Transfer function beloging to the Nyquist class (zero ISI at the sampling time) is called the raised-cosine filter. It can be express as
H(f)={1∣f∣<2W0−Wcos2(π4∣f∣+W−2W0W−W0)2W0−W<∣f∣<W0∣f>W∣H(f)={1∣f∣<2W0−Wcos2(π4∣f∣+W−2W0W−W0)2W0−W<∣f∣<W0∣f>W∣ size 12{H \( f \) = left lbrace matrix {
1 {} # lline f rline <2W rSub { size 8{0} } - W {} ##
"cos" rSup { size 8{2} } \( { {π} over {4} } { { lline f rline +W - 2W rSub { size 8{0} } } over {W - W rSub { size 8{0} } } } \) {} # 2W rSub { size 8{0} } - W< lline f rline <W {} ##
0 {} # lline f>W rline {}
} right none } {}(1a)
h(t)=2W0sinc(2W0t)cos[2π(W−W0)t]1−[4(W−W0)t]2h(t)=2W0sinc(2W0t)cos[2π(W−W0)t]1−[4(W−W0)t]2 size 12{h \( t \) =2W rSub { size 8{0} } "sin"c \( 2W rSub { size 8{0} } t \) { {"cos" \[ 2π \( W - W rSub { size 8{0} } \) t \] } over {1 - \[ 4 \( W - W rSub { size 8{0} } \) t \] rSup { size 8{2} } } } } {} (1b)
Where
WW size 12{W} {} is the absolute bandwidth.
W0=1/2TW0=1/2T size 12{W rSub { size 8{0} } =1/2T} {} represent the minimum bandwidth for the rectangular spectrum and the -6 dB bandwith (or half-amplitude point) for the raised-cosine spectrum.
W−W0W−W0 size 12{W - W rSub { size 8{0} } } {} is termed the "excess bandwith"
The roll-off factor is defined to be
r=W−W0W0r=W−W0W0 size 12{r= { {W - W rSub { size 8{0} } } over {W rSub { size 8{0} } } } } {} (2), where
0≤r≤10≤r≤1 size 12{0 <= r <= 1} {}
With the Nyquist constrain
W0=Rs/2W0=Rs/2 size 12{W rSub { size 8{0} } =R rSub { size 8{s} } /2} {} equation (2) can be rewriten as
W
=
1
2
(
1
+
r
)
R
s
W
=
1
2
(
1
+
r
)
R
s
size 12{W= { {1} over {2} } \( 1+r \) R rSub { size 8{s} } } {}
The raised-cosine characteristic is illustrate in figure 1 for
r=0,r=0.5,r=1r=0,r=0.5,r=1 size 12{r=0,r=0 "." 5,r=1} {}. When
r=1r=1 size 12{r=1} {}, the required excess bandwidth is 100 %, and the system can provide a symbol rate of
RsRs size 12{R rSub { size 8{s} } } {} symbols/s using a bandwidth of
RsRs size 12{R rSub { size 8{s} } } {} herts (twice the Nyquist minimum bandwidth), thus yielding asymbol-rate packing 1 symbols/s/Hz.
The lager the filter roll-off, the shorter will be the pulse tail. Small tails exhibit less sensitivity to timing errors and thus make for small degradation due to ISI.
The smaller the filter roll-off the smaller will be the excess bandwidth. The cost is longer pulse tails, larger pulse amplitudes, and thus, greater sensitivity to timing errors.
Recall that the raised-cosine frequency transfer function describes the composite
H(f)H(f) size 12{H \( f \) } {} including transmitting filter, channel filter and receiving filter. The filtering at the receiver is chosen so that the overall transfer function is a form of raised-cosine. Often this is accomplished by choosing both the receiving filter and the transmitting filter so that each has a transfer function known as a root raised cosine. Neglecting any channel-induced ISI, the product of these root-raised cosine functions yields the composite raised-cosine system transfer function.
