1
Dr. Uri Mahlab
INTRODUCTION
In order to transmit digital information over *
bandpass channels, we have to transfer
the information to a carrier wave of
.appropriate frequency
We will study some of the most commonly *
used digital modulation techniques wherein
the digital information modifies the amplitude
the phase, or the frequency of the carrier in
.discrete steps
2
Dr. Uri Mahlab
The modulation waveforms for transmitting
:binary information over bandpass channels
ASK
FSK
PSK
DSB
3
Dr. Uri Mahlab
OPTIMUM RECEIVER FOR BINARY
:DIGITAL MODULATION SCHEMS
The function of a receiver in a binary communication *
system is to distinguish between two transmitted signals
.S1(t) and S2(t) in the presence of noise
The performance of the receiver is usually measured *
in terms of the probability of error and the receiver
is said to be optimum if it yields the minimum
.probability of error
In this section, we will derive the structure of an optimum *
receiver that can be used for demodulating binary
.ASK,PSK,and FSK signals
4
Dr. Uri Mahlab
Description of binary ASK,PSK, and
: FSK schemes
-Bandpass binary data transmission system
Transmit
carrier
Clock pulses
Input
Binary
data
{bk}
5
Channel
(Hc(f
Modulator
(Z(t
Noise
(n(t
+
Local carrier
Clock pulses
Demodulator
(receiver)
+ּ
+
(V(t
Binary data output
{bk}
Dr. Uri Mahlab
:Explanation *
The input of the system is a binary bit sequence {bk} with a *
.bit rate r b and bit duration Tb
The output of the modulator during the Kth bit interval *
.depends on the Kth input bit bk
The modulator output Z(t) during the Kth bit interval is *
a shifted version of one of two basic waveforms S1(t) or S2(t) and
:Z(t) is a random process defined by
.1
6
s1[t  (k  1)Tb ] if b k  0
Z (t )  
 s2 [t  (k  1)Tb ] if b k  1
for : (k  1)Tb  t  kTb
Dr. Uri Mahlab
The waveforms S1(t) and S2(t) have a duration *
of Tb and have finite energy,that is,S1(t) and S2(t) =0
if
t [0, Tb ]
and
Tb
Energy
:Term
E1   [ s1 (t )]2 dt  
0
Tb
E2   [ s2 (t )]2 dt  
0
7
Dr. Uri Mahlab
:The received signal + noise
s1[t  (k  1)Tb t d ]  n(t )

V (t )  
or
(k  1)Tb t d  t  kTb  td
s [t  (k  1)T t ]  n(t )
b
d
2
8
Dr. Uri Mahlab
Choice of signaling waveforms for various types of digital*
modulation schemes
c
S1(t),S2(t)=0 for
t  [0, Tb ]; f c 
2
.The frequency of the carrier fc is assumed to be a multiple of rb
S1 (T );0  t  Tb
0
s2 (t );0  t  Tb
A cos wc t
(or A sin wc t )
 A cos wct
A cos wc t
(or  A sin wct )
( A sin wc t )
A cos{( wc  wd )t}
A cos{( wc  wd )t}
[(or A sin{( wc  wd )t}] [or A sin{( wc  wd )t}]
9
Type of
modulation
ASK
PSK
FSK
Dr. Uri Mahlab
:Receiver structure
(V0(t
Filter
(H(f
v(t )  z (t )  n(t )
10
Threshold
device or A/D
converter
output
Sample every
Tb seconds
Dr. Uri Mahlab
:{Probability of Error-{Pe*
The measure of performance used for comparing *
!!!digital modulation schemes is the probability of error
The receiver makes errors in the decoding process *
!!! due to the noise present at its input
The receiver parameters as H(f) and threshold setting are *
!!!chosen to minimize the probability of error
11
Dr. Uri Mahlab
:The output of the filter at t=kTb can be written as *
V0 (kTb )  s0 (kTb )  n0 (kTb )
12
Dr. Uri Mahlab
:The signal component in the output at t=kTb
s0 (kTb ) 

kTb
kTb
 Z ( )h(kT
b
 )d

Z
(

)
h
(
kT


)
d


ISI
terms
b

k 1)Tb
h( )is the impulse response of the receiver filter*
ISI=0*
s0 (kTb ) 
kTb
 Z ( )h(kT  )d
b
( k 1)Tb
13
Dr. Uri Mahlab
Substituting Z(t) from equation 1 and making*
change of the variable, the signal component
:will look like that
Tb
  s1 ( )h(Tb   )d  s01 ( kTb ) when b k  0
0
s0 (kTb )  Tb
 s ( )h(T   )d  s ( kT ) when b  1
b
01
b
k
 0 2
14
Dr. Uri Mahlab
:The noise component n0(kTb) is given by *
n0 (kTb ) 
kTb
n
(

)
h
(
kT


)
d

b


The output noise n0(t) is a stationary zero mean Gaussian random process
:The variance of n0(t) is*

N 0  E{n0 (t )}   Gn ( f ) H ( f ) df
2
2

:The probability density function of n0(t) is*
2

1
-n 
 ;    n  
f n0 (n) 
exp 
2N 0
 2N0 
15
The probability that the kth bit is incorrectly decoded*
:is given by
.2
Pe  P{bk  0 and V0 (kTb )  T0
or bk  1 and V0 (kTb )  T0 }
1
 P{V0 (kTb )  T0 | bk  0}
2
1
 P{V0 (kTb )  T0 | bk  1}
2
16
Dr. Uri Mahlab
:The conditional pdf of V0 given bk = 0 is given by*
.3
2

- (V0  s01 ) 
1
 , -   V0  
fV0 \bk 0 (V0 ) 
exp 
2N0
2N 0


:It is similarly when bk is 1*
 - (V0  s02 ) 2 
1
 , -   V0  
fV0 \bk 1 (V0 ) 
exp 
2N0
2N 0


17
Dr. Uri Mahlab
Combining equation 2 and 3 , we obtain an*
:expression for the probability of error- Pe as
.4

1
Pe  
2 T0
2

- (V0  S 01 ) 
1
dV0
exp 
2N0
2N 0


 - (V0  S 02 )
1
1
 
exp 
2  2N 0
2N0

T0
18
2

dV0

Dr. Uri Mahlab
:Conditional pdf of V0 given bk
f v0 bk  0
( v0 )
f v0 b k  1
( v0 )
:The optimum value of the threshold T0* is*
S
*
T 0  01
19
S
2
02
Dr. Uri Mahlab
Substituting the value of T*0 for T0 in equation 4*
we can rewrite the expression for the probability
:of error as


Pe 
 (V0  s01 )
1
exp  
2N0
2N 0

2

dV0

( s02  s01 ) / 2



( s02  s01 ) / 2 N 0
20
 Z
1
exp  
2
 2
2

dZ

Dr. Uri Mahlab
The optimum filter  is the filter that maximizes*
2
the ratio or the square of the ratio

(maximizing 2eliminates the requirement S01<S02)
S 02 (Tb )  S 01 (Tb )

N0
21
Dr. Uri Mahlab
:Transfer Function of the Optimum Filter*
The probability of error is minimized by an *
2
appropriate choice of h(t) which maximizes
Where
[ s02 (Tb )  s01 (Tb )]
 
N0
2
2
Tb
s02 (Tb )  s01 (Tb )   [ s2 ( )  s1 ( )]h(Tb   )d
0

And
22
2
N 0   Gn ( f ) H ( f ) df

Dr. Uri Mahlab
If we let P(t) =S2(t)-S1(t), then the numerator of the*
:quantity to be maximized is
S02 (Tb )  S01 (Tb )  P0 (Tb )
Tb

0

  P( )h(Tb   )d   P( )h(Tb   )d
Since P(t)=0 for t<0 and h( )=0
 for
:the Fourier transform of P0 is
<0*
P0 ( f )  P( f ) H ( f )

P0 (Tb ) 
 P( f ) H ( f ) exp( j 2fT )df
b

23
Dr. Uri Mahlab
:Hence  2can be written as*
2

 H ( f ) P( f ) exp( j 2fT )df
b
 

2
(*)

 H( f )
2
Gn ( f )df

We can maximize  by applying Schwarz’s*
:inequality which has the form
2
2

 X ( f )X
1
2
( f )df




24


2
X 1 ( f ) df
X
2
2
( f ) df
(**)

Dr. Uri Mahlab
Applying Schwarz’s inequality to Equation(**) with-
and
X 1 ( f )  H ( f ) Gn ( f )
P ( f ) exp( j 2fTb )
X2( f ) 
Gn ( f )
We see that H(f), which maximizes
2
,is given by-
P ( f ) exp(  j 2fTb )
H( f )  K
Gn ( f )
*
(***)
25
!!! Where K is an arbitrary constant
Dr. Uri Mahlab
Substituting equation (***) in(*) , we obtain2
:the maximum value of  as


2
max

P( f )
2
 G ( f ) df

n
:And the minimum probability of error is given by
Pe 


max/ 2
26
 Z
1
exp  
2
 2
2

  max 
dZ  Q

 2 

Dr. Uri Mahlab
:Matched Filter Receiver*

If the channel noise is white, that is, Gn(f)= /2 ,then the transfer :function of the optimum receiver is given by
H ( f )  P ( f ) exp(  j 2fTb )
*

From Equation (***) with the arbitrary constant K set equal to /2:The impulse response of the optimum filter is

h(t )   [ P* ( f ) exp( 2jfTb )] exp( 2jft)df

27
Dr. Uri Mahlab
Recognizing the fact that the inverse Fourier *
of P*(f) is P(-t) and that exp(-2  jfTb) represent
:a delay of Tb we obtain h(t) as
h(t )  p(Tb  t )
:Since p(t)=S1(t)-S2(t) , we have*
h(t )  S2 (Tb  t )  S1 (Tb  t )
The impulse response h(t) is matched to the signal *
:S1(t) and S2(t) and for this reason the filter is called
MATCHED FILTER
28
Dr. Uri Mahlab
:Impulse response of the Matched Filter *
(S2(t
(S1(t
1
0
2 \Tb
t
(a)
0
2 \Tb
1-
(P(t)=S2(t)-S1(t
(b)
t
2
2 \Tb
0
2
(c)
Tb
t
(P(-t
Tb-
2
(h(t)=p(Tb-t
29
0
0
t
(d)
(h(Tb-t)=p(t
2 \Tb
(e)
Tb
t
Dr. Uri Mahlab
:Correlation Receiver*
The output of the receiver at t=Tb*
Tb
V0 (Tb )   V ( )h(Tb   )d

Where
V( ) is the noisy input to the receiver

h( )  S2 (Tb   )  S1 (Tb   )
Substituting
and noting *
: that h( )  0 for   (0, Tb ) we can rewrite the preceding expression as
Tb
V0 (Tb )   V ( )[ S 2 ( )  S1 ( )]d
0
Tb
Tb
0
0
(# #)
  V ( ) S 2 ( )d   V ( ) S1 ( )d
30
Dr. Uri Mahlab
Equation(# #) suggested that the optimum receiver can be implemented *
as shown in Figure 1 .This form of the receiver is called
A Correlation Receiver
S1 (t )
integrator
Figure 1
Tb

 S1 (t )  n(t )

V (t )  
or
S (t )  n(t )
 2
0

+
Tb

Sample
every Tb
seconds
Threshold
device
(A\D)
0
31
S2 (t )
integrator
Dr. Uri Mahlab
In actual practice, the receiver shown in Figure 1 is actually *
.implemented as shown in Figure 2
In this implementation, the integrator has to be reset at the
- (end of each signaling interval in order to ovoid (I.S.I
!!! Inter symbol interference
:Integrate and dump correlation receiver
White
Gaussian
noise
(n(t
+
(Signal z(t
+
Filter
to
limit
noise
power
Figure 2
Closed every Tb seconds
c
R
S1 (t )  S2 (t )
Threshold
device
(A/D)
High gain
amplifier
The bandwidth of the filter preceding the integrator is assumed *
!!! to be wide enough to pass z(t) without distortion
32
Example: A band pass data transmission scheme
uses a PSK signaling scheme with
S 2 (t )  A cos wct , 0  t  Tb , wc  10 / Tb
S1 (t )   A cos wct , 0  t  Tb , Tb  0.2m sec
The carrier amplitude at the receiver input is 1 mvolt and
the psd of the A.W.G.N at input is 1011watt/Hz. Assume
that an ideal correlation receiver is used. Calculate the
.average bit error rate of the receiver
33
Dr. Uri Mahlab
:Solution
34
Dr. Uri Mahlab
:Solution Continue
=Probability of error = Pe *
35
Dr. Uri Mahlab
* Binary ASK signaling
The binary ASK
waveform can be described as
schemes:
s1[t  (k  1)Tb ] if b k  0

z (t )  
(k - 1)Tb  t  kTb
s [t  (k  1)T ] if b  1
b
k
 2
Where S2 (t )  A cos ct
and s1 (t )  0
We can represent
:Z(t) as
36
Z (t )  D(t )( A cos  c t )
Dr. Uri Mahlab
Where D(t) is a lowpass pulse waveform consisting of
.rectangular pulses
:The model for D(t) is
d (t ) 

 b g[t  (k  1)T ],
b k  0 or 1
k
b
1
g (t )  
0
0  t  Tb
k  
elswhere
D(t )  d (t  T )
37
Dr. Uri Mahlab
:The power spectral density is given by
A2
Gz ( f ) 
[GD ( f  f c )  GD ( f  f c )
4
The autocorrelation function and the power spectral density
:is given by

T 
1
b
 
4Tb
RDD ( )   4
0

38
for   Tb
for   Tb
sin 2 fTb 
1
GD ( f )    ( f )  2 2 
4
 f Tb 
Dr. Uri Mahlab
:The psd of Z(t) is given by
2
A
Gz ( f ) 
( ( f  f c )   ( f  f )
16
2
sin Tb ( f  f c )
 2
2
 Tb ( f  f c )
sin TB ( f  f c )

2
 Tb ( f  f c )
2
39

Dr. Uri Mahlab
If we use a pulse waveform D(t) in which the individual pulses
g(t) have the shape
 a 
 1  cos( 2rbt   ) 0  t  Tb
g (t )   2 
0
elsewere

40
Dr. Uri Mahlab
Coherent ASK
s2 (t )  A cos  ct and s1 (t )  0
We start with
The signal components of the receiver output at the
:of a signaling interval are
Tb
s01 (kTb )   s1 (t )[ s2 (t )  s1 (t )]dt  0
0
and
Tb
2
A
SO2 (kTb )   s2 (t )[ s2 (t )  s1 (t )]dt 
Tb
2
0
41
Dr. Uri Mahlab
:The optimum threshold setting in the receiver is
s01 (kTb )  s02 (kTb ) A
T 

Tb
2
4
2
*
0
:The probability of errorPe

2
max

A2Tb


pe 

1
 max
2
42
can be computed as
2
2




A
T
1
z
b 
dz  Q
exp 
 4 
2
 2 


Dr. Uri Mahlab
:The average signal power at the receiver input is given by
2
A
sav 
4
We can express the probability of error in terms of the
:average signal power
 S avTb
pe  Q






The probability of error is sometimes expressed in *
: terms of the average signal energy per bit P,e  Q Eav  as


43
 
Eav  ( sav )Tb
Dr. Uri Mahlab
Noncoherent ASK
:The input to the receiver is *
 A cos  ct  ni (t ) when b k  1
V (t )  
when b k  0
ni (t )
ni (t )  the noise at the receiver input
which is assumed to be zero mean,
44
Gaussian, and white.
Dr. Uri Mahlab
Noncoharent ASK Receiver
At the filter output we have :
Y (t )  Ak cos  c t  n(t ) 
 Ak cos  c t  nc (t ) cos  c t 
ns (t ) sin  c t
where A k  A when the kth
transmitte d bit b k  1 and A k  0
when n(t) is the noise at the output
of the bandpass filter
45
:The pdf is
 r2 
r
, r  0
f R|bk 0 (r ) 
exp  
N0
 2N0 
 r 2  A2 
r  Ar 
 exp  
, r  0
f R|bk 1 (r ) 
I 0 
N0  N0 
2N0 

N 0  noise power at the output of the
bandpass filter.
2
N 0  BT 
TB
46
1
I0 ( X ) 
2
2
exp(
x
cos(
u
))
du

0
Dr. Uri Mahlab
pdf’s of the envelope of the noise and the signal *
:pulse noise
47
Dr. Uri Mahlab
:The probability of error is given by
1
1
p (error | b k  0)  p (error | b k  1)
2
2
1
1
 pe 0  pe1
2
2
where
pe 

 r2 
 A2 
r
dr  exp  

pe 0  
exp  
A N0
 2N0 
 8N0 
2
and

 A
 (r  A) 
1
dr  Q
pe1  
exp  
2N0 
 2 N0

  2N 0

Using the approximat ion
A
2
48
2
  x2 

exp 
 2 
Q( x) 
x 2





Dr. Uri Mahlab
for large x, we can reduce pe1 to
pe1 

4N0
A2

exp

2

2A
 8N0




Hence,
1
pe 
1 
2
2


4N0
A
exp 

2 
 8N
2A 
0


1
A
 exp 


2
 8N0
2
49





2

 if A  N 0

Dr. Uri Mahlab
BINERY PSK SIGNALING
SCHEMES
:The waveforms are *
s1 (t )   A cos  ct for b k  1
s2 (t )  A cos  ct
for b k  0
:The binary PSK waveform Z(t) can be described by *
Z (t )  D(t )( A cos ct )
.D(t) - random binary waveform *
50
Dr. Uri Mahlab
:The power spectral density of PSK signal is
2
A
GZ ( f ) 
[GD ( f  f c )  GD ( f  f c )]
4
Where,
sin fTb
GD ( f )  2 2
 f Tb
2
51
Dr. Uri Mahlab
Coherent PSK
:The signal components of the receiver output are
s01 ( kTb ) 
kTb
 s (t )[ s (t )  s (t )]dt   A T
2
1
2
1
b
( k 1)Tb
s02 ( kTb ) 
kTb
 s (t )[ s (t )  s (t )]dt  A T
2
2
2
1
b
( k 1)Tb
52
Dr. Uri Mahlab
:The probability of error is given by
 
Pe  Q max 
 2 
where

2
max

2

Tb
 (2 A cos  t )
c
2
dt 
4 A2Tb
0

or
 A2T 
b 
pe  Q 
  


The average signal power sav end the
signal energy per bit Eav for the PSK
scheme are
A2
s av 
2
and
53
 A2 
Eav   Tb
 2 
Dr. Uri Mahlab
we can express the probabilit y of error :
 2 savTb
pe  






 2 Eav 

 Q




54
Dr. Uri Mahlab
DIFFERENTIALLY COHERENT *
:PSK
DPSK modulator
d k 
BINERY
SEQUENCE
LOGIC
NETWORK
d k 1
DELAY
o or 1
LEVEL
SHIFT
 A cos C t
1
Z(t)
A cos ct
Tb
55
Dr. Uri Mahlab
DPSK demodulator
n(t )
Z(t)

Lowpass
filter or
Filter to
limit noise
power
integrator
Delay
Tb
56
Threshold
device
(A/D)
b̂ 
k
sample
at kTb
Dr. Uri Mahlab
Differential encoding & decoding
Input
Sequence
1 1 0
1
0 0
0 1 1
Encoded
1 1 1 0
0
1 0
1 1 1
sequence
Transmit
Phase
57
0 0 0 pi pi 0 pi 0 0 0
Phase
Compari-son
output
+ + -
+
- -
- + +
Output
Bit
sequence
1 1 0
1
0 0
0 1 1
Dr. Uri Mahlab
* BINARY FSK SIGNALING
SCHEMES
: of FSK signaling
:The
waveforms
S1 (t )  A cos( ct   d t ) for b k  0
S 2 (t )  A cos( C t   d t ) for b k  1
:Mathematically it can be represented as


Z (t )  A cos  ct   d  D(t ' )dt ' 




58
 1
D(t )  
 1
for b k  1
for b k  0
Dr. Uri Mahlab
Power spectral density of FSK signals
wd
fd 
2
we
fe 
2
Power spectral density of a binary FSK signal
with 2 f  r
d
59
b
Dr. Uri Mahlab
Coherent FSK
:The local carrier signal required is
s2 (t )  s1 (t )  A cos( ct   d t )  A cos( ct   d t )
The input to the A/D converter at sampling time
t  kTb is s01 (kTb ) or s02 (kTb ) where
Tb
s02 (kTb )   s2 (t )[ s2 (t )  s1 (t )]dt
0
Tb
s01 (kTb )   s1 (t )[ s2 (t )  s1 (t )]dt
60
0
Dr. Uri Mahlab
The probability of error for the correlation receiver is
:given by
  max 
Pe  Q

 2 
where

2
max

2
Tb
[ s (t )  s (t )] dt


2
2
1
0
when
s2 (t )  A cos( c t   d t ) and
s1 (t )  A cos( c t   d t )
61
Dr. Uri Mahlab
.Which are usually encountered in practical system
:We now have

2 A Tb  sin 2 d Tb 
1 


 
2 d Tb 
2
2
max
:When
62
wcTb  1 , w c  wd
Dr. Uri Mahlab
Noncoherent FSK
Assuming that s1 (t )  A cos( c   d )t has been trans mitted during the kth signaling
interval, the pdf of the envelope R 1 (kTb ) of the bottom filter is :
 r12  A2 
r1  Ar1 
 exp  
, r1  0
f R1|s1 (t ) (r1 ) 
I 0 
N0  N0 
2n0 

and
 r22 
r2
,
f R2 |s1 (r2 ) 
exp  
N0
 2N0 
63
r2  0
Dr. Uri Mahlab
Noncoharenr demodulator of binary FSK
Bandpass
filter
ENVELOPE
DETECTOR
fc  fd
fc  fd
+

-
bandpass
filter
R2 (kTb )
ENVELOPE
DETECTOR
THRESHOLD
DEVICE
(A/D)
T 0*  0
R1 (kTb )
Z(t)+n(t)
 A2 
1

Pe  exp  
2
 4N0 
64
Dr. Uri Mahlab
Probability of error for binary digital modulation *
:schemes
65
Dr. Uri Mahlab
M-ARY SIGNALING
SCHEMES
:M-ARY coherent PSK
The M possible signals that would be transmitted
:during each signaling interval of duration Ts are
k 2

S k (t )  A cos  c t 
M


, k  0,1,...M  1, 0  t  Ts

:The digital M-ary PSK waveform can be represented

Z (t )  A  g (t  kTs ) cos( c t   k )
66
k  
Dr. Uri Mahlab


k  
k  
Z (t )  A cos  c t  (cos  k ) g (t  kTs )  A sin  c t  (sin  k ) g (t  kTs )
:In four-phase PSK (QPSK), the waveform are
S1 (t )  A cos  c t
S 2 (t )   A sin  c t
S3 (t )   A cos  c t
S 4 (t )  A sin  c t
67
for all 0  t  TS
Dr. Uri Mahlab
Phasor diagram for QPSK
A cos( c t  450 ) and A cos( c t  450 )
That are derived from a coherent local carrier
reference
A cos ct
68
If we assume that S 1 was the transmitted signal
:during the signaling interval (0,Ts),then we have

Ts
S 01 (Ts )   ( A cos  ct ) A cos( c t  )dt 
4
0
A2


Ts cos  L0
2
4


S 02 (Ts )   ( A cos  c t ) A cos  c t  dt
4

0
Ts
A


Ts cos  L0
2
4
2
69
Dr. Uri Mahlab
QPSK receiver scheme
A cos( c t  45 )
TS

Z(t)
70
0

n(t )
V01 (kTS )
TS

0
V02 (kTS )
A cos( c t  45 )
Dr. Uri Mahlab
:The outputs of the correlators at time t=TS are
V01 (Ts )  S 01 (Ts )  n01 (Ts )
V02 (Ts )  S 02 (Ts )  n02 (Ts )
where n01 (Ts ) & n02 (Ts ) are zero mean Gaussian random variables defined by
Ts
n01 (Ts )   n(t ) A cos( c t  450 )dt
0
TS
n02 (Ts )   n(t ) A cos( c t  450 )dt
0
71
Dr. Uri Mahlab
Probability of error of
QPSK:
Pec1  P(n01 (Ts )   L0 )
 P (n02 (Ts )  L0 )
 L0
 Q
 N
0

72
 A2T 

s 
  Q
 Pec 2

 2 



Dr. Uri Mahlab
Pc - The probabilit y that the transmitt ed signal is received correctly
Pc  (1  Pec1 )(1  Pec 2 )
Pe for the system is :
 A2T 
s 
Pe  1  Pc  2 Pec1  2Q
 2 


for M  4
 A2T


2
s

Pe  2Q
sin
 

M


73
Dr. Uri Mahlab
Phasor diagram for M-ary PSK ; M=8
74
Dr. Uri Mahlab
The average power requirement of a binary PSK
:scheme are given by
 Z 12 
( S av ) M
1


 2 
 
( S av ) b
 Z 2   sin 2 


M 
If Pe is very small & Z1  Z 2
( S av ) M

( S av ) b
75
1
2  
 sin 

M 
Dr. Uri Mahlab
* COMPARISION OF POWER-BANDWIDTH
:FOR M-ARY PSK
Pe  10 4
Value
of M
4
8
16
32
76
( Bandwidth ) M
( Bandwidth ) b
( S av ) m
( S av ) b
0.5
0.333
0.25
0.2
0.34 dB
3.91 dB
8.52 dB
13.52 dB
Dr. Uri Mahlab
* M-ary for four-phase
Differential PSK:
RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK
n(t )
Integrate
and dump
filter
Delay
V01 (t )
TS

90 0
Z(t)
phase
shift
Delay
TS
Integrate
and dump
filter
77
V02 (t )
Dr. Uri Mahlab
:The probability of error in M-ary differential PSK
 A2T


2
S
Pe  2Q
2 sin 
 
2
M







:The differential PSK waveform is
Z (t )  A g (t  kTS ) cos( ct  k )
k
78
Dr. Uri Mahlab
:Transmitter for differential PSK*
Binary
Data
rb  2400
Serial to
parallel
converter
rs  1200
M 4
Diff
phase
mod.
Envelope
modulator
BPF
(Z(t
f c  1800Hz
3
Clock
signal
2400 Hz
79
4
600 Hz
Dr. Uri Mahlab
* M-ary Wideband FSK
Schemas:
Let us consider an FSK scheme witch have the
: following properties
 A cos  i t
S i (t )  
0
0  t  Ts
elsewhere
and
 A2TS

TS
S i (t ) S j (t )   2
0
0

80
FOR i  j
FOR i  j
Dr. Uri Mahlab
:Orthogonal Wideband FSK receiver
gausian
noise
TS

n(t )
0
Y1 (t )
S1 (t )
Z(t)

TS

0
.
S2 (t )
Y2 (t )
MAXIMUM
SELECTOR
.
.
.
TS

0
81
SM (t )
YM (t )
Dr. Uri Mahlab
:The filter outputs are
Y j (Ts ) 
Ts
S
j
(t )[ n(t )  S1 (t )]dt ,
j  1,2,...., M
0

TSS
S
j
(t ) S1 (t ) dt 
0
TS
S
j
(t ) n(t ) dt
0
 S 0 j (Ts )  n j (Ts )
where
S 0 j (Ts ) - The signal component of the j - th filter output
n j (Ts ) - The noise component
82
Dr. Uri Mahlab
:N0 is given by
 
N 0  A Ts  
4
2
:The probability of correct decoding as
Pc1  p{Y2  Y1 , Y3  Y1 ,..., YM  Y1 | S1 sent}


s1 sent and
P
{
Y

y
,...,
Y

y
|
} f Y1 |S1 ( y1 ) dy1
M
1 Y1  y1
 2 1
-
:In the preceding step we made use of the identity

P( X  Y ) 
83
 P( X  y | Y  y ) f
Y
( y )dy

Dr. Uri Mahlab
The joint pdf of Y2 ,Y3 ,…,YM *
:is given by
M
fY 2...YM |S1:Y1  y1 ( y2 ,..., yM )   fYi ( yi )
i 2
84
Dr. Uri Mahlab
where
f Yi ( yi ) 

y 2i
1
exp  
2N 0
 2N0

 ,

   yi  
and
y
y


 1 1 M

Pc1     ...   f Yi ( yi ) dyi  f Y1 |S1 ( y1 ) dy1

 
    i  2


 y1

    f Y ( y ) dy 

- 


M 1
f Y1 |S1 ( y1 ) dy1
where
fY ( y ) 
f Y |S1 ( y1 ) 
1

1
y2
exp  
2N 0
 2N0

 ,

 ( y1  S 01 ) 2
1
exp  
2N0
2N 0

  y  

 ,

   y1  
and
85
 A2

N 0  
Ts 
 2
2
A2
S 01 
Ts
2
Dr. Uri Mahlab
Probability of error for M-ary orthogonal *
: signaling scheme
86
Dr. Uri Mahlab
The probability that the receiver incorrectly *
decoded the incoming signal S1(t) is
Pe1 = 1-Pe1
The probability that the receiver makes *
an error in decoding is
Pe = Pe1

rb  rs log 2 M  rs ( a positive inteegr )
e assume that
,
and
M 2
e can see that increasing values of M lead to smaller power
quirements and also to more complex transmitting
ceiving equipment.
87
Dr. Uri Mahlab
In the limiting case as M

the probability of error Pe satisfies
1 if Sav / rb  0.7

Pe  
0 if S / r  0.7
av
b

The maximum errorless rb at W data can be transmitted
using an M- ary orthogonal FSK signaling scheme
S av
S av
rb 

log 2 e
0.7 
88
The bandwidth of the signal set

as M

Dr. Uri Mahlab