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Performance Evaluation of an OFDM System under RayleighFading Environment
1 Gurpreet Kaur, 2 Partha Pratim Bhattacharya
Department of Electronics and Communication Engineering, Faculty of Engineering and TechnologyMody Institute of Technology & Science (Deemed University), Lakshmangarh , Dist. Sikar, Rajasthan,
Pin – 332311, India
ABSTRACT
The performance of an OFDM system is affected by parameters such as carrier frequency offset and phase noise. In the
presence of such parameters the performance of OFDM system degrades. Under Rayleigh fading environment the performance
fluctuates depending on the signal strength. In this paper the performance of an OFDM system is studied in the presence of
Rayleigh fading channel. Results show that the SINR of the overall system fluctuates due to the effect of Rayleigh fadingchannel.
Keywords: OFDM Wireless communication system, carrier frequency offset (CFO), Phase noise, signal to interference plus noise ratio(SINR), Rayleigh fading channel.
1. ORTHOGONAL FREQUENCY DIVISION
MULTIPLEXING (OFDM)
OFDM (Orthogonal Frequency Division
Multiplexing) has been developed to combat multipath
effects and make better use of the system.
An important parameter that should be carefullyconsidered while dealing with OFDM system is phase noise
because an accurate prediction of the tolerable phase noise
can allow the system to relax specifications. Theconsideration of phase noise in OFDM systems is important
with frequencies above 25 GHz, as suggested in some
European ACTS projects dealing with LMDS (Local-
Multipoint Distribution Systems) [1]. The effect of phase
noise in OFDM and the degradation caused by it have been
analyzed by several authors [2]–[5].
Carrier frequency offset (CFO) exist between user
terminals and the base station. OFDM systems are very
sensitive to CFO, which leads to performance degradation
by introducing inter-carrier-interference (ICI) [6]. The purpose of this paper is to analyze the performance of an
OFDM wireless communication system in the combined
effect of carrier frequency offset and phase noise in Rayleighfading environment.
OFDM is a block modulation scheme where a
block of N information symbols is transmitted on N
subcarriers in parallel. The time duration of an OFDM
symbol is N times larger than that of a single-carrier system.
An OFDM modulator can be implemented as an inversediscrete Fourier transform (IDFT) on a block of N
information symbols which is then followed by an analog-
to-digital converter (ADC). In order to mitigate the effects of
inter symbol interference (ISI) caused by channel time
spread, each block of N IDFT coefficients is typically
preceded by a cyclic prefix (CP) or a guard interval
consisting of G samples, such that the channel length is at
least equal to the length of CP. In this condition, a linear
convolution of the transmitted sequence and the channel is
converted to a circular convolution. As a result, the effectsof the ISI are easily and completely eliminated. This
approach enables the receiver to use fast signal processing
transforms such as a fast Fourier transform (FFT) for OFDM
implementation [7]. Similar techniques can be employed in
single-carrier systems as well, by preceding each transmitted
data block of length N by a CP of length G, while using
frequency domain equalization at the receiver.
One of the best ways to mitigate the effect of
multipath is to use OFDM communication systems. A
combination of OFDM and coding associated with
interleaving in the frequency domain (COFDM) can take
advantage from the diversity associated to multipath [8].
The following equation gives the N point complex
modulation sequence transmitted by OFDM signal for the
ℎ symbol:
() = () 2
−1
=0
(1)
where n ranges from 0 to + − 1.
2. PHASE NOISE AND CARRIER
FREQUENCY OFFSET
Phase noise must be carefully considered when
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dealing with any of these communication systems since an
accurate prediction of the tolerable phase noise can allow thesystem and RF designers to relax specifications. A
theoretical analysis of phase noise effects in OFDM signals
is carried out by many authors. The complex envelope of thetransmitted OFDM signal for a given OFDM symbol,
sampled with sampling frequency = , is:
() = . 2
−1
=0
(2),
where n=0, 1, …, N-1.
This symbol is actually extended with a Time
Guard in order to cope with multipath delay spread.
Assuming that the channel is flat, the signal is only affected by phase noise at the receiver:
(
) =
(
).
() (3)
The received signal is an Orthogonal Frequency
Division De-multiplexed (OFDD) signal by means of a
Discrete Fourier Transform. For the purpose of separating
the signal and noise terms, let us assume that Φ(n) is small,
so that:
() ≈ 1 + (). (4)
In this case, the de multiplexed signal is given as:
() ≈ +
−1
=0
(). 2 (− )
−1
=0
(5)
() ≈ + (6)
Thus we have an error term for each sub-carrier
and which results from some combination of all of them and
is added to the useful signal. The signal to noise ratio
degradation ( ) caused by phase noise is the same inOFDM and signal carrier systems, given that phase noise
variance is small (2 ≪ 1) it follows the expression [9] :
= 10. log1 + 2 0
. (7)
In this equation represents the symbol energy
and 0 is the power spectral density of additive whiteGaussian noise. One way to characterize oscillator’s phase
noise is the single-side-band phase noise power density
function ( ), which represents the ratio (in dBc; ‘c’ stands
for carrier) between the single-side-band noise power in a 1
Hz bandwidth at a distance from the carrier and thecarrier power.
This characterization is normally performed by
using a spectrum analyzer which provides the power spectral
density of the equipment’s phase noise ( 0) in relation to
the carrier power (C). Given that the phase noise has a zero
mean and that extends up to a frequency b (either because
phase noise is band-limited or due to the presence of
filtering in the receiver), its variance can be found as [10]:
2 = � (20
0
). (8)
The absolute value of the actual CFO ɛ , is either
an integer multiple or a fraction of △ , or the sum of them.
If ɛ is normalized to the subcarrier spacing △ , then the
resulting normalized CFO of the channel can be generallyexpressed as:
ɛ =
ɛ
△ = + є (9)
where δ is an integer and |є | ≤ 0.5. The impact of aninteger CFO on OFDM system is different from the
influence of a fractional CFO. In the event that δ≠0 and ε=0,
symbols transmitted on a certain subcarrier, e.g., subcarrier
k, will shift to another subcarrier , = + −1. As the ICI effect is focused, normalized CFO is
considered as given below,
ɛ =
ɛ
△ = є (10)
Since no ICI is caused by an integer CFO, relative
CFO (ε) to assumed to be a Gaussian process, statistically
independent of the input signal, with zero mean and
variance ɛ
2.
3. FORMULATION OF SINR
In the presence of CFO, phase noise, timing jitter
and Rayleigh fading the OFDM system performance can be
expressed by the following SINR expression [10]:
(ɛ, 2,, )
≥ 2 (1 − ){2(ɛ)}
1 + 2(1 − )[0.5947(sinɛ)2 + {22 2(ɛ) ∑ 1
2 ( )
}] + 2−1=1
; |ɛ|
≤0.5,|
|
≤1. (11)
where ɛ is the normalized CFO, 2 is the variance of phase
noise, denotes timing jitter and corresponds to the channelattenuation/gain parameter in Rayleigh fading environment.
is the input SNR, N is the number of sub-carriers in the
channel.
When the impulse response (; ) is modeled as a
zero mean complex valued guassian process, the envelope
|(; )| at any instant is Rayleigh distributed. In this case thechannel is said to be Rayleigh fading channel. In the event that
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there are fixed scatterers or signal reflectors in the medium in
addition to randomly moving scatterers (; ) can no longer be modeled to have zero mean.
SINR expression in the presence of phase noise and
CFO without timing jitter and considering a Rayleigh fading
environment can be expressed as:
(ɛ,2,)
≥ 2{2(ɛ)}
1 + 2[0.5947(sin ɛ)2 + {22 2(ɛ)∑ 1
2 ( )
}]−1=1
; |ɛ| ≤ 0.5 (12)
Table 1: System and channel parameters for simulation
Number of sub carriers (N) 64
Channel typeRayleigh fading
channel
Input SNR values 10 dB.
Channel attenuation/gain
parameter (α)
0.2,0.4,0.6,0.8,1,1.2,
1.4,1.6,1.8
Normalized CFO 0.05
Variance of phase noise 0.33
4. RESULT AND DISCUSSION
Simulation has been carried out using MATLAB.
Figure 1 and 2 shows the plot of SINR versus variance of
phase noise (2) for various values of attenuation/gain parameter, normalized CFO being 0.05. Results are plotted
for input SNR () of 10 dB.
Figure 1: SINR versus variance of phase noise as a function
of attenuation/gain parameter (CFO=0.05)
Figure 2: SINR versus variance of phase noise as a function
of attenuation/gain parameter (CFO=0.05)
Figure 3 and 4 show the variation of SINR versus
normalized CFO, for various values of attenuation/gain parameter. Variance of phase noise and input SNR () areconsidered to be 0.33 and 10 dB respectively.
Figure 3: SINR versus normalized CFO as a function of
attenuation/gain parameter (variance of phase noise=0.33)
Figure 4: SINR versus normalized CFO as a function of
attenuation/gain parameter (variance of phase noise=0.33)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-8
-6
-4
-2
0
2
4
6
8
10
variance of phase noise
S I N R
( i n d
B )
SINR versus variance of phase noise (CFO=0.05 and attenuation/gain parameter<=1)
attenuation/gain parameter=0.2
attenuation/gain parameter=0.4
attenuation/gain parameter=0.6
attenuation/gain parameter=0.8
attenuation/gain parameter=1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-6
-4
-2
0
2
4
6
8
10
12
14
variance of phase noise
S I N R
( i n
d B )
SINR versus variance of phase noise (CFO=0.05 and attenuation/gain parameter>=1)
attenuation/gain parameter=1
attenuation/gain parameter=1.2
attenuation/gain parameter=1.4
attenuation/gain parameter=1.6
attenuation/gain parameter=1.8
0.02 0.04 0.06 0.08 0.1 0.12 0.14
-8
-6
-4
-2
0
2
Normalized CFO
S I N R
( i n d B )
SINR versus normalized CFO (variance of phase noise=0.33 and attenuation/gain parameter<=1)
attenuation/gain parameter=0.2
attenuation/gain parameter=0.4
attenuation/gain parameter=0.6
attenuation/gain parameter=0.8
attenuation/gain parameter=1
0. 012 0. 014 0. 016 0. 018 0. 02 0. 022 0.024 0. 026 0. 028 0. 03 0. 032
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
Normalized CFO
S I N R
( i n d B )
SINR versus normalized CFO(variance of phase noise=0.33 and attenuation/gain parameter>=1)
attenuation/gain parameter=1
attenuation/gain parameter=1.2
attenuation/gain parameter=1.4
attenuation/gain parameter=1.6
attenuation/gain parameter=1.8
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It can be seen from Figure 1, that for an input SNR
of 10 dB as attenuation/gain parameter is decreased from 1to 0.2, the SINR values decreases and the gain parameter 1
indicates zero fading. Figure 2 show that for an input SNR
of 10 dB as attenuation/gain parameter increases from 1 to1.8, the SINR values increase and is highest for gain
parameter of 1.8. It can also be seen from Figure 3, that for an input SNR of 10 dB as attenuation/gain parameter
decreases from 1 to 0.2, the SINR values decreases. Figure 4
show that for an input SNR of 10 dB as attenuation/gain
parameter increases from 1 to 1.8 the SINR values increases.
So the SINR value increases or decreases from its value
when there is no fading depending on constructive or
destructive interference of the signals from multipath.
5. CONCLUSION
In this paper, performance evaluation of an OFDM
communication system is studied by taking normalized CFO
and variance of phase noise under consideration in Rayleigh
fading environment (α≠1). An SINR expression is taken intoconsideration for evaluating the same. Results show that as
compared to an ideal channel (attenuation/gain parameter=1)the SINR is less for an OFDM system under Rayleigh fading
environment when attenuation/gain parameter<1 whereas theSINR is more for an OFDM system under Rayleigh fading
environment when attenuation/gain parameter>1.
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