2. Dr. Uri Mahlab 2
Information
source
Transmitter Channel Receiver Decision
Communication system
3. Dr. Uri Mahlab 3
Information
source
Pulse
generator
Trans
filter
channel
(X(t
(XT(t
Timing
Receiver
filter
Clock
recovery
network
A/D
+
Channel noise
n(t)
Output
Block diagram of an Binary/M-ary signaling
scheme
+
HT(f)
HR(f)
Y(t)
Hc(f)
4. Dr. Uri Mahlab 4
Information
source
Pulse
generator
Trans
filter
(X(t (XT(t
Timing
Block diagram Description
HT(f)
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
Tb
Tb
)
t
(
pg
Tb
)
t
(
pg
Tb
For bk=1
For bk=0
;
"
0
"
a
;
"
1
"
a
ak
k
k
d
if
d
if
5. Dr. Uri Mahlab 5
Information
source
Pulse
generator
Trans
filter
(X(t (XT(t
Timing
Block diagram Description (Continue - 1)
HT(f)
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
Tb
Tb
)
t
(
pg
Tb
For bk=1
For bk=0
Transmitter
filter
Tb
)
t
(
pg
Tb
6. Dr. Uri Mahlab 6
Information
source
Pulse
generator
Trans
filter
(X(t (XT(t
Timing
Block diagram Description (Continue - 2)
HT(f)
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
Tb
k
b
g
k kT
t
p
a
t
X )
(
)
(
Tb
100110
2Tb
3Tb 4Tb
5Tb
t
6Tb
7. Dr. Uri Mahlab 7
Information
source
Pulse
generator
Trans
filter
(X(t (XT(t
Timing
Block diagram Description (Continue - 3)
HT(f)
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
Tb
k
b
g
k kT
t
p
a
t
X )
(
)
(
Tb
100110
2Tb
3Tb 4Tb
5Tb
t
6Tb
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
8. Dr. Uri Mahlab 8
Information
source
Pulse
generator
Trans
filter
(X(t
Timing
Block diagram Description (Continue - 4)
HT(f)
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
Channel noise n(t)
+
t
Receiver
filter
HR(f)
9. Dr. Uri Mahlab 9
Block diagram Description (Continue - 5)
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
t
k
b
d
r
k t
n
kT
t
t
p
A
t
Y )
(
)
(
)
( 0
10. Dr. Uri Mahlab 10
Information
source
Pulse
generator
Trans
filter
channel
(X(T (Xt(T
Timing
Receiver
filter
Clock
recovery
network
A/D
+
Channel noise
n(t)
Output
Block diagram of an Binary/M-ary signaling
scheme
+
HT(f)
HR(f)
Y(t)
Hc(f)
k
b
d
r
k t
n
kT
t
t
p
A
t
Y )
(
)
(
)
( 0
11. Dr. Uri Mahlab 11
Block diagram Description
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
t
1 0 0 0 1 0
1 0 0 1 1 0
t
12. Dr. Uri Mahlab 12
Information
source
Pulse
generator
Trans
filter
channel
(X(t
(XT(t
Timing
Receiver
filter
Clock
recovery
network
A/D
+
Channel noise
n(t)
Output
Block diagram of an Binary/M-ary signaling
scheme
+
HT(f)
HR(f)
Y(t)
Hc(f)
13. Dr. Uri Mahlab 13
Trans
filter
channel
Pg(t)
Receiver
filter
Explanation of Pr(t)
HT(f) HR(f)
Hc(f)
k
b
d
r
k t
n
kT
t
t
p
A
t
Y )
(
)
(
)
( 0
Pr(t)
HT(f) Hc(f) HR(f)
Pg(f) Pr(f)
1
)
0
(
pr
14. Dr. Uri Mahlab 14
The output of the pulse generator X(t),is given by
T
p
a b
g
k
k
k
t
t
X
Pg(t) is the basic pulse whose amplitude ak depends on
.the kth input bit
15. Dr. Uri Mahlab 15
For tm =mTb+td and td is the total time delay in the
system, we get.
t
t
t
tm
Y
t1
t2
t3
tm
The input to the A/D converter is
k
b
d
r
k t
n
kT
t
t
p
A
t
Y )
(
)
(
)
( 0
16. Dr. Uri Mahlab 16
t
n
T
p
A
A
t m
0
b
r
m
K
k
m
m
k
m
Y
t
tm
Y
t1
t2
t3
tm
The output of the A/D converter at the sampling time
tm =mTb+td
k
b
d
r
k t
n
kT
t
t
p
A
t
Y )
(
)
(
)
( 0
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
17. Dr. Uri Mahlab 17
t
n
T
p
A
A
t m
0
b
r
m
K
k
m
m
k
m
Y
t
tm
Y
t1
t2
t3
tm
ISI - Inter Symbol
Interference
18. Dr. Uri Mahlab 18
Trans
filter
channel
Pg(t)
Receiver
filter
Explanation of ISI
HT(f) HR(f)
Hc(f)
Pr(t)
Pg(f) Pr(f)
Trans
filter
channel
Receiver
filter
HT(f) HR(f)
Hc(f)
t
f
Fourier
Transform
BandPass
Filter
f
Fourier
Transform
t
19. Dr. Uri Mahlab 19
Explanation of ISI - Continue
t
f
Fourier
Transform
BandPass
Filter
f
Fourier
Transform
t
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
20. Dr. Uri Mahlab 20
-The pulse generator output is a pulse waveform
k
b
g
k kT
t
p
a
t
X )
(
)
(
a
a
a
p
k
g 1
)
0
(
If kth input bit is 1
if kth input bit is 0
k
b
d
r
k t
n
kT
t
t
p
A
t
Y )
(
)
(
)
( 0
-The A/D converter input Y(t)
21. Dr. Uri Mahlab 21
Data
rate
Error
rate
Transmitted
power
Noise
power
Noise
Spectral
density
System
complexity
TypeofImportantParametersInvolvedInTheDesign
OfaPAMSystem
22. Dr. Uri Mahlab 22
5.2 BASEBAND BINARY PAM SYSTEMS
Pulse shapes
pg(t)
pr(t) HR(f) HT(t)
Design of a baseband
binary PAM system
- minimize the combined effects of inter symbol
interference and noise in order to achieve minimum
probability of error for given data rate.
23. Dr. Uri Mahlab 23
0
n
for
0
0
n
for
1
)
nT
(
p b
r
5.2.1 Baseband pulse shaping
The ISI can be eliminated by proper choice
of received pulse shape pr (t).
Doe’s not Uniquely Specify Pr(t) for all
values of t.
24. Dr. Uri Mahlab 24
0
n
for
0
0
n
for
1
)
nT
(
p
Then
T
2
/
1
f
for
T
)
T
k
f
(
P
if
b
r
k
b
b
b
r
k
T
2
/
)
1
k
2
(
T
2
/
)
1
k
2
(
r
r
r
r
b
b
df
)
ft
2
j
exp(
)
f
(
p
)
t
(
p
df
)
ft
2
j
exp(
)
f
(
p
)
t
(
p
Theorem
Proof
To meet the constraint, Fourier Transform Pr(f) of Pr(t), should
satisfy a simple condition given by the following theorem
25. Dr. Uri Mahlab 25
b
b
b
b
T
2
/
1
T
2
/
1
k b
b
r
b
r
k
T
2
/
1
T
2
/
1
b
b
r
b
r
df
)
fnT
2
j
exp(
))
T
k
f
(
p
(
)
nT
(
p
'
df
)
nT
'
f
2
j
exp(
)
T
k
'
f
(
p
)
nT
(
p
k
T
k
T
k
b
r
b
r
b
b
df
t
fnT
j
f
p
nT
p
2
/
)
1
2
(
2
/
)
1
2
(
)
2
exp(
)
(
)
(
b
b
T
2
/
1
T
2
/
1
b
b
b
r
n
)
n
sin(
df
)
fnT
2
j
exp(
T
)
nT
(
p
Which verify that the Pr(t) with a transform Pr(f)
Satisfy ZERO ISI
26. Dr. Uri Mahlab 26
The condition for removal of ISI given in the theorem is called
Nyquist (Pulse Shaping) Criterion
m
k
m
0
b
r
k
m
m )
t
(
n
)
T
)
k
m
((
P
A
A
)
t
(
Y
0
n
for
0
0
n
for
1
)
nT
(
p b
r
Tb 2Tb
-Tb
-2Tb
1
n
)
n
sin(
)
nT
(
p b
r
27. Dr. Uri Mahlab 27
The Theorem gives a condition for the removal of ISI using a Pr(f) with
a bandwidth larger then rb/2/.
ISI can’t be removed if the bandwidth of Pr(f) is less then rb/2.
HT(f) Hc(f) HR(f)
Pg(f) Pr(f)
Tb 2Tb
3Tb 4Tb
5Tb
t
6Tb
28. Dr. Uri Mahlab 28
Rate of decay
of pr(t)
Shaping filters
pr(t)
Particular choice of Pr(t) for a
given application
The smallest values near Tb, 2Tb, …
In such that timing error (Jitter)
will not Cause large ISI
Shape of Pr(f) determines the ease
with which shaping filters can be
realized.
29. Dr. Uri Mahlab 29
A Pr(f) with a smooth roll - off characteristics is preferable
over one with arbitrarily sharp cut off characteristics.
Pr(f) Pr(f)
30. Dr. Uri Mahlab 30
In practical systems where the bandwidth available for
transmitting data at a rate of rb bitssec is between rb2 to rb
Hz, a class of pr(t) with a raised cosine frequency
characteristic is most commonly used.
A raise Cosine Frequency spectrum consist of a flat amplitude portion and a roll off
portion that has a sinusoidal form.
t
r
t
r
sin
)
t
4
(
1
t
2
cos
)
t
(
P
)
f
(
P
FT
2
r
f
2
r
2
/
r
f
,
0
),
2
r
f
(
4
cos
T
2
/
r
f
,
T
)
f
(
P
b
b
2
r
r
1
b
b
b
b
2
b
b
b
r
31. Dr. Uri Mahlab 31
raised cosine frequency characteristic
32. Dr. Uri Mahlab 32
The BW occupied by the pulse spectrum is B=rb/2+.
The minimum value of B is rb/2 and the maximum value is rb.
Larger values of imply that more bandwidth is required for a
given bit rate, however it lead for faster decaying pulses, which
means that synchronization will be less critical and will not
cause
large ISI.
=rb/2 leads to a pulse shape with two convenient properties.
The half amplitude pulse width is equal to Tb, and there are zero
crossings at t=3/2Tb, 5/2Tb…. In addition to the zero crossing
at Tb, 2Tb, 3Tb,…...
Summary
33. Dr. Uri Mahlab 33
5.2.2
Optimum transmitting and receiving
filters
pulse shaping noise immunity
HT ,HR
The transmitting and receiving filters are chosen to provide
a proper
34. Dr. Uri Mahlab 34
)
2
2
exp(
)
(
)
(
)
(
)
( d
r
c
R
T
g ft
j
f
P
K
f
H
f
H
f
p
-One of design constraints that we have for selecting the filters
is the relationship between the Fourier transform of pr(t) and
pg(t).
In order to design optimum filter Ht(f) & Hr(f), we will assume that Pr(f),
Hc(f) and Pg(f) are known.
Where td, is the time delay Kc normalizing constant.
Portion of a baseband PAM system
35. Dr. Uri Mahlab 35
If we choose Pr(t) {Pr(f)} to produce Zero ISI we are left
only to be concerned with noise immunity, that is will choose
HT(f) Hc(f) HR(f)
Pg(f) Pr(f)
effects
noise
of
minimum
)
f
(
H
and
)
f
(
H R
T
36. Dr. Uri Mahlab 36
Noise Immunity
Problem definition:
For a given :
•Data Rate - rb
•Transmission power - ST
•Noise power Spectral Density - Gn(f)
•Channel transfer function - Hc(f)
•Raised cosine pulse - Pr(f)
Choose
effects
noise
of
minimum
)
f
(
H
and
)
f
(
H R
T
37. Dr. Uri Mahlab 37
Error probability Calculations
At the m-th sampling time the input to the A/D is:
m
k
m
0
b
r
k
m
m )
t
(
n
)
T
)
k
m
((
P
A
A
)
t
(
Y
We decide:
0
)
t
(
Y
"
0
"
0
)
t
(
Y
"
1
"
m
m
if
if
"1"
sent
To
sent
was
"1"
"0"
sent
To
sent
was
"0"
ob
Pr
0
)
t
(
Y
ob
Pr
ob
Pr
0
)
t
(
Y
ob
Pr
P
m
m
error
38. Dr. Uri Mahlab 38
"
0
"
)
t
(
A
)
t
(
Y
"
1
"
)
t
(
A
)
t
(
Y
m
m
m
m
if
n
if
n
0
0
A=aKc
5
.
0
ob
Pr
ob
Pr
"1"
sent
To
"0"
sent
To
A
)
t
(
n
ob
Pr
A
)
t
(
n
ob
Pr
2
1
P m
0
m
0
error
The noise is assumed to be zero mean Gaussian at the receiver input
then the output should also be Zero mean Gaussian with variance No
given by:
df
)
f
(
H
)
f
(
G
N
2
R
n
0
39. Dr. Uri Mahlab 39
0
2
0
2
N
2
/
)
A
n
0
N
2
/
n
0
e
N
2
1
e
N
2
1
0 A
b
N
2
/
)
z
0
error dz
e
N
2
1
b
n
ob
Pr
P 0
2
y(tm)
b
40. Dr. Uri Mahlab 40
0
2
m
0
2
m N
2
/
)
A
)
t
(
y
0
N
2
/
)
A
)
t
(
y
0
e
N
2
1
e
N
2
1
-A A
0
)
t
(
Y
ob
Pr
P m
error
0
)
t
(
Y
ob
Pr m
y(tm)
0
y(tm)
41. Dr. Uri Mahlab 41
0
2
m
0
2
m N
2
/
)
A
)
t
(
y
0
N
2
/
)
A
)
t
(
y
0
e
N
2
1
e
N
2
1
-A A
y(tm)
Vreceived
VTransmit
42. Dr. Uri Mahlab 42
u
2
0
N
/
A
2
e
A
N
/
x
z
0
2
0
A
x
0
2
0
e
dz
2
/
z
exp
2
1
u
Q
N
A
Q
dz
2
/
z
exp
2
1
P
dx
N
2
/
x
exp
N
2
1
dx
N
2
/
x
exp
N
2
1
2
/
1
P
0
0
43. Dr. Uri Mahlab 43
u
2
0
N
/
A
2
e
dz
2
/
z
exp
2
1
u
Q
N
A
Q
dz
2
/
z
exp
2
1
P
0
U
Q(u)
u
u
2
dz
2
/
z
exp
2
1
u
Q
dz=
44. Dr. Uri Mahlab 44
Ratio
Noise
to
Signal
N
A
0
Perror decreases as 0
N
/
A increase
Hence we need to maximize the signal
to noise Ratio
Thus for maximum noise immunity the filter transfer functions HT(f)
and HR(f) must be xhosen to maximize the SNR
45. Dr. Uri Mahlab 45
Optimum filters design calculations
We will express the SNR in terms of HT(f) and HR(f)
We will start with the signal:
k
b
g
k kT
t
p
a
t
X )
(
)
(
b
2
g
2
2
k
b
g
X
T
)
f
(
p
a
a
E
T
)
f
(
p
)
f
(
G
The psd of the transmitted signal is given by::
)
f
(
G
)
f
(
H
)
f
(
G X
2
T
X
46. Dr. Uri Mahlab 46
df
)
f
(
H
)
f
(
P
T
K
A
S
df
)
f
(
H
)
f
(
P
T
a
S
2
T
2
g
b
2
c
2
T
a
K
A
a
K
A
2
T
2
g
b
2
T
c
k
c
k
And the average transmitted power ST is
df
)
f
(
H
)
f
(
P
T
K
S
A
2
T
2
g
b
2
c
T
2
The average output noise power of n0(t) is given by:
df
)
f
(
H
)
f
(
G
N
2
R
n
o
47. Dr. Uri Mahlab 47
The SNR we need to maximize is
)
f
(
H
)
f
(
H
)
f
(
H
)
f
(
P
where
df
)
f
(
H
)
f
(
H
)
f
(
P
df
)
f
(
H
)
f
(
G
T
S
N
A
T
R
c
r
2
R
c
2
r
2
R
n
b
T
o
2
Or we need to minimize
2
2
R
c
2
r
2
R
n min
df
)
f
(
H
)
f
(
H
)
f
(
P
df
)
f
(
H
)
f
(
G
min
48. Dr. Uri Mahlab 48
Using Schwartz’s inequality
2
2
2
df
)
f
(
W
)
f
(
V
df
)
f
(
W
df
)
f
(
V
The minimum of the left side equaity is reached when
V(f)=const*W(f)
If we choose :
)
f
(
H
)
f
(
H
)
f
(
P
)
f
(
W
)
f
(
G
)
f
(
H
)
f
(
V
c
R
r
2
/
1
n
R
49. Dr. Uri Mahlab 49
when
minimized
is
2
constant
positive
arbitrary
an
K
)
f
(
P
)
f
(
H
K
)
f
(
G
)
f
(
P
K
)
f
(
H
)
f
(
G
)
f
(
H
)
f
(
P
K
)
f
(
H
2
g
c
2
/
1
n
r
2
c
2
T
2
/
1
n
c
r
2
R
The filter should have alinear phase response in a total time delay of td
50. Dr. Uri Mahlab 50
Finally we obtain the maximum value of the SNR to be:
max
o
2
error
2
c
2
/
1
n
r
b
T
max
o
2
N
A
Q
P
df
)
f
(
H
)
f
(
G
)
f
(
P
T
S
N
A
51. Dr. Uri Mahlab 51
For AWGN with
and
pg(f) is chosen such that it does not change much over the
bandwidth of interest we get.
)
f
(
H
)
f
(
p
K
)
f
(
H
)
f
(
H
)
f
(
p
K
)
f
(
H
c
r
2
2
T
c
r
1
2
R
Rectangular pulse can be used at the input of HT(f).
elsewhere
0
T
;
2
/
t
for
1
)
t
(
p
b
g
2
/
)
f
(
Gn
52. Dr. Uri Mahlab 52
5.2.3 Design procedure and Example
The steps involved in the design procedure.
Example:Design a binary baseband PAM system to
transmit data at a a bit rate of 3600 bits/sec with a bit
error probability less than .
10 4
The channel response is given by:
elewhere
f
for
f
Hc
_
0
2400
10
)
(
2
The noise spectral density is Hz
watt
f
Gn /
10
)
( 14
53. Dr. Uri Mahlab 53
Solution:
Hz
watt
f
G
Hz
B
p
bits
r
n
e
b
/
10
)
(
2400
10
sec
/
3600
4
4
If we choose a braised cosine pulse spectrum with
600
6
/
b
r
2400
1200
),
1200
(
2400
cos
2400
,
0
3600
1
1200
,
3600
1
)
( 2
f
f
f
f
f
pr
54. Dr. Uri Mahlab 54
We choose a pg(t)
973
.
0
)
2400
(
,
)
0
(
)
sin
(
)
(
)
10
)(
28
.
0
(
10
/
;
,
0
1200
,
1
)
( 4
g
g
g
b
g
p
p
f
f
f
p
T
elsewhere
t
t
p
2
/
1
2
/
1
1
)
(
)
(
)
(
)
(
f
P
f
H
f
p
K
f
H
r
R
r
T
We choose
)
(
)
(
)
(
)
(
)
(
)
10
)(
3600
( 3
1
f
p
f
H
f
H
f
H
f
p
K
r
R
c
T
g
55. Dr. Uri Mahlab 55
Plots of Pg(f),Hc(f),HT(f),HR(f),and Pr(f).
56. Dr. Uri Mahlab 56
To maintain a 4
10
e
P
2
4
14
2
2
/
1
max
0
2
max
0
2
max
0
2
4
max
0
2
max
0
2
)
(
10
10
)
06
.
14
)(
3600
(
)
(
)
(
)
(
)
(
1
06
.
14
)
/
(
75
.
3
)
/
(
10
)
/
(
(
)
/
(
df
f
P
df
f
H
f
G
f
P
N
A
T
S
N
A
N
A
N
A
Q
N
A
r
c
n
r
b
T
For Pr(f) with raised cosine shape
1
)
( df
f
Pr
And hence dBm
ST 23
)
10
)(
3600
)(
06
.
14
( 10
Which completes the design.
