ELEC3028 Digital Transmission – MODEM
S Chen
Digital Communication System
• Purpose: communicate information at certain rate between geographically
separated locations reliably (quality)
Important point: rate, quality ↔ spectral bandwidth requirement
• Major components: CODEC, MODEM and channel (transmission medium)
input
source
encoding
channel
encoding
modulation
channel
output source
decoding
channel
decoding
CODEC
demodulation
MODEM
Medium
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ELEC3028 Digital Transmission – MODEM
S Chen
Digital Communication System (continue)
• A pair of transmitter (coder, modulator) and receiver (demodulator, decoder) is
called transceiver
• Information theory provides us basic communication theory for communication
system design, including CODEC and MODEM
• Detailed practical CODEC design, including source coding and channel coding, will
be covered latter by the other lecturer
• This part considers MODEM (modulation/demodulation)
• The purpose of MODEM: transfer the bit stream at certain rate over the
communication medium reliably
• Why carrier communication (modulation): low frequency signal cannot travel far,
also most spectral resource (channels) are in RF
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ELEC3028 Digital Transmission – MODEM
S Chen
Digital Modulation
• In the old day, communications were analogue, analogue modulation techniques
include amplitude modulation (AM), frequency modulation (FM), and phase
modulation (PM)
• Communications today are mostly all digital, equivalent digital modulation forms
exist: amplitude shift keying (ASK), frequency shift keying (FSK), or phase shift
keying (PSK)
Sin waveform A sin (2πfc t + θ): amplitude A, frequency fc , phase θ → three kinds of modulation
• A large number of other digital modulations are in use, and often combinations are
employed
• We will consider quadrature amplitude modulation (QAM), which is a combination
of ASK and PSK
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ELEC3028 Digital Transmission – MODEM
S Chen
Quadrature Amplitude Modulation
cos (ω t)
x i ( k)
bit
stream
const.
map x ( k)
q
s/p
q
g( t )
s (t )
g( t )
Σ δ( t-kT s )
QAM symbol
generation
x i (t )
D/A conversion
x q( t )
sin (ω t)
QAM modulation
Note: e.g., odd bits go to form xi (k) and even bits to form xq (k); xi (k) and xq (k) are in-phase and quadrature components of the
xi (k) + jxq (k) QAM symbol; xi (k) and xq (k) are M -ary symbols
D/A conversion is not “correct full name”, should be called transmit filter, part of pulse shaping filter pair
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ELEC3028 Digital Transmission – MODEM
S Chen
Quadrature Amplitude Demodulation
cos (ω t)
LP
x i (t )
g(-t )
s (t )
LP
sin (ω t)
QAM demodulation
x q( t )
g(-t )
x i ( k)
const.
xq ( k) map
p/s
bit
stream
q
Σ δ( t-kT s )
symbol detection
bit recovery
Note: in-phase and quadrature branches are “identical”; many issues, such as design
of Tx/Rx filters g(t)/g(−t), carrier recovery, synchronisation, can be studied using
one branch
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ELEC3028 Digital Transmission – MODEM
S Chen
Channel (Medium) I
• Between modulator and demodulator is medium (channel)
• Passband channel and baseband (remove modulator/demodulator) equivalence:
Hb(f)
−B
Hp(f)
carrier modulation
B
f
f
−fc
fc
2B
Baseband channel bandwidth B ↔ passband channel bandwidth 2B
• Communication is at passband channel but
for analysis and design purpose one can
consider equivalent baseband channel
• Channel has finite bandwidth, ideally phase
is linear and amplitude is flat:
phase
amplitude
f
channel bandwidth
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ELEC3028 Digital Transmission – MODEM
S Chen
Channel (Medium) II
• Bandwidth is a prime consideration, and another consideration is noise level
• Channel noise: AWGN with a constant power
spectrum density (PSD)
N0 /2
• Power is the area under PSD, so WN has infinitely
large power
f
0
• But communication channels are bandlimited, so noise is also bandlimited and has
a finite power:
n(t)
n(t)
Tx filter
x(k)
channel
x(t)
Σ
B
Rx filter
y(t)
y(k)
nB(t)
Σ
channel
y(t)
y(k)
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ELEC3028 Digital Transmission – MODEM
S Chen
Pulse Shaping I
• Unless transmission symbol rate fs is very low, one cannot use impulse, narrow pulse or rectangular
pulse to transmit data symbols, and discrete samples have to be pulse shaped
– {x[k]}: transmitted symbols
P
–
δ(t − kTs ): pulse clock (every Ts s a symbol is transmitted)
– r(t): combined impulse response of Tx/Rx filters, and channel
r(t) = g(−t) ⋆ c(t) ⋆ g(t) or R(f ) = GR (f ) · C(f ) · RT (f )
x [k ]
r (t )
x (t )
Σ δ( t-kT )
– Baseband (received) signal, assuming no noise
Z X
”
“X
x[k]δ(t − kTs ) =
r(t − τ ) · x[k]δ(τ − kTs ) dτ
x(t) = r(t) ⋆
=
+∞
X
x[k] · r(t − kTs )
k=−∞
• A number of choices for r(t) would allow to retrieve the original data sample x[k] from x(t):
what are the requirements for r(t)?
• To transmit at symbol rate fs needs certain bandwidth BT and BT depends on which pulse shaping
used — does the channel bandwidth B enough to accommodate BT ?
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ELEC3028 Digital Transmission – MODEM
S Chen
Pulse Shaping II — Time Domain
sinc
square pulse
raised cosine
filter impulse responses
1
0.8
0.6
0.4
0.2
0
−0.2
−10
−8
−6
−4
−2
0
time t/T
2
4
6
8
10
s
• sinc: assume t → ±∞; square: last one Ts; and raised cosine: truncate to 8 Ts s
• All these filters have regular zero-crossing at symbol-rate spacing except t = 0 (Nyquist system),
but they have different time supports
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ELEC3028 Digital Transmission – MODEM
S Chen
Pulse Shaping III — Frequency Domain
sinc
square pulse
raised cosine
filter magnitude responses / [dB]
0
−10
−20
−30
−40
−50
−60
−70
0
0.5
1
1.5
2
2.5
3
frequency 2f/f s
3.5
4
4.5
5
• Square pulse produces considerable excess bandwidth beyond the symbol rate fs;
sinc impractical to realize; truncated raised cosine easy to realize
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ELEC3028 Digital Transmission – MODEM
S Chen
Pulse Shaping IV
• Example: binary (±1) x[k], each is transmitted as a sinc pulse; the peak of different
shifted sinc functions coincide with zero crossings of all other sincs:
1.5
1
x(t)
0.5
0
−0.5
−1
−1.5
−5
−4
−3
−2
−1
0
time t/T
1
2
3
4
5
• At receiver, sampling at correct symbol rate enables recovery of transmitted x[k]
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ELEC3028 Digital Transmission – MODEM
S Chen
Transmit and Receive Filters
• Pulse shaping fulfils two purposes: limit the transmission bandwidth, and enable
to recover the correct sample values of transmitted symbols; such a pulse shaping
r(t) is called a Nyquist system
1. (Infinite) sinc has a (baseband) bandwidth BT = fs/2, (infinite) raised cosine
has fs/2 ≤ BT ≤ fs depending on roll-off factor
2. A Nyquist time pulse have regular zero-crossing at symbol-rate spacings to avoid
interference with neighboring pulses at correct sampling instances
• Nyquist system r(t) is separated into transmit filter g(t) and receive filter g(−t)
(square-root Nyquist systems)
1. The filter g(−t) in the receiver is also called a matched Filter (to g(t)); g(t) and
g(−t) are basically identical (square-root of r(t))
2. This division of r(t) enables suppression of out-of-band noise and results in the
maximum received SNR
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ELEC3028 Digital Transmission – MODEM
S Chen
Summary
• Revisit major blocks of a digital communication system
• MODEM: responsible for transferring the bit stream at a given rate over the
communication medium reliably
• Transmission channel (medium) has finite bandwidth and introduces noise, these
are two factors that has to be considered in design
• Purpose of pulse shaping, how to design transmit and receive filters
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