Asymmetric factorization of generalized raised cosine filters for improved selectivity
An apparatus to transmit and receive wireless communications is disclosed in which the transmit circuitry includes a square root raised cosine filter to pulse shape modulate signals and the receive circuitry includes a higher order Nyquist receive filter coupled to receive the input signals and remove the pulse shaping modulation. The cascaded combination of the transmit and receive filters has a frequency response equivalent to a higher order generalized raised cosine filter response.
1. An apparatus to receive wireless communications from a transmitter transmitting signals using a root-raised cosine transmit filter, the apparatus comprising:
an antenna;
receive circuitry coupled to receive input signals from the antenna, the input signals having been pulse shaped modulated by the root-raised cosine transmit filter; and
a receive filter within the receive circuitry coupled to receive the input signals and to remove the pulse shaping modulation, wherein the receive filter is an asymmetric partition of a higher order (n>=1) generalized Nyquist filter;
wherein a cascaded combination of the transmit and receive filters has a frequency response equivalent to a higher order generalized raised cosine filter response.
2. The apparatus of claim 1 , wherein:
the asymmetric partition of the higher order generalized Nyquist filter at the receive filter has a frequency response at the receiver of R n (z) given by:
R
n
(
f
)
=
{
T
2
,
f
≤
(
1
-
β
)
/
2
T
R
n
(
z
)
,
1
-
β
2
T
<
f
≤
1
+
β
2
T
0
,
else
;
wherein:
R
n
(
z
)
=
{
0
,
z
=
π
T
/
2
2
cos
(
z
/
2
)
[
A
0
+
A
1
cos
(
z
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
z
)
]
;
n represents the order of the higher order generalized Nyquist receive filter and is greater than or equal to one;
z
=
π
T
β
[
f
-
1
-
β
2
T
]
;
β represents a roll-off factor that defines a roll-off transition region wherein:
1
-
β
2
T
<
f
≤
1
+
β
2
T
;
ƒ represents frequency; and
T is a symbol period.
3. The apparatus of claim 2 , wherein the coefficients (A n ) for the frequency response R n (z) of the receive filter satisfy a matrix M(n) for any given integer n, wherein:
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
]
.
4. The apparatus of claim 1 , further comprising a decoder within the receive circuitry coupled to receive filtered signals from the receive filter and to output decoded signals.
5. The apparatus of claim 1 , further comprising MAC (media access control) circuitry coupled to receive decoded signals from a decoder and to output processed signals through an external interface.
6. The apparatus of claim 1 , wherein the receive filter comprises a linear combination of higher order Nyquist subfilters.
7. The apparatus of claim 1 , wherein the receive filter comprises a combination of nth-order sub-filters, each sub-filter having a frequency response R k (z) in a roll-off transition region, and the combination of sub-filters has a frequency response R C (z), in which these terms are represented by the following equations and relationships:
R
C
(
z
)
=
∑
k
=
0
m
α
k
R
k
(
z
)
for a combination of m sub-filters;
α 0 through α m are coefficients that represent weights of the asymmetric partitions of different order Nyquist filters at the receiver, each having a value greater than or equal to 0, that add up to one; and
R k (z) represent frequency responses of the sub-filters.
8. The apparatus of claim 7 , wherein the frequency responses of the sub-filters are characterized by the equation:
R
k
(
f
)
=
{
T
2
,
f
≤
(
1
-
β
)
/
2
T
R
n
(
z
)
,
1
-
β
2
T
<
f
≤
1
+
β
2
T
0
,
else
;
wherein:
R
k
(
z
)
=
{
0
,
z
=
π
T
/
2
2
cos
(
z
/
2
)
[
A
0
+
A
1
cos
(
z
)
+
A
3
cos
(
3
z
)
+
…
+
A
2
k
+
1
cos
(
(
2
k
+
1
)
z
)
]
;
each value k uniquely designates one of the nth-order sub-filters;
A 0 , A 1 , A 3 , . . . A 2k+1 represent cosine coefficients for each sub-filter;
z
=
π
T
β
[
f
-
1
-
β
2
T
]
;
ƒ represents frequency;
T is a symbol period;
β represents a roll-off factor; and
a roll-off transition region is defined by:
1
-
β
2
T
<
f
≤
1
+
β
2
T
.
9. The apparatus of claim 5 , wherein the cosine coefficients A 0 , A 1 , . . . A 0 for each sub-filter satisfy the matrix M(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
]
.
10. The apparatus of claim 1 , wherein the receive filter is implemented in a time domain.
11. The apparatus of claim 1 , wherein the receive filter is implemented in a frequency domain.
12. A higher order generalized raised cosine cascaded filter comprising:
a square root raised cosine transmit filter;
a higher order receive filter;
wherein:
n represents the order of the higher order receive filter and n is greater than or equal to one;
a frequency response X(z) of the square root raised cosine transmit filter is given by the following equation:
X
(
f
)
=
{
T
/
2
,
|
f
|
≤
(
1
-
β
)
/
2
T
T
/
2
cos
(
z
/
2
)
,
1
-
β
2
T
<
f
≤
1
+
β
2
T
0
,
else
,
wherein: ƒ represents a frequency of a signal transmitted through the higher order generalized raised cosine cascaded filter;
T is a symbol period;
β represents a roll-off factor; and
z is given by the following equation:
z
=
π
T
β
[
f
-
1
-
β
2
T
]
.
13. A higher order generalized raised cosine cascaded filter comprising:
a square root raised cosine transmit filter;
a higher order receive filter;
wherein:
n represents the order of the higher order receive filter and n is greater than or equal to one;
in a roll-off transition region
1
-
β
2
T
<
f
≤
1
+
β
2
T
,
a frequency response H(z) of the higher order generalized cosine cascaded filter is given by the following equation:
H ( z )= X ( z )* R ( z );
ƒ represents frequency;
T is a symbol period; and
β represents a roll-off factor;
X(z) is a frequency response of the root raised cosine transmit filter;
R(z) is a frequency response of the n-order receive filter; and
z is given by the following equation:
z
=
π
T
β
[
f
-
1
-
β
2
T
]
.
14. A higher order generalized raised cosine cascaded filter comprising:
a square root raised cosine transmit filter; and
a higher order receive filter;
wherein:
n represents the order of the higher order receive filter and n is greater than or equal to one; and
a frequency response of the root raised cosine transmit filter is given by the following equation:
X
(
z
)
=
τ
2
cos
(
z
2
)
;
wherein
:
T is a symbol period; and
z is given by the following equation:
z
=
π
T
β
[
f
-
1
-
β
2
T
]
.
15. A higher order generalized raised cosine cascaded filter comprising:
a square root raised cosine transmit filter; and
a higher order receive filter;
wherein:
n represents the order of the higher order receive filter and n is greater than or equal to one;
R
(
z
)
=
{
0
,
z
=
π
T
/
2
2
cos
(
z
/
2
)
[
A
0
+
A
1
cos
(
z
)
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
z
)
]
,
else
;
and
the values of the coefficients A n for the receive filter 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
]
.
16. A higher order generalized raised cosine cascaded filter comprising:
a square root raised cosine transmit filter; and
a higher order receive filter;
wherein:
the root raised cosine transmit filter has a frequency response H(ƒ) given by the following equation:
H
(
f
)
=
{
T
2
,
f
≤
(
1
-
β
)
/
2
T
H
(
z
)
,
1
-
β
2
T
<
f
≤
1
+
β
2
T
0
,
else
;
ƒ represents a frequency of a signal transmitted through the higher order generalized raised cosine cascaded filter;
T is a symbol period;
β represents a roll-off factor that defines a roll-off transition region
1
-
β
2
T
<
f
≤
1
+
β
2
T
;
H(z) is a Fourier series that represents the frequency response of the cascaded filter in the roll-off transition region
1
-
β
2
T
<
f
≤
1
+
β
2
T
;
z is given by the following equation:
z
=
π
T
β
[
f
-
1
-
β
2
T
]
;
and
H(z) is characterized by the following formula:
H
(
z
)
=
T
4
[
A
0
+
…
+
A
2
n
+
1
cos
(
(
2
n
+
1
)
z
)
]
.
17. A method of wireless communication comprising:
receiving signals that have been pulse-shape modulated by a root-raised cosine transmit filter of a transmitter; and
filtering the signals with a receive filter to remove the pulse shaping modulation;
wherein the receive filter is a higher order Nyquist filter, and a cascaded combination of the transmit and receive filters has a frequency response equivalent to a higher order generalized raised cosine filter response.
18. The method of claim 17 , further comprising
applying pulse-shaping modulation to pre-transmitted signals using the root-raised cosine transmit filter to generate transmit signals having the pulse shaping modulation; and
transmitting the signals through a transmit antenna.