Systems and methods of Efficient Fractional Delay Filtering
A system may include an analog front-end circuit configured to receive signals from an antenna array and a digital circuit coupled to the analog front end. The digital circuit may include a digital beamforming circuit configured to include a Nyquist fractional delay filter that is piecewise continuous in the frequency domain. The Nyquist fractional delay filter may be implemented as a Gaussian Nyquist filter, a generalized raised cosine Nyquist filter, or another Nyquist filter. The fractional delay filter may be critically sampled and evaluated numerically or with a closed-form time-domain expression. The fractional delay filter may be part of a digital beamforming phased array antenna system.
1 . A system comprising:
an analog front-end circuit configured to receive signals from an antenna array; and
a digital circuit coupled to the analog front end, the digital circuit including a digital beamforming circuit configured to include a Nyquist fractional delay filter that is piecewise continuous in the frequency domain.
2 . The system of claim 1 , wherein the Nyquist fractional delay filter comprises a critically sampled generalized raised cosine Nyquist filter.
3 . The system of claim 2 , wherein the Nyquist fractional delay filter has a frequency response represented by frequency response equations:
H
n
(
f
)
=
T
;
for
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
≤
1
-
β
2
T
;
H
n
(
f
)
=
T
2
{
A
0
+
A
1
cos
(
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
+
A
3
cos
(
3
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
}
for
1
-
β
2
T
<
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
≤
1
+
β
2
T
;
H
n
(
f
)
=
0
;
for
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
>
1
+
β
2
T
;
where T represents a symbol period, β represents a roll-off factor, A n represents filter coefficients, and f represents a signal frequency.
4 . The system of claim 1 , wherein the Nyquist fractional delay filter comprises a Gaussian Nyquist filter.
5 . The system of claim 4 , wherein the Nyquist fractional delay filter has a frequency response represented by frequency response equation:
h
(
i
+
τ
)
=
sin
c
(
i
+
τ
)
e
-
π
2
(
i
+
τ
)
2
2
σ
2
,
wherein i represents an integer value within a sample space, t represents a fractional delay, and σ parameterizes the gaussian expression.
6 . The system of claim 1 , wherein:
the system comprises a phased array antenna system; and
the fractional delay filter defines a plurality of filter coefficients, each coefficient of the plurality of filter coefficients corresponding to a fractional delay corresponding to an element of the phased array antenna system.
7 . The system of claim 1 , wherein the fractional delay filter includes one of a numerically computed expression or a closed-form expression for one of a time-domain response or a frequency response.
8 . The system of claim 7 , wherein the closed-form expression for the time-domain response is represented by time response equation:
h
n
(
i
+
τ
)
=
sin
(
π
(
i
+
τ
)
(
1
-
β
)
)
π
i
+
cos
(
π
(
i
+
τ
)
)
sin
(
πβ
(
i
+
τ
)
)
π
i
+
∑
k
=
0
k
=
n
A
2
k
+
1
4
β
2
t
sin
(
π
(
i
+
τ
)
)
cos
(
π
β
(
i
+
τ
)
)
(
2
k
+
1
)
2
π
-
4
π
β
2
(
i
+
τ
)
2
,
where i represents an integer value within a sample space, τ represents a fractional delay, β represents a roll-off factor, and A n represents coefficients, and k represents an index value.
9 . A system comprising:
an analog front-end circuit configured to receive signals from an antenna array; and
a digital circuit coupled to the analog front end, the digital circuit including a digital beamforming circuit including a fractional delay filter implemented as a Nyquist filter that is critically sampled.
10 . The system of claim 9 , wherein the fractional delay filter comprises a critically sampled generalized raised cosine Nyquist filter.
11 . The system of claim 10 , wherein the fractional delay filter has a frequency response represented by frequency response equations:
H
n
(
f
)
=
T
;
for
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
≤
1
-
β
2
T
;
H
n
(
f
)
=
T
2
{
A
0
+
A
1
cos
(
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
+
A
3
cos
(
3
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
}
for
1
-
β
2
T
<
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
≤
1
+
β
2
T
;
H
n
(
f
)
=
0
;
for
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
>
1
+
β
2
T
;
where T represents a symbol period, β represents a roll-off factor, A n represents filter coefficients, and f represents a signal frequency.
12 . The system of claim 9 , wherein the fractional delay filter includes one of a numerically computed expression or a closed-form expression for one of a time-domain response or a frequency response.
13 . The system of claim 9 , wherein the fractional delay filter comprises a Gaussian Nyquist filter having a frequency response represented by time response equation:
h
(
i
+
τ
)
=
sin
c
(
i
+
τ
)
e
-
π
2
(
i
+
τ
)
2
2
σ
2
,
wherein i represents an integer value within a sample space, τ represents a fractional delay, and σ parameterizes the gaussian expression.
14 . The system of claim 9 , wherein the system comprises a phased array antenna system.
15 . The system of claim 9 , wherein the fractional delay filter includes a closed-form expression for a time-domain response that is represented by time response equation:
h
n
(
i
+
τ
)
=
sin
(
π
(
i
+
τ
)
(
1
-
β
)
)
π
i
+
cos
(
π
(
i
+
τ
)
)
sin
(
πβ
(
i
+
τ
)
)
π
i
+
∑
k
=
0
k
=
n
A
2
k
+
1
4
β
2
t
sin
(
π
(
i
+
τ
)
)
cos
(
π
β
(
i
+
τ
)
)
(
2
k
+
1
)
2
π
-
4
π
β
2
(
i
+
τ
)
2
,
where i represents an integer value within a sample space, τ represents a fractional delay, β represents a roll-off factor, and A n represents coefficients, and k represents an index value.
16 . A system comprising:
an analog front-end circuit configured to receive signals from an antenna array;
a digital circuit coupled to the analog front end, the digital circuit including a digital beamforming circuit including a fractional delay filter implemented as one of a Gaussian Nyquist filter or a generalized raised cosine Nyquist filter; and
wherein the fractional delay filter is critically sampled.
17 . The system of claim 16 , wherein the generalized raised cosine Nyquist filter has a frequency response represented by time response equations:
H
n
(
f
)
=
T
;
for
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
≤
1
-
β
2
T
;
H
n
(
f
)
=
T
2
{
A
0
+
A
1
cos
(
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
+
A
3
cos
(
3
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
π
T
β
[
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
-
1
-
β
2
T
]
)
}
for
1
-
β
2
T
<
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
≤
1
+
β
2
T
;
H
n
(
f
)
=
0
;
for
❘
"\[LeftBracketingBar]"
f
❘
"\[RightBracketingBar]"
>
1
+
β
2
T
;
where T represents a symbol period, β represents a roll-off factor, A n represents filter coefficients, and f represents a signal frequency.
18 . The system of claim 16 , wherein the Gaussian Nyquist filter has a frequency response represented by time response equation:
h
(
i
+
τ
)
=
sin
c
(
i
+
τ
)
e
-
π
2
(
i
+
τ
)
2
2
σ
2
,
wherein i represents an integer value within a sample space, τ represents a fractional delay, and σ parameterizes the gaussian expression.
19 . The system of claim 16 , wherein the fractional delay filter includes a closed-form expression for a time-domain response that is represented by time response equation:
h
n
(
i
+
τ
)
=
sin
(
π
(
i
+
τ
)
(
1
-
β
)
)
π
i
+
cos
(
π
(
i
+
τ
)
)
sin
(
πβ
(
i
+
τ
)
)
π
i
+
∑
k
=
0
k
=
n
A
2
k
+
1
4
β
2
t
sin
(
π
(
i
+
τ
)
)
cos
(
π
β
(
i
+
τ
)
)
(
2
k
+
1
)
2
π
-
4
π
β
2
(
i
+
τ
)
2
,
where i represents an integer value within a sample space, t represents a fractional delay, β represents a roll-off factor, and A n represents coefficients, and k represents an index value.
20 . The system of claim 16 , wherein:
the system comprises a phased array antenna system; and
the fractional delay filter defines a plurality of filter coefficients, each coefficient of the plurality of filter coefficients corresponding to a fractional delay corresponding to an element of the phased array antenna system.