1. Lecture 1.6. Modulation

2. Bandpass Signalling Definitions Complex Envelope Representation
Representation of Modulated Signals
Spectrum of Bandpass Signals
Power of Bandpass Signals
Examples

3. Time Waveform of Bandpass Signal
Bandpass Signals
Energy spectrum of a bandpass signal is concentrated around the carrier frequency fc.
Bandpass Signal Spectrum
A time portion of a bandpass signal. Notice the carrier and the baseband envelope.
Time Waveform of Bandpass Signal

4. DEFINITIONS
The Bandpass communication signal is obtained by modulating a baseband analog or digital signal onto a carrier.
Definitions:
A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin ( f=0) and negligible elsewhere.
A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency where fc>>0. fc-Carrier frequency
Modulation is process of imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude or phase perturbations or both.
This bandpass signal is called the modulated signal s(t), and the baseband source signal is called the modulating signal m(t).
Transmission
medium
(channel)
Carrier
circuits
Signal
processing
Information m
input
Communication System

5. Complex Envelope Representation
The waveforms g(t) , x(t), R(t), and are all baseband waveforms. Additionally all of them except g(t) are real and g(t) is the Complex Envelope.
g(t) is the Complex Envelope of v(t)
x(t) is said to be the In-phase modulation associated with v(t)
y(t) is said to be the Quadrature modulation associated with v(t)
R(t) is said to be the Amplitude modulation (AM) on v(t)
(t) is said to be the Phase modulation (PM) on v(t)
In communications, frequencies in the baseband signal g(t) are said to be heterodyned up to fc
THEOREM: Any physical bandpass waveform v(t) can be represented as below
where fc is the CARRIER frequency and c=2 fc

6. Generalized transmitter using the AM–PM generation technique.

7. Generalized transmitter using the quadrature generation technique.

8. Complex Envelope Representation
THEOREM: Any physical bandpass waveform v(t) can be represented by
where fc is the CARRIER frequency and c=2 fc
PROOF: Any physical waveform may be represented by the Complex Fourier Series
The physical waveform is real, and using , Thus we have:
cn - negligible magnitudes for n in the vicinity of 0 and, in particular, c0=0
Introducing an arbitrary parameter fc , we get
v(t) – bandpass waveform with non-zero spectrum concentrated near f=fc => cn – non-zero for ‘n’ in the range
=> g(t) – has a spectrum concentrated near f=0 (i.e., g(t) - baseband waveform)

9. Complex Envelope Representation
Equivalent representations of the Bandpass signals:
Converting from one form to the other form
Inphase and Quadrature (IQ) Components.
Envelope and Phase Components

10. Complex Envelope Representation
The complex envelope resulting from x(t) being a computer generated voice signal and y(t) being a sinusoid. The spectrum of the bandpass signal generated from above signal.

11. Representation of Modulated Signals
Modulation is the process of encoding the source information m(t) into a bandpass signal s(t). Modulated signal is just a special application of the bandpass representation. The modulated signal is given by:
The complex envelope g(t) is a function of the modulating signal m(t) and is given by: g(t)=g[m(t)] where g[• ] performs a mapping operation on m(t).
The g[m] functions that are easy to implement and that will give desirable spectral properties for different modulations are given by the TABLE 4.1
At receiver the inverse function m[g] will be implemented to recover the message.
Mapping should suppress as much noise as possible during the recovery.

12. Bandpass Signal Conversion
On off Keying (Amplitude Modulation) of a unipolar line coded signal for bandpass conversion. 1 X
Unipolar
Line Coder
cos(ct)
g(t) Xn

13. Bandpass Signal Conversion
Binary Phase Shift keying (Phase Modulation) of a polar line code for bandpass conversion. 1 X
Polar
Line Coder
cos(ct)
g(t) Xn

14. Mapping Functions for Various Modulations

15. Envelope and Phase for Various Modulations

16. Spectrum of Bandpass Signals
Theorem: If bandpass waveform is represented by
Where
is PSD of g(t)
Proof:
Thus,
Using and the frequency translation property:
We get,

17. PSD of Bandpass Signals
PSD is obtained by first evaluating the autocorrelation for v(t):
Using the identity
where
and
We get
- Linear operators
=> or
but
AC reduces to
PSD =>

18. Evaluation of Power
Theorem: Total average normalized power of a bandpass waveform v(t) is
Proof:
But
So, or
But is always real
So,

19. Example : Amplitude-Modulated Signal
Evaluate the magnitude spectrum for an AM signal:
Complex envelope of an AM signal:
Spectrum of the complex envelope:
AM signal waveform:
AM spectrum:
Magnitude spectrum:

20. Example : Amplitude-Modulated Signal
Spectrum of AM signal.

21. Example : Amplitude-Modulated Signal
Total average power:

22. Study Examples SA4-1.Voltage spectrum of an AM signal
Properties of the AM signal are:
g(t)=Ac[1+m(t)]; Ac=500 V; m(t)=0.8sin(21000t); fc=1150 kHz;
Fourier transform of m(t):
Spectrum of AM signal:
Substituting the values of Ac and M(f), we have

23. Study Examples SA4-2. PSD for an AM signal
Autocorrelation for a sinusoidal signal (A sin w0t )
Autocorrelation for the complex envelope of the AM signal is
Thus
Using
PSD for an AM signal:

24. Study Examples SA4-3. Average power for an AM signal
Normalized average power
Alternate method: area under PDF for s(t)
Actual average power dissipated in the 50 ohm load:
SA4-4. PEP for an AM signal
Normalized PEP:
Actual PEP for this AM voltage signal with a 50 ohm load:

25. Bandpass Signalling Bandpass Filtering and Linear Distortion
Bandpass Sampling Theorem
Bandpass Dimensionality Theorem
Amplifiers and Nonlinear Distortion
Total Harmonic Distortion (THD)
Intermodulation Distortion (IMD)

26. Bandpass Filtering and Linear Distortion
Equivalent Low-pass filter: Modeling a bandpass filter by using an equivalent low pass filter (complex impulse response)
Bandpass filter
H(f) = Y(f)/X(f)
Input bandpass waveform
Output bandpass waveform
Impulse response of the bandpass filter
Frequency response of the bandpass filter

27. Bandpass Filtering

28. Bandpass Filtering Theorem:
The complex envelopes for the input, output, and impulse response of a bandpass filter are related by
g1(t) – complex envelope of input
k(t) – complex envelope of impulse response
Also,
Proof:
Spectrum of the output is
Spectra of bandpass waveforms are related to that of their complex enveloped
But

29. Bandpass Filtering Thus, we see that
Taking inverse fourier transform on both sides
Any bandpass filter may be described and analyzed by using an equivalent low-pass filter.
Equations for equivalent LPF are usually much less complicated than those for bandpass filters & so the equivalent LPF system model is very useful.

30. Linear Distortion
For distortionless transmission of bandpass signals, the channel transfer function H(f) should satisfy the following requirements:
The amplitude response is constant
A- positive constant
The derivative of the phase response is constant
Tg – complex envelope delay
Integrating the above equation, we get
Are these requirements sufficient for distortionless transmission?

31. Linear Distortion

32. Bandpass filter delays input info by Tg , whereas the carrier by Td
Linear Distortion
The channel transfer function is
If the input to the bandpass channel is
Then the output to the channel (considering the delay Tg due to ) is
Using
Bandpass filter delays input info by Tg , whereas the carrier by Td
Modulation on the carrier is delayed by Tg & carrier by Td

33. Bandpass Sampling Theorem
If a waveform has a non-zero spectrum only over the interval , where the transmission bandwidth BT is taken to be same as absolute BW, BT=f2-f1, then the waveform may be reproduced by its sample values if the sampling rate is
Quadrature bandpass representation
Let fc be center of the bandpass:
x(t) and y(t) are absolutely bandlimited to B=BT/2
The sampling rate required to represent the baseband signal is
Quadrature bandpass representation now becomes
Where and samples are independent , two sample values are obtained for each value of n
Overall sampling rate for v(t):

34. Bandpass Dimensionality Theorem
Assume that a bandpass waveform has a nonzero spectrum only over a frequency interval , where the transmission bandwidth BT is taken to be the absolute bandwidth given by BT=f2-f1 and BT<<f1.
The waveform may be completely specified over a T0-second interval by N Independent pieces of information. N is said to be the number of dimensions required to specify the information.

35. Received Signal Pulse The signal out of the transmitter
Transmission
medium
(Channel)
Carrier
circuits
Signal
processing
Information m
input
The signal out of the transmitter
g(t) – Complex envelope of v(t)
If the channel is LTI , then received signal + noise
n(t) – Noise at the receiver input
Signal + noise at the receiver input
A – gain of the channel
- carrier phase shift caused by the channel, Tg – channel group delay.
Signal + noise at the receiver input

36. Nonlinear Distortion Amplifiers Non-linear Linear
Circuits with memory and circuits with no memory
Memory - Present output value ~ function of present input + previous input values
- contain L & C
No memory - Present output values ~ function only of its present input values.
Circuits : linear + no memory – resistive ciruits
- linear + memory – RLC ciruits (Transfer function)

37. Nonlinear Distortion
Assume no memory  Present output as a function of present input in ‘t’ domain
If the amplifier is linear
K- voltage gain of the amplifier
In practice, amplifier output becomes saturated as the amplitude of the input signal is increased.
output-to-input characteristic (Taylor’s expansion):
Where
- output dc offset level
- 1st order (linear) term
- 2nd order (square law) term

38. 2nd Harmonic Distortion with
Nonlinear Distortion
Harmonic Distortion associated with the amplifier output:
Let the input test tone be represented by
To the amplifier input
Then the second-order output term is
2nd Harmonic Distortion with =
In general, for a single-tone input, the output will be
Vn – peak value of the output at the frequency nf0
The Percentage Total Harmonic Distortion (THD) of an amplifier is defined by

39. Nonlinear Distortion
Intermodulation distortion (IMD) of the amplifier:
If the input (tone) signals are
Then the second-order output term is
IMD
Harmonic distortion at 2f1 & 2f2
Second-order IMD is:

40. Main Distortion Products
Nonlinear Distortion
Third order term is
The second term (cross-product) is
The third term is
Intermodulation terms at nonharmonic frequencies
For bandpass amplifiers, where f1 & f2 are within the pasband, f1 close to f2, the distortion products at 2f1+f2 and 2f2+f1 ~ outside the passband
Main Distortion Products

41. Bandpass Circuits Limiters Mixers, Upconverters and Downconverters
Detectors, Envelope Detector, Product Detector
Phase Locked Loops (PLL)

42. Limiters
Limiter is a nonlinear circuit with an output saturation characteristic.
It rejects envelope variations but preserves the phase variations.
Ideal limiter characteristic with illustrative input and unfiltered output waveforms.

43. Mixers
Ideal mixer is a mathematical multiplier of two input signals. One of the signals is sinusoidal generated by a local oscillator. Mixing results in frequency translation.
SSB mixer

44. Mixers

45. Choosing LO Frequency of Mixers
Up-conversion
Down-conversion
Bandpass Filter
Baseband/bandpass
Filter (fc-f0)
If (fc- f0) = 0  Low Pass Filter gives baseband spectrum
If (fc- f0 )> 0  Bandpass filter  Modulation is preserved
Filter Output:
If fc>f0  modulation on the mixer input is preserved
‘’ needs to be positive
If fc<f0 
Complex envelope is conjugated ~ sidebands are exchanged

46. Mixers (Up Converter and Down Converter)
Complex envelope of an Up Converter:
- Amplitude is scaled by A0/2
Complex envelope of a Down Converter:
i.e., f0<fc  down conversion with low-side injection
- Amplitude is scaled by A0/2
i.e., f0>fc  down conversion with high-side injection
- Sidebands are reversed from those on the input
- Amplitude is scaled by A0/2

47. Mixer Realizations Without Multipliers
Multiplication operation needed by mixers can be obtained by using a nonlinear device together with a summer.
Nonlinear device used as a mixer.

48. Mixer Realizations Without Multipliers
Multiplication operation needed by mixers can also be obtained by using an analog switch.
Linear time-varying device used as a mixer.

49. Mixer Realizations Without Multipliers
Analysis of a double-balanced mixer circuit.

50. Frequency Multiplier
Frequency Multipliers consists of a nonlinear device together with a tuned circuit. The frequency of the output is n times the frequency of the input.

51. Detector Circuits
Detectors convert input bandpass waveform into an output baseband waveform.
Detector circuits can be designed to produce R(t), Θ(t), x(t) or y(t).
Envelope Detector
Product Detector
Frequency Modulation Detector
Transmission
medium
(Channel)
Carrier
circuits
Signal
processing
Information m
input
Detector Circuits

52. Diode Envelope Detector Circuit
Ideal envelope detector: Waveform at the output is a real envelope R(t) of its input
Bandpass input:
K – Proportionality Constant
Envelope Detector Output:
Diode Envelope Detector Circuit

53. Envelope Detector
The Time Constant RC must be chosen so that the envelope variations can be followed.
In AM, detected DC is used for Automatic Gain Control (AGC)

54. Product Detector
Product Detector is a Mixer circuit that down converts input to baseband.
fc- Freq. of the oscillator
θ0- Phase of the oscillator
Output of the multiplier:
LPF passes down conversion component:
Where g(t) is the complex envelope of the input and x(t) & y(t) are the quadrature components of the input:

55. Different Detectors Obtained from Product Detector
Oscillator phase synchronized with the in-phase component
We obtain INPHASE DETECTOR.
We obtain QUADRATURE PHASE DETECTOR
We obtain ENVELOPE DETECTOR If the input has no angle modulation and reference phase (θ0) =0
We obtain PHASE DETECTOR If an angle modulated signal is present at the input and reference phase (θ0) =90
The product detector output is or
If the phase difference is small
The output is proportional to the Phase difference (Sinusoidal phase characteristics)

56. Frequency Modulation Detector
A ideal FM Detector is a device that produces an output that is proportional to the instantenous frequency of the input.
Frequency demodulation using slope detection.
The DC output can easily be blocked

57. Frequency Detector Using Freq. to Amplitude Conversion
Figure 4–16 Slope detection using a single-tuned circuit for frequency-to amplitude conversion.

58. Balanced Discriminator

59. Balanced zero-crossing FM detector.

60. Phase Locked Loop (PLL)
PLL can be used to Track Phase and Frequency of the carrier component of the incoming signal
Three basic components:
- Phase Detector : Multiplier (phase comparator)
- VCO : Voltage Controlled Oscillator
- Loop filter: LPF
Operation is similar to a feedback system
Basic PLL.

61. PLL, Voltage Controlled Oscillator (VCO)
Oscillator frequency is controlled by external voltage
Oscillation frequency varies linearly with input voltage
If e0(t) – VCO input voltage, then its output is a sinusoid of frequency
(t)=c+ce0(t)
c - free-running frequency of the VCO.
The multiplier output is further low-pass-filtered & then input to VCO
This voltage changes the frequency of the oscillator & keeps it locked.

62. Phase Locked Loop (PLL)
Let input signal be :
Let the VCO output be:
The phase detector output v1(t) is given by :
The sum frequency term is rejected by LPF so the filter output v2(t) is:
e(t) is called the Phase Error. The Phase Error voltage characteristics is SINUSOIDAL.
A PLL can track the incoming frequency only over a finite range  Lock/hold-in range
The frequency range over which the input will cause the loop to lock  pull-in/capture range

63. Phase Locked Loop (PLL)
Various types of Phase Detector characteristics used in PLL’s.

64. Figure 4–24 PLL used for coherent detection of AM.
Aplications of PLL
PLL used for coherent detection of AM signals.
A synchronized carrier signal is generated by the PLL.
VCO locks with 90 phase difference so a -90 extra phase shift is needed.
The generated carrier is used with a product detector to recover the envelope
Figure 4–24 PLL used for coherent detection of AM.

65. Figure 4–25 PLL used in a frequency synthesizer.
Aplications of PLL
PLL used as a frequency synthesizer.
Frequency dividers use integer values of M and N.
For M=1 frequency synthesizer acts as a frequency multiplier.
Figure 4–25 PLL used in a frequency synthesizer.

66. Transmitters and Receivers
Generalized Transmitters
AM PM Generation
Inphase and Quadrature Generation
Superheterodyne Receiver
Frequency Division Multiplexing

67. Generalized Transmitters
Modulating signal
Modulated signal
Transmitter
Any type of modulated signal can be represented by
The complex envelope g(t) is a function of the modulating signal m(t)
Example:

68. Generalized Transmitters
Two canonical forms for the generalized transmitter:
1. AM- PM Generation Technique: Envelope and phase functions are generated to modulate the carrier as
Figure 4–27 Generalized transmitter using the AM–PM generation technique.
R(t) and θ(t) are functions of the modulating signal m(t) as given in TABLE 4.1

69. Generalized Transmitters
2. Quadrature Generation Technique: Inphase and quadrature signals are generated to modulate the carrier as
Fig. 4–28 Generalized transmitter using the quadrature generation technique.
x(t) and y(t) are functions of the modulating signal m(t) as given in TABLE 4.1

70. IQ (In-phase and Quadrature-phase) Detector

71. Generalized Receivers
Two types of receivers:
Tuned Radio Frequency (TRF) Receiver:
Composed of RF amplifiers and detectors.
No frequency conversion
It is not often used.
Difficult to design tunable RF stages.
Difficult to obtain high gain RF amplifiers
Receivers
Superheterodyne Receiver:
Downconvert RF signal to lower IF frequency
Main amplifixcation takes place at IF

72. Tuned Radio Frequency (TRF) Receivers
Composed of RF amplifiers and detectors.
No frequency conversion. It is not often used.
Difficult to design tunable RF stages.
Difficult to obtain high gain RF amplifiers
Active
Tuning
Circuit
Detector
Local
Oscillator
Bandpass
Filter
Baseband
Audio Amp

73. Heterodyning All Incoming Fixed Frequencies Intermediate Frequency
(Upconversion/
Downconversion)
Subsequent
Processing
(common)
All
Incoming
Frequencies
Fixed
Intermediate
Frequency

74. Superheterodyne Receivers
Superheterodyne Receiver Diagram

75. Superheterodyne Receiver

76. Superheterodyne Receivers
The RF and IF frequency responses H1(f) and H2(f) are important in providing the required reception characteristics.

77. Superheterodyne Receivers
fIF
RF Response
IF Response

78. Superheterodyne Receivers

79. Superheterodyne Receiver Frequencies

80. Superheterodyne Receiver Frequencies

81. Frequency Conversion Process

82. Image frequency not a problem.
Image Frequencies
Image frequency is also received
Image frequency not a problem.

83. AM Radio Receiver

84. Superheterodyne Receiver Typical Signal Levels

85. Double-conversion block diagram.

86. Transmitter and Receiver Applications
AM and FM Radios
GSM and CDMA
TV Transmitters and Receivers
Direct Broadcast Satellite (DBS) TV

87. Complete 88- to 108-MHz stereo receiver.

88. Complete 88- to 108-MHz stereo receiver.

89. The GSM and CDMA networks.
MSC = mobile switching center; BST = base transceiver station;
BSC = base station controller.

90. MTSO linking two mobile users.

91. GSM Mobile Phone
Transmitter
Receiver

92. Mobile Phone Transmitter
CDMA Cell Phone
Mobile Phone Transmitter

93. TV receiver block diagram.

94. VHF/UHF tuner block diagram.

95. Video section block diagram.

96. Composite color TV transmission.

97. Direct Broadcast Satellite (DBS) TV

98. Direct Broadcast Satellite (DBS) TV

99. DBS-TV footprint and rain regions
150o 120o o o o

100. DBS TV Uplink Transmitter
QPSK IF amplifier upconverter LPA HPA Multiplexer
modulator M
MUX
17 GHz
uplink
antenna ~ ~
Other RF
signals
70 MHz LO
RF LO
MPEG-2
encoder
Digital
multiplexer
Reed-Soloman
encoder
Convolutional
encoder
Interleaver
MPEG 2
R-S
Coder
Conv
Coder
ADC
MUX I
Analog video
and audio
signals
Other digital
signals

101. DBS TV Receiver 12.2 – 12.7 GHz DBS-TV signal Image rejection BPF
MHz
IF amplifier
LNA
Mixer
Low noise
block converter
mounted on
antenna feed
Coaxial cable
to set top
receiver
Local oscillator
11.3 GHz
Ku-band
antenna
MHz
IF amplifier
70 MHz
IF amplifier
QPSK
demod
Baseband
amplifier
Mixer D
Tuned BPF
input
Select
polarization
Tuned LO
Select
transponder
Frequency
synthesizer
Microprocessor
Video
Inner
decoder
Outer
decoder
Demux
MPEG-2
decoder DI
D/A
Audio
Deinterleaver

102. DBS TV Receiver Front End
LNA
Mixer
MHz
IF amplifier
Coaxial cable
to set top
receiver
Ku-band
antenna
Image rejection
BPF LO
11.3 GHz
LNB mounted
on antenna feed
Select
polarization

103. DBS TV Receiver IF Section
Coaxial cable from
antenna and LNB
Baseband
amplifier
MHz
IF amplifier
70 MHz
IF amplifier
QPSK
demod
Mixer D in
Tuned BPF F P
to select
transponder
Frequency
synthesizer
Tuned LO
Microprocessor

104. DBS TV Receiver Baseband Processing
Deinterleaver
bits CD DI
R-SD
Inner convolutional
decoder
Outer R-S
decoder
Video
DeMux
MPEG
D/A
Audio
Analog
outputs
Digital
demux
MPEG-2
decoder

105. END