1. DIGITAL
CARRIER
MODULATION
SCHEMES
Dr.Uri Mahlab 1
Dr. Uri Mahlab

2. 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

3. The modulation waveforms for transmitting :binary information over bandpass channels
ASK
FSK
PSK
The modulation scheme using bandpass pulse
,shaping followed by analog modulation
DSB or VSB- requires the minimum transmission
bandwidth. However ,the equipment required to generate , transmit ,and demodulate the waveform is .quite complex
DSB 3
Dr. Uri Mahlab

4. 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

5. Description of binary ASK,PSK, and
: FSK schemes
-Bandpass binary data transmission system
Transmit
carrier
Local carrier
Noise
n(t)
Clock pulses
Clock pulses +
Input
Modulator
Channel
Hc(f) ּ+
Demodulator
(receiver)
Binary
data +
V(t)
Z(t)
{bk}
Binary data output
{bk} 5
Dr. Uri Mahlab

6. :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 6
Dr. Uri Mahlab

7. 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 7
Dr. Uri Mahlab

8. :The received signal + noise
The shape of the waveforms depends on the type of the modulation used .The output of the modulator passes through a bandpass channel Hc(f) which
for purposes of analysis is assumed to be an ideal channel with adequate bandwidth so the signal passes through without suffering any distortion other then propagation delay .the channel nosie n(t) is assumed to be a zero mean stationary, Gaussian random process with a known power spectral .(density Gn(f 8
Dr. Uri Mahlab

9. Type of modulation ASK PSK FSK 9
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 9
Dr. Uri Mahlab

10. :Receiver structure V0(t) Threshold Filter device or A/D Hr(f)
converter
Filter
Hr(f)
output
Sample every
Tb seconds
The receiver has to determine which of the two known waveforms s1(t) or
.s2(t) was present at its input during each signaling interval
The sampled value is compared against a predetermined threshold value
T0 and the transmitted bit is decoded (with occasional errors) as 1 or 0
.depending on whether V0(kTb) is greater or less then threshold T0 10
Dr. Uri Mahlab

11. :{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
We will assume that bk is an equiprobable,independent sequence *
of bits.the accurrence of s1(t)or s2(t) during a bit interval dose not influence the occurrrnce of s1(t) or s2(t) during any other
non-overlapping bit interval ;further , s1(t) and s2(t) are .equiprobable
The channel noise will be assumed to be a zero mean Gaussian*
.(random process with a power spectral density Gn(f
.We will assume that the ISI generated by the filter is minimum* 11
Dr. Uri Mahlab

12. :The output of the filter at t=kTb can be written as *12
Dr. Uri Mahlab

13. :The signal component in the output at t=kTb
h( ) is the impulse response of the receiver filter
ISI=0 13
Dr. Uri Mahlab

14. Substituting Z(t) from equation 1 and making
change of the variable, the signal component
:will look like that 14
Dr. Uri Mahlab

15. 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 15
Dr. Uri Mahlab

16. .2 16 The probability that the kth bit is incorrectly decoded
is given by .2 16
Dr. Uri Mahlab

17. The conditional pdf of V0 given bk = 0 is given by.3
:It is similarly when bk is 1 17
Dr. Uri Mahlab

18. Combining equation 2 and 3 , we obtain an
expression for the probability of error- Pe as .4 18
Dr. Uri Mahlab

19. Conditional pdf of V0 given bk
The optimum value of the threshold T0* is 19
Dr. Uri Mahlab

20. Substituting the value of T*0 for T0 in equation 4
we can rewrite the expression for the probability
of error as 20
Dr. Uri Mahlab

21. The optimum filter is the filter that maximizes
the ratio or the square of the ratio
(maximizing eliminates the requirement S01<S02)
S01,S02 and depend on the choice of the
filter impulse response or the transfer function 21
Dr. Uri Mahlab

22. Transfer Function of the Optimum Filter
The probability of error is minimized by an
appropriate choice of h(t) which maximizes
Where
The receiver has to determine which of the two known waveforms*
.s1(t) and s2(t) was present at its input during each signaling interval
The optimum receiver distinguishes between s1(t) and s2(t) from the
!noisy versions of s1(t) and s2(t) with minimum probability of error
And 22
Dr. Uri Mahlab

23. 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 23
Dr. Uri Mahlab

24. (*) (**) :Hence can be written as
We can maximize by applying Schwarz’s*
:inequality which has the form
Where X1(f) and X2(f) are arbitrary complex functions of a common variable f. The equal sign in ** applies when X1(f)=KX*2(f) ,where K is an arbitrary constant and X*2(f) is
.(the complex conjugate of X2(f
(**) 24
Dr. Uri Mahlab

25. (***) Applying Schwarz’s inequality to Equation(**) with- and
We see that H(f), which maximizes ,is given by-
(***)
!!! Where K is an arbitrary constant 25
Dr. Uri Mahlab

26. Substituting equation (***) in(*) , we obtain-
:the maximum value of as
:And the minimum probability of error is given by- 26
Dr. Uri Mahlab

27. Exrecises 27

28. Matched Filter Receiver
Exrecise - 1
Matched Filter Receiver

29. Exrecise - 1 27 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 27
Dr. Uri Mahlab

30. 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)=S2(t)-S1(t) , we have
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

31. Impulse response of the Matched Filter
S2(t) 1 t
2 \Tb
(a)
S1(t)
2 \Tb t 1-
(b) 2
P(t)=S2(t)-S1(t)
2 \Tb t Tb 2
(c)
(a).S1(t)
(b).S2(t)
(c).p(t)=S1(t)-S2(t)
(d).p(-t)
(e).h(t)=p(Tb-t)
(P(-t t
(d)
Tb- 2
h(Tb-t)=p(t)
h(t)=p(Tb-t)
2 \Tb t 29
(e) Tb
Dr. Uri Mahlab

32. Exrecise - 2
Correlation Receiver

33. Exrecise - 2 Correlation Receiver 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
(# #) 30
Dr. Uri Mahlab

34. A Correlation Receiver
Equation(# #) suggested that the optimum receiver can be implemented
as shown in Figure 1 .This form of the receiver is called
A Correlation Receiver
integrator
Figure 1
Threshold
device
(A\D) - +
It must be pointed out that Equation # # and the receiver*
shown in Figure 1 require that the integration operation
.be ideal with zero initial conditions
Sample
every Tb
seconds
integrator 31
Dr. Uri Mahlab

35. !!! Inter symbol interference
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
Closed every Tb seconds
(n(t
Filter to
limit
noise
power
Threshold
device
(A/D) + + c R
(Signal z(t
If RC >>Tb ,the circuit shown in Figure 2 very closely
approximates an ideal integrator and operates with the
same probability of error as the ideal receiver shown
.in Figure 1
The sampling and discharging of the capacitor
. dumping) must be carefully synchronized
Furthermore ,the local reference signal
S1(t)-S2(t) must be in “phase” with the signal
component at the receiver input,that is, the
!correlation receiver performs coherent demodulation
High gain
amplifier
Figure 2
The bandwidth of the filter preceding the integrator is assumed
to be wide enough to pass z(t) without distortion 32
Dr. Uri Mahlab

36. Exrecise - 3
PSK 27

37. Exrecise - 3 Example: A band pass data transmission scheme 33
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 33
Dr. Uri Mahlab

38. :Solution 34 Data rate =5000 bit/sec Receiver impulse response
Threshold setting is 0 and 34
Dr. Uri Mahlab

39. =Probability of error = Pe
:Solution Continue
=Probability of error = Pe
From the table of Gaussian probabilities ,we
get Pe and
Average error rate (rb) pe /sec = 4 bits/sec 35
Dr. Uri Mahlab

40. Exrecise - 4
Binary ASK 27

41. Binary ASK signaling schemes
Exrecise - 4
Binary ASK signaling schemes
The binary ASK waveform can be described as
Where and
We can represent Z(t) as: 36
Dr. Uri Mahlab

42. Where D(t) is a lowpass pulse waveform consisting of
.rectangular pulses
:The model for D(t) is 37
Dr. Uri Mahlab

43. Exrecise - 5
Power Spectrum 27

44. The power spectral density is given by
Exrecise - 5
The power spectral density is given by
The autocorrelation function and the power spectral density
is given by 38
Dr. Uri Mahlab

45. The psd of Z(t) is given by39
Dr. Uri Mahlab

46. 40 If we use a pulse waveform D(t) in which the individual pulses
g(t) have the shape 40
Dr. Uri Mahlab

47. Exrecise - 6
Coherent ASK 27

48. Coherent ASK Exrecise - 6 41 We start with
The signal components of the receiver output at the
:of a signaling interval are 41
Dr. Uri Mahlab

49. The optimum threshold setting in the receiver is
The probability of error can be computed as 42
Dr. Uri Mahlab

50. 43 The average signal power at the receiver input is given by
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 43
Dr. Uri Mahlab

51. Exrecise - 7
Noncoherent ASK

52. Noncoherent ASK The input to the receiver is Exrecise-7 44
Dr. Uri Mahlab

53. Non-coherent ASK Receiver
Exrecise -8
Non-coherent
ASK Receiver 27

54. Non-coherent ASK Receiver
Exrecise-8 45
Dr. Uri Mahlab

55. :The pdf is 46
Dr. Uri Mahlab

56. pdf’s of the envelope of the noise and the signal
pulse noise 47
Dr. Uri Mahlab

57. The probability of error is given by48
Dr. Uri Mahlab

58. 49
Dr. Uri Mahlab

59. BINARY PSK SIGNALING SCHEMES
Exrecise-9
BINARY PSK SIGNALING SCHEMES 27

60. BINARY PSK SIGNALING SCHEMES
Exrecise-9
BINARY PSK SIGNALING SCHEMES
The waveforms are
The binary PSK waveform Z(t) can be described by
.D(t) - random binary waveform * 50
Dr. Uri Mahlab

61. :The power spectral density of PSK signal is51
Dr. Uri Mahlab

62. Exrecises-10
Coherent PSK

63. Coherent PSK Exrecise-10
:The signal components of the receiver output are 52
Dr. Uri Mahlab

64. :The probability of error is given by53
Dr. Uri Mahlab

65. 54
Dr. Uri Mahlab

66. DIFFERENTIALLY COHERENT
PSK 27

67. DIFFERENTIALLY COHERENT *
:PSK
DPSK modulator
BINERY
SEQUENCE
LOGIC
NETWORK
LEVEL
SHIFT
Z(t)
DELAY 55
Dr. Uri Mahlab

68. DPSK demodulator 56 Z(t) Filter to Lowpass limit noise filter or power
integrator
Threshold
device
(A/D)
Z(t)
Delay 56
Dr. Uri Mahlab

69. Differential encoding & decoding57
Dr. Uri Mahlab

70. BINARY FSK SIGNALING SCHEMES27

71. BINARY FSK SIGNALING SCHEMES :
:The waveforms of FSK signaling
:Mathematically it can be represented as 58
Dr. Uri Mahlab

72. Power spectral density of FSK signals
Power spectral density of a binary FSK signal
with 59
Dr. Uri Mahlab

73. Exrecise-11
Coherent FSK

74. Coherent FSK :The local carrier signal required is
The input to the A/D converter at sampling time 60
Dr. Uri Mahlab

75. 61 The probability of error for the correlation receiver is :given by
Dr. Uri Mahlab

76. :We now have .Which are usually encountered in practical system :When62
Dr. Uri Mahlab

77. Exrecise-12
Noncoherent
FSK

78. Noncoherent FSK 63
Dr. Uri Mahlab

79. Noncoherent demodulator of binary FSK
ENVELOPE
DETECTOR +
THRESHOLD
DEVICE
(A/D) -
ENVELOPE
DETECTOR
Z(t)+n(t) 64
Dr. Uri Mahlab

80. Probability of error for binary digital modulation *
:schemes 65
Dr. Uri Mahlab

81. M-ARY SIGNALING
SCHEMES 66
Dr. Uri Mahlab

82. M-ARY
coherent
PSK 66
Dr. Uri Mahlab

83. :The digital M-ary PSK waveform can be represented
:M-ARY coherent PSK
The M possible signals that would be transmitted
:during each signaling interval of duration Ts are
:The digital M-ary PSK waveform can be represented 66
Dr. Uri Mahlab

84. :In four-phase PSK (QPSK), the waveform are67
Dr. Uri Mahlab

85. Phasor diagram for QPSK
That are derived from a coherent local carrier
reference 68
Dr. Uri Mahlab

86. 69 If we assume that S 1 was the transmitted signal
:during the signaling interval (0,Ts),then we have 69
Dr. Uri Mahlab

87. QPSK receiver scheme
Z(t) 70
Dr. Uri Mahlab

88. :The outputs of the correlators at time t=TS are71
Dr. Uri Mahlab

89. Probability of error of QPSK:72
Dr. Uri Mahlab

90. 73
Dr. Uri Mahlab

91. Phasor diagram for M-ary PSK ; M=874
Dr. Uri Mahlab

92. The average power requirement of a binary PSK
:scheme are given by 75
Dr. Uri Mahlab

93. * COMPARISION OF POWER-BANDWIDTH
:FOR M-ARY PSK
Value
of M 4 8 16 32
0.5
0.333
0.25
0.2 dB dB dB
13.52 dB 76
Dr. Uri Mahlab
Dr. Uri Mahlab

94. M-ary for four-phase
Differential PSK

95. * M-ary for four-phase Differential PSK:
RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK
Integrate
and dump
filter
Z(t)
Integrate
and dump
filter 77
Dr. Uri Mahlab

96. :The differential PSK waveform is
:The probability of error in M-ary differential PSK
:The differential PSK waveform is 78
Dr. Uri Mahlab

97. :Transmitter for differential PSK*
Serial to
parallel
converter
Diff
phase
mod.
Envelope
modulator
BPF
Z(t)
Clock
signal
2400 Hz
600 Hz 79
Dr. Uri Mahlab

98. M-ary Wideband FSK Schemas
Exrecise 13
M-ary Wideband FSK Schemas 27

99. M-ary Wideband FSK Schemas:
Let us consider an FSK scheme witch have the
: following properties 80
Dr. Uri Mahlab

100. :Orthogonal Wideband FSK receiver
MAXIMUM
SELECTOR
Z(t) 81
Dr. Uri Mahlab

101. :The filter outputs are82
Dr. Uri Mahlab

102. 83 :N0 is given by :The probability of correct decoding as
:In the preceding step we made use of the identity 83
Dr. Uri Mahlab

103. The joint pdf of Y2 ,Y3 ,…,YM *
:is given by 84
Dr. Uri Mahlab

104. where 85
Dr. Uri Mahlab

105. Probability of error for M-ary orthogonal * : signaling scheme86
Dr. Uri Mahlab

106. Pe1 = 1-Pc1 87 The probability that the receiver incorrectly *
decoded the incoming signal S1(t) is
Pe1 = 1-Pc1
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. 87
Dr. Uri Mahlab

107. 88 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 88
Dr. Uri Mahlab

108. :Synchronization Methods
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 89
Dr. Uri Mahlab

109. Open loop carrier recovery scheme
(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 90
Recovered carrier
cos (wct)
Dr. Uri Mahlab