S-72.1140 Transmission Methods in
Telecommunication Systems (5 cr)
Digital Bandpass Transmission
Digital Bandpass Transmission

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
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
2
CW detection techniques
– Coherent
– Non-coherent
– Differentially coherent
Examples of coherent and non-coherent detection error rate
analysis (OOK)
A method for ‘analyzing’ PSK error rates
Effect of synchronization and envelope distortion (PSK)
Comparison: Error rate describing
– reception sensitivity Pe  f1 ( Eb / N 0 )
– bandwidth efficiency Pe  f 2 (rb / BT )
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Overview
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Bandpass Transmission
Link
Transmitted power;
bandpass/baseband
signal BW
Information:
- analog:BW &
dynamic range
- digital:bit rate
Maximization of
information
transferred
Message protection &
channel adaptation;
convolution, block
coding
Information
source
Message
Message
estimate
Source
encoder
Information
sink
Source
decoder
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
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Received signal
(may contain errors)
wireline/wireless
constant/variable
linear/nonlinear
CW Binary Waveforms
ASK
FSK
PSK
DSB
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Carrier Wave Communications



Carrier wave modulation is used to transmit messages over a
distance by radio waves (air, copper or coaxial cable), by optical
signals (fiber), or by sound waves (air, water, ground)
CW transmission allocates bandwidth around the applied
carrier that depends on
– message bandwidth and bit rate
– number of encoded levels (word length)
– source and channel encoding methods
Examples of transmission bandwidths for certain CW
techniques:



MPSK, M-ASK BT  r  rb / n  rb / log 2 M ( M  2n )
Binary FSK (fd=rb/2) BT  rb
MSK (CPFSK fd=rb/4), QAM: BT  rb / 2
FSK: Frequency shift keying
CPFSK: Continuous phase FSK
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Spectral Shaping by
Data Pre-filtering

Continuous Phase Modulator (CPM)
data
filter
modulated
signal
Making phase changes continuos in time domain by filtering
(before modulator) results spectral narrowing & some ISI - often
so small that making BT smaller is more important
raised cos
Gaussian MSK
tamed FM
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VCO
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
CW Detection Types
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
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8
Number of allocated signaling levels determines constellation
diagram (=lowpass equivalent of the applied digital modulation
format)
At the receiver, detection can be
– coherent (carrier phase information used for detection)
– non- coherent (no carrier phase used for detection)
– differentially coherent (‘local oscillator’ synthesized from
received bits)
CW systems characterized by bit or symbol error rate (number
of decoded errors(symbols)/total number of bits(symbols))
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Coherent Detection by Integrate and Dump /
Matched Filter Receiver


Coherent detection utilizes carrier phase information and requires inphase replica of the carrier at the receiver (explicitly or implicitly)
It is easy to show that these two techniques have the same
performance:
y (t )
h(t )  s (  t )
v (T )
v(t )  s (  t )  y (t )
 0 s (t   ) y ( )d

y (t )
v (T )
ss(t t ))
v(t )  0 s(t   ) y( )d

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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Non-coherent Detection
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

Base on filtering signal energy on allocated spectra and using
envelope detectors
Has performance degradation of about 1-3 dB when compared
to coherent detection (depending on Eb/N0)
Examples:
2-ASK
2-FSK
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Differentially coherent PSK (DPSK)

This methods circumvents usage of coherent local oscillator and
can achieve almost the same performance as PSK:

After the multiplier the signal is
xC (t )  2 xC (t  Tb )  2 AC2 cos( ct    ak )
 cos  C (t  Tb )    ak 1 
 AC2 cos(ak  ak 1 )
 cos  2 C t  2  (ak  ak 1 ) 
and the decision variable after the LPF is
2
 AC , ak  ak 1
z (tk )   2
 AC , ak  ak 1
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Differential Encoding and Decoding

Differential encoding and decoding:
mk

start, say with ak  1
ak
if mk  1,set ak  ak 1

if mk  0,set ak  ak 1
Decoding is obtained by the simple rule:
d k  ak 1  ak


12
that is realized by the circuit shown
right.
Note that no local oscillator is required
How would you construct the encoder?
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
XOR
ABY
001
010
100
111
Coherent Detection
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example: Optimum Binary Detection
m  0,1
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

Received signal consist of bandpass filtered signal and noise
that is sampled at the decision time instants tk yielding decision
variable: Y  y (tk )  zm  n
Quadrature presentation of the signaling waveform is
sm (t )  AC I k pi (t )cos(C t )  Qk pq (t )sin(C t )
Assuming that the BPF has the impulse response h(t), signal
component at the sampling instants is then expressed by
zm  sm (t  kTb )  h(t ) t t
k


Tb
0
sm ( )h(Tb   )d 

( k 1) Tb
kTb
sm (  kTb )h(tk   )d 
( x  y (t ) 
 x( ) y(t   )d  )
A
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Optimum Binary Detection - Error Rate

Assuming ‘0’ and ‘1’ reception is equally likely, error happens
when H0 (‘0’ transmitted) signal hits the dashed region or for H1
error hits the left-hand side of the decision threshold that is at
Vopt  ( z1  z0 ) / 2
x
Q

  
For optimum performance
we have the maximized
SNR that is obtained
by matched filtering/
integrate and dump receiver
 z1  z0 / 2 
Errors for ‘0’ or/and ‘1’ are equal alike, for instance for ‘0’:
1
pe 0 
 2

 Vopt  z0 
2
2


exp    z0  / 2  d   Q 




Vopt

pe  ( pe 0  pe1 ) / 2  Q  z1  z0 / 2 
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
2
Optimum Binary Detection (cont.)
Express energy / bit embedded in signaling waveforms by


Tb
0
 s (  )  s (  )  d    s ( ) d 
Tb
2
1
2
0
0
1
E1


Tb
0
s0 ( )d   2
2

Tb
0
s0 ( ) s1 ( )d 
E0
pe  Q  z1  z0 / 2 

z1  z0
4 2
E10
Note that the signaling waveform
correlation greatly influences the SNR!
Therefore, for coherent CW we have the SNR and error rate
2
 2  / 2
 E1  E0  2 E10
E1  E0  2 E10

 pe  Q 
2
2

SNRmax 
16

 Eb  E10 
  Q

 


Eb
E
 b  No   2   / 2
No  / 2
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Timing and Synchronization


s (t )
h(t )
z (t )
Performance of coherent detection is greatly
dependent on how successful local carrier recovery is
Consider the bandpass signal s(t) with a rectangular pulses
pTb(t), that is applied to the matched filter h(t):
s (t )  AC pTb (t ) cos( C t ),
 z ( )  s(t )  h(t )
h(t )  Ks (Tb  t )  KpTb (Tb  t ) cos( C t    )
   Tb 
2
 KA  
cos
 / Tc   1/ 2


 Tb 
   Tb 
 KA  

T
 b 
  1/ 2  cos  c  1/ 2 
  s (  )  s (  )  d    s ( ) d    s (  ) d   2  s (  ) s (  ) d 
Tb
0
2
1
0
Tb
Tb
2
0
1
0
E1
17
Tb
2
0
0
E0
0
1
E10
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Tb
E   s 2 (t )dt
0
tk
Tc
 AC2Tb / 2

Therefore, due to phase mismatch
at the receiver, the error rate is
degraded to
Analyzing phase
error by Mathcad
 Eb  E10

2
pe  Q 
cos   



    c
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example

Assume data rate is 2 kbaud/s and carrier is 100 kHz for an
BPSK system. Hence the symbol duration and carrier period are
TS  1/ 2kbaud/s = 0.5ms
TC  1/ fC  10  s
therefore the symbol duration is in radians
10 s 2

 x  314.2rad(or carrier cycles)
0.5ms x

Assume carrier phase error is 0.3 % of the symbol duration.
Then the resulting carrier phase error is
  0.003x  0.94 rad  54
o
and the error rate for instance for   8  9dB is
pe  Q( 16cos 2 54)  102

that should be compared to the error rate without any phase
errors or
pe  Q( 16)  3 105
Hence, phase synchronization is a very important point to
remember in coherent detection
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example: Coherent Binary On-off Keying (OOK)
For on-off keying (OOK) the signaling waveforms are

s1 (t )  AC pTb (t )cos C t , s0 (t )  0
and the optimum coherent receiver can be sketched by
SR
SR
S RTb Eb
b 



W  (1/ Tb ) 

E1 

Tb
0
s1 ( )d   ACTb / 2, E0  0, E10 
2
2

Tb
0
s1 ( ) s0 ( )d   0
 Eb  E10
Eb  ( E0  E1 ) / 2  ACTb / 4 pe  Q 


2
20

 Eb 
  Q
Q

  
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
 
b
011
111
nq
Error rate for M-PSK

001
d
000
101
010
100

In general,PSK error rate can be expressed by
 d
pe  nnQ 
 2

a

n
Q
 n  

 
110
decision region
where d is the distance between constellation points (or a=d/2 is
the distance from constellation point to the decision region
border) and nn is the average number of constellation points in
the immediate neighborhood. Therefore
 d
pe  2Q 
 2

 2 A sin( / 2) 
A


2
Q

2
Q
sin(

/
M
)





2







Note that for matched filter reception
A

21

2E

, E  nEb  log 2 ( M ) Eb
A: RMS signal voltage
at the moment of sampling
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
M  2n
2

M
Error rate for M-QAM, example 16-QAM
 d
pe  nnQ 
 2

a
  nnQ  

 
4 4  83  4 2
nn 
3
48 4
4  2a 2  8 10a 2  4 18a 2
2
A 
 10a 2
16
 A2
a
pe  3Q    3Q 
 10 2
 

2  3a 
2
 18a 2
a2
2a 2
 3a   a
2
2
 10a 2

 2E 
  3Q 
 symbol error rate

 10 N 0 

Constellation follows from 4-bit words and therefore
3  2E 
3  4 Eb 
pb  Q 
 Q


4  10 N 0 
4  5N0 
E 4 Eb
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
n  log 2 M , E  nEb

 pe  p( E ) / n

 A /   2E / N0
Envelope distortion and QPSK



QPSK is appealing format, however requires constant envelope
Passing constellation figure via (0,0) gives rise to envelope -> 0
Prevention:
– Gray coding
– Offset - QPSK
– Pi/4 QPSK
Delay I and Q channels
by symbol time
Apply two /4 offset
QPSK constellations by turns
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Non-coherent Detection
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example: Non-coherent On-off Keying (OOK)


Bandpass filter is matched to the signaling waveform (not to
carrier phase), in addition fc>>fm, and therefore the energy for ‘1’
2
is simply E1  Tb ( AC / 2)
Envelopes follow Rice and Rayleigh distributions for ‘1’ and ‘0’
respectively:
distribution for ”0”"
distribution for "1"
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Non-coherent Binary Systems: Noisy Envelopes

AWGN plus carrier signal have the envelope whose probability
distribution function is
– For nonzero, constant carrier component Ac, Rician
distributed:
2
2
 x  AC   xAC 
pY (Y | H1 )  2 exp  
 I0  2  , x  0
2

 2    
x
– For zero carrier component Rayleigh distributed:
2
 x 
pY ( y | H 0 )  2 exp   2  , x  0

 2 
x

For large SNR (AC >> ) the Rician envelope simplifies to
  x  AC 
x
pY (Y | H1 ) 
exp  
2
2
2 AC
2



26
2

, x  0

Therefore in this case the received envelope is then essentially
Gaussian with the variance  2 and mean equals pA ( x)  AC
Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Envelope Distributions with different
Carrier Component Strengths
Rayleigh distribution
Rice distribution
A  AC
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Noncoherent OOK Error Rate
The optimum threshold is at the intersection of Rice and
Rayleigh distributions (areas are the same on both sides)
Usually high SNR is assumed and hence the threshold is
approximately at the half way and the error rate is the average
of '0' and '1' reception probabilities


Pe  1

2
Pe 0  Pe1 
probability to detect "0" in error
2
2
P 
p
(
Y
|
H
)
dy

exp

A
/
8


  exp   b / 2 
0
C
 e 0 A/2 Y
probability to detect "1" in error

A /2
P 
0 pY (Y | H1 )dy  Q( AC / 2 )  Q(  b )
 e1
C
C

Therefore, error rate for non-coherent OOK equals
Pe  1 exp( b / 2)  Q(  b )   1 exp( b / 2),  b  1
2
2
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Comparison
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Error Rate Comparison
a: Coherent BPSK
b: DPSK
c:Coherent OOK
d: Noncoherent FSK
e: noncoherent OOK
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Comparison of Quadrature Modulation
Methods
Note that still the performance is good, envelope is not
constant. APK (or M-QASK) is used in cable modems
(pe=10-4)
APK=MQASK
M-APK: Amplitude Phase Shift Keying
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Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen