Maximum likelihood estimation of the channel coefficients and of the DC offset in a digital baseband signal of a radio receiver using the SAGE algorithm
View Patent ↗The proposed iterative method concurrently calculates maximum likelihood estimated values for the channel coefficients (h 0 , . . . , h L ; θ h ) and the DC offset (d; θ d ) on the basis of a training sequence (TSC) in a TDMA mobile communications system. The SAGE algorithm leads to two recursion formulae, by means of which the iterative calculation of estimated values for the channel coefficients (h 0 , . . . , h L ; θ h ) and for the DC offset (d; θ d ) is provided.
1. A method for estimation of the channel coefficients (h 0 . . . h L ; θ h ) and of the DC offset (d; θ d ) in a digital baseband signal in a radio receiver using a training sequence, comprising:
a) generating a received signal vector y from a number of received training sequence data symbols (y(k)) and a number of known training sequence data symbols (t(k));
b) producing a maximum likelihood estimate of a function Λ(θ) which contains the channel coefficients (θ h ) and the DC offset (θ d ), in such a way that a first likelihood function Q h (θ h ; {circumflex over (θ)}′) is produced for the channel coefficients (θ h ), and a second likelihood function Q d (θ d ; {circumflex over (θ)}′) is produced for the DC offset (θ d ); and
c) using the SAGE algorithm to produce a convergent sequence of expected values of the first and second likelihood functions Q h (θ h ; {circumflex over (θ)}′) and Q d (θ d ; {circumflex over (θ)}′), respectively, by means of at least one expected value determination step, and at least one maximization step with a parameter vector {circumflex over (θ)}′ comprising initial estimated values of the maximized values {circumflex over (θ)} h ′ and {circumflex over (θ)} d ′.
2. The method of claim 1 , further comprising iterating the expected value determination step and the maximization step until a predetermined convergence criteria is reached.
3. The method of claim 1 , wherein the received signal vector y further comprises noise data symbols (n(k)) associated with the received training sequence data symbols (y(k)).
4. The method of claim 1 , wherein the function Λ(θ) comprises:
Λ(θ):=−( y−A θ) H ( y−A θ),
wherein y comprises the received signal vector for signal samples of y(k) in the sample interval y(L) . . . y(N−1), and [.] H denotes a conjugate transposition, and wherein the first and second likelihood functions comprise:
Q
h
(
θ
h
;
θ
^
′
)
=
1
2
(
y
^
h
-
T
θ
h
)
H
(
y
^
h
-
T
θ
h
)
Q
d
(
θ
d
;
θ
^
′
)
=
1
2
(
y
^
d
-
a
θ
d
)
H
(
y
^
d
-
a
θ
d
)
,
where
ŷ h =ŷ−a{circumflex over (θ)}′ d
ŷ d =y−T{circumflex over (θ)} h ′, where
{circumflex over (θ)} h ″=( T H T ) −1 T H ( y−a{circumflex over (θ)} d ′)
{circumflex over (θ)} d ″=( a H a ) −1 a H ( y−T{circumflex over (θ)} h ″)
y=[y ( L ), . . . , y ( N −1)] T
a:=[e −jΦL , . . . , e −jΦ(N-L) ] T
T
=
[
t
(
L
)
⋯
t
(
0
)
⋮
⋰
⋯
t
(
N
-
1
)
⋯
t
(
N
-
L
-
1
)
]
A:=[T, a]
wherein T defines a channel transmission matrix, and a defines a DC offset transmission matrix, and wherein the likelihood functions for θ h and θ d are maximized in the maximization step.
5. The method of claim 4 , wherein the expected value determination step and the maximization step are combined to form the following recursion formulae:
{circumflex over (θ)} h ″=( T H T ) −1 T H ( y−a{circumflex over (θ)} d ′)
{circumflex over (θ)} d ″=( a H a ) −1 a H ( y−T{circumflex over (θ)} h ″).
6. The method of claim 5 , further comprising:
using the recursion formulae to run through an iteration loop in which an initial value {circumflex over (θ)} h ′ is predetermined and a check is carried out after each recalculation of {circumflex over (θ)} h ″ and {circumflex over (θ)} d ″ to determine whether a convergence has been reached on the basis of predetermined criteria; and
terminating the iteration process when convergence is reached.
7. The method of claim 5 , wherein in the iteration steps, T is replaced by one of its orthogonal matrix elements T short and y is replaced by a corresponding vector element y short , such that the first iteration equation is simplified to:
θ
^
h
″
=
T
short
H
(
y
short
-
a
short
θ
^
d
′
)
=
θ
^
h
′
-
T
short
H
a
short
θ
^
d
′
.
8. The method of claim 7 , wherein
θ
^
d
′
=
1
a
2
-
a
H
T
(
T
H
T
)
-
1
T
a
(
a
H
y
-
a
H
T
(
T
H
T
)
-
1
T
y
)
.
θ
^
h
″
=
T
short
H
(
y
short
-
a
short
θ
^
d
′
)
=
θ
^
h
′
-
T
short
H
a
short
θ
^
d
′
is used for the initial estimated values, such that only one iteration step is required for estimation of the channel coefficients.
9. The method of claim 1 , wherein the function Λ(θ) and the first and second likelihood functions comprise:
Λ
(
θ
)
:=
-
(
y
-
A
θ
)
H
Π
-
1
(
y
^
n
-
T
θ
n
)
Q
h
(
θ
h
;
θ
^
′
)
=
1
2
(
y
^
h
-
T
θ
h
)
H
Π
-
1
(
y
^
h
-
T
θ
h
)
Q
d
(
θ
d
;
θ
^
′
)
=
1
2
(
y
^
d
-
a
θ
d
)
H
Π
-
1
(
y
^
d
-
a
θ
d
)
,
where
Π −1 is a covariance matrix of a noise vector N:=[n(L), . . . , n(N-1)] T associated with the training sequence data symbols (y(k)).