DIGITAL CARRIER MODULATION SCHEMES. Dr.Uri Mahlab. 1. Dr. Uri Mahlab. תוכן עניינים :. Introduction of Binary Digital Modulation Schemes 210 Probability of error 1121 Transfer function of the optimum filter 2226 Matched filter receiver 2729 Correlation receiver 3032
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Introduction of Binary Digital Modulation Schemes 210
Probability of error 1121
Transfer function of the optimum filter 2226
Matched filter receiver 2729
Correlation receiver 3032
Example (the BER average of PSK) 3335
Binary ASK Signaling Schemes 3640
Coherent ASK 4143
Noncoherent ASK 4449
Binary PSK signaling schemes 5051
Coherent PSK 5254
Differentially Coherent PSK 5557
Binary FSK signaling schemes 5859
Coherent FSK 6062
Noncoherent FSK 6364
Comparison of digital modulation schemes 65
Mary signaling schemes 6685
Probability of error of Mary orthogonal signaling scheme 8688
Synchronization Methods 8990
1.א
Dr. Uri Mahlab
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
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Dr. Uri Mahlab
The modulation waveforms fortransmitting :binary information over bandpass channels
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Dr. Uri Mahlab
: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
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Dr. Uri Mahlab
Description of binary ASK,PSK, and
: FSK schemes
Bandpass binary data transmission system
+
Input
ּ+
+
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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
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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
and
Energy
:Term
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Dr. Uri Mahlab
Choice of signaling waveforms for various types of digital*
modulation schemes
S1(t),S2(t)=0 for
.The frequency of the carrier fc is assumed to be a multiple of rb
Type of
modulation
ASK
PSK
FSK
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
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Dr. Uri Mahlab
:The signal component in the output at t=kTb
h( ) is the impulse response of the receiver filter*
ISI=0*
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Dr. Uri Mahlab
Substituting Z(t) from equation 1 and making*
change of the variable, the signal component
:will look like that
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Dr. Uri Mahlab
:The noise component n0(kTb) is given by *
.The output noise n0(t) is a stationary zero mean Gaussian random process
:The variance of n0(t) is*
:The probability density function of n0(t) is*
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Dr. Uri Mahlab
:The conditional pdf of V0 given bk = 0 is given by*
.3
:It is similarly when bk is 1*
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Dr. Uri Mahlab
Combining equation 2 and 3 , we obtain an*
:expression for the probability of error Pe as
.4
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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
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Dr. Uri Mahlab
The optimum filter is the filter that maximizes*
the ratio or the square of the ratio
(maximizing eliminates the requirement S01<S02)
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Dr. Uri Mahlab
:Transfer Function of the Optimum Filter*
The probability of error is minimized by an *
appropriate choice of h(t) which maximizes
Where
And
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Dr. Uri Mahlab
If we let P(t) =S2(t)S1(t), then the numerator of the*
:quantity to be maximized is
Since P(t)=0 for t<0 and h( )=0 for <0*
:the Fourier transform of P0 is
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Dr. Uri Mahlab
(*)
We can maximize by applying Schwarz’s*
:inequality which has the form
(**)
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Dr. Uri Mahlab
Applying Schwarz’s inequality to Equation(**) with
and
We see that H(f), which maximizes ,is given by
(***)
!!! Where K is an arbitrary constant
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Dr. Uri Mahlab
Substituting equation (***) in(*) , we obtain
:the maximum value of as
:And the minimum probability of error is given by
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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
From Equation (***) with the arbitrary constant K set equal to /2
:The impulse response of the optimum filter is
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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
:Since p(t)=S1(t)S2(t) , we have*
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Dr. Uri Mahlab
:Impulse response of the Matched Filter *
1
t
0
2 \Tb
(a)
0
2 \Tb
t
1
(b)
2
2 \Tb
0
t
Tb
2
(c)
t
(d)
0
2
2 \Tb
0
t
(e)
Tb
Dr. Uri Mahlab
The output of the receiver at t=Tb*
Where V( ) is the noisy input to the receiver
Substituting and noting *
: that we can rewrite the preceding expression as
(# #)
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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

+
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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
+
c
Figure 2
The bandwidth of the filter preceding the integrator is assumed *
!!! to be wide enough to pass z(t) without distortion
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Dr. Uri Mahlab
Example actually *: A band pass data transmission scheme
uses a PSK signaling scheme with
The carrier amplitude at the receiver input is 1 mvolt and
the psd of the A.W.G.N at input is watt/Hz. Assume
that an ideal correlation receiver is used. Calculate the
.average bit error rate of the receiver
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Dr. Uri Mahlab
:Solution actually *
Data rate =5000 bit/sec
Receiver impulse response
Threshold setting is 0 and
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Dr. Uri Mahlab
* Binary ASK signaling schemes: actually *
The binary ASK waveform can be described as
Where and
We can represent :Z(t) as
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Dr. Uri Mahlab
Where D(t) is a lowpass pulse waveform consisting of actually *
.rectangular pulses
:The model for D(t) is
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Dr. Uri Mahlab
:The power spectral density is given by actually *
The autocorrelation function and the power spectral density
:is given by
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Dr. Uri Mahlab
If we use a pulse waveform D(t) in which the individual pulses
g(t) have the shape
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Dr. Uri Mahlab
We start with
The signal components of the receiver output at the
:of a signaling interval are
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Dr. Uri Mahlab
:The optimum threshold setting in the receiver is pulses
:The probability of error can be computed as
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Dr. Uri Mahlab
:The average signal power at the receiver input is given by pulses
We can express the probability of error in terms of the
:average signal power
The probability of error is sometimes expressed in *
: terms of the average signal energy per bit , as
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Dr. Uri Mahlab
49 pulses
Dr. Uri Mahlab
:The waveforms are *
:The binary PSK waveform Z(t) can be described by *
.D(t)  random binary waveform *
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Dr. Uri Mahlab
54 pulses
Dr. Uri Mahlab
DPSK demodulator pulses
Filter to
limit noise
power
Lowpass
filter or
integrator
Threshold
device
(A/D)
Delay
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Dr. Uri Mahlab
:The waveforms of FSK signaling
:Mathematically it can be represented as
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Dr. Uri Mahlab
Power spectral density of FSK signals pulses
Power spectral density of a binary FSK signal
with
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Dr. Uri Mahlab
:The local carrier signal required is
The input to the A/D converter at sampling time
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Dr. Uri Mahlab
Noncoharenr demodulator of binary FSK pulses
ENVELOPE
DETECTOR
+
THRESHOLD
DEVICE
(A/D)

ENVELOPE
DETECTOR
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Dr. Uri Mahlab
MARY SIGNALING SCHEMES pulses
:MARY coherent PSK
The M possible signalsthat would be transmitted
:during each signaling interval of duration Ts are
:The digital Mary PSK waveform can be represented
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Dr. Uri Mahlab
Phasor diagram for QPSK pulses
That are derived from a coherent local carrier
reference
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Dr. Uri Mahlab
If we assume that S pulses 1 was the transmitted signal
:during the signaling interval (0,Ts),then we have
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Dr. Uri Mahlab
73 pulses
Dr. Uri Mahlab
* COMPARISION OF POWERBANDWIDTH pulses
:FOR MARY PSK
Value
of M
4
8
16
32
0.5
0.333
0.25
0.2
0.34 dB
3.91 dB
8.52 dB
13.52 dB
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Dr. Uri Mahlab
Dr. Uri Mahlab
* Mary for fourphase pulsesDifferential PSK:
RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK
Integrate
and dump
filter
Z(t)
Integrate
and dump
filter
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Dr. Uri Mahlab
:The probability of error in Mary differential PSK pulses
:The differential PSK waveform is
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Dr. Uri Mahlab
:Transmitter for differential PSK* pulses
Serial to
parallel
converter
Diff
phase
mod.
Envelope
modulator
BPF
(Z(t
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Dr. Uri Mahlab
* Mary Wideband FSK pulsesSchemas:
Let us consider an FSK scheme witch have the
: following properties
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Dr. Uri Mahlab
:N pulses0 is given by
:The probability of correct decoding as
:In the preceding step we made use of the identity
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Dr. Uri Mahlab
The probability that the receiver incorrectly * pulses
decoded the incoming signal S1(t) is
Pe1 = 1Pe1
The probability that the receiver makes *
an error in decoding is
Pe = Pe1
We assume that , and
We can see that increasing values of M lead to smaller power
requirements and also to more complex transmitting
receiving equipment.
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Dr. Uri Mahlab
In the limiting case as M the probability of error Pe satisfies
The maximum errorless rb at W data can be transmitted
using an M ary orthogonal FSK signaling scheme
The bandwidth of the signal set as M
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Dr. Uri Mahlab
:Synchronization Methods error P
For optimum demodulation of ASK ,FSK ,and PSK waveforms
timing information is needed at the receiver
There are three general methods used for synchronization in
:digital nodulation schemes
.Use of primary or secondary time standard
.Utilization of a separate synchronization signal
Extraction of clock information from the modulated waveform
.itself , referred to as self  synchronization
.1
.2
.3
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Dr. Uri Mahlab
(Extraction of local carrier for coherent demodulation of PSK signals)
Open loop carrier recovery scheme
Squaring
circuit
BPF
Frequency
divider
Closed loop carrier recovery scheme
Squaring
circuit
Loop
Filter
VCO
Frequency
doubler
Recovered carrier
cos (wct)
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Dr. Uri Mahlab