Transcript Chapter3_Lect4.ppt

Digital Signaling
 Digital Signaling
 Vector Representation
 Bandwidth Estimation
 Binary Signaling
 Multilevel Signaling
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Digital Signaling
 How do we mathematical represent the waveform of a digital signal?
 How do we estimate the bandwidth of the waveform?
N
w  t    wkk  t 
k 1
0  t  T0
wk Digital Data, w  t  Waveform of PCM word,
k (t ) k=1, 2, 3 N N Orthogonal functions,
N Number of dimensions required to describe w(t )
T0 Message time span
 Example: Message ‘X’ for ASCII computer keyboard - code word “0001101”
 What is the data rate?
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Digital Signaling
 Baud (Symbol Rate) :
D = N/T0 symbols/sec ;
N- number of dimensions used in T0 sec.
 Bit Rate :
R = n/T0 bits/sec ;
n- number of data bits sent in T0 sec.
Binary (2) Values
Binary signal
More than 2 Values
Multilevel signal
wk
 How to detect the data at the receiver?
1
wk 
Kk

T0
0
w  t k*  t  dt ; k  1,2...N
w  t   Waveform at the receiver input
Matched Filter Detection
k*  t   Orthogonal function
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Vector Representation
 Orthogonal function space corresponds to orthogonal vector space :
N
w  t    wkk  t  0  t  T0
Orthogonal Function Space
k 1
N
w
 w    w , w , w ,..., w 
j
j
1
2
3
N
Orthogonal Vector Space
j 1
w is an N-dimensional vector, w   w1 , w2 , w3 ,..., wN 
  Orthogonal set of N-dimensional vectors
j
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Vector Representation of a Binary Signal
 Examine the representation in next slide for the waveform of a 3-bit (binary)
signal. This signal can be directly represented by,
 
1   N 3
s  t    d j p t   j   T    d j p j (t )
2   j 1
j 1
 
N 3
 
1 
p j (t )  p  t   j   T 
2 
 

 p (t ) is not normalized
The Pulse Shape p(t ),
j
Vector d  d1 , d 2 , d3   1, 0,1
.
 Orthogonal function approach
M 3
 t  is the set of Orthonormal functions
s(t )   s j j  t 
j
j 1
 j t  
p j t 
Kj

p j t 

T0
0
p 2j  t  dt

p j t 
25T
Or
 1
,

 j t    T
 0,

The Orthonormal series coefficients are:
( j  1)T  t  jT
t
j =1, 2, 3
Otherwise
s1 , s2 , s3  5
T , 0, 5 T

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Vector Representation of a Binary Signal
A 3 bit Signal waveform
Bit shape pulse
Orthogonal Function Set
Vector Representation of the 3 bit signal
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Bandwidth Estimation
 The lower bound for the bandwidth of the waveform w(t) is given by the Dimensionality
Theorem
B
sinx 
N 1

Lower Bound for  k (t) 
,
B

 D

x 
2T0 2

N 1
 D (Hertz)
2T0 2
 Binary Signaling:
N
Waveform:
wt    wk k t 
k 1
0  t  T0
wk takes only BINARY values
Example: Binary signaling from a digital source: M=256 distinct messages
M = 2n = 28 = 256  Each message ~ 8-bit binary words
T0=8 ms – Time taken to transmit one message; Code word: 01001110
w1= 0, w2= 1, w3= 0, w4= 0, w5= 1, w6= 1, w7= 1, w8= 0

Case 1: Rectangular Pulse Orthogonal Functions:
 k t  : unity-amplitude rectangular pulses;
T0
Tb   1 msec Time taken to send 1 bit of data
n
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Bandwidth Estimation (Binary Signaling)
 Receiver end: How are we going to detect data?
Orthogonal series coefficients wk are needed. Sample anywhere in the bit interval
1
wk 
Kk

T0
0
w  t k*  t  dt ; k  1,2...N
w  t   Waveform at the receiver input
Bit Rate R 
n
 1 Kbits/s
T0
Matched Filter Detection
k*  t   Orthogonal function
Baud Rate (Symbol Rate) D 
N
 1 Kbaud
T0
Same as BINARY Signaling
The Lower Bound : B 
1
D  500 Hz
2
The actual Null Bandwidth: B 
1
 D  1000 Hz
Ts
Bandwidth:  Null BW > lower bound BW
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Binary Signaling
0
1
0
0
1
1
1
To recover the digital data at the receiver, we sample received wavform at the
right time instants (SYNCHRONIZATION) and from the sample values a
decision is made about the value of the transmitted bit at that time instant.
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Binary Signaling
Which wave shape
gives lower bound
BW?
0
1
0
0
1
1
1
Individual Pulses
Total Waveform
 

sin
t

kT


 
s 
N
  Ts


k  t    
 w  t    wkk  t 
k 1

 t  kTs  
 Ts



0  t  T0
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Binary Signaling Using Sa Shape
1
0
0
1
0
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Binary Signaling Using Raised Cosine Shape
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 Case 2:
Binary Signaling
sin(x)/x Pulse Orthogonal Functions
Minimum Bandwidth
 

 sin  t  kTs  
  Ts
  Where T =T for the case of Binary signaling.
 k t    

s
b

t  kTs  
 Ts

 Receiver end: How are we going to detect data?
Orthogonal series coefficients wk are needed. Sample at MIDPOINT of each interval
n
N
Bit Rate R   1 Kbits/s
Baud Rate (Symbol Rate) D   1 Kbaud
T0
T0
Same as BINARY Signaling
The ABSOLUTE Bandwidth: B 
Lower bound BW: B 
N
2T0
1
 500 Hz
2Ts
LOWER BOUND bandwidth
For N=8 pulses, T0=8 ms => B=500Hz.
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Multilevel Signaling
 B Reduces, if N Reduces: So wk should take more than 2 values ( 2- binary signaling)
 If wk’s have L>2 values  Resultant waveform – Multilevel signal
 Multilevel data : Encoding l-bit binary data  into L-level : DAC
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Multilevel Signaling (Example)
M=256-message source ; L=4; T0=8 ms
Encoding Scheme: A 2-Bit Digital-to-Analog Converter
Binary Input
Output Level
(l=2 bits)
(V)
11
+3
10
+1
00
-1
01
-3
Binary code word - 01001110
w1= -3, w2= -1, w3= +3, w4= +1
Bit rate : R 
n
1
T0
Baud ( symbol rate):
Relation :
R  lD
k bits/second
N 1
D    0.5 k baud
T0 Ts
Different
Where l  log2 ( L)
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Multilevel Signaling - Example
B=1/Ts=D=500 Hz
B=N/2T0=250Hz
 How can the data be detected at the receiver?
 Sampling at midpoint of Ts=2 ms interval for either case (T=1, 3, 5, 7 ms)
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Multilevel Signaling - Example
0
1
-3
Individual Pulses
1
0
+1
1
1
+3
1
0
+1
Total Waveform
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Binary-to-multilevel polar NRZ Signal Conversion
 Binary to multilevel conversion is used to reduce the bandwidth required by the binary
signaling.
• Multiple bits (l number of bits) are converted into words having SYMBOL durations
Ts=lTb where the Symbol Rate or the BAUD Rate D=1/Ts=1/lTb.
• The symbols are converted to a L level (L=2l ) multilevel signal using a l-bit DAC.
• Note that now the Baud rate is reduced by l times the Bit rate R (D=R/l).
• Thus the bandwidth required is reduced by l times.
Ts: Symbol Duration
L: Number of M ary levels
Tb: Bit Duration
l: Bits per Symbol
L=2l
D=1/Ts=1/lTb=R/l
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Binary-to-multilevel Polar NRZ Signal Conversion
(c) L = 8 = 23 Level Polar NRZ Waveform Out
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