Digital Pulse Amplitude Modulation PAM

1
Digital Pulse AmplitudeModulation (PAM)
2
Outline

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

3
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
• Bit rate is J fsym , where symbol rate fsym 1 /
  Tsym
• Uniformly spaced amplitudes
• Multiple ways to map symbols to amplitudes

PulseshapergT(t)
Serial/Parallel
Map to PAM constellation
Impulsemodulator
ai
1
J
s(t)
bit stream
symbol amplitude
J bits per symbol
transmitted waveform
samplingrate
4
Pulse Shaping

• Without pulse shaping, one would modulate using
  an impulse train, which uses infinite bandwidth
• Limit bandwidth by pulse shapingfilter with
  impulse response gT(t)
• Summation has finite number of terms
• Related to discrete-time convolution
• Sample let t n L m

Non-overlapping impulses
At each time t, index k is indexed over number of
pulses overlapping in one symbol period
L samples per symbol periodn is symbol indexm
is sample index in symbol(i.e. m 0, 1, , L-1)
5
Pulse Shaping Example
2-PAM with Raised Cosine Pulse Shaping
6
Pulse Shaping Block Diagram

• 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 gTsymm
• Fills in zero values generated by upsampler
• Multiplies by zero most of time (L-1 out of every
  L times)

an
s(t)
gTsymm
D/A
Transmit Filter
L
sampling rate
sampling rate
analog
analog
symbol rate
7
Digital Interpolation Example

• Upsampling by 4 (denoted by 4)
• Output input sample followed by 3 zeros
• Four times the samples on output as input
• Increases sampling rate by factor of 4
• FIR filter performs interpolation
• Lowpass filter with stopband frequency wstopband
  ? p / 4
• For fsampling 176.4 kHz, w p / 4 corresponds
  to 22.05 kHz

8
Pulse Shaping Filter Bank Example

• L 4 samples per symbol
• Pulse shape gm lasts for 2 symbols (8 samples)

bits
a2a1a0
000a1000a0
encoding
?4
gm
xm
sm
sm xm gm
s0 a0 g0s1 a0 g1s2 a0 g2s3
a0 g3
s4 a0 g4 a1 g0s5 a0 g5 a1
g1s6 a0 g6 a1 g2s7 a0 g7 a1
g3
Filter Bank
9
Pulse Shaping Filter Bank

• Simplify by avoiding multiplication by zero
• Split the long pulse shaping filter into L short
  polyphase filters operating at symbol rate

an
gTsymm
D/A
Transmit Filter
L
sampling rate
sampling rate
analog
analog
symbol rate
gTsym,0n
s(Ln)
D/A
Transmit Filter
gTsym,1n
s(Ln1)
an
Filter Bank Implementation
gTsym,L-1n
s(Ln(L-1))
10
Pulse Shaping Filter Bank Example

• Pulse length 24 samples and L 4 samples/symbol
• Derivation in Tretter's manual,
• Define mth polyphase filter
• Four six-tap polyphase filters (next slide)

Six pulses contribute to each output sample
11
Pulse Shaping Filter Bank Example
24 samples in pulse
gTsym,0n
gTsym,0n
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
Polyphase filter 0 has only one non-zero sample.
12
Pulse Shaping Filter Bank Example
24 samples in pulse
gTsym,1n
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
13
Pulse Shaping Filter Bank Example
24 samples in pulse
gTsym,2n
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
Pulse Shaping Filter Bank Example
24 samples in pulse
gTsym,3n
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
15
Design Tradeoffs

• Direct pulse shaping

• Polyphase filter bank

16
Symbol Clock Recovery
Optional

• 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

17
Symbol Clock Recovery
Optional

• 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
Eak am a2 dk-m
Periodic with period Tsym
18
Symbol Clock Recovery
Optional

• Fourier series representation of E p(t)
• In terms of g1(t) and using Parsevals relation
• Fourier series representation of E z(t)

where
19
Symbol Clock Recovery
Optional

• 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 ½wsym
• H(w) is bandpass filter with center frequency wsym

20
Communication Systems
Optional

• Digital subscriber line (DSL) modems
• Business High-speed DSL
• Home Asymmetric DSL and Very High-Speed DSL
• Cable modems
• Wireless local area networks (Wi-Fi)
• Wireless metropolitan area networks (WiMax)
• Cell phones
• First generation (1G) AMPS
• Second generation (2G) GSM, IS-95 (CDMA)
• Third generation (3G) cdma2000, WCDMA
• A whole lot of communicating going on!

Analog
Digital