Nyquist Pulse Shaping Criteria

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					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

				
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posted:2/20/2011
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