D. Qu et al. EM-based noise plus interference estimation
ing to the kth symbol as shown in Reference [9]
λ(c
i
k
= b) = log p(c
i
k
= b|y
k
) ∝ log
S
k
∈χ
i
b
p(y
k
|x
k
= S
k
)
≈ max
S
k
∈χ
i
b
log p(y
k
|x
k
= S
k
)
= min
S
k
∈χ
i
b
|y
k
− α
k
S
k
|
2
σ
2
k
+ const (2)
where χ
i
b
={µ(c
1
k
, ···c
i
k
, ···c
m
k
)|c
i
k
= b} is the signal sub-
set with the ith bit c
i
k
equaling b. Then, the bit metric of each
demodulated bit can be represented by its log-likelihood
ratio (LLR)
(c
i
k
) = λ(c
i
k
= 1) − λ(c
i
k
= 0) (3)
For the conventional decoding scheme, the noise plus
interference variance σ
2
k
is assumed to be identical for all
the symbols, i.e., σ
2
k
= σ
2
, and thus can be dropped in the bit
metric calculation. When narrowband interference exists,
the noise variance is not a constant for all the subcarriers,
and dropping the noise plus interference variance σ
2
k
in the
bit metric calculation results in metric mismatch. Knowing
the position of the interfered subcarriers and the variance
of the noise plus interference, the optimal decoder, i.e.,
maximum-likelihood decoder (MLD), weights each sym-
bol differently depending on whether the symbol is hit by
background Gaussian noise (σ
2
k
= σ
2
G
) or noise plus inter-
ference (σ
2
k
= σ
2
G
+ I
2
k
) , where I
2
k
is the interference power
on the kth symbol.
For OFDM-based CR systems with narrowband inter-
ference, it is difficult to estimate the position and power
of interference with a limited number of OFDM symbols.
Therefore, we propose the IED scheme, which is based
on EM algorithm, to effectively estimate the narrowband
interference.
3. EM-BASED NOISE PLUS
INTERFERENCE POWER
ESTIMATION
The EM algorithm is an iterative method for maximum
likelihood (ML) estimation of parameters when the obser-
vations can be viewed as ‘incomplete’ data [11,12,19]. The
key idea of the EM is to augment the observed data with
latent data, in which the latent data can be either missing
data or parameter values. Therefore, the likelihood function
conditioned on ‘complete’ data (the observed data and the
latent data) has a form that is easy to manipulate. In this
section, we propose a scheme based on EM algorithm to
estimate the individual σ
2
k
(1 ≤ k ≤ K), separately.
3.1. Proposed scheme of noise plus
interference power estimation
Since the noise plus interference is assumed to be Gaussian
distributed, the probability density function (PDF) of y
k
can
be written as follows when x
k
and σ
2
k
are given
f (y
k
|x
k
,σ
2
k
) =
1
πσ
2
k
exp
−
|y
k
− α
k
x
k
|
2
σ
2
k
(4)
Therefore, when L consecutive OFDM symbols are
used for the estimation of noise plus interference esti-
mation, we can define the received signal vector y
k
=
[y
1
k
, ···,y
l
k
, ···,y
L
k
], the transmitted signal vector x
k
=
[x
1
k
, ···,x
l
k
, ···,x
L
k
], and the fading factor vector α
k
=
[α
l
k
, ···,α
l
k
, ···,α
L
k
] for the kth subcarrier over L symbols,
where the subscript l denotes the symbol index and 1 ≤ l ≤
L. In this paper, we call y
k
and (y
k
, x
k
) as ‘incomplete’ and
‘complete’ data, respectively, following the terminology of
the EM algorithm.
Suppose the noise plus interference is independent across
symbols, the conditional PDF of the incomplete data can be
expressed as
f (y
k
|x
k
,σ
2
k
) =
L
l=1
f (y
l
k
|x
l
k
,σ
2
k
) (5)
Therefore, the log likelihood function of the incomplete
data is
log f (y
k
|x
k
,σ
2
k
) =
L
l=1
log f (y
l
k
|x
l
k
,σ
2
k
) (6)
For the conventional ML estimation, the optimal esti-
mation of σ
2
k
is found to maximize f (y
k
|σ
2
k
). However,
it is not easy to manipulate log f (y
k
|σ
2
k
) (summation of
several exponential functions). Therefore, we resort to the
EM algorithm in this paper. The iteration of the EM algo-
rithm consists of an expectation step (E-step) followed by a
maximization step (M-step). For p-th (p = 0, 1, 2 ...,P)
iteration, where P is the maximum number of the iterations,
the E-step and M-step are given as
E-step:
Q(σ
2
k
|
ˆ
σ
2
k,p
) = E
x
k
log f (y
k
, x
k
|σ
2
k
)|y
k
,
ˆ
σ
2
k,p
= E
x
k
log f (y
k
|x
k
,σ
2
k
)f (x
k
|y
k
,
ˆ
σ
2
k,p
)
=
L
l=1
M
j=1
log
f (y
l
k
|x
i
k
= S
j
,σ
2
k
)
· (7)
p(x
l
k
= S
j
|y
l
k
,
ˆ
σ
2
k,p
)
M-step:
ˆ
σ
2
k,p+1
= arg max
σ
2
k
Q(σ
2
k
|
ˆ
σ
2
k,p
) (8)
Wirel. Commun. Mob. Comput. (2010) © 2010 John Wiley & Sons, Ltd.
DOI: 10.1002/wcm