2.1 Real-valued system with PAM signaling
We first consider a real-valued system with PAM signaling. The received signal is given by
(1)
where
s
n
is the source signal and
v
n
is the ambient noise. We assume that
v
n
is zero-mean white Gaussian noise with variance
σ2.
A stands for the channel gain/attenuation, and
I is the deterministic interference. We assume that
s
n
is generated randomly by a
M-ary PAM source with equiprobable constellation points at ±(2
m−1),
m=1,2,⋯,
M/2, with
M=2
p
and
. Therefore, the average power of a
M-ary PAM signal is given by
. As a result, the SNR of the received signal is defined as
(2)
We note that in the open literature [
1], the SINR, defined as
has also been applied as the performance measure. On top of that, cancellation of the DC offset is discussed in [
3] and [
9], where the SNR is used as the major figure of merit (it was argued that the DC offset has a less significant impact on the system performance than noise [
3]). In the current paper, the constant deterministic interference (i.e. the DC-offset) was estimated and suppressed from the received signal, and then the SNR is applied as the performance measure. Our theoretical analysis further shows that varying the value of deterministic interference has no effect on the accuracy of parameter estimation (i.e. the estimation of
A,
I and
σ2; see also Section 3). Hence, we argue that the SNR is a more proper performance measure in the presence of deterministic interference.
We shall separately estimate {
A,
I,
σ2} and then use these estimates to calculate
γ. Without loss of generality,
A is assumed to be a positive number in this paper. The PDF of
x
n
is given by
(3)
Assume there are
N available samples, denoted as
x= [
x1,
x2,⋯,
x
N
]. Since the transmitted symbols are assumed to be independent and identically distributed (i.i.d.) and the additive noise is white, then the corresponding received samples are independent and their joint probability density function (PDF) is simply the product of the PDF of each sample. Consequently, the log-likelihood function of
x is given by
(4)
where
is given as
(5)
By forcing the derivatives
,
,
to zero, we obtain the subsequent relationships:
(6)
(7)
(8)
where
(9)
(10)
(11)
It is not an easy task to obtain closed form solutions for these equations. Hence, we propose an iterative approach, which has to be initialized properly. For this purpose, we consider the initialization based on the moments of the received signal.
By directly calculating the first-order sample raw moment of the received signal, we obtain the following initialization of
I:
(12)
The second- and fourth-order central moments of the signal population are respectively given as,
(13)
(14)
where
and
are the second- and fourth-order raw moments of the
M-ary PAM signal {
s
n
}, respectively. And
(15)
According to (13) and (14), one has an equation for
A as
(16)
Equating the population moments with the sample moments, one moment-based estimator for
A can be derived as
(17)
where
and
are the second- and fourth-order sample central moments of the received signal and are given by, respectively,
(18)
When
and
are available,
σ2 can be estimated by exploiting the second-order central moment of the received signal, as follows:
(19)
As a result, the moment-based estimator for
γ can be expressed as
(20)
These moment-based estimators are only valid in the high SNR range. Unfortunately, they usually require a large quantity of samples to obtain accurate estimation, which is not realistic in practical time-varying systems. In order to obtain a more accurate estimation and speed up the convergence, we propose that the iterative method in (6) to (8) is initialized by the closed form estimates given by (12), (17) and (19). The procedure stops when a predefined maximum of iterations is achieved or when the error is lower than a specified value. The algorithm is outlined in Algorithm 1, where
K is a predefined maximal number of iterations,
ε is a predefined small constant, and the iterative error
ε
k
is defined as
(21)
2.2 Complex-valued system with QAM signaling
Considering the complex-valued signal system, the signal model is revised as
(22)
We assume s
n
=a
n
+j b
n
is a M-ary square QAM source, that is, , where M=22p for any natural number p, G=A+j B is the complex channel gain, I=C+j D and w
n
=u
n
+j v
n
are respectively the complex interference and complex noise. The average power of the square QAM constellation power is given by .
The SNR of the received signal is defined as
(23)
We shall separately estimate the unknown parameters of the vector {A,B,C,D,σ2} and then use these estimates to calculate γ. Without loss of generality, A and B are assumed to be positive numbers in this paper.
The joint PDF of the real and image parts of
r
n
is given by:
(24)
According to the symmetry of the square QAM constellation,
can be written as
(25)
where
(26)
(27)
(28)
(29)
Assume there are
N available received samples, denoted as
r= [
r1,
r2,⋯,
r
N
]. As a result, the logarithmic likelihood function, define as
is given by
(30)
By forcing the derivatives
,
,
,
,
(see Appendix) to zero, we obtain the following equations:
(31)
(32)
(33)
(34)
(35)
where
(36)
(37)
(38)
(39)
Similar to the case of PAM signaling, we adopt the moment-based estimators as initialization and then apply iterative estimation based on (31) to (35) to refine the estimated results. At first, the moment estimate of
I is given as
(40)
The second- and fourth-order central moments of the signal’s real and imaginary parts are respectively given as,
(41)
(42)
(43)
(44)
(45)
where
and
are the second- and fourth-order raw moments of the real/imaginary part of the
M-ary QAM signal {
s
n
}, respectively. And
(46)
According to (41) to (45), one has two equations for |
G|
2=
A2+
B2 as
(47)
(48)
Equating the population moments with the sample moments, one moment-based estimator for |
G|
2=
A2+
B2 can be derived as
(49)
where
,
,
,
and
are the second- and fourth-order sample central moments of the real/imaginary part of received signal, and are given by, respectively,
(50)
(51)
(52)
When
and
are available,
σ2 can be estimated by exploiting the second-order central moment of the received signal, as follows:
(53)
As a result, the moment-based estimator for
γ can be expressed as
(54)
It can be seen from (41) to (45) that
A and
B are asymmetric such that they cannot be decoupled from these equations. Therefore, they are temporally assumed to be equal and then they are refined by the iteration. That is, the rough estimates of
A and
B are given as
(55)
The iterative algorithm is similar to the case of PAM signaling and will not be shown here for saving space.