Multiplexing
•
Multiplexing is the name given to techniques, which allow more
than one message to be transferred via the same
communication channel. The channel in this context could be a
transmission line, e.g. a twisted pair or co-axial cable, a radio
system or a fibre optic system etc.
• A channel
will offer a specified bandwidth, which is available for
a time t, where t may  . Thus, with reference to the channel
there are 2 ‘degrees of freedom’, i.e. bandwidth or frequency
and time.
1
Multiplexing
CHANNEL
BL
BH
freq
BH
Multiplexing is a technique which allows k
users to occupy the
channel for the duration in time that the
channel is available.
BL
Frequency
Time t
Now consider a signal
vs (t )  Amp cos(t   )
The signal is characterised by amplitude, frequency, phase and time.
2
Multiplexing
• Various multiplexing methods are possible in terms of the channel bandwidth and time,
and the signal, in particular the frequency, phase or time. The two basic methods are:
1) Frequency Division Multiplexing FDM
FDM is derived from AM techniques in which the signals occupy the same physical
‘line’ but in different frequency bands. Each signal occupies its own specific band of
frequencies all the time, i.e. the messages share the channel bandwidth.
2) Time Division Multiplexing TDM
TDM is derived from sampling techniques in which messages occupy all the channel
bandwidth but for short time intervals of time, i.e. the messages share the channel time.
• FDM – messages occupy narrow bandwidth – all the time.
• TDM – messages occupy wide bandwidth – for short intervals of time.
3
Multiplexing
These two basic methods are illustrated below.
time
time
M1
BL
M2
B
BL
M3
M4
M1
M5
M4 M5
M2 M3
t
BH
BH
freq
freq
t
BH
BL
M1
M2
B
M1
M3
M2
M3
M4
M5
M4
M5
BL
BH
FDM
t
t
TDM
4
Frequency Division Multiplexing FDM
• FDM is widely used in radio and television systems (e.g.
broadcast radio and TV) and was widely used in
multichannel telephony (now being superseded by digital
techniques and TDM).
• The multichannel telephone system illustrates some
important aspects and is considered below. For speech,
a bandwidth of  3kHz is satisfactory.
• The physical line, e.g. a co-axial cable will have a
bandwidth compared to speech as shown next
5
Frequency Division Multiplexing FDM
3kHz
freq
GHz
From AM we have noted:
m(t)
freq
m(t)
DSBSC
carrier
cos( c t )
B
DSBSC
freq
fc
6
Frequency Division Multiplexing FDM
In order to use bandwidth more effectively, SSB is used i.e.
SSB
Filter
m(t)
SSBSC
carrier
cos( c t )
freq
fc
We have also noted that the message signal m(t) is usually band limited, i.e.
Speech
Band
Limiting
Filter
300Hz – 3400Hz
m(t)
SSB
Filter
cos( c t )
SSBSC
7
Frequency Division Multiplexing FDM
The Band Limiting Filter (BLF) is usually a band pass filter with a pass band 300Hz to
3400Hz for speech. This is to allow guard bands between adjacent channels.
f
f
300Hz
3400Hz
f
300Hz
3400Hz
10kHz
Speech
m(t)
Convention
8
Frequency Division Multiplexing FDM
For telephony, the physical line is divided (notionally) into 4kHz bands or channels, i.e.
the channel spacing is 4kHz. Thus we now have:
Guard Bands
Bandlimited
Speech
f
4kHz
Note, the BLF does not have an ideal cut-off – the guard bands allow for filter ‘roll off’
in order to reduce adjacent channel crosstalk.
9
Frequency Division Multiplexing FDM
Consider now a single channel SSB system.
DSBSC
m(t)
BLF
The spectra will be
SSB
Filter
SSBSC
fc
m(t)
300Hz
3400Hz
freq
DSBSC
freq
fc
freq
fc
10
Frequency Division Multiplexing FDM
Consider now a system with 3 channels
m1(t)
f
SSB
Filter
BLF
fc1
m2(t)
f1
SSB
Filter
BLF

FDM
Signal
M(t)
f
fc2
f2
SSB
Filter
BLF
m3(t)
fc3
f3
f
Bandlimited
FDM Transmitter
or Encoder
11
Frequency Division Multiplexing FDM
Each carrier frequency, fc1, fc2 and fc3 are separated by the channel spacing
frequency, in this case 4 kHz, i.e. fc2 = fc1 + 4kHz, fc3 = fc2 + 4kHz.
The spectrum of the FDM signal, M(t) will be:
4kHz
4kHz
M(t)
4kHz
Shaded areas are to
show guard bands.
f1
fc1
f3
f2
fc2
fc3
freq
12
Frequency Division Multiplexing FDM
Note that the baseband signals m1(t), m2(t), m3(t) have been multiplexed into adjacent
channels, the channel spacing is 4kHz. Note also that the SSB filters are set to select
the USB, tuned to f1, f2 and f3 respectively. A receiver FDM decoder is illustrated below:
SSB
Filter
f1
M(t)
FDM
Signal
LPF
fc1
SSB
Filter
f2
Band
Limited
LPF
m2(t)
Back to
baseband
fc2
SSB
Filter
f3
m1(t)
LPF
fc3
m3(t)
13
Frequency Division Multiplexing FDM
• The SSB filters are the same as in the encoder, i.e. each one
centred on f1, f2 and f3 to select the appropriate sideband and reject
the others. These are then followed by a synchronous demodulator,
each fed with a synchronous LO, fc1, fc2 and fc3 respectively.
• For the 3 channel system shown there is 1 design for the BLF (used
3 times), 3 designs for the SSB filters (each used twice) and 1
design for the LPF (used 3 times).
• A co-axial cable could accommodate several thousand 4 kHz
channels, for example 3600 channels is typical. The bandwidth used
is thus 3600 x 4kHz = 14.4Mhz. Potentially therefore there are 3600
different SSB filter designs. Not only this, but the designs must
range from kHz to MHz.
14
Frequency Division Multiplexing FDM
For ‘designs’ around say 60kHz, Q 
60kHz
= 15 which is reasonable.
4 kHz
However, for designs to have a centre frequency at around say 10Mhz,
10,000kHz
gives a Q = 2500 which is difficult to achieve.
Q
4 kHz
To overcome these problems, a hierarchical system for telephony used the FDM
principle to form groups, supergroups, master groups and supermaster groups.
15
Basic 12 Channel Group
The diagram below illustrates the FDM principle for 12 channels (similar to 3 channels)
to a form a basic group.
m1(t)
m2(t)
m3(t)
Multiplexer
freq
m12(t)
12kHz
60kHz
i.e. 12 telephone channels are multiplexed in the frequency band 12kHz  60 kHz in
4kHz channels  basic group.
16
Basic 12 Channel Group
A design for a basic 12 channel group is shown below:
Band Limiting Filters
SSB Filter
DSBSC
4kHz
CH1
m1(t)
8.6  15.4kHz
300Hz
12.3  15.4kHz
3400kHz
f1 = 12kHz
4kHz
12.6  19.4kHz
CH2
m2(t)
300Hz
16.3  19.4kHz
3400kHz
f1 = 16kHz

Increase in 4kHz steps
FDM OUT
12 – 60kHz
4kHz
52.6  59.4kHz
CH12
m12(t)
300Hz
56.3  59.4kHz
3400kHz
f12 = 56kHz
17
Super Group
These basic groups may now be multiplexed to form a super group.
12
Inputs
BASIC
GROUP
12 – 60kHz
SSB
FILTER
420kHz
12
Inputs
BASIC
GROUP
12 – 60kHz
SSB
FILTER
468kHz
12
Inputs
BASIC
GROUP
12 – 60kHz
SSB
FILTER

516kHz
12
Inputs
BASIC
GROUP
12 – 60kHz
SSB
FILTER
564kHz
12
Inputs
BASIC
GROUP
12 – 60kHz
SSB
FILTER
612kHz
18
Super Group
5 basic groups multiplexed to form a super group, i.e. 60 channels in one super group.
Note – the channel spacing in the super group in the above is 48kHz, i.e. each carrier
frequency is separated by 48kHz. There are 12 designs (low frequency) for one basic
group and 5 designs for the super group.
612kHz
 12 - which is reasonable
The Q for the super group SSB filters is Q 
48kHz
Hence, a total of 17 designs are required for 60 channels. In a similar way, super groups
may be multiplexed to form a master group, and master groups to form super master
groups…
19
Time Division Multiplexing TDM
TDM is widely used in digital communications, for example in the form of pulse code
modulation in digital telephony (TDM/PCM). In TDM, each message signal occupies
the channel (e.g. a transmission line) for a short period of time. The principle is
illustrated below:
1
1
m1(t)
2
m2(t)
m3(t)
m4(t)
m5(t)
m1(t)
2
3
Tx
4
5
3
Rx
SW2
SW1
Transmission
Line
m2(t)
4
5
m3(t)
m4(t)
m5(t)
Switches SW1 and SW2 rotate in synchronism, and in effect sample each message
input in a sequence m1(t), m2(t), m3(t), m4(t), m5(t), m1(t), m2(t),…
The sampled value (usually in digital form) is transmitted and recovered at the ‘far end’
to produce output m1(t)…m5(t).
20
Time Division Multiplexing TDM
For ease of illustration consider such a system with 3 messages, m1(t), m2(t) and m3(t),
each a different DC level as shown below.
m1(t)
V1
t
0
m2(t)
V2
0
m3(t)
t
V3
0
t
SW1
‘Sample’
t
Position
1
2
3
1
2
3
21
Time Division Multiplexing TDM
V3
V2
V1
t
m1(t)
m2(t)
m3(t)
m1(t)
1
2
3
1
m2(t)
m3(t)
m1(t)
Channel
Time
Slots
2
3
1
t
Time slot
22
Time Division Multiplexing TDM
•
In this illustration the samples are shown as levels, i.e. V1, V2 or V3.
Normally, these voltages would be converted to a binary code before
transmission as discussed below.
•
Note that the channel is divided into time slots and in this example, 3
messages are time-division multiplexed on to the channel. The sampling
process requires that the message signals are a sampled at a rate fs  2B,
where fs is the sample rate, samples per second, and B is the maximum
frequency in the message signal, m(t) (i.e. Sampling Theorem applies). This
sampling process effectively produces a pulse train, which requires a
bandwidth much greater than B.
•
Thus in TDM, the message signals occupy a wide bandwidth for short
intervals of time. In the illustration above, the signals are shown as PAM
(Pulse Amplitude Modulation) signals. In practice these are normally
converted to digital signals before time division multiplexing.
23
Time Division Multiplexing TDM
A schematic diagram to illustrate the principle for 3 message signals is shown below.
m1(t)
S/H
BLF
‘PAM’
1
fs1
m2(t)
S/H
BLF
‘PAM’
2
Multiplexing
Analogue
To
Digital
Convertor
Serial output
Binary digital
data d(t)
fs2
m3(t)
S/H
BLF
‘PAM’
3
fs3
Band limiting
Filter 0  B Hz
Sample and Hold
Sample rate fs
fs  2B Hz
Multiplexing ADC
Converts each input
in turn to an n bit code.
Again for simplicity, each message input is assumed to be a DC level.
24
Time Division Multiplexing TDM
25
Time Division Multiplexing TDM
26
Time Division Multiplexing TDM
• Each sample value is converted to an n bit code by the ADC. Each n bit code ‘fits into’
the time slot for that particular message. In practice, the sample pulses for each
message input could be the same. The multiplexing ADC could pick each input
(i.e. a S/H signal) in turn for conversion.
• For an N channel system, i.e. N message signals, sampled at a rate fs samples per
second, with each sample converted to an n bit binary code, and assuming no
additional bits for synchronisation are required (in practice further bits are required) it is
easy to see that the output bit rate for the digital data sequence d(t) is
Output bit rate = Nnfs bits/second.
27
School of Electrical, Electronics and
Computer Engineering
University of Newcastle-upon-Tyne
Baseband digital Modulation
Prof. Rolando Carrasco
Lecture Notes
University of Newcastle-upon-Tyne
2005
Baseband digital information
Bit-rate, Baud-rate and
Bandwidth
 B denotes the duration of the 1 bit
Hence Bit rate =
1
B
bits per second
All the forms of the base band signalling shown transfer data at the same bit rate.
E
denotes the duration of the shortest signalling element.
Baud rate is defined as the reciprocal of the duration of the shortest signalling element.
Baud Rate =
E
baud
Baud Rate ≠ Bit Rate
In general
For
1
NRZ :
RZ :
Bi-Phase:
AMI:
Baud Rate = Bit Rate
Baud Rate = 2 x Bit Rate
Baud Rate = 2 x Bit Rate
Baud Rate = Bit Rate
Non Return to Zero (NRZ)
The highest frequency occurs when the data is 1010101010…….
i.e.
This sequence produces a square wave with periodic time   2 E
Fourier series for a square wave,
If we pass this signal through a LPF then the maximum bandwidth would be 1/T
Hz, i.e. to just allow the fundamental (1st harmonic) to pass.
Non Return to Zero (NRZ)
(Cont’d)
The data sequence 1010……
could then be completely
recovered
Hence the minimum channel bandwidth
1
1 Baud Rate
1
Bmin  

Since
 Baud Rate
T 2 E
2
E
Return to Zero (RZ)
Considering RZ signals, the max frequency occurs when continuous 1’s are transmitted.
.
This produces a square wave with periodic time
Bmin
  2 E
Baud Rate
 fU 
2
If the sequence was continuous 0’s, the signal would be –V continuously, hence
f L  ' DC '
Bi-Phase
Maximum frequency occurs when continuous
1’s or 0’s transmitted.
This is similar to RZ with
Baud Rate =
1
E
= 2 x Bit rate
Baud Rate
2
The minimum frequency occurs when the sequence is 10101010…….
e.g.
Bmin  f U 
In this case
B = E
Baud Rate = Bit rate
Bmin  f L 
Baud Rate
2
Digital Modulation and
Noise
The performance of Digital Data Systems is dependent on the bit error rate, BER, i.e.
probability of a bit being in error.
Prob. of Error or BER,
No of Errors E
P
as N  
Total bits N
Digital Modulation
There are four basic ways of sending
digital data
The BER (P) depends on several factors
• the modulation type, ASK FSK or
PSK
• the demodulation method
• the noise in the system
• the signal to noise ratio
Digital Modulation and
Noise
Amplitude Shift Keying ASK
Digital Modulation and
Noise
Frequency Shift Keying FSK
Digital Modulation and
Noise
Phase Shift Keying PSK
System Block diagram for
Analysis
DEMODULATOR – DETECTOR – DECISION
For ASK and PSK
Demodulator-Detector-Decision
FOR FSK
Demodulator
Demodulator Cont’d)
1
VIN dt

RC
Hence design RC T
Vout  
Detector-Decision
V1 - V 0 is the voltage difference
between a ‘1’ and ‘0’.
(VREF 
 V1  V2

)
2
2
Detector-Decision (Cont’d)
ND is the noise at the Detector input.
Probability of Error,
1 

  1  erf
2 
2 2 ND
Hence




Probability density of binary signal
v0
0
v1
v
P(v0)
-
0 
vn
Probability density function of noise
P0 (vn ) 
1 1
e
2  2

 2  ND
( v0  v1 ) 2
2 2
P1 (vn )

Pe1 
vn
v1
v0

v0  v1
2
Using the change of variable

1
 2
x
e
( v n  v0 ) 2
2 2
v n  v0
2
dv n
(*)
This becomes
Pe1 

1

e
 x 2 dx
(**)
v1  v0
2 2
The incomplete integral cannot be evaluated analytically but can be recast as a
complimentary error function, erfc(x), defined by
erfc( z ) 
Equations (*) and (**) become
1
 v1  v0 
Pe1  erfc

2
 2 2 
2


e
z
 x2
dx
erfc( z )  1  erf ( z )
Pe1 
Pe 0 
1
 v1  v0 
1

erf



2
2

2


v0  v1
2



1
e
 2
( vn  v1 ) 2
2 2
dvn
It is clear from the symmetry of this problem that Pe0 is identical to Pe1 and the
probability of error Pe, irrespective of whether a ‘one’ or ‘zero’ was transmitted, can
be rewritten in terms of v = v1 – v0
1
 v 
Pe  1  erf 

2
 2 2 
for unipolar signalling (0 and v)
v
for polar signalling (symbol represented by voltage 
2
Detector-Decision (Cont’d)
ASK
1
e  1  erf
2
S IN
4 N IN
1
 e  1  erf
2
S IN
2 N IN
OOK
FSK
PSK
PRK
1
e  1  erf
2
S IN
N IN




 
 

 




 
 

 







For Optimum ASK , FSK , PSK
SNR in watt
10 SNR in dB/10
Linear gain
ASK
0
1.00
2
FSK
PSK
0.2398
0.1587
0.0786
1.5849
0.1867
0.104
0.0375
4
2.5119
0.1312
0.0565
0.0125
6
3.9811
0.0791
0.023
0.0024
8
6.3096
0.0379
0.006
0.0002
10
10.00
0.0127
0.0008
0
12
15.8489
0.0024
0
0
SNR in dB
Pe
Pe
Pe
Detector-Decision (Cont’d)
Probability of Symbol Error
1.00E+00
Probability of Symbol Error
1.00E-01
ASK
FSK
PSK
1.00E-02
1.00E-03
1.00E-04
0
2
4
6
8
SNR in dB
10
12
14
FM/ FSK Demodulation
One form of FM/FSK demodulator is shown below
In general VIN (t) will be
VIN (t )  Vc Cos IN t
Where  IN is the input frequency (rad/sec) IN  2  f IN 
V x  V IN t V IN t   
V x  Vc Cos  IN t .Vc Cos  IN (t   )
Since CosA CosB 
1
Cos A  B   Cos A  B 
2
Vc2
Cos  IN t      IN t   Cos  IN t      IN t 
Vx 
2
FM/ FSK Demodulation (Cont’d)
i.e
Vc2
Cos  IN t   IN    IN t   Cos  IN t   IN    IN t 
Vx 
2
Vc2
Cos 2  IN    IN t   Cos  IN  
Vx 
2
Thus there are two components
Vc2
Cos 2  IN t  
           (1)
2
2
Vc2
and
Cos IN t
           ( 2)
2




Component (1) is at frequency 2 fIN Hz and component (2) is effectively a ‘DC’ voltage if
 IN is constant.
The cut-off frequency for the LPF is designed so that component (1) is removed and
component (2) is passed to the output.
VOUT
Vc2

CosIN t
2
FM/ FSK Demodulation (Cont’d)
The V/F characteristics and inputs are shown below
Analogue FM
f c   Vm
ym xc
f out   VIN  f 0
VIN VDC  m(t )
VIN VDC  Vm Cos mt
i.e. f out   VDC   Vm Cos mt  f 0
f c   VDC ,
Tc 
Modulation Index  
1
fc
f c Vm

fm
fm
FM/ FSK Demodulation (Cont’d)
The spectrum of the analogue FM signal depends on


and is given by
FM  Vs (t )  Vc  J n (  ) Cos c  n  m t
n 1
Digital FSK
ym xc
f out   V IN  f 0
V IN V DC  m(t )
V IN V DC  V1
for 1' s
V IN V DC  V0
for 0' s
f 1   V DC   V1  f 0
for 1' s
f 0   V DC   V0  f 0
for 0' s
f c   V DC ,
Tc 
1
fc
Normalized frequency Deviation ratio
h
f1  f 0
Rb
i.e. Modulus f1  f 0
The spectrum of FSK depends on h
Digital FSK (Cont’d)
FM/ FSK Demodulation (Cont’d)
Consider again the output from the demodulator VOUT
The delay

Hence
VOUT
is set to Tc
4
1
where Tc 
fc
 2  f IN
Vc2

Cos 
2
 4 fc



VOUT
and
Vc2

Cos IN 
2
fc
is the nominal carrier frequency
  f IN
Vc2

Cos 
2
 2 fc



FM/ FSK Demodulation (Cont’d)
The curve shows the demodulator F/V characteristics which in this case is non linear.
Practical realization of F/V process
The comparator is LIMITER – which is a zero crossing detector to give a ‘digital’ input to
the first gate.
This is form of ‘delay and multiply’ circuit where the delay

= CR

is set by C and R with
Practical realization of F/V process (Cont’d)
Practical realization of F/V process (Cont’d)
Consider now
f IN ≠ f c
Practical realization of F/V process (Cont’d)
VOUT
AE f IN

4 fc
Plotting Vout versus
f IN
(Assuming A=1)