Nyquist Pulse Shaping Criteria

Document Sample

```					Nyquist Pulse Shaping Criteria
Pulse Shaping
A      simple baseband transmission can be
demonstrated using rectangular pulses.
 There are two problems in transmitting such
pulses:
 They require infinite bandwidth.
 When transmitted over bandlimited channels become
time unlimited on the other side, and spread over
adjacent symbols, resulting in Inter-Symbol-
Interference (ISI).
Nyquist-Criterion for Zero ISI
 Use a pulse that has the following characteristics

￬ 1              t =0
p (t ) = ￬
￬ 0 t = ￬ T b , 2T b , 3T b ,K

 One such pulse is the sinc function.
The Sinc Pulse
p(t)
1

t
-6Tb -5Tb -4Tb -3Tb -2Tb   -Tb          Tb   2Tb 3Tb   4Tb   5Tb    6Tb

Note that such pulse has a bandwidth of Rb/2 Hz.                     P(f)
Therefore, the minimum channel bandwidth
required for transmitting pulses at a rate
of Rb pulses/sec is Rb/2 Hz
-1/(2Tb)            1/(2Tb)
f
Zero ISI

1

0
-2    -1        0   1   2   3   4

-1
More on Pulse Shaping
 The sinc pulse has the minimum bandwidth
among pulses satisfying Nyquist criterion.
 However, the sinc pulse is not fast decaying;
 Misalignment in sampling results in significant ISI.
 Requires long delays for realization.
 There is a set of pulses that satisfy the Nyquist
criterion and decay at a faster rate. However,
they require bandwidth more than Rb/2.
Raised-Cosine Pulses
￬ 1�      � { ω − ( ωb / 2 ) }
π                     �
�        ωb
￬ �− sin �
1                           �
�   ω−      < ωx
￯2 �    �     2ωx              �
�         2
�    �                      �
�
￯
￯                                     ωb
P (ω ) = ￬               0                      ω >   + ωx
￯                                      2
￯                                     ω
￯           1                      ω < b + ωx
2
￯
￬

where ω b is 2π Rb and ω x is the excess bandwidth. It defines how much
bandwidth required above the minimum bandwidth of a sinc pulse,
where                  ω
0 ￬ ωx ￬ b
2
Spectrum of Raised-Cosine Pulses

P( ω)          ωx            ωx

ω
ω b/2
= π/Tb
ωb/2 – ωx            ωb /2 + ωx
Extremes of Raised-Cosine Spectra
P( ω)                     ωx = ωb/2
ωx = 0
“Sinc”
ωb /2                  ωb /2

ω
ωb /2                   ωb
= π/Tb                =2π/T b

Excess Bandwidth   ωx    2ωx
r=                   =     =
Minimum Bandwidth ωb / 2 ωb
Raised-Cosine Pulses

ωx = 0              p(t)
“Sinc”
ωx = ωb/2
1

t
–3T b    –2T b   –T b              Tb         2Tb   3Tb

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 540 posted: 2/20/2011 language: English pages: 10