# 2.5.3

Mobile - Wireless

Nov 21, 2013 (5 years and 3 months ago)

358 views

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

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