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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
• 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
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)
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)
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)