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Pulse Shaping to Reduce ISI

Module by: Ha Ta-Hong, Tuan Do-Hong. E-mail the authorsEdited By: Ha Ta-Hong, Tuan Do-HongTranslated By: Ha Ta-Hong, Tuan Do-Hong

The Raised-Cosine Filter

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)={1f<2W0Wcos2(π4f+W2W0WW0)2W0W<f<W0f>WH(f)={1f<2W0Wcos2(π4f+W2W0WW0)2W0W<f<W0f>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[(WW0)t]1[4(WW0)t]2h(t)=2W0sinc(2W0t)cos[(WW0)t]1[4(WW0)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. WW0WW0 size 12{W - W rSub { size 8{0} } } {} is termed the "excess bandwith"

The roll-off factor is defined to be r=WW0W0r=WW0W0 size 12{r= { {W - W rSub { size 8{0} } } over {W rSub { size 8{0} } } } } {} (2), where 0r10r1 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} } } {}

Figure 1: Raised-cosine filter characteristics. (a) System transfer function. (b) System impulse response
Figure 1 (graphics1.png)

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.

The Root Raised-Cosine Filter

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.

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