1. Digital Pulse Amplitude Modulation (PAM)

2. Outline Introduction Pulse shaping Pulse shaping filter bank
Design tradeoffs
FIR 0
FIR 1
FIR L-1
Symbol recovery

3. 4-level PAM Constellation Map
Introduction
4-level PAM Constellation Map d d
3 d
3 d 00 01 10 11
symbol of bits
symbol ampl.
Convert bit stream into pulse stream
Group stream of bits into symbols of J bits
Represent symbol of bits by unique amplitude
Scale pulse shape by that amplitude
M-level PAM or simply M-PAM (M = 2J)
Symbol period is Tsym and bit rate is J fsym
Impulse train has impulses separated by Tsym
Pulse shape may last one or more symbol periods
Serial/ Parallel
Map to PAM
constellation an 1 J
bit stream
J bits per symbol
Pulse shaper gTsym(t)
s*(t)
symbol amplitude
baseband waveform
Impulse modulator
impulse train
d is a voltage

4. Pulse Shaping Without pulse shaping
One impulse per symbol period
Infinite bandwidth used (not practical)
k is a symbol index
Limit bandwidth by pulse shaping (FIR filtering)
Convolution of discrete-time signal ak and continuous-time pulse shape
For pulse shape lasting Ng Tsym seconds, Ng pulses overlap in each symbol period
See lecture slide 7-5
Serial/ Parallel
Map to PAM
constellation an 1 J
bit stream
J bits per symbol
Pulse shaper gTsym(t)
s*(t)
symbol amplitude
baseband waveform
Impulse modulator
impulse train

5. 2-PAM Transmission 2-PAM example (right) Highest frequency ½ fsym
Truncated raised cosine pulse with peak value of 1
What are d and Tsym ?
How does maximum amplitude relate to d?
Highest frequency ½ fsym
Alternating symbol amplitudes +d, -d, +d, …
Serial/ Parallel
Map to PAM
constellation an 1 J
bit stream
J bits per symbol
Pulse shaper gTsym(t)
s*(t)
symbol amplitude
baseband waveform
Impulse modulator
impulse train

6. Baseband PAM Transmission Code
% Discrete-time baseband PAM signal
mmax = N*L;
v = zeros(1,mmax);
v(1:L:end) = symAmp; % interpolation
s = conv(v, g); % pulse shaping
slength = length(s); % trim result
s = s(midpt+1:slength-midpt+1);
% Interpretation in continuous time
Tsym = 4; % Symbol period in sec
fsym = 1/Tsym; % Symbol rate in Hz
fs = L*fsym; % Sampling rate in Hz
Ts = 1/fs; % Sampling time in sec
% Plots
Mmax = length(s);
m = 0 : (Mmax-1);
t = m*Ts;
Nmax = Mmax / L;
n = 0 : (Nmax-1);
figure;
plot(t,s);
hold on;
stem(n*Tsym,symAmp);
hold off;
xlim( [0 (Nmax-(Ng/2))*Tsym-Ts] );
ymax = Ng * (M-1) * d / 2;
ylim( [-ymax ymax] );
xlabel('Time (ms)');
title('Baseband PAM Signal s(t)');
% Baseband PAM Signal Generation
% by Prof. Brian L. Evans
% The University of Texas at Austin
% Spring 2018 %
% m is sample index
% n is symbol index
% Simulation parameters
N = 25; % Number symbol periods
% Pulse shape g[m]
Ng = 4; % Number symbol periods
L = 20; % Samples/symbol period
f0 = 1/L;
midpt = Ng*L/2;
m = (-midpt) : (midpt-1);
g = sinc(f0*m);
% Adjust for group delay
N = N + (Ng/2);
% M-level PAM symbol amplitudes
d = 1;
M = 2;
ioffset = M + 1;
symAmp = (2*randi(M,[1,N]) - ioffset)*d;

7. 4-PAM Transmission 4-PAM example (right) Highest frequency ½ fsym
Truncated raised cosine pulse with peak value of 1
What are d and Tsym ?
How does maximum amplitude relate to d?
Highest frequency ½ fsym
Alternating symbol amplitudes +d, -d, +d, ... or +3d, -3d, +3d, ...
Serial/ Parallel
Map to PAM
constellation an 1 J
bit stream
J bits per symbol
Pulse shaper gTsym(t)
s*(t)
symbol amplitude
baseband waveform
Impulse modulator
impulse train

8. PAM Transmission Transmitted signal
Sample at sampling time Ts : let t = (n L + m) Ts
L samples per symbol period Tsym i.e. Tsym = L Ts
n is the index of the current symbol period being transmitted
m is a sample index within nth symbol (i.e., m = 0, 1, …, L-1)
Pulse shaper gTsym[m]
Serial/ Parallel
Map to PAM
constellation L
D/A 1 J an
s*(t)
bit stream
J bits per symbol
symbol amplitude
impulse train
baseband waveform
baseband waveform

9. Pulse Shaping Block Diagram
Upsampling by L denoted as L
Outputs input sample followed by L-1 zeros
Upsampling by L 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) an
s*(t) L
gTsym[m]
D/A
Transmit
Filter
symbol rate
sampling rate
sampling rate
cont. time
cont. time

10. Digital Interpolation Example
Upsampling by 4 (denoted by L)
Output input sample followed by 3 zeros
Four times the samples on output as input
Increases sampling rate by factor of 4
FIR interpolation filter (lowpass)
Prior to upsampling, fmax is kHz
At filter input, wmax = 2 p fmax / fsampling = p / 4 = p / L
Filter specifications: wstop < p / 4 and wpass = 0.9 wstop
Digital 4x Oversampling Filter
16 bits 44.1 kHz
28 bits kHz 4
FIR Filter
16 bits kHz 1 2
Input to Upsampler by 4 n n’
Output of Upsampler by 4 1 2 3 4 5 6 7 8 1 2
Output of FIR Filter 3 4 5 6 7 8 n’

11. 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[1] = a0 g[1] s[2] = a0 g[2] s[3] = a0 g[3]
s[4] = a0 g[4] + a1 g[0] s[5] = a0 g[5] + a1 g[1] s[6] = a0 g[6] + a1 g[2] s[7] = a0 g[7] + a1 g[3]
L polyphase filters operating at symbol rate
{g[0],g[4]}
{g[1],g[5]}
{g[2],g[6]}
{g[3],g[7]}
s[m]
…,s[4],s[0]
…,s[5],s[1]
…,s[6],s[2]
…,s[7],s[3]
…,a1,a0
m=0
Commutator (Periodic)
Filter Bank

12. Pulse Shaping Filter Bank
Simplify by avoiding multiplication by zero
Split long pulse shaping filter into L short polyphase filters operating at symbol rate an L
gTsym[m]
D/A
Transmit
Filter
symbol rate
sampling rate
sampling rate
cont. time
cont. time
gTsym,0[n]
s(Ln)
gTsym,1[n]
D/A
Transmit
Filter an
s(Ln+1)
Filter Bank Implementation
gTsym,L-1[n]
s(Ln+(L-1))

13. Pulse Shaping Filter Bank Example
Pulse length 24 samples and L = 4 samples/symbol
Derivation: let t = (n + m/L) Tsym
Define mth polyphase filter
Four six-tap polyphase filters (next slide)
Six pulses contribute to each output sample

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

15. Pulse Shaping Filter Bank Example
24 samples in pulse
gTsym,1[n]
4 samples per symbol
Polyphase filter 1 impulse response is second sample of the pulse shape and every fourth sample after that

16. Pulse Shaping Filter Bank Example
24 samples in pulse
gTsym,2[n]
4 samples per symbol
Polyphase filter 2 impulse response is third sample of the pulse shape and every fourth sample after that

17. Pulse Shaping Filter Bank Example
24 samples in pulse
gTsym,3[n]
4 samples per symbol
Polyphase filter 3 impulse response is fourth sample of the pulse shape and every fourth sample after that

18. Pulse Shaping Design Tradeoffs
Computation in MACs/s
Memory size in words
Memory reads in words/s
Memory writes in words/s
Direct structure (slide 13-9)
(L Ng)(L fsym)
Filter bank structure (slide 13-12)
L Ng fsym
fsym symbol rate L samples/symbol Ng duration of pulse shape in symbol periods

19. Pulse Shaping Design Tradeoffs
FIR filter with N coefficients to compute output
Read current input value x[n]
Read pointer to newest input sample in circular buffer
Read previous N-1 input values
Read N filter coefficients h
Compute the output value y[n]
Write current input value to circular buffer to oldest sample
Write (update) the newest pointer in circular buffer
Write output value
N multiplications-additions, 2N + 1 reads, 3 writes

20. Pulse Shaping Design Tradeoffs
Direct structure (slide 13-9)
FIR filter has L Ng coefficients (N = L Ng)
FIR filter: N multiplications-additions, 2N + 1 reads, 3 writes
FIR filter runs at sampling rate L fsym
Upsampler reads at symbol rate and writes at sampling rate
Filter bank structure (slide 13-12)
L polyphase FIR filters with Ng coefficients each
FIR filter: Ng multiplications-additions, 2Ng + 1 reads, 3 writes
Each polyphase FIR filter runs at symbol rate fsym
Commutator reads and writes at sampling rate L fsym

21. Optional
Symbol Clock Recovery
Transmitter and receiver normally have different oscillator circuits
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

22. s*(t) is transmitted signal Periodic with period Tsym
Optional
Symbol Clock Recovery
g1(t) is impulse response of LTI composite channel of pulse shaper, noise-free channel, receive filter
s*(t) is transmitted signal
g1(t) is deterministic
E{ak am} = a2 d[k-m]
Periodic with period Tsym
Receive
B(w)
Squarer
BPF
H(w)
PLL
x(t)
q(t)
q2(t)
p(t)
z(t)

23. Symbol Clock Recovery Fourier series representation of E{ p(t) }
Optional
Symbol Clock Recovery
Fourier series representation of E{ p(t) }
In terms of g1(t) and using Parseval’s relation
Fourier series representation of E{ z(t) }
where
Receive
B(w)
Squarer
BPF
H(w)
PLL
x(t)
q(t)
q2(t)
p(t)
z(t)

24. Symbol Clock Recovery With G1(w) = X(w) B(w)
Optional
Symbol Clock Recovery
With G1(w) = X(w) B(w)
Choose B(w) to pass  ½wsym  pk = 0 except k = -1, 0, 1
Choose H(w) to pass wsym  Zk = 0 except k = -1, 1
B(w) is lowpass filter with wpassband = ½ wsym
H(w) is bandpass filter with center frequency wsym
Receive
B(w)
Squarer
BPF
H(w)
PLL
x(t)
q(t)
q2(t)
p(t)
z(t)