S-72.1140 Transmission Methods in
Telecommunication Systems (5 cr)
Digital Baseband Transmission
Digital Baseband Transmission
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2
Why to apply digital transmission?
Symbols and bits
Binary PAM Formats
Baseband transmission
– Binary error probabilities in baseband transmission
Pulse shaping
– minimizing ISI and making bandwidth adaptation - cos roll-off
signaling
– maximizing SNR at the instant of sampling - matched
filtering
– optimal terminal filters
Determination of transmission bandwidth as a function of pulse
shape
– Spectral density of Pulse Amplitude Modulation (PAM)
Equalization - removing residual ISI - eye diagram
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Why to Apply Digital Transmission?
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Digital communication withstands channel noise, interference
and distortion better than analog system. For instance in PSTN
inter-exchange STP*-links NEXT (Near-End Cross-Talk)
produces several interference. For analog systems interference
must be below 50 dB whereas in digital system 20 dB is
enough. With this respect digital systems can utilize lower
quality cabling than analog systems
Regenerative repeaters are efficient. Note that cleaning of
analog-signals by repeaters does not work as well
Digital HW/SW implementation is straightforward
Circuits can be easily configured and programmed by DSP
techniques
Digital signals can be coded to yield very low error rates
Digital communication enables efficient exchange of SNR to
BW-> easy adaptation into different channels
The cost of digital HW continues to halve every two or three
years
STP: Shielded twisted pair
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Transmitted power;
bandpass/baseband
signal BW
Information:
- analog:BW &
dynamic range
- digital:bit rate
Digital
Transmission
Information
source
Message
Maximization of
information
transferred
Message protection &
channel adaptation;
convolution, block
coding
Source
encoder
Message
estimate
Information
sink
Source
decoder
In baseband systems
these blocks are missing
Channel
Encoder
Channel
decoder
Interleaving
Deinterleaving
Modulator
Demodulator
Fights against burst
errors
M-PSK/FSK/ASK...,
depends on channel
BW & characteristics
Transmitted
signal
Channel
Noise
Interference

Received signal
(may contain errors)
wireline/wireless
constant/variable
linear/nonlinear
‘Baseband’ means that no carrier wave modulation is used for transmission
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Symbols and Bits – M-ary PAM
D (symbol rate r  1/ D baud)
n  log 2 M
Tb (bitrate rb  1/ Tb bits/s)
1
1
0
1
1
0
0
0
1
1
1
1
s (t )
M  2n
Generally:
s (t )   ak p (t  kD) (a PAM* signal)
k
For M=2 (binary signalling):
s (t )   ak p(t  kTb )
k
For non-Inter-Symbolic Interference (ISI), p(t) must
satisfy:
1, t  0
unipolar,
p (t )  
2-level pulses
0, t   D, 2 D...
n : number of bits
 M : number of levels


 D : Symbol duration
Tb : Bit duaration
This means that at the instant of decision, received signal
component is
s (t )   ak p (t  kD)  ak
k
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*Pulse Amplitude Modulation
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Binary PAM Formats (1)
Bit stream
Unipolar RZ and NRZ
Polar RZ and NRZ
Bipolar NRZ or
alternate mark inversion
(AMI)
Split-phase Manchester
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Binary PAM Formats (2)
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Unipolar RZ, NRZ:
– DC component
• No information, wastes power
• Transformers and capacitors in route block DC
– NRZ, more energy per bit, synchronization more difficult
Polar RZ, NRZ:
– No DC term if ´0´and ´1´ are equally likely
Bipolar NRZ
– No DC term
Split-phase Manchester
– Zero DC term regardless of message sequence
– Synchronization simpler
– Requires larger bandwidth
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Baseband Digital Transmission Link
Unipolar PAM
tD
original message bits
received wave y(t)
y (t )   ak p(t  td  kD)  n(t )
decision instances
k
message reconstruction at
tK  KD  td yields
y (t K )  ak   ak p ( KD  kD )  n(t )
k K
message
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ISI
Gaussian bandpass noise
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Baseband Unipolar Binary Error Probability
Assume binary & unipolar x(t)
The sample-and-hold circuit yields:
r.v. Y : y (tk )  ak  n(tk )
Establish H0 and H1 hypothesis:
H 0 : ak  0, Y  n
pY ( y | H 0 )  pN ( y )
and
H 1 : ak  1, Y  A  n
pY ( y | H 1 )  pN ( y  A)
pN(y): Noise probability density
function (PDF)
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Determining Decision Threshold
H 0 : ak  0, Y  n
pY ( y | H 0 )  pN ( y )
H 1 : ak  1, Y  A  n
pY ( y | H 1 )  pN ( y  A)
The comparator implements decision rule:
pe1  P(Y  V | H1 )   pY ( y | H1 )dy
V
Choose Ho (ak=0) if Y<V
Choose H1 (ak=1) if Y>V
peo  P(Y  V | H 0 )  V pY ( y | H 0 )dy

Average error error probability:
Pe  P0 Pe 0  PP
1 e1
P0  P1  1/ 2  Pe  12 ( Pe 0  Pe1 )
Channel noise is Gaussian with the pfd:
2
1
 x 
pN ( x ) 
exp   2 
 2
 2 
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Transmitted ‘0’
but detected as ‘1’
Error rate and Q-function
pe 0  V pN ( y)dy

2
1

 x 
pe 0 
V exp   2  dx
 2
 2 
This can be expressed by using the Q-function
Q(k )
by
2
1 
  
k exp   d 
 2
2

V 
pe 0  V pN ( y )dy  Q  
 
and also
V
 A V 
Pe1   pN ( y  A)dy  Q 

  
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Baseband Binary Error Rate
in Terms of Pulse Shape and g
setting V=A/2 yields then
 A
pe  12 ( pe 0  pe1 )  pe 0  pe1  pe  Q 

2



for unipolar, rectangular NRZ [0,A] bits
1
1 2
2
S R  ( A)  (0)  A2 / 2
2
2
Probability of occurrence
for polar, rectangular NRZ [-A/2,A/2] bits
1
1
2
S R  ( A / 2)  ( A / 2) 2  A2 / 4
2
2
and hence
2
A
A2  S R /(2 N R ), unipolar




 
 2  4 N R  S R / N R , polar
2
g b  Eb / N 0  S R / N 0 rb  A  2g b N 0 rb /(2 N 0 rb )  g b , unipolar


 
 2  2g b N 0 rb / N 0 rb  2g b , polar
 N R  N 0 rb / 2
Note that N R  N 0 BN  N 0 rb / 2 (lower limit with sinc-pulses)
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Pulse Shaping and Band-limited Transmission
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In digital transmission signaling pulse shape is chosen to satisfy
the following requirements:
– yields maximum SNR at the time instance of decision
(matched filtering)
– accommodates signal to channel bandwidth:
• rapid decrease of pulse energy outside the main lobe in
frequency domain alleviates filter design
• lowers cross-talk in multiplexed systems
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Signaling With Cosine Roll-off Signals
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Maximum transmission rate can be obtained with sinc-pulses
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However, they are not time-limited. A more practical choice is
the cosine roll-off signaling:
 p(t )  sinc  rt   sinc  t / D 


1 f
 P( f )  F [ p(t )]  r   r 
 

for raised cos-pulses =r/2
P ( f )  r / 2
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1
2  f
 cos
 ( f / 2r )
r
2r
cos 2 t
p (t ) 
sinc rt
2
1  (4  t )
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example
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By using   r / 2 and polar signaling, the following waveform
is obtained:
Note that the zero crossing are spaced by D at
t  0.5D, 1.5D, 2.5D,....

(this could be seen easily also in eye-diagram)
The zero crossing are easy to detect for clock recovery.
Note that unipolar baseband signaling involves performance
penalty of 3 dB compared to polar signaling:
 Q( g b ), unipolar [0 /1]
pe  
Q( 2g b ), polar[  1]
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g E /N S /N r
b
b
0
N R  N 0 rb / 2
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
R
0 b
Matched Filtering
Gn ( f ) 
 xR (t )  AR p(t  t0 )

 X R ( f )  AR P( f )exp( j t0 )
A  F 1[ H ( f ) X R ( f )] t t t
0
d
xR (t )

2
+
yD (t )
H(f)
Peak amplitude to be maximized

 A  H ( f ) P( f )exp  jt df
R
d


   H ( f ) G ( f )df 
2
2
n


2

2
 H ( f ) df


 H ( f ) P ( f ) exp  j td df
 A
2 
   AR
2

 
 H ( f ) df

2
Post filter noise
2
2
Using Schwartz’s inequality
Should be maximized

2

 H ( f )  KP( f )exp( jtd )  h(t )  Kp(td  t )
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
 V ( f )W * ( f )df   W ( f ) df  V ( f ) df



Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
2
2
Optimum terminal filters
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 Px : transmitting waveform
 H : transmitter shaping filter
 T

 H C : channel transfer function
 R : receiver filter
Assume
– arbitrary TX pulse shape x(t)
– arbitrary channel response hc(t)
– multilevel PAM transmission
What kind of filters are required for TX and RX to obtain matched
filtered, non-ISI transmission?
R( f )
Nyqvist shaped
pulse in y(t)
T( f )

The following condition must be fulfilled:
Px ( f ) H T ( f ) H C ( f ) R( f ) P( f )exp( j t d )
that means that undistorted transmission is obtained
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Avoiding ISI and enabling band-limiting in
radio systems
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Two goals to achieve: band limited transmission & matched filter
reception
data
TX
filt.
T( f )
RX
filt.
Decision
device
noise R( f )
T ( f )R ( f )  C N ( f ), raised-cos shaping

T ( f )  R *( f ), matched filtering

R( f )

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T( f )
CN ( f )
Hence at the transmitter and receiver
alike root-raised cos-filters
must be applied
raised cos-spectra CN(f)
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Determining Transmission Bandwidth for an
Arbitrary Baseband Signaling Waveform
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Determine the relation between r and B when p(t)=sinc2 at
First note from time domain that
1, t  0
sinc at  
ra
0, t  1/ a, 2 / a...
2

hence this waveform is suitable for signaling
There exists a Fourier transform pair
1 f
sinc 2 at    
a a

From the spectra we note that BT  a
and hence it must be that
for baseband
 BT  r
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
1/ a
a
f
a
PAM Power Spectral Density (PSD)

PSD for PAM can be determined by using a general expression
Amplitude autocorrelation
1
2 
Gx ( f )  P ( f )  Ra (n) exp(  j 2 nfD)
n 
D

For uncorrelated message bits
2
2
 a  ma , n  0
Ra (n)  
2
m
,n  0

a
Total power
DC power
and therefore


 Ra (n)exp( j 2 nfD)   n  ma  exp( j 2 nfD)
2
2
n
n
1  
n
exp(

j
2

nfD
)


f



on the other hand n

 and r  1/ D
n 
D
D



Gx ( f )   a r P( f )  ma r  P(nr )  ( f  nr )
n
2
20
2
2
2
2
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example

For unipolar binary RZ signal:
1
f
P( f ) 
sinc
2rb
2rb

Assume source bits are equally likely and independent, thus
ma  ak  A / 2, ak2  A2 / 2, a2  A2 / 4,
A2
f
A2 
2
2 n
 Gx ( f ) 
sinc

  ( f  nrb )sinc
n 
16rb
2rb 16
2
A  1 
rb  
4  2rb 
2
2
Gx ( f )   a2 r P ( f )

2
 ma r  P (nr )  ( f  nr )
n 
2
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
2
2
Equalization: Removing Residual ISI

Consider a tapped delay line equalizer with
pN (t )  cN p(t  2 ND)
p N (t )  c N p(t )
p0 (t )  c0 p(t  ND)

Search for the tap gains cN such that the output equals zero at
sample intervals D except at the decision instant when it should
be unity. The output is (think for instance paths c-N, cN or c0)
N
peq (t )   cn p(t  nD  ND)
n N
that is sampled at tk  kD  ND yielding
peq (tk )  peq (kD  ND)   cn p(kD  nD)   cn p  D(k  n)
n N
n N
N
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
N
Tapped Delay Line:
Matrix Representation

p (t )
peq (tk )   cn p  D(k  n)    cn p
n  N
n  N

N
p0 c n  p1c n1  ...  p2 n cn  0
p1c n  p0 c n1  ...  p2 n1cn  0
pn c n  pn1c n1  ...  p n cn  1
p2 n c n  p2 n1c n1  ...  p0 cn  0
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c n c n1
cn
k n
peq (t )
1, k  0

0, k  1, 2,...,  N
That leads into (2N+1)x(2N+1) matrix where (2N+1) tap
coefficients can be solved:
taps
time
p2 n
+
At the instant of decision:
N
p0 p1
 p0


 pN 1
p
 N
 pN 1


 p2 N
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
...
...
...
...
...
...
p2 N  c N  0 
   
   
p N 1   c1  0 
p N   c0   1 
 
p N 1   c1  0 
   
   
p0   cN  0 
Example of Equalization

p0
Read the distorted pulse values into
matrix from fig. (a)
0.1 0.0 c1  0
 1.0
 0.2 1.0 0.1  c   1 

 0   
 0.1 0.2 1.0   c1  0
p2
p1
p2
p 1
and the solution is
c1   0.096 
 c    0.96 
 0 

 c1   0.2 
Question: what does these
zeros help because they don’t
exist at the sampling instant?
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Zero forced values
Monitoring Transmission Quality
by Eye Diagram
Required minimum bandwidth is
BT  r / 2
Nyqvist’s sampling theorem:
Given an ideal LPF with the
bandwidth B it is possible to
transmit independent
symbols at the rate:
BT  r / 2  1/(2Tb )
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen