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									EE445S Real-Time Digital Signal Processing Lab   Spring 2011


         Digital Pulse Amplitude
           Modulation (PAM)

               Prof. Brian L. Evans
  Dept. of Electrical and Computer Engineering
        The University of Texas at Austin


                          Lecture 14
                      Outline
• Introduction
• Pulse shaping
• Pulse shaping filter bank
• Design tradeoffs
• Optional sections
  Symbol recovery
  Communication system examples


                                  14 - 2
                                    Introduction
   • Each data symbol contains J bits of information
   • Modulate one of M = 2J discrete messages onto
     amplitude of waveform in each symbol period Tsym
          Known as M-level PAM or M-PAM for short
                                                                                 01            3d
          Bit rate is J fsym , where symbol rate fsym = 1 / Tsym
          Uniformly spaced amplitudes                                            00            d
                                          M                 M
             ai  d (2i  1) where i    1 , ...,0 , ...,                      10        d
                                          2                 2
          Multiple ways to map symbols to amplitudes                                       3 d
                                                                                 11
                                                                 Pulse
         Serial/        Map to PAM            Impulse
   1     Parallel   J   constellation   ai   modulator
                                                                shaper    4-PAM
                                                                 gT(t)   s*(t)
  bit           J bits per         symbol          sampling transmitted Constellation
stream           symbol           amplitude              rate      waveform           14 - 3
                                    Pulse Shaping
• Without pulse shaping, one would modulate using
  an impulse train, which uses infinite bandwidth
                 
   s (t ) 
    *
                 a
              k  
                          k    (t  k Tsym )                       Non-overlapping impulses

• Limit bandwidth by pulse shaping                                    At each time t, index k is
  filter with impulse response gT(t)
           
                                                                      indexed over number of

                 a               gTsym t  k Tsym 
                                                                         pulses overlapping in
    s * (t )                 k                                             one symbol period
                 k  
   Summation has finite number of terms
   Related to discrete-time convolution
                                                                  L samples per symbol period
• Sample: let t = n L + m                                                       n is symbol index
                              
  s * Ln  m           a            gTsym L (n  k )  m     m is sample index in symbol
                          k  
                                    k
                                                                          (i.e. m = 0, 1, …, L-1)
                                                                                        14 - 4
 Pulse Shaping Example




2-PAM with Raised Cosine Pulse Shaping   14 - 5
        Pulse Shaping Block Diagram
   an                                  s*(t)   Transmit
            L      gTsym[m]     D/A             Filter

  symbol    sampling     sampling     analog              analog
   rate       rate         rate

• Upsampling by L denoted as L
   Outputs input sample followed by L-1 zeros
   Upsampling by converts symbol rate to sampling rate
• Pulse shaping (FIR) filter gTsym[m]
   Fills in zero values generated by upsampler
   Multiplies by zero most of time (L-1 out of every L times)
                                                               14 - 6
       Digital Interpolation Example
                                                             Input to Upsampler by 4
 16 bits            16 bits                  28 bits
                4             FIR Filter
 44.1 kHz           176.4 kHz                176.4 kHz
                                                                                          n
                                                         0           1                2
            Digital 4x Oversampling Filter
                                                             Output of Upsampler by 4

• Upsampling by 4 (denoted by 4)
                                                                                          n’
   Output input sample followed by 3 zeros               0 1 2 3 4 5 6 7 8

   Four times the samples on output as input                 Output of FIR Filter


   Increases sampling rate by factor of 4                                                 n’

• FIR filter performs interpolation                      0 1 2 3 4 5 6 7 8


   Lowpass filter with stopband frequency wstopband  p / 4
   For fsampling = 176.4 kHz, w = p / 4 corresponds to 22.05 kHz
                                                                             14 - 7
  Pulse Shaping Filter Bank Example
• L = 4 samples per symbol
• Pulse shape g[m] lasts for 2 symbols (8 samples)
 bits                           …a2a1a0           …000a1000a0
            encoding                         ↑4                    g[m]
                                                      x[m]                 s[m]
s[m] = x[m] * g[m]               s[0] = a0 g[0]     s[4] = a0 g[4] + a1 g[0]
                                 s[1] = a0 g[1]     s[5] = a0 g[5] + a1 g[1]
                                 s[2] = a0 g[2]     s[6] = a0 g[6] + a1 g[2]
                                 s[3] = a0 g[3]     s[7] = a0 g[7] + a1 g[3]
        L polyphase filters
                                    …,s[4],s[0]
                  {g[0],g[4]}                                m=0
                                    …,s[5],s[1]
…,a1,a0           {g[1],g[5]}                                      s[m]        Filter
                                    …,s[6],s[2]
                  {g[2],g[6]}                                                  Bank
                                    …,s[7],s[3]              Commutator        14 - 8
                  {g[3],g[7]}                                (Periodic)
            Pulse Shaping Filter Bank
     an                                               Transmit
               L       gTsym[m]           D/A          Filter

  symbol        sampling        sampling        analog           analog
   rate           rate            rate
• Simplify by avoiding multiplication by zero
          Split long pulse shaping filter into L short polyphase filters
                                               operating at symbol rate
              gTsym,0[n] s(Ln)
                                                            Transmit
              gTsym,1[n]    s(Ln+1)             D/A          Filter
an
                                                   Filter Bank
             gTsym,L-1[n]                        Implementation           14 - 9
                            s(Ln+(L-1))
 Pulse Shaping Filter Bank Example
• Pulse length 24 samples and L = 4 samples/symbol
                      n 3
  s * L n  m      a      k   g T L ( n  k )  m               Six pulses contribute
                     k n2                                          to each output sample
• Derivation in Tretter's manual,
              m      n 3              m             
  s *  nTsym  Tsym    ak gT  nTsym  Tsym  kTsym                        m  0, 1,...,L  1
              L      k n2            L             
• Define mth polyphase filter
                               m     
  gTsym ,m [n]  gTsym  nTsym  Tsym                     m  0, 1,...,L  1
                               L     
• Four six-tap polyphase filters (next slide)
                     n 3
  s *  nTsym  Tsym    ak gTsym ,m n  k 
               m
              L      k n2                                                          14 - 10
Pulse Shaping Filter Bank Example
                                                   24 samples
                                    gTsym,0[n]      in pulse
                                                    4 samples
                                                   per symbol

                                 Polyphase filter 0 response
                                    is the first sample of the
                                      pulse shape plus every
                                   fourth sample after that
                                                    x marks
                                                   samples of
                                                   polyphase
                                                      filter
                                                      14 - 11
Polyphase filter 0 has only one non-zero sample.
Pulse Shaping Filter Bank Example
                                    24 samples
                     gTsym,1[n]      in pulse
                                     4 samples
                                    per symbol

                   Polyphase filter 1 response
                   is the second sample of the
                        pulse shape plus every
                      fourth sample after that
                                     x marks
                                    samples of
                                    polyphase
                                       filter
                                       14 - 12
Pulse Shaping Filter Bank Example
                                     24 samples
                     gTsym,2[n]       in pulse
                                      4 samples
                                     per symbol

                   Polyphase filter 2 response
                     is the third sample of the
                        pulse shape plus every
                      fourth sample after that
                                      x marks
                                     samples of
                                     polyphase
                                        filter
                                       14 - 13
Pulse Shaping Filter Bank Example
                                     24 samples
                     gTsym,3[n]       in pulse
                                      4 samples
                                     per symbol

                   Polyphase filter 3 response
                    is the fourth sample of the
                         pulse shape plus every
                       fourth sample after that
                                      x marks
                                     samples of
                                     polyphase
                                        filter
                                       14 - 14
      Pulse Shaping Design Tradeoffs
               Computation   Memory       Memory     Memory
                in MACs/s     size in     reads in   writes in
                              words       words/s    words/s
Direct
structure
(slide 14-6)
Filter bank
structure
(slide 14-9)

fsym symbol rate
L samples/symbol
Ng duration of pulse shape in symbol periods            14 - 15
                             Optional


          Symbol Clock Recovery
• Transmitter and receiver normally have different
  crystal oscillators
• Critical for receiver to sample at correct time
  instances to have max signal power and min ISI
• Receiver should try to synchronize with
  transmitter clock (symbol frequency and phase)
  First extract clock information from received signal
  Then either adjust analog-to-digital converter or interpolate
• Next slides develop adjustment to A/D converter
• Also, see Handout M in the reader
                                                           14 - 16
                                                           Optional


                  Symbol Clock Recovery
• g1(t) is impulse response of LTI composite channel
  of pulse shaper, noise-free channel, receive filter
                                           
  q (t )  s (t )  g1 (t ) 
            *
                                         a
                                         k  
                                                    k   g1 (t  kTsym )        s*(t) is transmitted signal
                                   

                           
                                                                                             g1(t) is
  p(t )  q 2 (t )                       ak am g1 (t  kTsym ) g1 (t  mTsym )
                          k   m  
                                                                                          deterministic
                        
  E{ p(t )}      E{a
                k   m  
                                    k   am } g1 (t  kTsym ) g1 (t  mTsym )        E{ak am} = a2 [k-m]
                        
           a     2
                       g (t  kT
                     k  
                            1
                             2
                                          sym   )                              Periodic with period Tsym
                                                        p(t)
            Receive                                            BPF
  x(t)                                  Squarer                              PLL
             B(w)                                              H(w)
                                                                                                14 - 17
                             q(t)                   q2(t)             z(t)
                                                   Optional


            Symbol Clock Recovery
• Fourier series representation of E{ p(t) }
                     

                 
                                  j k w sym t                1            Tsym                  j k w sym t
  E{ p (t )} 
                 k  
                          pk e                   where pk 
                                                            Tsym      0
                                                                                 E{ p (t )}e                   dt

• In terms of g1(t) and using Parseval’s relation
                                                             

                                             G1 w G1 kw sym  w dw
         a2                           a2
                 g12 t e sym dt 
                           jkw t
   pk 
        Tsym                     2p Tsym 

• Fourier series representation of E{ z(t) }
                                                                  
   zk  pk H kw sym   H kw sym                            G1 w G1 kw sym  w dw
                                                          2
                                                         a
                                                       2pTsym 
                                                p(t)
          Receive                                      BPF
  x(t)                           Squarer                              PLL
           B(w)                                        H(w)
                                                                                                    14 - 18
                          q(t)              q2(t)             z(t)
                                               Optional


            Symbol Clock Recovery
• With G1(w) = X(w) B(w)
   Choose B(w) to pass  ½wsym  pk = 0 except k = -1, 0, 1
                                               
   Z k  pk H kw sym   H kw sym           G1 w G1 kw sym  w dw
                                        a2
                                      2p Tsym 
   Choose H(w) to pass wsym  Zk = 0 except k = -1, 1
   Ezt    Z k e
                           jkw symt         jw symt        jw symt
                                      e               e                    w
                                                                       2 cos( symt )
               k
• B(w) is lowpass filter with wpassband = ½wsym
• H(w) is bandpass filter with center frequency wsym
                                           p(t)
          Receive                                  BPF
  x(t)                      Squarer                                      PLL
           B(w)                                    H(w)
                                                                                        14 - 19
                    q(t)               q2(t)                    z(t)

								
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