BER Analysis for OFDM with ZF-FDE inNakagami-m Fading Channels

Juan J. Sanchez-Sanchez, Unai Fernandez Plazaola and Mari Carmen Aguayo-TorresDepartamento de Ingenierıa de Comunicaciones,

Universidad de Malaga, Spain,{jjsanch,unai,aguayo}@ic.uma.es

Abstract—In this paper we study the statistical distributionof the enhanced noise after zero forcing frequency domainequalization in an OFDM system transmitting over a Nakagami-m fading channel. With this purpose, we obtain the expressionof the density function of the ratio between the modulus of acomplex Gaussian random variable (i.e., a Rayleigh distributedrandom variable) and that of a 𝑛-dimensional Gaussian randomvariable (i.e., a Nakagami-m distributed random variable). Fromthis expression, we derive the density and the distribution of theresulting noise term after zero forcing equalization. Lastly, wepresent an analytical expression for BER in the scenario understudy which is validated through simulations.

I. INTRODUCTION

High-data-rate transmissions over wireless channels mustcope with impairments due to the limitations in bandwidth andpower. Orthogonal Frequency-Division Multiplexing (OFDM)has proven itself as a suitable candidate to overcome thesedifficulties in an efficient fashion. OFDM is inherently im-mune to multi-path effects and provides tolerance to interfer-ence in non-Line Of Sight (non-LOS) conditions. OrthogonalFrequency-Division Multiple Access (OFDMA) (i.e., the mul-tiple access scheme based on OFDM [1]) also has many inter-esting properties as its scalability and multi-path robustnessinherited from OFDM or its multiple-input multiple-output(MIMO) compatibility [1]). Due to these nice properties, ithas been adopted in 802.16 (WiMAX) standards [2] and inthe Long Term Evolution (LTE) for the Universal MobileTelecommunications System (UMTS) [3] specifications.

Nakagami-m fading model has greater flexibility and accu-racy in matching various experimental data more general thanRayleigh, log-normal, or Rician distributions [4]. To the bestof our knowledge, the literature regarding OFDM performancein Nakami-m fading channels is not abundant and its mainlyfocused in semi-analytical expressions (see [5] and [6]). Inthis paper we present a performance analysis of OFDM basedon the stochastic study of zero-forcing enhanced noise in aNakagami-m fading channel model. Based on the results ofthis study, we present an approximate closed-form expression,for BER with Binary Phase Shift Keying (BPSK) and squareMulti-level Quadrature Amplitude Modulation (M-QAM).

The rest of this article is organized as follows. Section IIcontains the description of the general model for an OFDMsystem over a Nakagami-m fading channel. This system isused in section III to study the stochastic nature of the

equivalent noise after zero forcing equalization. Results ob-tained in the previous section serve to derive in section IVthe aforementioned BER expression. Lastly, the validity ofthe proposed expression is checked in section V throughsimulations whereas concluding remarks are presented in VI.

II. SYSTEM AND CHANNEL MODELS

A. System Model

The block diagram for a simple OFDM system compoundedby transmitter and receiver-ends is depicted in Fig. 1. Thesequence of bits to be transmitted is mapped into a sequence ofcomplex symbols x′ according to the modulation scheme used(e.g., QAM). These complex symbols are mapped into the setof allocated sub-carriers that may be adjacent (i.e., LocalizedOFDMA) or may be distributed across the total bandwidth(i.e., Interleaved OFDMA).

Non-allocated sub-carriers are set to zero and an inverse FastFourier Transform (IFFT) operation converts the frequency-domain symbol into a time-domain symbol. A cyclic prefix oflength greater than the channel response is added as a guardperiod to reduce temporal dispersion and eliminate intersym-bol interference (ISI) [7]. This prefix also helps to preserve theorthogonality among sub-carriers, thereby avoiding intercarrierinterference (ICI) as well. The addition of the cyclic prefix alsotransforms the convolution between the OFDM symbols in thetime domain and the channel response in circular. This circularconvolution becomes in the frequency domain a point-wisemultiplication between the complex symbol allocated in eachsub-carrier and the corresponding channel frequency responseand, thereby it is possible to perform the equalization in thefrequency domain.

The main advantage of OFDM is its behavior to bad channelconditions, such as fading caused by multi-path propagation,because if the bandwidth of the transmitted signal is less thanthe coherence bandwidth of channel, the channel response ineach carrier can be considered flat. Thus, the channel can bemodelled as a set of narrowband fading channels, one for eachsingle sub-carrier. Besides, thanks to its inherent immunityto multipath effects, it provides multipath and interferencetolerance in non-Line Of Sight (non-LOS) conditions. Besides,OFDM can be also employed for multi-user transmission, thatis, as a multiple access technique as described below.

At the receiver the cyclic prefix is suppressed and an FFToperation converts the time-domain symbols into a frequency-

978-1-4244-6952-9/10/$26.00 ©2010 IEEE

S/P M-IFFT CPX’

Channel

CPM-FFTEqualizer S/PDetector

Fig. 1: Transmitter and receiver for OFDM.

domain symbol. The received signal is a version of thetransmitted signal transformed by channel response and con-taminated with thermal noise

y = Hx+ 𝜂 (1)

where H is a 𝑁𝑥𝑁 diagonal matrix whose elements 𝐻𝑘

with 𝑘 = 1, . . . , 𝑁 are the channel frequency responses, 𝜂 isa vector with the Additive White Gaussian Noise (AWGN)component for each sub-carrier.

Finally, a frequency equalization is carried out. In the caseunder study, ideal estimation of channel frequency response isassumed and zero forcing equalization is applied

x = x+ (H𝐻H)−1H𝐻𝜂 (2)

Thus, the expression for the 𝑘-th received symbol is

��𝑘 = 𝑥𝑘 +𝜂𝑘𝐻𝑘

= 𝑥𝑘 + 𝜂𝑘. (3)

B. Channel Model

In the scenario under analysis a frequency-selectiveNakagami-m channel model is considered. The channel im-pulse response is modeled as a finite impulse response (FIR)filter whose taps ℎ(𝑘) follow a Nakagami-m distribution where𝑘 = 1, . . . , 𝐿−1 being 𝐿 the number of taps. This probabilitydensity function was first introduced in [8] and it is oftenexpressed as

𝑓∣ℎ(𝑘)∣(𝑥) =2

Γ (𝑚)

(𝑚𝜔

)𝑚𝑥2𝑚−1𝑒−

𝑥2𝑚𝜔 𝑥 ∈ ℝ 𝑎𝑛𝑑 𝑥 ≥ 0,

(4)

where𝑚

𝜔=

1

2𝜎. The parameter 𝜔 controls the spread

whereas 𝑚 determines the shape and, therefore, the intensityof the fading.

Its mean and variance are 𝐸[∣ℎ(𝑘)∣] = Γ(𝑚+ 12 )

Γ(𝑚)

√𝜔𝑚 and

𝑉 𝑎𝑟(∣ℎ(𝑘)∣) = 𝜔 − 𝜔𝑚(

Γ(𝑚+ 12 )

Γ(𝑚+1)

)2

.

In the case under study the channel is assumed to haveunitary power 𝜔 = 11.

Kang et al. show in [4] how, for frequency-selectiveNakagami-m fading channels, the magnitudes of the channelfrequency responses can be approximated as Nakagami-m

1𝜔 = 𝐸[∣ℎ(𝑘)∣2] =∫∞−∞ 𝑥2𝑓∣ℎ(𝑘)∣(𝑥)𝑑𝑥.

distributed random variables with fading and mean powerparameters as explicit functions of the fading and meanpower parameters of the channel impulse responses. Thus,the magnitude of the channel frequency response for a givensubcarrier is distributed with the following probability densityfunction (PDF)

𝑓∣𝐻∣(𝑥) =2

à (��)

(��

��𝑓

)��

𝑥2��−1𝑒−𝑥2���� 𝑥 ∈ ℝ 𝑎𝑛𝑑 𝑥 ≥ 0,

(5)with �� and �� as defined in [4]. This allows us to calculate

the density of the zero forcing enhanced noise in the followingsection.

III. ENHANCED NOISE ANALYSIS

The enhanced noise term after zero-forcing equalization fora Nakagami-m fading channel can be described as the ratioof two random variables. For any given pair of independentrandom variables, the density of their ratio can be computedas follows.

Definition 1. Let 𝑋 and 𝑌 be continuous random variableswith a joint density 𝑓𝑋,𝑌 (𝑥, 𝑦). Let 𝑉 = 𝑋/𝑌 . Then thedensities of 𝑈 and 𝑉 are given by

𝑓𝑉 (𝑣) =

∫ ∞

−∞𝑓𝑋,𝑌 (𝑣𝑦, 𝑦)∣𝑦∣𝑑𝑦, (6)

In the case under study, the numerator corresponds to thecomplex noise at the receiver, that is, a complex Gaussianrandom variable. This is a circularly symmetric random vari-able, thereby its phase and modulus are independent randomvariables. The former follows a uniform distribution between0 and 2𝜋 whereas the latter has the a Rayleigh PDF whoseexpression is

𝑓𝑅𝜂(𝑥) =

𝑥

𝜎2𝑒−

𝑥2

2𝜎2 𝑥 ∈ ℝ 𝑎𝑛𝑑 𝑥 ≥ 0, (7)

with 𝜎2 =𝑁0

𝐸𝑠where 𝑁0 is the noise power spectral density

and 𝐸𝑠 is the energy per symbol.The denominator corresponds to the frequency channel re-

sponse and it is another circularly symmetric complex randomvariable. Its modulus follows the Nakagami-m distribution in5 and its phase is assumed to be uniform [0, 2𝜋]. Formally,the ratio of the two complex random variable can be derivedas follows.

Definition 2. Let 𝑋 be a complex Gaussian random variableswith zero-mean and variance 𝜎𝑋 . Let 𝑌 be a 𝑛-dimensionalGaussian random vector with zero-mean and variance 𝜎𝑌 .Let 𝑅𝑋 and 𝑅𝑌 be the random variables for their respectivemodulus. The former follows a Rayleigh distribution (eq. (7))whereas the latter has a Nakagami-m distribution (eq. (4))where 𝑚 = 𝑛/2. Lastly, let 𝜃𝑋 and 𝜃𝑌 be random variableswith a uniform distribution in [0, 2𝜋]. Then, the randomvariable 𝑅𝑍 is the ratio of 𝑅𝑋 and 𝑅𝑌 and its density iscalculated applying eq. (6) as

𝑓𝑅𝑍(𝑟) =

2𝑚𝜎2𝑌 𝜎2𝑚𝑋 𝑟

(𝑟2𝜎2𝑌 + 𝜎2𝑋)𝑚+1 𝑟 ∈ ℝ , 𝑟 ≥ 0, (8)

whereas the variable 𝜃𝑍 has a uniform random in [0, 2𝜋].The modulus and the phase of 𝑍 are independent randomvariables, thereby its density yields

𝑓𝑍 (𝑟) =1

2𝜋

2𝑚𝜎2𝑌 𝜎2𝑚𝑋 𝑟

(𝑟2𝜎2𝑌 + 𝜎2𝑋)𝑚+1 𝑟 ∈ ℝ , 𝑟 ≥ 0, (9)

and hence 𝑍 is also a circular symmetric random variable.

Since the enhanced noise 𝜂 is a random variable resultingof the ratio of two complex circularly symmetric random vari-ables, it also has circularly symmetry. Hence, its modulus andphase are independent random variables and the probabilityvalues depend only on the radius of the circle on which theylie [9] as shown in 2b. In fact, its phase is a uniform randomvariable in [0, 2𝜋] whereas its modulus has the followingdensity,

𝑓𝜂 (𝑟) =2��𝜎2��𝑟

(𝑟2 + 𝜎2)��+1

𝑟 ∈ ℝ , 𝑟 ≥ 0. (10)

With a simple change of variable, eq. (10) can be trans-formed into the joint PDF for real and imaginary componentsof the resulting complex random variable. Thus, the joint PDFin Cartesian coordinates coordinates is

𝑓𝜂𝑟,𝜂𝑖(𝑥, 𝑦) =

��𝜎2��

𝜋 (𝜎2 + (𝑥2 + 𝑦2))1+��

𝑥, 𝑦 ∈ ℝ, (11)

where 𝜂𝑟 and 𝜂𝑖 are the real and imaginary components ofthe elementary noise term.

In this case it is possible to compute each correspondingmarginal PDF

𝑓𝜂𝑟(𝑥) =

Γ (��+ 1/2)𝜎2��

√𝜋Γ(��) (𝑥2 + 𝜎2)

12+��

𝑥 ∈ ℝ, (12)

where Γ(��) is the Gamma function [10].The mean for this PDF exists for �� > 1/2 and in that case

it is zero. Similarly, the variance is finite only if �� > 1 and

then its value is��𝜎2

2(−1 + ��)The corresponding Cumulative Distribution Function (CDF)

of 𝜂𝑟 results

𝐹𝜂𝑟(𝑥) =

(−1)−��

2√𝜋

Γ(��+ 1

2

)Γ(𝑚)

𝐵− 2��𝜎2

𝑥2

(��,

1

2− ��

), (13)

where 𝐵 is the incomplete Beta function [10].From eq. (13) and with �� = 1, it is possible to obtain an

analytical expression of the BER for BPSK in OFDM overNakagami-m fading channels as

𝑝𝑒 = 𝐹��𝑟(0) =

1

2

(1−

√𝐸𝑠

𝐸𝑠 +𝑁0

), (14)

which is the same expression obtained by the PDF methodin Chapter 14.3 in [11].

IV. BER ANALYSIS

The expression (13) is used here to derive BER expressionsfor square M-QAM constellations. Under the assumption ofindependent bit-mapping for in-phase and quadrature (e.g,Gray mapping) and reminding that the resultant noise 𝜂𝑘 iscircularly symmetric, the BER can be expressed, using theapproach of [12], as

𝐵𝐸𝑅 =∑𝐿−1

𝑛=1𝑤(𝑛)𝐼(𝑛) (15)

where 𝐼(𝑛) are called components of error probability(CEP), the 𝑤(𝑛) are coefficients dependent on the constellationmapping and 𝐿 = 2 for BPSK and 𝐿 =

√𝑀 for M-QAM.

The CEPs are defined as

𝐼(𝑛) = 𝑃𝑟{ℜ{𝜂𝑟} > (2𝑛− 1)𝑑} = 1− 𝐹𝜂𝑟((2𝑛− 1)𝑑) (16)

where 𝑑 is the minimum distance between the symbol andthe decision boundary and can be expressed as a function ofthe constellation energy 𝐸𝑠 as 𝑑 =

√𝐸𝑠 for BPSK and as 𝑑 =√

3𝐸𝑠

2(𝑀−1) for M-QAM. The coefficients 𝑤(𝑛) are obtainedusing the method described in [12].

Plugging eq. (16) into the sum (15), we obtain the finalBER expression as

𝐵𝐸𝑅 =∑𝐿−1

𝑛=1𝑤(𝑛) (1− 𝐹𝜂𝑟

((2𝑛− 1)𝑑)) (17)

where 𝐹𝜂𝑟is computed with eq. (13).

V. SIMULATIONS AND NUMERICAL RESULTS

In order to validate the presented expression, results fromeq. (17) for signal to noise ratio (SNR) values ranging from0 to 30 dB are compared with values obtained by means ofsimulations. In order to simplify the validation process, we di-rectly generate channel frequency responses whose magnitudeis distributed according to a Nakagami-m distribution withsome integers and half-integer values of ��. In Fig. 3 showsthe numerical evaluation of the BER for a BPSK transmissionfor different values of ��. Note that for �� = 1 the channel hasa Rayleigh fading. If �� < 1 BER values worsen (i.e., it is aworse than Rayleigh scenario) whereas for 𝑚 > 1 we obtainlower values of BER. Similar conclusions can be extractedfrom Fig. 4.

(a) 3D Plot

0.02

0.04

0.06

0.08

0.1

0.12

0.14

�2 �1 0 1 2�2

�1

0

1

2

x

y

(b) Contour plot

Fig. 2: Example of density of the Rayleigh/Nakagami-m ratio.

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR(dB)

BE

R

BER for Nakagami−m fading channels (BPSK)

SimulatedAnalytical

m = 2

m = 0.5

m = 1

m = 1.5

Fig. 3: BER values for different Nakagami-m fading channels.

0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR(dB)

BE

R

BER for Nakagami−m fading channels (QPSK)

SimulatedAnalytical

m = 1.5

m = 1

m = 0.5

m = 2

Fig. 4: BER values for different Nakagami-m fading channels.

VI. CONCLUSION

In this paper we study the enhanced noise after zero forcingequalization for a OFDM signal transmitted over a Nakagami-m fading channel was undertaken. Following the work byKang et al. [4], we assume the magnitude of their frequencyresponse to be approximately distributed as Nakagami-m ran-dom variable. Under this assumption, we derive the expressionof the density function of the ratio between a Rayleigh anda Nakagami-m random variable and, from there, we to obtainthe density of the enhanced noise in the case under study. Notethat, whereas results based on semi-analytical expressions canbe found in the literature [6], we provided an exact closed-form expression which is validated through simulations.

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial supportof the Junta de Andalucıa (Proyecto de Excelencia P07-TIC-03226) and express their gratitude to the Spanish Governmentand the European Union (TEC2007-67289/TCM) for theirpartial support.

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