Pulse or digital communications – Equalizers – Automatic
Reexamination Certificate
2000-09-12
2004-06-22
Chin, Stephen (Department: 2634)
Pulse or digital communications
Equalizers
Automatic
C375S229000
Reexamination Certificate
active
06754264
ABSTRACT:
BACKGROUND OF THE INVENTION
The present invention relates to digital equalisation of signals. It finds a major application in the field of radio communications.
The method is applicable when the received signal results from a superimposition of different signals, sent from the same transmitter and/or different transmitters, for which the responses of the transmission channels between transmitter and receiver have been determined beforehand. One of the major problems arising in this situation is to strike a compromise between performance of the equaliser and its complexity. A complete maximum likelihood estimation of all the symbols of the overlaid signals is theoretically possible, using the Viterbi algorithm for example (see G. D. Forney Jr.: “The Viterbi Algorithm”, Proc. of the IEEE, Vol. 61, No. 3, March 1973, pages 268-278). However, as the impulse response of the channels becomes long or the number of overlaid signals is high, the exponential complexity of these methods makes them impractical.
Spread spectrum radio communications using code division multiple access (CDMA) illustrate this problem.
Let us assume that K synchronous logical channels, denoted here by an index k (1≦k≦K), share the same physical channel, i.e. use the same carrier frequency at the same time, and are differentiated by spreading sequences c
k
respectively assigned to them. Each channel is used to transmit a respective signal made up of successive sequences d
k
, or frames, of n symbols d
p
k
(1≦p≦n). The symbols d
p
k
have discrete values:binary (±1) in the case of a BPSK modulation, quaternary (±1±j) in the case of a QPSK modulation, . . .
Let us also assume that the spreading sequences c
k
have a rate Q times greater than that of the symbols d
p
k
, Q being an integer representing a spreading factor (the extreme situation Q=1 corresponds to no spreading), and a periodicity equal to the period of a symbol. The complex-valued samples (generally ±1 or ±1±j) of the spreading sequences c
k
or “chips” are written c
q
k
(1≦q≦Q). The sequences c
k
are selected so as to be mutually orthogonal
(
∑
q
=
1
Q
c
q
k
·
c
q
k
′
=
0
where
k
≠
k
′
)
in order to make it easier to distinguish between the channels. However, the propagation conditions, particularly the multiple paths, do not generally preserve this orthogonality, particularly in the case of short spreading sequences.
After base band conversion, digitisation and adapted filtering, a vector Y of the received signal reflecting the symbols emitted over a frame period is expressed as follows:
Y
=
(
y
1
y
2
⋮
y
p
⋮
y
L
)
=
∑
k
□
=
1
K
(
r
1
k
0
0
⋯
0
r
2
k
r
1
k
r
2
k
⋰
⋮
⋮
⋰
⋮
⋰
0
0
r
W
k
r
1
k
0
0
r
W
k
r
2
k
r
1
k
0
0
r
2
k
⋮
⋮
⋰
⋮
r
W
k
0
0
⋯
0
r
W
k
)
(
c
1
k
·
d
1
k
c
2
k
·
d
1
k
⋮
c
Q
k
·
d
1
k
c
1
k
·
d
2
k
⋮
c
q
k
·
d
p
k
⋮
c
Q
k
·
d
n
k
)
+
Y
N
(
1
)
where W is the length, in number of chips, of the estimated impulse response of the channels, r
k
=(r
1
k
, . . . , r
w
k
) is the estimated impulse response of the k-th channel, the r
j
k
being complex numbers such that r
j
k
=0 if j≦0 or j>W, y
p
is the p-th complex sample received (at the chip frequency) where 1≦p≦L=nQ+W−1, and Y
N
is a vector of size L made up of additive noise samples. The estimated impulse response r
k
takes account of the propagation channel, shaping of the signal by the transmitter and filtering on reception.
The size of the problem can be reduced by a factor Q by integrating the codes c
k
in the responses r
k
, i.e. by calculating the following convolution products for 1≦j≦L′=Q+W−1:
b
j
k
=
∑
q
=
1
Q
r
j
+
1
-
q
k
·
c
q
k
(
2
)
Expression (1) then becomes:
Y
=
∑
k
=
1
K
(
b
1
k
0
⋯
0
⋮
⋮
b
Q
k
0
b
Q
+
1
k
b
1
k
⋮
⋮
⋮
b
Q
k
b
Q
+
1
k
⋰
0
b
Q
+
W
-
1
k
b
1
k
0
⋮
⋮
b
Q
k
b
Q
+
W
-
1
k
b
Q
+
1
k
⋮
0
⋰
⋮
0
⋯
0
b
Q
+
W
-
1
k
)
(
d
1
k
d
2
k
⋮
d
n
k
)
+
Y
N
=
∑
k
=
1
K
B
k
·
D
k
+
Y
N
(
3
)
The matrices B
k
of size L×n have a structure of the Toeplitz type along a diagonal having a slope Q (along the main diagonal if Q=1), i.e. if &bgr;
i,j
K
denotes the term located on the i-th row and j-th column of a matrix B
k
, then &bgr;
ki+Q,j+1
=&bgr;
i,j
k
where 1≦i≦L−Q and 1≦j≦n−1. The terms of the matrix B
k
are given by: &bgr;
1,j
k
=0 where 1≦j≦n (B
k
therefore has only zeros above its main diagonal, and even above its diagonal of slope Q); &bgr;
i,1
k
=0 where Q+W≦i≦L (band-matrix structure); and &bgr;
i,1
k
=b
i
k
where 1≦i<Q+W.
In simple equalisers, the aim is merely to estimate the contribution of one channel, the contributions of the other channels being assimilated with noise. In other words, the receiver considers that K=1 after having determined the relevant response B
1
. The linear system (
3
) reduces to Y=B
1
D
1
+Y
N
and can then be processed, using a method such as the Viterbi algorithm, which maximises the likelihood of the symbol estimations D
1
, using an equaliser of reasonable complexity provided the length W of the channel response is not too big.
If the contributions of several channels have to be estimated, one option is to use several simple equalisers of the type described above, operating in parallel with the responses of different channels. Obviously, however, it is preferable to resolve the system (
3
) in one go where K>1 by trying to maximise the likelihood of the symbol estimates d
p
k
as far as possible. This is referred to as joint equalisation.
A matrix A of L=nQ+W−1 rows and N=nQ columns and a vector D of N discrete components are defined in order to express the vector Y in the form:
Y=AD+Y
N
(4)
One possible way of expressing the matrix and the vector D is as follows
A
=
(
B
K
⋯
B
2
B
1
)
,
and
D
=
(
D
K
⋮
D
2
D
1
)
(
5
)
In this organisation of the matrix A and vector D, the components are arranged channel by channel.
The matrix A and vector D of the linear system (
4
) can also be organised so that the components are arranged symbol by symbol. The matrix A will then have a blockwise Toeplitz structure, with a same block B of L′=Q+W−1 rows and K columns repeated n times with a offset of Q zero lines between adjacent blocks B, and the vector D is divided into corresponding blocks D
i
of a size K (1≦i≦n), i.e.
where
B
=
(
b
1
1
b
1
2
⋯
b
1
k
b
2
1
b
2
2
⋯
b
2
k
⋮
⋮
⋮
b
Q
+
W
-
1
1
b
Q
+
W
-
1
2
⋯
b
Q
+
W
-
1
K
)
and
D
i
=
(
d
i
1
d
i
2
⋮
d
i
K
)
(
7
)
One conventional solution for resolving a system such as (
4
) is the least square method, by means of which the vector {circumflex over (D)} is determined with N continuous components, which minimises the quadratic error &egr;=∥A{circumflex over (D)}−Y∥
2
. Discrete values of the components of the vector {circumflex over (D)} relating to each channel are then obtained, often through a channel decoder. The solution {circumflex over (D)} within the least square meaning is given by: {circumf
Ben Rached Nidham
Boumendil Sarah
Ahn Sam K.
Chin Stephen
Nortel Matra Cellular
Piper Rudnick LLP
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