Analysis of phase noise on OFDM signal
 An OFDM baseband signal generated from a inverse discrete
Fourier (IDFT) can be written as:
N 1
rn
s[n]   d r exp( j 2 )
N
r 0
 The received baseband signal is given by:
r[n]  s[n] exp( j[n])
where [n ] is a random process representing phase noise.
 After OFDM demodulated (equivalent to DFT), the signal in
the kth subcarrier y[k ] is given by:
1
y[k ]
N
N 1
N 1
m
exp( j[m])  d r exp( j 2  (r  k ) )

N
m 0
r 0
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Analysis of phase noise on OFDM signal
 To simplify analysis, the term exp( j[n])  1  j[n] is
assumed, hence above Eq. can be approximated to be :
1
y[k ] 
N
N 1
N 1
m
d r  exp( j 2  (r  k ) )

N
r 0
m 0
j N 1 N 1
m
  d r  [m] exp( j 2  (r  k ) )
N r 0 m 0
N
 d k  ek
where d k represents the correct information in the kth
subcarrier, and ek is the error caused by phase noise.
 The phase error ( ek ) is considered in two conditions. One is
the common phase error, the other introduces ICI.
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Common phase error
1. Common phase error: r = k:
The phase error ek
j
Nc 
N
N 1
r k
is expressed as N c ,
N 1
m
j N 1
d r  [m] exp( j 2  (r  k ) ) r  k  d k  [m]

N
N m 0
r 0
m 0
 Hence one can observe that each sub-carrier constellation has the same
rotation called the common phase error, where the rotational angle is equal
to
N 1
1
Tan (  [m] / N )
.
m 0
 Since this phase error is constant for every sub-carrier, it can be corrected
by the information from the pilot signal.
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ICI error
2. Loss of orthogonality (or inter-carrier interference; ICI): r≠k
ek
r k
is expressed as N i ,
j N 1 N 1
m
N i   d r  [m] exp( j 2  (r  k ) ) r  k
N r 0 m 0
N
 N i is the summation of the information of the other N-1 sub-carriers
multiplied by some complex function of phase noise.
 If the number of sub-carriers is large enough and the data point in each
sub-carrier is statistically independent, according to the central limit
theorem, N i can be treated as additive Gaussian noise that could not be
compensated.
 Intuitively, the value of N i will increases with the number of sub-carriers.
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ICI error
 The ICI effect can be treated as a white noise with zero mean. The average
power of the white noise can be expressed as [2]
 Ni
2
fs / 2 S ( f ) sin( Nf / f )
 fs / 2

2
S
 Ps 
S ( f )df  
(
) df 
 fs / 2
 fs / 2 N 2
sin( f / f S )


where PS is the average signal power, S  ( f ) denotes the PSD (power
spectral density) of phase noise, N is the number of sub-carrier, and f s
is the DFT sampling rate. From above Eq. , the variance of N i increases
with N and PS .
 As a special case, in conventional single-carrier modulation (N=1) the
variance of N i approaches zero since there is no other sub-carrier to cause
inter-channel interference.
 If the common phase error can be corrected completely, the phase noise
effect on an OFDM signal is equivalent to the increase of the white noise
due to ICI.
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Lorentzian spectrum
 In general, the PSD of an oscillator signal with phase noise
can be modeled by a Lorentzian spectrum, whose two-sided
spectrum is given by
Phase Noise
0
1  f  f c / f12
2
where f1 is the 3-dB
linewidth of the
oscillator signal.
-100 dBc/Hz @100kHz
-95 dBc/Hz @100kHz
-90 dBc/Hz @100kHz
-85 dBc/Hz @100kHz
-10
-20
-30
-40
PSD [dBc/Hz]
Sd ( f ) 
1 / f1
-50
-60
-70
-80
-90
-100
0
10
1
10
2
3
10
10
4
10
5
10
Hz
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