Nyquist and square root nyquist filters for pulse shaping in wireless communications
View Patent ↗Nyquist filters for pulse shaping and related matched filters in wireless communications are disclosed that provide improved performance. The disclosed embodiments recognize that the second derivative of the raised cosine function is discontinuous in the frequency domain and that the first derivative of the square root raised cosine is discontinuous in the frequency domain. As such, a generalization for the raised cosine filter is applied, and improvements can be made to the raised cosine function time-frequency localization and ultimately to tradeoffs between inter-symbol interference and adjacent channel interference by introducing smoothness to the higher order derivatives of the frequency response.
1. An apparatus to receive wireless communications, comprising:
an antenna;
receive circuitry coupled to receive input signals from the antenna, the input signals having a pulse shaping modulation; and
a matched filter within the receive circuitry coupled to receive the input signals and to remove the pulse shaping modulation;
wherein the matched filter is a generalized Nyquist filter that has a frequency response (H(f)) represented by frequency response equations:
H
(
f
)
=
T
;
for
f
≤
(
1
-
β
)
/
2
T
H
(
f
)
=
T
2
{
A
0
+
A
1
cos
(
1
π
T
β
[
f
-
1
-
β
2
T
]
)
+
A
3
cos
(
3
π
T
β
[
f
-
1
-
β
2
T
]
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
π
T
β
[
f
-
1
-
β
2
T
]
)
}
;
for
1
-
β
2
T
<
f
≤
1
+
β
2
T
H
(
f
)
=
0
;
for
f
>
(
1
+
β
)
/
2
T
,
where f represents frequency, A n represent coefficients, T represents a symbol period, n represents an integer, and β represents a roll-off factor.
2. The apparatus of claim 1 , further comprising a decoder within the receive circuitry coupled to receive filtered signals from the matched filter and to output decoded signals.
3. The apparatus of claim 2 , further comprising MAC (media access control) circuitry coupled to receive the decoded signals from the decoder and to output processed signals through an external interface.
4. The apparatus of claim 1 , wherein the coefficients (A n ) satisfy a matrix (M(n)) for any given integer (n) represented by:
M
(
n
)
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
1
…
1
1
2
3
2
…
(
2
n
+
1
)
2
⋮
⋮
⋮
⋮
1
2
n
3
2
n
…
(
2
n
+
1
)
2
n
]
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
0
⋮
0
]
.
5. The apparatus of claim 4 , wherein n=2, A 1 =9/8, and A 3 =−1/8; or wherein n=3, A 1 =75/64, A 3 =−25/128, and A 5 =3/128.
6. The apparatus of claim 4 , wherein the matched filter comprises a plurality of sub-filters combined to form a linear combination of Nyquist filters each having a frequency response (H(f)) represented by the frequency response equations in claim 1 and satisfying the matrix (M(n)) for any given integer (n).
7. An apparatus to transmit wireless communications, comprising:
an antenna;
transmit circuitry coupled to provide transmit signals to the antenna, the transmit signals having a pulse shaping modulation; and
a pulse shape filter within the transmit circuitry coupled to receive output signals and to apply the pulse shaping modulation to the output signals for the transmit circuitry;
wherein the pulse shape filter is a generalized Nyquist filter that has a frequency response (H(f)) represented by frequency response equations:
H
(
f
)
=
T
;
for
f
≤
(
1
-
β
)
/
2
T
H
(
f
)
=
T
2
{
A
0
+
A
1
cos
(
1
π
T
β
[
f
-
1
-
β
2
T
]
)
+
A
3
cos
(
3
π
T
β
[
f
-
1
-
β
2
T
]
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
π
T
β
[
f
-
1
-
β
2
T
]
)
}
;
for
1
-
β
2
T
<
f
≤
1
+
β
2
T
H
(
f
)
=
0
;
for
f
>
(
1
+
β
)
/
2
T
where f represents frequency, A n represent coefficients, T represents a symbol period, n represents an integer, and β represents a roll-off factor.
8. The apparatus of claim 7 , further comprising a modulator within the transmit circuitry coupled to provide modulated output signals to the pulse shape filter.
9. The apparatus of claim 8 , further comprising MAC (media access control) circuitry coupled to process signals received through an external interface and to provide processed signals to the modulator.
10. The apparatus of claim 7 , wherein the coefficients (A n ) satisfy a matrix (M(n)) for any given integer (n) represented by:
M
(
n
)
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
1
…
1
1
2
3
2
…
(
2
n
+
1
)
2
⋮
⋮
⋮
⋮
1
2
n
3
2
n
…
(
2
n
+
1
)
2
n
]
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
0
⋮
0
]
.
11. The apparatus of claim 10 , wherein n=2, A 1 =9/8, and A 3 =−1/8; or wherein n=3, A 1 =75/64, A 3 =−25/128, and A 5 =3/128.
12. The apparatus of claim 10 , wherein the pulse shape filter comprises a plurality of sub-filters combined to form a linear combination of Nyquist filters each having a frequency response (H(f)) represented by the frequency response equations in claim 7 and satisfying the matrix (M(n)) any given integer (n).
13. A method to receive wireless communications, comprising:
receiving input signals through an antenna, the input signals having a pulse shaping modulation;
filtering the input signals circuitry with a matched filter to remove the pulse shaping modulation; and
outputting the filtered signals for further processing by receive circuitry;
wherein the matched filter is a generalized Nyquist filter that has a frequency response (H(f)) represented by the frequency response equations:
H
(
f
)
=
T
;
for
f
≤
(
1
-
β
)
/
2
T
H
(
f
)
=
T
2
{
A
0
+
A
1
cos
(
1
π
T
β
[
f
-
1
-
β
2
T
]
)
+
A
3
cos
(
3
π
T
β
[
f
-
1
-
β
2
T
]
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
π
T
β
[
f
-
1
-
β
2
T
]
)
}
;
for
1
-
β
2
T
<
f
≤
1
+
β
2
T
H
(
f
)
=
0
;
for
f
>
(
1
+
β
)
/
2
T
where f represents frequency, A n represent coefficients, T represents a symbol period, n represents an integer, and β represents a roll-off factor.
14. The method of claim 13 , further comprising decoding the filtered signals to output decoded signals for the receive circuitry.
15. The method of claim 14 , further comprising processing the decoded signals with MAC (media access control) circuitry and outputting processed signals through an external interface.
16. The method of claim 13 , wherein the coefficients (A n ) satisfy a matrix (M(n)) for any given integer (n) represented by:
M
(
n
)
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
1
…
1
1
2
3
2
…
(
2
n
+
1
)
2
⋮
⋮
⋮
⋮
1
2
n
3
2
n
…
(
2
n
+
1
)
2
n
]
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
0
⋮
0
]
.
17. The method of claim 16 , wherein n=2, A 1 =9/8, and A 3 =−1/8; or wherein n=3, A 1 =75/64, A 3 =−25/128, and A 5 =3/128.
18. The method of claim 16 , wherein the filtering comprises filtering the input signals with a matched filter including a plurality of sub-filters combined to form a linear combination of Nyquist filters each having a frequency response (H(f)) represented by the frequency response equations in claim 13 and satisfying the matrix (M(n)) any given integer (n).
19. A method to transmit wireless communications, comprising:
receiving output signals to be transmitted;
filtering the output signals with a pulse shape filter to apply a pulse shaping modulation and to generate transmit signals having the pulse shaping modulation; and
transmitting the transmit signals through an antenna;
wherein the pulse shape filter is a generalized Nyquist filter that has a frequency response (H(f)) represented by frequency response equations:
H
(
f
)
=
T
;
for
f
≤
(
1
-
β
)
/
2
T
H
(
f
)
=
T
2
{
A
0
+
A
1
cos
(
1
π
T
β
[
f
-
1
-
β
2
T
]
)
+
A
3
cos
(
3
π
T
β
[
f
-
1
-
β
2
T
]
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
π
T
β
[
f
-
1
-
β
2
T
]
)
}
;
for
1
-
β
2
T
<
f
≤
1
+
β
2
T
H
(
f
)
=
0
;
for
f
>
(
1
+
β
)
/
2
T
where f represents frequency, A n represent coefficients, T represents a symbol period, n represents an integer, and β represents a roll-off factor.
20. The method of claim 19 , further comprising modulating the output signals prior to the filtering.
21. The method of claim 20 , further comprising receiving external signals from an external interface with MAC (media access control) circuitry and providing processed signals for the modulating.
22. The method of claim 19 , wherein the coefficients (A n ) satisfy a matrix (M(n)) for any given integer (n) represented by:
M
(
n
)
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
1
…
1
1
2
3
2
…
(
2
n
+
1
)
2
⋮
⋮
⋮
⋮
1
2
n
3
2
n
…
(
2
n
+
1
)
2
n
]
[
A
1
A
3
⋮
A
2
n
+
1
]
=
[
1
0
⋮
0
]
.
23. The method of claim 22 , wherein n=2, A 1 =9/8, and A 3 =−1/8; or wherein n=3, A 1 =75/64, A 3 =−25/128, and A 5 =3/128.
24. The method of claim 22 , wherein the filtering comprises filtering the output signals with a pulse shape filter including a plurality of sub-filters combined to form a linear combination of Nyquist filters each having a frequency response (H(f)) represented by the frequency response equations in claim 19 and satisfying the matrix (M(n)) any given integer (n).