Compensation of mismatch errors in a time-interleaved analog-to-digital converter
A method for the compensation of frequency-response mismatch errors in M-channel time-interleaved ADCs. The compensation is done through an M-periodic time-varying filter h n (k)=h n mod M (k) (2), or, equivalently, a set of M time-invariant filters h n (k), n=0, 1, . . . , M−1. The overall compensation system is constructed by determining the M filter impulse responses h n (k) through M separate matrix inversions, where the size of the matrices equals the filter impulse response length. Also, a compensated M-channel time-interleaved ADC based on and performing the method.
1. A method for compensating frequency response mismatch errors in an M-channel time-interleaved analog-to-digital converter comprising M analog-to-digital converters (1 1 -1 M ) in an array and a filter connected to the output of the time-interleaved analog-to-digital converter, the method comprising:
determining M frequency responses Q r from said M-channels, calculating separately M channel impulse responses h r for said M-channels, calculating matrix elements of M matrices S r and vectors c r being factors of error power functions P r ,
minimizing said power functions P r by inverting separately said M matrices S r ,
forming M output subsequences utilizing convolutions of the impulse responses being calculated separately and the output of the time-interleaved analog-to-digital converter, and
forming an overall output from the compensated time-interleaved analog-to-digital converter by interleaving output subsequences y r of the channels.
2. The method according to claim 1 , further comprising:
performing said compensation utilizing an M-periodic time-varying filter h n (k)=h n mod M (k), or equivalently utilizing a set of M time-invariant filters h r (k), r=0, 1, . . . , M−1, wherein h r (k) are the M separately calculated impulse responses.
3. The method according to claim 2 , wherein said calculation of said M impulse responses, h r (n) r=0, 1, . . . , M−1, includes:
minimizing suberror sequences e r (m) being the differences between the corresponding output subsequences y r (m) and desired uniform subsequences x r (m)=x a (mMT+rT) containing desired uniform samples for the time instances t=mMT+rT.
4. The method according to claim 3 , further comprising:
minimizing subsequence errors e r (m) wherein:
e
r
(
m
)
=
1
2
π
∫
-
ω
0
T
ω
0
T
(
A
r
(
j
w
)
-
1
)
X
(
ⅇ
j
ω
T
)
ⅇ
j
ω
T
(
mM
+
r
)
ⅆ
(
ω
T
)
,
wherein
A
r
(
j
ω
)
=
∑
k
=
-
N
N
h
r
(
k
)
Q
<
r
-
k
>
M
(
jω
)
ⅇ
-
j
ω
Tk
.
5. The method according to claim 3 , further comprising:
the size of said matrices being equal to the filter impulse response length.
6. The method according to claim 2 , further comprising:
forming said output subsequences y r (m) using the convolution formula:
y
r
(
m
)
=
∑
k
=
-
N
N
v
(
mM
+
r
-
k
)
h
r
(
k
)
;
r
=
0
,
1
,
…
,
M
-
1
,
wherein h r (k) is the impulse response of a non-casual 2Nth order M-periodic time varying FIR-filter (Finite-length Impulse Response) h n (k)=h <n> M (k).
7. The method according to claim 6 , further comprising:
forming said overall output from the compensated time-interleaved analog-to-digital converter by interleaving the output subsequences y r (m) according to: y(n)=y <n> M (└n/M┘).
8. The method according to claim 6 , further comprising:
computing y r (m)-samples when the corresponding analog-to-digital converter outputs v r (m) arrive.
9. The method according to claim 6 , further comprising:
providing an M-rate synthesis filter bank of time-invariant synthesis filters (2 1 -2 M ), wherein channel sequences v r (m) are upsampled by M and then fed into said filters, the outputs of which are added to form the overall output y(n).
10. The method according to claim 1 , further comprising:
denoting a desired frequency response by Q des (jω), and
setting Q des (jω)=1.
11. The method according to claim 10 , further comprising:
determining frequency responses Q r (jω)r=0, 1, . . . , M−1, in each analog-to-digital converter (1 1 -1 M ) of the M channels performed by any one of:
a measurement of grid frequency points,
predetermining a finite-order polynomial model of each Q r (jω) followed by a determination of parameters of M polynomials representative of the frequency response of the respective channel.
12. The method according to claim 1 , further comprising:
minimizing error power functions P r , r=0, 1, . . . , M−1, written in matrix form as:
P r =h r T S r h r +c r T h r +C,
wherein
h r =[h r (− N ) h r (− N+ 1) . . . h r ( N )] T
and
c r =[c r,−N c r,−N+1 . . . c r,N ] T ,
with c r,k , k=−N, −N+1, . . . , N, being
c
r
,
k
=
-
1
π
∫
-
ω
0
T
ω
0
T
Q
<
r
-
k
>
M
(
jω
)
×
cos
(
-
ω
Tk
+
arg
{
Q
<
r
-
k
>
M
(
jω
)
}
)
ⅆ
(
ω
T
)
wherein S r are 2N+1×2N+1 symmetric and positive definite matrices with entries s r,kp , p=−N, −N+1, . . . ,N given by
s
r
,
kp
=
-
1
π
∫
-
ω
0
T
ω
0
T
Q
<
r
-
k
>
M
(
jω
)
❘
Q
<
r
-
p
>
M
(
jω
)
×
cos
(
ω
T
(
p
-
k
)
+
arg
{
Q
<
r
-
k
>
M
(
jω
)
}
+
arg
{
Q
<
r
-
k
>
M
(
jω
)
}
)
ⅆ
(
ω
T
)
and wherein the constant C is given by
C
=
ω
0
T
π
.
13. The method according to claim 12 , further comprising:
obtaining for each r the values of h r (k)that minimizes the corresponding function P r by setting the partial derivatives of with respect to h r (k)to zero and solving for h r (k), wherein the impulse responses h r are obtained from
h r =−0.5 S r −1 c r .
14. The method according to claim 1 , further comprising:
denoting said desired frequency response by Q des (jω), and
setting Q des (jω)=Q r ref (jω).
15. The method according to claim 14 , further comprising:
determining frequency responses
Q
r
(
jω
)
Q
r
ref
(
jω
)
,
wherein r=0, 1, . . . , M−1, r≠r ref in each analog-to-digital converter (1 1 -1 M ) of the M channels performed by any one of:
a measurement of grid frequency points,
predetermining a finite-order polynomial model of each
Q
r
(
jω
)
Q
r
ref
(
jω
)
followed by a determination of parameters of M polynomials representative of the frequency response of the respective channel.
16. The method according to claim 15 , wherein said calculation of said M impulse responses, h r (n)=r=0, 1, . . . , M−1 includes:
minimizing suberror sequences e r (m) being the differences between the corresponding output subsequences y r (m) and desired uniform subsequences v ar ref (mMT+rT) containing desired uniform samples for the time instances
t=mMT+rT.
17. The method according to claim 16 , further comprising:
minimizing subsequence errors e r (m) through:
e
r
(
m
)
=
1
2
π
∫
-
ω
0
T
ω
0
T
(
A
r
(
j
w
)
-
1
)
X
(
ⅇ
j
ω
T
)
ⅇ
j
ω
T
(
mM
+
r
)
ⅆ
(
ω
T
)
,
wherein
A
r
(
j
ω
)
=
∑
k
=
-
N
N
h
r
(
k
)
Q
′
(
j
ω
)
ⅇ
-
j
ω
Tk
,
and wherein
Q
′
=
Q
′
Q
des
(
jω
)
;
r≠r ref ; h r ref (k)=δ(k); δ(k)=the unit impulse sequence.
18. The method according to claim 17 , further comprising:
performing said determining of the M−1 filter impulse responses h n (k) through M−1 separate matrix inversions of the matrices S r , where the size of the matrices equals the filter impulse response length.
19. The method according to claim 1 , further comprising:
minimizing said error power functions P r , r=0, 1, . . . , M−1, written in matrix form as:
P r =h r T S r h r +c r T h r +C,
wherein
h r =[h r (− N ) h r (− N+ 1) . . . h r ( N )] T
and
c r =[c r,−N c r,−N+1 . . . c r,N ] T ,
with c r,k , k=−N, −N+1, . . . , N, being
c
r
,
k
=
-
1
π
∫
-
ω
0
T
ω
0
T
Q
′
(
jω
)
×
cos
(
-
ω
Tk
+
arg
{
Q
′
(
jω
)
}
)
ⅆ
(
ω
T
)
wherein S r are 2N+1×2N+1 symmetric and positive definite matrices with entries s r,kp , k, p=−N, −N+1, . . . ,N given by
s
r
,
kp
=
-
1
π
∫
-
ω
0
T
ω
0
T
Q
′
(
jω
)
❘
Q
′
(
jω
)
×
cos
(
ω
T
(
p
-
k
)
+
arg
{
Q
<
r
-
k
>
M
(
jω
)
}
+
arg
{
Q
′
(
jω
)
}
)
ⅆ
(
ω
T
)
and wherein the constant C is given by
C
=
ω
0
T
π
.
20. The method according to claim 19 , further comprising:
obtaining for each r, r≠r ref , the values of h r (k) that minimize the corresponding function P r by setting the partial derivatives of P r with respect to h r (k) to zero and solving for h r (k).
21. The method according to claim 1 , further comprising:
feeding channel output sequences into polyphase subfilters working at lowest sample rate, 1/(MT),
combining and interleaving outputs of said subfilters for forming the overall output y(n).
22. A compensated analog-to-digital time-interleaved converter comprising:
a number of M channels,
each channel including an analog-to-digital converter (1 1 -1 M ), and
a filter (2 1 -2 M ),
wherein the converter is configured to perform a method for compensating frequency response mismatch errors in the converter comprising
determining M frequency responses Q r from said M-channels,
calculating separately M channel impulse responses h r for said M-channels,
calculating matrix elements of M matrices S r and vectors c r being factors of error power functions P r ,
minimizing said power functions P r by inverting separately said M matrices S r ,
forming M output subsequences utilizing convolutions of the impulse responses being calculated separately and the output of the interleaved analog-to-digital converter, and
forming an overall output from the compensated interleaved analog-to-digital converter by interleaving output subsequences y r of the M channels.
23. The compensated analog-to-digital converter of claim 22 , wherein said filter (2 1 -2 M ) is one of:
an M-periodic time-varying filter h n (k)=h n mod M (k),
a set of M time-invariant filters h r (k), r=0, 1, . . . , M−1,
an M-rate synthesis filter bank of time-invariant synthesis filters (2 1 -2 M ), or
a set of polyphase subfilters working at lowest sample rate, 1/(MT).