Coded data generation or conversion – Analog to or from digital conversion – Analog to digital conversion
Reexamination Certificate
2001-10-01
2002-11-05
Young, Brian (Department: 2819)
Coded data generation or conversion
Analog to or from digital conversion
Analog to digital conversion
Reexamination Certificate
active
06476754
ABSTRACT:
This application claims priority under 35 U.S.C. §§119 and/or 365 to 0003549-3 filed in Sweden on Oct. 2, 2000; the entire content of which is hereby incorporated by reference.
TECHNICAL FIELD OF THE INVENTION
The present invention generally relates to field of sampling, and more specifically, to methods and apparatus for reconstruction of nonuniformly sampled bandlimited signals, to methods and apparatus for compensation of time skew in time-interleaved analog-to-digital converters (ADCs), and to a computer program product for performing said methods of reconstruction.
DESCRIPTION OF RELATED ART AND BACKGROUND OF THE INVENTION
In uniform sampling, a sequence x(n) is obtained from an analog signal x
a
(t) by sampling the latter equidistantly at t=nT, −∞<n<∞, i.e., x(n)=x
a
(nT), T being the sampling period, as illustrated in
FIG. 1
a
. In this case, the time between two consecutive sampling instances is always T. In nonuniform sampling, on the other hand, the time between two consecutive sample instances is dependent on the sampling instances. The present invention deals with the situation where the samples can be separated into N subsequences x
k
(m), k=0, 1, . . . , N−1, where x
k
(m) is obtained by sampling x
a
(t) with the sampling rate 1/(MT) at t=nMT+t
k
, i.e., x
k
(m)=x
a
(nMT+t
k
), M being a positive integer. This sampling scheme is illustrated in
FIG. 1
b
for N=2 and M=2. Such nonuniformly sampled signals occur in, e.g., time-interleaved analog-to-digital converters (ADCs) due to time skew errors.
The question that arises is how to form a new sequence y(n) from x
k
(m) such that y(n) is either exactly or approximately (in some sense) equal to x(n). For conventional time-interleaved ADCs, N=M and, ideally, t
k
=kT. In this case, y(n)=x(n) is obtained by simply interleaving x
k
(m). However, in practice, t
k
is not exactly equal to kT due to time skew errors which introduces aliasing components into Y(e
j&ohgr;T
), Y(e
j&ohgr;T
) being the Fourier transform of y(n). This means that y(n)≠x(n), and thus the information in y(n) is no longer the same as that in x(n).
It should be noted that it is well known that, if the t
k
's are distinct such that all samples are separated in time, then x
a
(t) is uniquely determined by the samples in the x
k
(m)'s. It is also well known how to retain x
a
(t) from the x
k
(m)'s using analog interpolation functions. However, these functions are not easily, if at all possible, achievable in practical implementations, which thus call for other solutions.
SUMMARY OF THE INVENTION
Accordingly, it is an object of the present invention to provide a method and an apparatus, respectively, for reconstruction of a nonuniformly sampled bandlimited analog signal x
a
(t), said nonuniformly sampled signal comprising N subsequences x
k
(m), k=0, 1, . . . , N−1, N≧2, obtained through sampling at a sampling rate of 1/(MT) according to x
k
(m)=x
a
(nMT+t
k
), where M is a positive integer, and t
k
=kMT/N+&Dgr;t
k
, &Dgr;t
k
being different from zero, which are capable of forming a new sequence y(n) from said N subsequences x
k
(m) such that y(n) at least contains the same information as x(n)=x
a
(nT), i.e. x
a
(t) sampled with a sampling rate of 1/T, in a frequency region lower than &ohgr;
0
(and possibly including &ohgr;
0
), &ohgr;
0
being a predetermined limit frequency.
A further object of the present invention is to provide such method and apparatus, respectively, which are efficient, fast, simple, and of low cost.
Still a further object of the present invention is to provide such method and apparatus, respectively, which are capable of reducing noise such as e.g. quantization noise.
Those objects among others are attained by a method and an apparatus, respectively, which perform the steps of:
(i) upsampling each of the N subsequences x
k
(m), k=0, 1, . . . , N−1, by the factor M;
(ii) filtering each of the upsampled N subsequences x
k
(m), k=0, 1, . . . , N−1, by a respective digital filter; and
(iii) adding the N digitally filtered subsequences to form y(n).
Preferably, the respective digital filter is a fractional delay filter and has a frequency response G
k
=a
k
e
(−j&ohgr;sT)
, k=0, 1, . . . , N−1, in the frequency band |&ohgr;T|≦&ohgr;
0
T, a
k
being a constant and s being different from an integer, and particularly s equals d+t
k
, d being an integer.
If &ohgr;
0
T is a fixed value less than &pgr;, such that the original analog signal comprises frequency components of a higher frequency than &ohgr;
0
, regional perfect reconstruction is achieved, i.e. y(n) contains the same information as x(n)=x
a
(nT), i.e. x
a
(t) sampled with a sampling rate of 1/T, only in a frequency region |&ohgr;|≦&ohgr;
0
. Regionally perfect reconstruction is of particular interest in oversampled systems where the lower frequency components carry the essential information, whereas the higher frequency components contain undesired components (e.g., noise) to be removed by digital and/or analog filters.
Here, the fractional delay filters have a frequency response G
k
=a
k
A
k
(e
j&ohgr;T
), k=0, 1, . . . , N−1, in the frequency band &ohgr;
0
T<|&ohgr;T|≦&pgr;, where A
k
(e
j&ohgr;T
) is an arbitrary complex function.
If on the other hand &ohgr;
0
does include all frequency components of the original analog signal (i.e. &ohgr;
0
T includes all frequencies up to &pgr;) perfect reconstruction is achieved, i.e. y(n) is identical with x(n).
In either case two different situations arise: (1) 2K
0
+1=N and (2) 2K
0
+1<N, wherein K
0
is given by
K
0
=
⌈
M
(
ω
0
T
+
ω
1
T
)
2
π
⌉
-
1
for regionally perfect reconstruction, wherein ┌x┐ should be read as the smallest integer larger than or equal to x and [−&ohgr;
1
, &ohgr;
1
] being the frequency band wherein said bandlimited analog signal x
a
(t) is found, respectively, and by
K
0
=M
−1
for perfect reconstruction.
In situation (1) the a
k
's are calculated as
a=B
−1
c,
a being the a
k
's in vector form given by
a=[a
0
a
1
. . . a
N−1
]
T
,
B
−1
being the inverse of B as given by
B
=
[
u
0
-
K
0
u
1
-
K
0
⋯
u
N
-
1
-
K
0
u
0
-
(
K
0
-
1
)
u
1
-
(
K
0
-
1
)
⋯
u
N
-
1
-
(
K
0
-
1
)
⋮
⋮
⋮
u
0
K
0
u
1
K
0
⋯
u
N
-
1
K
0
]
,
wherein
u
k
=
ⅇ
-
j
2
π
MT
t
k
,
and c being
c=[c
0
c
1
. . . c
2K
0
]
T
,
wherein
c
k
=
{
M
,
k
=
K
0
0
,
k
=
0
,
1
,
…
,
2
K
0
,
k
≠
K
0
.
In situation (2) the a
k
's are calculated as
a={circumflex over (B)}
−1
ĉ,
a being defined as
a=[a
u
a
fix
]
T
wherein a
u
and a
fix
contain (2K
0
+1) unknown a
k
's and L=N−2K
0
−1 fixed constant a
k
's, {circumflex over (B)}
−1
being the inverse of {circumflex over (B)} as given by
B
^
=
[
B
S
]
,
wherein B is given by
B
=
[
u
0
-
K
0
u
1
-
K
0
⋯
u
N
-
1
-
K
0
u
0
-
(
K
0
-
1
)
u
1
-
(
K
0
-
1
)
⋯
u
N
-
1
-
(
K
0
-
1
)
⋮
⋮
⋮
u
0
K
0
u
1
K
0
⋯
u
N
-
1
K
0
]
,
wherein
u
k
=
ⅇ
-
j
2
π
MT
t
k
,
S is given by
S=[S
z
S
d
],
wherein
S
z
=
[
0
0
⋯
0
0
0
⋯
0
⋮
⋮
⋮
0
0
⋯
0
]
and
S
d
=diag[1 1 . . . 1],
and
ĉ being
ĉ=[c a
fix
]
T
,
wherein
c is given by
c=[c
0
c
1
. . . c
2k
0
]
T
,
wherein
c
k
=
{
M
,
k
=
K
0
0
,
k
=
0
,
1
,
…
,
2
K
0
,
k
≠
K
0
.
Thus L a
k
's can be arbitrarily chosen.
Johansson Håkan
Löwenborg Per
Burns Doane Swecker & Mathis L.L.P.
Telefonaktiebolaget L M Ericsson (publ)
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