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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Pulse-shaping filter in digital modulation
The raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form (
β
=
1
{\displaystyle \beta =1}
) is a cosine function, 'raised' up to sit above the
f
{\displaystyle f}
(horizontal) axis.
Mathematical description
[ edit ]
Frequency response of raised-cosine filter with various roll-off factors
Impulse response of raised-cosine filter with various roll-off factors
The raised-cosine filter is an implementation of a low-pass Nyquist filter , i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd symmetry about
1
2
T
{\displaystyle {\frac {1}{2T}}}
, where
T
{\displaystyle T}
is the symbol-period of the communications system.
Its frequency-domain description is a piecewise -defined function , given by:
H
(
f
)
=
{
1
,
|
f
|
≤
1
−
β
2
T
1
2
[
1
+
cos
(
π
T
β
[
|
f
|
−
1
−
β
2
T
]
)
]
,
1
−
β
2
T
<
|
f
|
≤
1
+
β
2
T
0
,
otherwise
{\displaystyle H(f )={\begin{cases}1,&|f|\leq {\frac {1-\beta }{2T}}\\{\frac {1}{2}}\left[1+\cos \left({\frac {\pi T}{\beta }}\left[|f|-{\frac {1-\beta }{2T}}\right]\right)\right],&{\frac {1-\beta }{2T}}<|f|\leq {\frac {1+\beta }{2T}}\\0,&{\text{otherwise}}\end{cases}}}
or in terms of havercosines :
H
(
f
)
=
{
1
,
|
f
|
≤
1
−
β
2
T
hvc
(
π
T
β
[
|
f
|
−
1
−
β
2
T
]
)
,
1
−
β
2
T
<
|
f
|
≤
1
+
β
2
T
0
,
otherwise
{\displaystyle H(f )={\begin{cases}1,&|f|\leq {\frac {1-\beta }{2T}}\\\operatorname {hvc} \left({\frac {\pi T}{\beta }}\left[|f|-{\frac {1-\beta }{2T}}\right]\right),&{\frac {1-\beta }{2T}}<|f|\leq {\frac {1+\beta }{2T}}\\0,&{\text{otherwise}}\end{cases}}}
for
0
≤
β
≤
1
{\displaystyle 0\leq \beta \leq 1}
and characterised by two values;
β
{\displaystyle \beta }
, the roll-off factor , and
T
{\displaystyle T}
, the reciprocal of the symbol-rate.
The impulse response of such a filter[1] is given by:
h
(
t
)
=
{
π
4
T
sinc
(
1
2
β
)
,
t
=
±
T
2
β
1
T
sinc
(
t
T
)
cos
(
π
β
t
T
)
1
−
(
2
β
t
T
)
2
,
otherwise
{\displaystyle h(t )={\begin{cases}{\frac {\pi }{4T}}\operatorname {sinc} \left({\frac {1}{2\beta }}\right),&t=\pm {\frac {T}{2\beta }}\\{\frac {1}{T}}\operatorname {sinc} \left({\frac {t}{T}}\right){\frac {\cos \left({\frac {\pi \beta t}{T}}\right)}{1-\left({\frac {2\beta t}{T}}\right)^{2}}},&{\text{otherwise}}\end{cases}}}
in terms of the normalised sinc function . Here, this is the "communications sinc"
sin
(
π
x
)
/
(
π
x
)
{\displaystyle \sin(\pi x)/(\pi x)}
rather than the mathematical one.
Roll-off factor
[ edit ]
The roll-off factor,
β
{\displaystyle \beta }
, is a measure of the excess bandwidth of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of
1
2
T
{\displaystyle {\frac {1}{2T}}}
.
Some authors use
α
=
β
{\displaystyle \alpha =\beta }
.[2]
If we denote the excess bandwidth as
Δ
f
{\displaystyle \Delta f}
, then:
β
=
Δ
f
(
1
2
T
)
=
Δ
f
R
S
/
2
=
2
T
Δ
f
{\displaystyle \beta ={\frac {\Delta f}{\left({\frac {1}{2T}}\right)}}={\frac {\Delta f}{R_{S}/2}}=2T\,\Delta f}
where
R
S
=
1
T
{\displaystyle R_{S}={\frac {1}{T}}}
is the symbol-rate.
The graph shows the amplitude response as
β
{\displaystyle \beta }
is varied between 0 and 1, and the corresponding effect on the impulse response . As can be seen, the time-domain ripple level increases as
β
{\displaystyle \beta }
decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.
β = 0
[ edit ]
As
β
{\displaystyle \beta }
approaches 0, the roll-off zone becomes infinitesimally narrow, hence:
lim
β
→
0
H
(
f
)
=
rect
(
f
T
)
{\displaystyle \lim _{\beta \rightarrow 0}H(f )=\operatorname {rect} (fT )}
where
rect
(
⋅
)
{\displaystyle \operatorname {rect} (\cdot )}
is the rectangular function , so the impulse response approaches
h
(
t
)
=
1
T
sinc
(
t
T
)
{\displaystyle h(t )={\frac {1}{T}}\operatorname {sinc} \left({\frac {t}{T}}\right)}
. Hence, it converges to an ideal or brick-wall filter in this case.
β = 1
[ edit ]
When
β
=
1
{\displaystyle \beta =1}
, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:
H
(
f
)
|
β
=
1
=
{
1
2
[
1
+
cos
(
π
f
T
)
]
,
|
f
|
≤
1
T
0
,
otherwise
{\displaystyle H(f )|_{\beta =1}=\left\{{\begin{matrix}{\frac {1}{2}}\left[1+\cos \left(\pi fT\right)\right],&|f|\leq {\frac {1}{T}}\\0,&{\text{otherwise}}\end{matrix}}\right.}
or
H
(
f
)
|
β
=
1
=
{
hvc
(
π
f
T
)
,
|
f
|
≤
1
T
0
,
otherwise
{\displaystyle H(f )|_{\beta =1}=\left\{{\begin{matrix}\operatorname {hvc} \left(\pi fT\right),&|f|\leq {\frac {1}{T}}\\0,&{\text{otherwise}}\end{matrix}}\right.}
Bandwidth
[ edit ]
The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero frequency-positive portion of its spectrum, i.e.:
B
W
=
R
S
2
(
β
+
1
)
,
(
0
<
β
<
1
)
{\displaystyle BW={\frac {R_{S}}{2}}(\beta +1),\quad (0<\beta <1 )}
As measured using a spectrum analyzer, the radio bandwidth B in Hz of the modulated signal is twice the baseband bandwidth BW (as explained in [1 ]), i.e.:
B
=
2
B
W
=
R
S
(
β
+
1
)
,
(
0
<
β
<
1
)
{\displaystyle B=2BW=R_{S}(\beta +1),\quad (0<\beta <1 )}
Auto-correlation function
[ edit ]
The auto-correlation function of raised cosine function is as follows:
R
(
τ
)
=
T
[
sinc
(
τ
T
)
cos
(
β
π
τ
T
)
1
−
(
2
β
τ
T
)
2
−
β
4
sinc
(
β
τ
T
)
cos
(
π
τ
T
)
1
−
(
β
τ
T
)
2
]
{\displaystyle R\left(\tau \right)=T\left[\operatorname {sinc} \left({\frac {\tau }{T}}\right){\frac {\cos \left(\beta {\frac {\pi \tau }{T}}\right)}{1-\left({\frac {2\beta \tau }{T}}\right)^{2}}}-{\frac {\beta }{4}}\operatorname {sinc} \left(\beta {\frac {\tau }{T}}\right){\frac {\cos \left({\frac {\pi \tau }{T}}\right)}{1-\left({\frac {\beta \tau }{T}}\right)^{2}}}\right]}
The auto-correlation result can be used to analyze various sampling offset results when analyzed with auto-correlation.
Application
[ edit ]
Consecutive raised-cosine impulses, demonstrating zero-ISI property
When used to filter a symbol stream, a Nyquist filter has the property of eliminating ISI, as its impulse response is zero at all
n
T
{\displaystyle nT}
(where
n
{\displaystyle n}
is an integer), except
n
=
0
{\displaystyle n=0}
.
Therefore, if the transmitted waveform is correctly sampled at the receiver, the original symbol values can be recovered completely.
However, in many practical communications systems, a matched filter is used in the receiver, due to the effects of white noise . For zero ISI, it is the net response of the transmit and receive filters that must equal
H
(
f
)
{\displaystyle H(f )}
:
H
R
(
f
)
⋅
H
T
(
f
)
=
H
(
f
)
{\displaystyle H_{R}(f )\cdot H_{T}(f )=H(f )}
And therefore:
|
H
R
(
f
)
|
=
|
H
T
(
f
)
|
=
|
H
(
f
)
|
{\displaystyle |H_{R}(f )|=|H_{T}(f )|={\sqrt {|H(f )|}}}
These filters are called root-raised-cosine filters.
Raised cosine is a commonly used apodization filter for fiber Bragg gratings .
References
[ edit ]
Glover, I.; Grant, P. (2004). Digital Communications (2nd ed.). Pearson Education Ltd. ISBN 0-13-089399-4 .
Proakis, J. (1995). Digital Communications (3rd ed.). McGraw-Hill Inc. ISBN 0-07-113814-5 .
Tavares, L.M.; Tavares G.N. (1998) Comments on "Performance of Asynchronous Band-Limited DS/SSMA Systems" . IEICE Trans. Commun., Vol. E81-B, No. 9
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Raised-cosine_filter&oldid=1189773695 "
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