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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 n2 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 n2 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 n2 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
Ezt 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)
