+σ4 res+EΔε4

n r,

4

⎞

⎠

⎤

⎦

+2(Nt −1)π2Es

3Nt

"

σ2

+σ2 res

#

σ2

ΔH,

(14)

respectively After averaging out frequency oﬀset ε n r,n t, frequency oﬀset estimation error Δε n r,n t, and channel estima-tion errorΔH(n)

n,n for all (nr,nt), the average SINR ofrn,n[n]

Trang 6

(parameterized by onlyH n(n) r,n t) is

γ n r,n t

n | H(n)

n r,n t

-

Es/Nt m(n n,n) r,n t H n(n) r,n txi[n]2

E-

η(n n) r,n t+Δλ(n)

n r,n t+Δξ(n)

n r,n t+wn r,n t[n]2

∼ Es/Nt · σ m2 ·H(n)

n r,n t

2

H n(n) r,n t2

·Var

α(n n) r,n t

+ν,

ν = π2σ2

resEs/3Nt

−Var

α(n)

n r,n t

+E-

Δλ(n)

n r,n t

2

+E-

Δξ(n)

n r,n t

2 +σ w2

(15)

whereσ2

m = E{| m(n n,n) r,n t |2} ∼ =1− π2σ2

res/3 + π4E{ Δε4

n r,} /36 and

ν, independent of (nr,nt,n).

For signal demodulation in MIMO-OFDM, signal

received in multiple receive antennas can be exploited to

improve the receive SINR In the following, equal gain

combining (EGC) and maximal ratio combining (MRC) are

considered

3.2 SINR Analysis with EGC at Receive Antennas In order

to demodulate the signal transmitted by the ntth transmit

antenna, theNrreceived signals are cophased and combined

to improve the receiving diversity Therefore, the EGC output

is given by

rEGCn t [n] =

N r

n r =1

e − jθ(nr ,nt n) rn r,n t[n]

=

N r

n r =1

Es

Nt e

− jθ(nr ,nt n) m(n,n)

n r,n t H(n)

n r,n txn t[n]

+

N r

n r =1

e − jθ(nr ,nt n)

η(n)

n r,n t+Δλ(n)

n r,n t+Δξ(n)

n r,n t+wn r,n t[n]

, (16)

where θ(n n) r,n t = arg{ m(n n,n) r,n t H n(n) r,n t } After averaging outεn r,n t,

Δε n,n, and ΔH(n)

n,n for each (nr,nt), the average SINR of

rEGC

n t [n] is given by

γEGC

n t

n | H1,(n) n t, , H N(n) r,n t

-

N r

n r =1

Es/Nt e − jθ(nr ,nt n) m(n n,n) r,n t H n(n) r,n txn t[n]2

E-

N r

n r =1e − jθ(nr ,nt n)

η n(n) r,n t+Δλ(n)

n r,n t+Δξ(n)

n r,n t+wn r,n t[n]2

∼

Es/Nt · σ2

m ·

'

N r

n r =1H(n)

n r,n t2

+

n r = / l

H n(n) r,n t ·H(n)

l,n t

(

N r

n r =1H(n)

n r,n t2

·Var

α(n n) r,n t

(17)

WhenNris large enough, (17) can be further simplified as

γEGCn t

n | H1,(n) n t, , H N(n) r,n t

2

m ·N r

n r =1H(n)

n r,n t2

+Nr(Nr −1)π/4

N r

n r =1H(n)

n r,n t2

·Var

α(n n) r,n t

(18)

3.3 SINR Analysis with MRC at Receive Antennas In a

MIMO-OFDM system withNr receive antennas, based on the channel estimation H (n)

n r,n t = H n(n) r,n t +ΔH(n)

n r,n t for each (nr,nt,n), the received signal at all the Nr receive antennas can be combined by using MRC, and therefore the combined output is given by

rMRCn t [n]

=

N r

n r =1ωn r,n trn r,n t[n]

N r

n r =1ωn r,n t2

=

Es/NtN r

n r =1H(n)

n r,n t

2m(n,n)

n r,n t

2xn t[n]

N r

n r =1ωn r,n t2

+

Es/NtN r

n r =1ΔH(n)H

n r,n t H n(n) r,n tm(n,n)

n r,n t2

xn t[n]

N r

n r =1ωn r,n t2

+

N r

n r =1ωn r,n t

η(n n) r,n t+Δλ(n)

n r,n t+Δξ(n)

n r,n t+wn r,n t[n]

N r

(19)

Trang 7

where ωn r,n t = (H (n)

n r,n t m(n n,n) r,n t)∗ After averaging out εn r,n t,

Δε n r,n t, and ΔH(n)

n r,n t for each (nr,nt), the average SINR of

r M

n t[n] is

γMRCn t

n | H1,(n) n t, , H N(n) r,n t

E

1

Es/NtAm(n,n)

n r,n t

2xn t[n]

2 0

E

1

Es/NtN r

n r =1ΔH(n) ∗

n r,n t H n(n) r,n tm(n,n)

n r,n t2

xn t[n]

2

0 +ℵ

A−n r = / lAH(n)

l,n t

2/AVar

α(n n) r,n t

+ν +Nr · ν · σ ΔH2 /A,

A=

N r

n r =1

H(n)

n r,n t

2

(20) where we have definedν =[ν + (Es/Nt+ Var{ α(n n) r,n t })σ2

ΔH],

and the noise part can be represented as ℵ =

E{|N r

n r =1ω ∗ n r,n t(η(n n) r,n t+Δλ(n)

n r,n t+Δξ(n)

n r,n t+ wn r,n t[n]) |2}

Wh-enNris large enough, (20) can be further simplified as

γMRCn t

n | H1,(n) n t, , H N(n) r,n t

A−n r = / lH(n)

n r,n t

2H(n) l,n t

2/AVar

α(n n) r,n t

+ν +Nr · ν · σ ΔH2 /A

(A−(Nr −1)) Var

α(n n) r,n t

+ν +ν · σ ΔH2 .

A=

N r

n r =1

H(n)

n r,n t

2

(21)

4 BER Performance

The BER as a function of SINR in MIMO-OFDM is derived

in this section We considerM-ary square QAM with Gray

bit mapping In the work of Rugini and Banelli [11], the BER

of SISO-OFDM with frequency oﬀset is developed The BER

analysis in [11] is now extended to MIMO-OFDM

As discussed in [11,28,29], the BER for thentth transmit

antenna with the input constellation being M-ary square

QAM (Gray bit mapping) can be represented as

PBER

"

γn t

#

=

√

M −1

i =1

a M

i erfc2

b M

i γi

wherea M i andb M i are specified by signal constellation,γn t is

the average SINR of thentth transmit antenna, and erfc(x) =

(2/ √

π)3∞

x e − u2

du is the error function (Please refer to [28]

for the meaning ofa Mandb M.)

Note that in MIMO-OFDM systems, the SINR at each subcarrier is an RV parameterized by the frequency oﬀset and channel attenuation In order to derive the average SINR

of MIMO-OFDM systems, (22) should be averaged over the distribution ofγias

PBER

"

γn t

#

=

√

M −1

i =1

a M i

4

γ nterfc

2

b M

i γn t

f"

γn t

#

dγn t

=

√

M −1

i =1

a M i

4

Ent

4

Φnt

erfc2

b M

i γn t

· f"

Hn t#

f"

En t

#

f"

vn t#

× f"

Φn t

#

dHn tdEn tdvn tdΦn t,

(23)

where Hn t = [H1,n t, , HN r,n t], En t = [ε1,n t, , εN r,n t]T,

vn t =[Δε1,n t, , Δε N r,n t]T, andΦn t =[ΔH1,n t, ,ΔHN r,n t] Since obtaining a close-form solution of (23) appears impos-sible, an infinite-series approximation ofPBERis developed

In [11], the average is expressed as an infinite series of generalized hypergeometric functions

From [30, page 939], erfc(x) can be represented as an

infinite series:

erfc(x) = √2

π

∞

m =1

(−1)(m+1) x(2m −1)

(2m −1)(m −1)!. (24) Therefore, (23) can be rewritten as

PBER

"

γn t

#

= √2

π

√

M −1

i =1

a M i

∞

m =1

(−1)(m+1)

b i M

(m −1/2)

(2m −1)(m −1)! · Dn t;m,

Dn t;m =

4

Ent

4

Φnt

"

γn t

#(m −1/2)

f"

Hn t

#

× f"

En t

#

f"

vn t

#

f"

Φn t

#

dHidEn tdvn tdΦn t

(25) whereDn t;mdepends on the type of combining Note thatγn t

has been derived inSection 3and that for thenth subcarrier

(0≤ n ≤ N −1),εn r,n t,Δε n r,n t andΔH(n)

n r,n t for each (nr,nt) have been averaged out Therefore,γn tin (25) can be replaced

subcarrier n (0 ≤ n ≤ N −1), and finally PBER can be simplified as

PBER

γ n t(n)

= √2

π

√

M −1

i =1

a M i

∞

m =1

(−1)(m+1)

b M i

(m −(1/2))

(2m −1)(m −1)! · Dn t;m,

(26) whereDn t;mis based onγ n t(n) instead of γn t We first define

 = Es/Nt · σ2

mandμ =Var{ α(n n) r,n t }, which will be used in the following subsections We next give a recursive definition for

Trang 8

Dn t;mfor the following reception methods: (1) demodulation

without combining, (2) EGC, and (3) MRC

Note that the SINR for each combining scenario (i.e.,

without combining, EGC, or MRC) is a function of the

second-order statistics of the channel and frequency oﬀset

estimation errors (although the interference also comprises

the fourth-order statistics of the frequency oﬀset estimation

errors, they are negligible as compared to the

second-order statistics for small estimation errors) Any probability

distribution with zero mean and the same variance will result

in the same SINR Therefore, the exact distributions need

not be specified However, when the BER is derived by using

an infinite-series approximation, the actual distribution of

the frequency oﬀset estimation errors is required In [31], it

is shown that both the uniform distribution and Gaussian

distribution are amenable to infinite-series solutions with

closed-form formulas for the coeﬃcients In the following

sections, the frequency oﬀset estimation errors are assumed

to be i.i.d Gaussian RVs with mean zero and variance σ2

[10]

4.1 BER without Receiving Combining The BER measured

at thenrth receive antenna for thentth transmit antenna can

be approximated by (25) withD n r

n t;m instead ofDn t;m being used here; that is,

P n r

BER

γ n r,n t

n | H n(n) r,n t

= √2

π

√

M −1

i =1

a M i

∞

m =1

(−1)(m+1)

b M i

(m −1/2) (2m −1)(m −1)! · D n r

n t;m.

(27)

Whenm > 2, we have D n r

n t;m = [(2m −3)μ + ν]/μ2(m −3/2) ·

D n r

n t;m −1− 2/μ2· D n r

i;m −2, as derived inAppendix A The initial condition is given by

D n r

n t;1=

4∞ 0

1/2 h1/2

"

4.2 BER with EGC For a MIMO-OFDM system with EGC

reception, the average BER can be approximated by (25) with

n t;minstead ofDn t;mbeing used here; that is,

PEGCBER

γEGCn t

n | H1,(n) n t, , H N(n) r,n t

= √2

π

√

M −1

i =1

a M i

∞

m =1

(−1)(m+1)

b M i

(m −1/2) (2m −1)(m −1)! · DEGCn t;m

(29)

Definingν E = Nrν, σ2

EGC=(Nr!)2/8[(Nr −(1/2)) · · ·1/2]2,

ν E = ν E − μNr(Nr −1)π/4, and μ=2σ2

EGC· μ, when m > 2,

we have

n t;m =2σEGC2 !

(2m + Nr −4)μ(Nr −1)! +ν E$

μ2(m −3/2)(Nr −1)!

· DEGC

n t;m −1−

"

2σ2 EGC#2

(m + Nr −5/2)

n t;m −2

(30)

Table 1: Parameters for BER simulation in MIMO-OFDM systems

σ2

MIMO parameters (Nt =1, 2;N r =1, 2, 4) Receiving combining Without combining; EGC; MRC

as derived inAppendix B The initial condition is given by

DEGCn t;1 =

"

2σ2 EGC#1/2

(Nr −1)!

4∞ 0

h(N r −1/2)

"

μh +ν E#1/2 e − hdh. (31)

4.3 BER with MRC For a MIMO-OFDM system with

channel knowledge at the receiver, the receiving diversity can

be optimized by using MRC, and the average BER can be approximated by (25) withDMRC

n t;m instead ofDn t;mbeing used here; that is,

PMRCBER

γMRCn t

n | H1,(n) n t, , H N(n) r,n t

= √2

π

√

M −1

i =1

a M i

∞

m =1

(−1)(m+1)

b i M

(m −1/2) (2m −1)(m −1)! · DMRC

n t;m

(32)

By definingν M = ν +ν · σ2

ΔH,DMRC

n t;m withm > 2 is given by

DMRCn t;m = 

! (2m + Nr −4)μ(Nr −1)! +ν M$

μ2(m −3/2)(Nr −1)! · DMRCn t;m −1

− 2(m + Nr −5/2)e −(N r −1)

n t;m −2,

(33)

as derived inAppendix C The initial condition is given by

n t;1 = e −(N r −1)1/2

(Nr −1)!

4∞ 0

h(N r −1/2)

"

μh +ν M#1/2 e − hdh. (34)

4.4 Complexity of the Infinite-Series Representation of BER.

Infinite-series BER expression (27), (29), or (32) must be truncated in practice The truncation error is negligible

if the number of terms is large enough: Reference [31] shows that when the number of terms is as large as 50, the finite-order approximation is good In this case, a total of

151√

M summation operations

are needed to calculate the BER for each combining scheme

5 Numerical Results

Quasistatic MIMO wireless channels are assumed; that is, the channel impulse response is fixed over one OFDM symbol period but changes across the symbols The simulation parameters are defined inTable 1

The SINR degradation due to the residual frequency oﬀsets is shown inFigure 2forσ2

The SINR degradation increases withσ2

res Because of IAI due

to the multiple transmit antennas, the SINR performance of

Trang 9

8

9

10

11

12

2

13

σ2 res

SISO

EGC (N t= 2,N r= 2)

MRC (N t= 2,N r= 2)

EGC (N t= 2,N r= 4) MRC (N t= 2,N r= 4)

× 10−3

σ2

ΔH= 0.01; ε= 0.1; SNR= 10 dB

Figure 2: SINR reduction by frequency oﬀset in MIMO-OFDM

systems

10−4

10−3 10−2

10−1

10 0

10−5 10−4

10−3

10−2

σ2 res

E b /N0 = 10 dB;ε= 0.1; σ2

H= 10−3

QPSK:N t=N r= 1

16QAM:N t=N r= 1

EGC (QPSK:N t= 2,N r= 2)

MRC (QPSK:N t= 2,N r= 2)

EGC (16QAM:N t= 2,N r= 2)

MRC (16QAM:N t= 2,N r= 2)

EGC (QPSK:N t= 2,N r= 4)

MRC (QPSK:N t= 2,N r= 4)

EGC (16QAM:N t= 2,N r= 4)

MRC (16QAM:N t= 2,N r= 4)

Figure 3: BER degradation due to the residual frequency oﬀset in

MIMO-OFDM systems

10−3

10−2

10−1

10 0

E b /N0(dB) Simulation:σ2

res = 10−4 Theory:σ2

res = 10−4

Simulation:σ2

res = 10−3 Theory:σ2

res = 10−3

σ2

ΔH= 10−4 ;N t= 1,N r= 1

Figure 4: BER with QPSK when (Nt =1,N r =1)

10−3

10−2

10−1

10 0

E b /N0(dB)

σ2

ΔH= 10−4 ;N t= 1,N r= 1

Simulation: without combining;σ2

res = 10−4

Theory: without combining;σ2

res = 10−4

res = 10−3

Figure 5: BER with 16QAM when (Nt =1,N r =1)

MIMO-OFDM with (Nt = 2,Nr = 2) is worse than that

of SISO-OFDM, even though EGC or MRC is applied to exploit the receiving diversity IAI in MIMO-OFDM can be suppressed by increasing the number of receive antennas

In this simulation, when Nr = 4, the average SINR with

Trang 10

10−3

10−2

10−1

10 0

res = 10−4

res = 10−3

res = 10−4

res = 10−3

E b /N0(dB)

σ2

ΔH= 10−4 ;N t= 2,N r= 2

res = 10−4

res = 10−3

Figure 6: BER with QPSK when (Nt =2,N r =2)

either EGC or MRC will be higher than that of SISO-OFDM

system For each MIMO scenario, MRC outperforms EGC

The BER degradation due to the residual frequency

oﬀsets is shown inFigure 3 forσ2

ΔH = 10−3 andEb/N0 =

10 dB (Eb/N0is the bit energy per noise per Hz) The BER

for 4-phase PSK (QPSK) or 16QAM subcarrier modulation

is considered Just as with the case of SINR, the BER degrades

with large σ2

res For example, when (Nt = 2,Nr = 2) and

σ2

res =10−5 for QPSK (16QAM), a BER of 7×10−3(2.5 ×

10−2) or 6×10−3(2×10−2) is achieved with EGC or MRC

at the receiver, respectively Whenσ2

resis increased to 10−2, a BER of 2×10−2(6×10−2) or 1×10−2(5.5 ×10−2) can be

achieved with EGC or MRC, respectively

Figures 4 to 9 compare BERs of QPSK and 16QAM

with diﬀerent combining methods Figures4and5consider

SISO-OFDM The BER is degraded due to the frequency

oﬀset and channel estimation errors For a fixed channel

estimation variance errorσ ΔH2 , a larger variance of frequency

oﬀset estimation error, that is, σ2

res, implies a higher BER For example, ifσ ΔH2 =10−4,Eb/N0=20 dB andσ2

res=10−4, the BER with QPSK (16QAM) is about 1.8 ×10−3(5.5 ×10−3);

whenσ2

resincreases to 10−3, the BER with QPSK (16QAM)

increases to 4.3 ×10−3(1.5 ×10−2)

10−4

10−3

10−2

10−1

10 0

Simulation: EGC;σ2

res = 10−4

Theory: EGC;σ2

res = 10−4

Simulation: EGC;σ2

res = 10−3

Theory: EGC;σ2

res = 10−3

Simulation: MRC;σ2

res = 10−4

Theory: MRC;σ2

res = 10−4

Simulation: MRC;σ2

res = 10−3

Theory: MRC;σ2

res = 10−3

E b /N0(dB)

σ2

ΔH= 10−4 ;N t= 2,N r= 2

res = 10−4

res = 10−3

Figure 7: BER with 16QAM when (Nt =2,N r =2)

IAI appears with multiple transmit antennas, and the BER will degrade as IAI increases Note that since IAI cannot

be totally eliminated in the presence of the frequency oﬀset and channel estimation errors, a BER floor occurs at the high SNR IAI can be reduced considerably by exploiting the receiving diversity by using either EGC or MRC, as shown

in Figures 6,7,8, and 9 Without receiver combining, the BER is much worse than that in SISO-OFDM, simply because

of the SINR degradation due to IAI For example, when

Nt = Nr =2 andσ2

ΔH =10−4, the BER with QPSK is about

5.5 ×10−3 whenσ2

res = 10−4, which is three times of that

of SISO-OFDM (which is about 1.8 ×10−3), as shown in

Figure 6 For a given number of receive antennas, MRC can achieve a lower BER than that achieved with EGC, but the receiver requires accurate channel estimation For example,

in Figure 7, when σ ΔH2 = 10−4 with Nt = Nr = 2 and 16QAM, the performance improvement of EGC (MRC) over that without combining is about 5.5 dB (6 dB), and that performance improvement increases to 7.5 dB (8.5 dB) ifσ2

res

is increased to 10−3 By increasing the number of receive antennas to 4, this performance improvement is about 8.2 dB (9 dB) for EGC (MRC), withσ2

ΔH =10−4, or 11 dB (13.9 dB) for EGC (MRC), withσ2

ΔH =10−3, as shown inFigure 9

oﬀset and channel estimation errors For a fixed channel

estimation variance error< i>σ ΔH2 , a larger variance of frequency

oﬀset estimation error, that... statistics of the channel and frequency oﬀset

estimation errors (although the interference also comprises

the fourth-order statistics of the frequency oﬀset estimation

errors,... function of SINR in MIMO-OFDM is derived

in this section We considerM-ary square QAM with Gray

bit mapping In the work of Rugini and Banelli [11], the BER

of SISO-OFDM with

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