Wireless Communication Technologies
1
Analysis of phase noise on OFDM signal
An OFDM baseband signal generated from a inverse discrete
Fourier (IDFT) can be written as:
The received baseband signal is given by:
where is a random process representing phase noise.
After OFDM demodulated (
equivalent to DFT
), the signal in
the
k
th subcarrier is given by:
2.5.3
)
2
exp(
]
[
1
0
N
r
r
N
rn
j
d
n
s
])
[
exp(
]
[
]
[
n
j
n
s
n
r
]
[
n
]
[
k
y
1
0
1
0
)
)
(
2
exp(
])
[
exp(
1
]
[
N
m
N
r
r
N
m
k
r
j
d
m
j
N
k
y
Wireless Communication Technologies
2
Analysis of phase noise on OFDM signal
To simplify analysis, the term is
assumed, hence above Eq. can be approximated to be :
where represents the correct information in the
k
th
subcarrier, and is the error caused by phase noise.
The phase error ( ) is considered in two conditions. One is
the
common phase error
, the other
introduces ICI
.
2.5.3
]
[
1
])
[
exp(
n
j
n
j
k
k
N
m
N
r
r
N
m
N
r
r
e
d
N
m
k
r
j
m
d
N
j
N
m
k
r
j
d
N
k
y
1
0
1
0
1
0
1
0
)
)
(
2
exp(
]
[
)
)
(
2
exp(
1
]
[
k
d
k
e
k
e
Wireless Communication Technologies
3
Common phase error
1.
Common phase error: r = k:
The phase error is expressed as
,
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
.
Since
this
phase
error
is
constant
for
every
sub

carrier,
it
can
be
corrected
by
the
information
from
the
pilot
signal
.
2.5.3
k
r
k
e
c
N
1
0
1
0
1
0
]
[
)
)
(
2
exp(
]
[
N
m
k
k
r
N
m
N
r
r
c
m
d
N
j
N
m
k
r
j
m
d
N
j
N
)
/
]
[
(
1
0
1
N
m
Tan
N
m
Wireless Communication Technologies
4
ICI error
2.
Loss of orthogonality (or inter

carrier interference; ICI): r≠k
is expressed as
,
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
, can be treated as
additive Gaussian noise
that could not be
compensated.
Intuitively, the value of will increases with the number of sub

carriers.
2.5.3
k
r
k
e
i
N
k
r
N
m
N
r
r
i
N
m
k
r
j
m
d
N
j
N
1
0
1
0
)
)
(
2
exp(
]
[
i
N
i
N
i
N
Wireless Communication Technologies
5
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]
where is the average signal power, denotes the
PSD
(power
spectral density) of phase noise,
N
is the number of sub

carrier, and
is the DFT sampling rate. From above Eq. , the variance of increases
with
N
and .
As a special case, in conventional single

carrier modulation (
N
=1) the
variance of 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
.
2.5.3
S
P
)
(
f
S
s
f
S
P
2
/
2
/
2
2
2
/
2
/
2
)
)
/
sin(
)
/
sin(
(
)
(
)
(
fs
fs
S
S
fs
fs
s
i
df
f
f
f
Nf
N
f
S
df
f
S
P
N
i
N
i
N
Wireless Communication Technologies
6
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
where is the 3

dB
linewidth of the
oscillator signal.
2.5.3
1
f
10
0
10
1
10
2
10
3
10
4
10
5
100
90
80
70
60
50
40
30
20
10
0
Hz
PSD [dBc/Hz]
Phase Noise
100 dBc/Hz @100kHz
95 dBc/Hz @100kHz
90 dBc/Hz @100kHz
85 dBc/Hz @100kHz
2
1
2
1
/
1
/
1
)
(
f
f
f
f
f
S
c
d
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