14/09/2019

Linear Modulation

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Index (1/2)
• Motivation for signal modulation
• Representation of the information or message by an analog signal
o bandpass signals
o Representation of the bandpass signal in baseband format
o signal and channel representation in lowpass equivalent format
• Amplitude modulation (AM)
o Information or message, carrier, waveform, bandwidth, power, spectral and energy
• Double sideband amplitude modulation (DSB)
o Spectral efficiency and bandwidth, power and energy efficiency
• Modulator for AM and DSB
o Product modulator, square law modulator for AM signal, balanced modulator, ring modulator,
switched modulator
• Frequency conversion and demodulation
• Coherent or synchronous demodulation
o Double sideband band signal, loss of synchronism in phase and frequency, single sideband
signal

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Index (2/2)
• Non-coherent demodulation
o Rectification and filtering, envelope detector
• Spectral efficiency
o Quadrature amplitude modulation (QAM)
o Single side band modulation (SSB)
• Modulators for SSB
o Method of filters and modified version
• Carrier acquisition
o Sensitivity for frequency and phase deviation, Phase lock loop (PLL), Costas receiver
• Pulse Amplitude Modulation (PAM)
• Time Division Multiplexing (TDM)

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Motivation for Signal Modulation (1/2)

• Before the information or message is transmitted through a communication channel, modulation is
performed do produce a signal that can be easily accommodated by the channel
• The modulation basically consists on the translation of the baseband signal bearing the information,
usually referred to as the message signal, to a new spectral location depending upon the intended
frequency for transmission
• We can give two motivations for the use of modulation in communication systems:
1. Frequency translation is necessary whenever the signal is to be transmitted through the atmosphere
or free space, in order to raise the signal spectrum to a frequency band in which it can be radiated
efficiently, with antennas of reasonable size
2. Modulation is used in multiplexing systems whenever two or more signals utilizes a channel, as
modulation allows translation of different signals to different spectral locations, thus allowing the
receiver to select the desired signal; multiplexing allows message signals to be transmitted by a
single transmitter and received by a single receiver simultaneously

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Motivation for Signal Modulation (2/2)

• For example: we want to transmit an audio signal (bandwidth of roughly 3 kHz) over the air:
o In the baseband format (without modulation) the electromagnetic wavelength would be 100 000
m, which forbids the physical implementation of antennas with such huge dimensions
o The signal is translated to a higher portion of the electromagnetic spectrum, for example, at the
carrier 300 MHz, which would decrease the dimension of the antenna to a much more
reasonable 1 meter length
• Modulation also overcomes limitations in hardware:
o The design of linear band pass analog amplifiers, which minimize signal distortion over its
bandpass, is a complex task, namely for modulated signals whose bandwidth 𝐵 is much higher
than the frequency 𝑓 of the carrier
o A parameter which is relevant in the dimensioning of the hardware is the fractional
bandwidth,(𝐵/𝑓 ), which relates the bandwidth 𝐵 with the carrier frequency 𝑓 , and which
should be kept in the range
𝐵
0,01 < < 0,1
𝑓

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Mapping of Information into an Analog Signal

• In Communications Systems we are going to study the transmission of energy or power signals over
channels affected by the presence of additive thermal noise, which degrades the performance of the
transmission
• The signal transmitted through the channel:
o Conveys the information (or message 𝑚 𝑡 ) generated by the source of information
o It is analytically represented by a waveform of arbitrary shape 𝑚 𝑡 whose amplitude varies
continuously with time 𝑡
• It is relevant here two enumerate the two essential features in the characterization of the signal 𝑚 𝑡 :
𝑀 𝑓
o In terms of energy, its spectra is quantified by the signal’s
∠𝑀 𝑓
bandwidth 𝐵, in in such a way that it is not relevant to consider the
energy of the signal for the range of frequencies 𝑓 > 𝐵
−𝐵 𝐵 𝑓

o The amplitude of the signal (voltage or current) is typically normalized by the average value of
its energy (in case it is an energy signal) or by its average power (in case it is a power signal)
 Maximum amplitude of 1: 𝑚 𝑡 ≤1
 Normalization by the average power: 𝑃 = 𝑥 𝑡 ≤1

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Bandpass Signals (1/3)

• 𝑥 𝑡 is a real energy signal with spectrum 𝑋 𝑓

𝑓 <𝑓 −𝐵 𝐵≪𝑓
𝑋 𝑓 = 0;
𝑓 >𝑓 +𝐵

𝑋 𝑓

𝜙 𝑓 = ∠𝑋 𝑓 𝑋 𝑓

0
−𝑓 𝑓 −𝐵 𝑓 𝑓 +𝐵 𝑓

𝑥 𝑡
1/𝑓
𝐴 𝑡

• Representation of the real signal’s waveform 𝑥 𝑡 in the polar format (envelope and phase)
o 𝐴 𝑡 : envelope of signal 𝑥 𝑡
𝑥 𝑡 = 𝐴 𝑡 cos 𝜔 𝑡 + 𝜙 𝑡
o 𝜃 𝑡 : phase of signal 𝑥 𝑡

• It is worth mentioning here that both the amplitude 𝐴 𝑡 as well as the phase 𝜙 𝑡 of signal 𝑥 𝑡 can
vary with time

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Bandpass Signals (2/3)

Representation of bandpass signals in polar format
• Representation of the bandpass signal 𝑥 𝑡 as a complex exponential in the complex plane

𝑥 𝑡 𝐴 𝑡 is the length of the vector

𝐴 𝑡 𝑥 𝑡
𝐴 𝑡  
𝐴 𝑡 = 𝑥 𝑡 +𝑥 𝑡
𝜔 𝑡+𝜙 𝑡 𝑥 𝑡
𝜙 𝑡
𝜙 𝑡 is the angle between the x axis and the vector
𝑥 𝑡 𝑥 𝑡
• The in-phase and quadrature components of the bandpass signal 𝜙 𝑡 = tan
𝑥 𝑡
result from the complex exponential notation
𝑥 𝑡 = 𝐴 𝑡 cos 𝜙 𝑡 o 𝑥 𝑡 is the in-phase component
𝑥 𝑡 = 𝐴 𝑡 sin 𝜙 𝑡 o 𝑥 𝑡 is the quadrature component

• Both components 𝑥 𝑡 e 𝑥 𝑡 are real baseband signals (bandwidth 𝐵), and orthogonal to each other
(its inner product is zero)
Quadrature-carrier description of a bandpass signal
• The bandpass signal 𝑥 𝑡 is represented as the sum of the in-phase and quadrature components

𝑥 𝑡 = 𝑥 𝑡 cos 𝜔 𝑡 − 𝑥 𝑡 sin 𝜔 𝑡
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Bandpass Signals (3/3)

• In the frequency domain, and with reference to the properties of the Fourier transform

𝑥 𝑡 ⟺𝑋 𝑓
1 j
𝑋 𝑓 = 𝑋 𝑓−𝑓 +𝑋 𝑓+𝑓 + 𝑋 𝑓−𝑓 −𝑋 𝑓+𝑓 with
2 2 𝑥 𝑡 ⟺𝑋 𝑓

• From the analytic expression of the spectrum 𝑋 𝑓 we are sure that the signal 𝑥 𝑡 is effectively a
bandpass one whenever the in-phase 𝑥 𝑡 and quadrature 𝑥 𝑡 components are baseband signals

𝑋 𝑓 = 𝑋 𝑓 = 0, 𝑓 >𝐵

• The spectra 𝑋 𝑓 results from the association of two baseband signals, translated to frequency 𝑓 of
the carrier, and for this reason, the signal 𝑥 𝑡 resulting from this association is bandpass
• It is important to mention that the quadrature-carrier description of bandpass signals is only possible
as long as the bandwidth of the bandpass signal (2𝐵) is smaller than the frequency 𝑓 of the carrier
(and this is typically true in realistic bandpass communication systems)
• In modulation and demodulation operations this condition asserts that the positive and negative
parts of the spectrum do not interfere to each other
• If there would be interference it would be very difficult, or even impossible, for the receiver to
identify and separate both components of the bandpass signal
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Bandpass to Lowpass Conversion (1/3)

Lowpass equivalent or complex form representation of a bandpass signal
• Representation of the bandpass signal 𝑥 𝑡 as lowpass equivalent in the frequency domain 𝑋 𝑓

𝑢 𝑓 is the unit step function in the frequency domain

1
𝑋 𝑓 = 𝑋 𝑓 + j𝑋 𝑓 =𝑋 𝑓+𝑓 𝑢 𝑓+𝑓
2 1, 𝑓≥0
𝑢 𝑓 =
0, 𝑓<0
𝑋 𝑓
o 𝑋 𝑓 stems from the positive part of the spectrum of the
𝑋 𝑓
bandpass signal 𝑋 𝑓 , after translation to DC (𝑓 = 0)
∠𝑋 𝑓
o Contrary to the bandpass spectrum 𝑋 𝑓 (the bandpass
signal is a real one and therefore has Hermitian symmetry),
−𝐵 0 𝐵 𝑓
the representation in lowpass format results in a complex
signal (it has no symmetry)
o The waveform in time domain is a complex function

• In the time domain, the envelope of the band pass signal 𝑥 𝑡 , represented as lowpass equivalent
𝑥 𝑡 , stems from the application of the inverse Fourier transform
1 1
𝑥 𝑡 = 𝑥 𝑡 + j𝑥 𝑡 = 𝐴 𝑡 e
2 2
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Bandpass to Lowpass Conversion (2/3)

Lowpass (complex) equivalent and real bandpass signal representations
• We can change from the representation in bandpass (real) format 𝑥 𝑡 to the representation on
lowpass (complex) equivalent format 𝑥 𝑡 , and vice-versa

1
• In the time domain 𝑥 𝑡 = Re 𝐴 𝑡 e = 2 Re 𝐴 𝑡 𝑒 𝑒
2

𝑥 𝑡

𝑥 𝑡 = 2 Re 𝑥 𝑡 e

• In the frequency domain, using the property of complex numbers

Note:

2Re 𝑧 𝑡 = 𝑧 𝑡 + j𝑧 ∗ 𝑡
𝑋 𝑓 = 𝑋 𝑓 − 𝑓 + 𝑋 ∗ −𝑓 − 𝑓

• Signals are real functions (or waveforms) in real communication systems, and thus it is possible to
use the Hermitian symmetry in the spectrum 𝑋 𝑓 of bandpass signals to use the simplified analytical
expression
𝑋 𝑓 =𝑋 𝑓−𝑓 , 𝑓>0

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Bandpass to Lowpass Conversion (3/3)

𝑥 𝑡
Block diagram of the receiver
cos 𝜔 𝑡
• It generates at the output the in-phase and
quadrature baseband components from the
𝑥 𝑡 𝑓
bandpass signal at the input

sin 𝜔 𝑡
𝑥 𝑡

cos 𝜔 𝑡
𝑥 𝑡
Block diagram of the linear modulator at the
emitter
• It generates the bandpass signal at the output 𝑥 𝑡
from the in phase and quadrature baseband
components at the input
𝑥 𝑡
sin 𝜔 𝑡

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Equivalent Representations of Signal and Channel (1/3)

Bandpass systems
• It is possible to analyse the transmission of bandpass signals through the bandpass channel using the
lowpass equivalent representation, for both the signal as well as the impulse response of the channel

𝑥 𝑡 = 2 Re 𝑥 𝑡 e 𝑦 𝑡 = 2 Re 𝑦 𝑡 e
𝑥 𝑡 𝑦 𝑡
𝐻 𝑓 𝐻 𝑓

• In effect, the analysis in time domain using the lowpass equivalent representation is simpler from the
analytical point of view

𝑌 𝑓 =𝐻 𝑓 𝑋 𝑓

• 𝐻 𝑓 is the transfer function of the channel in the lowpass equivalent representation

𝐻 𝑓 = 𝐻 𝑓+𝑓 𝑢 𝑓+𝑓

• Applying the inverse Fourier transform results the signal at the output of the channel in the time
domain
𝑦 𝑡 =ℱ 𝑌 𝑓 𝑋 𝑓 =ℱ 𝑌 𝑓 𝐻 𝑓

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Equivalent Representations of Signal and Channel (2/3)

• Making the conversion to the bandpass representation
1
𝑦 𝑡 = Re 𝐴 𝑡 e = 2 Re 𝐴 𝑡 e e = 2 Re 𝑦 𝑡 e
2

• The in-phase 𝑦 𝑡 and quadrature 𝑦 𝑡 components, as well as the envelope 𝐴 𝑡 and phase 𝜃 𝑡
components of the signal y 𝑡 at the output of the channel can be obtained directly from the
waveform representation of the signal in lowpass format 𝑦 𝑡

𝑦 𝑡 = 2 Re 𝑦 𝑡 𝐴 𝑡 =2𝑦 𝑡

𝑦 𝑡 = 2 Im 𝑦 𝑡 𝜙 𝑡 = ∠𝑦 𝑡
Time domain analysis
• Taking into consideration the representation of the input and output signals in the lowpass
equivalent format
𝑦 𝑡 =ℎ 𝑡 ∗𝑥 𝑡 = 𝑥 𝑡 𝑥 𝑡 − 𝜏 𝑑𝜏

• and the decomposition of the respective in-phase and quadrature components

𝑦 𝑡 + j𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 − ℎ 𝑡 ∗ 𝑥 𝑡 +j ℎ 𝑡 ∗𝑥 𝑡 +ℎ 𝑡 ∗𝑥 𝑡

𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 = ℎ 𝑡 + jℎ 𝑡 ∗ 𝑥 𝑡 + 𝑗𝑥 𝑡

= ℎ 𝑡 ∗𝑥 𝑡 −ℎ 𝑡 ∗𝑥 𝑡 +j ℎ 𝑡 ∗𝑥 𝑡 +ℎ 𝑡 ∗𝑥 𝑡
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Equivalent Representations of Signal and Channel (3/3)

Time domain analysis
• We group the real and imaginary components from both sides of the equation

𝑦 𝑡 + j𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 − ℎ 𝑡 ∗ 𝑥 𝑡 +j ℎ 𝑡 ∗𝑥 𝑡 +ℎ 𝑡 ∗𝑥 𝑡

𝑥 𝑡 ℎ 𝑡
𝑦 𝑡 =ℎ 𝑡 ∗𝑥 𝑡 −ℎ 𝑡 ∗𝑥 𝑡 𝑦 𝑡

𝑦 𝑡 =ℎ 𝑡 ∗𝑥 𝑡 +ℎ 𝑡 ∗𝑥 𝑡 ℎ 𝑡

• Block diagram of the equivalent model for the ℎ 𝑡

transmission of the signal through the bandpass
𝑦 𝑡
channel, with centre frequency 𝑓 (carrier) 𝑥 𝑡 ℎ 𝑡

• From this model we can determine the in-phase and quadrature components of the bandpass signal
at the output of the channel, as well as the waveforms which compose the models for real signals in
time domain and lowpass representation
• This model has application in the simulation of communications systems and in signal processing, as
it avoids arithmetic operations of addition and multiplication of complex numbers, i.e., all arithmetic
operations are executed with real numbers in time domain
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Amplitude or Linear Modulation (1/8)

Information or message
• We want to transmit the information bearing signal (or message) 𝑚 𝑡 through the channel
• 𝑚 𝑡 is a baseband signal as its energy is concentrated at the lower part of the electromagnetic
spectrum, i.e., close to DC
o Some examples of useful information signals:
 Voice in analog format (telephone): 0 - 3,5 kHz
 Video in analog format (television): 0 - 4,3 MHz
Carrier
• The information is mapped into the sinusoidal waveform 𝑐 𝑡 (named the carrier), which can be a
voltage or current signal

o 𝐴 : carrier amplitude (V)

𝑐 𝑡 = 𝐴 cos 𝜔 𝑡 + 𝜃 = 𝐴 cos 2𝜋𝑓 𝑡 + 𝜃 o 𝜔 = 2𝜋𝑓 : angular frequency (rad/s)

o 𝑓 : real frequency (Hz)
o 𝜃 : carrier’s initial phase (rads)

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Amplitude or Linear Modulation (2/8)

Amplitude modulation (AM)
• In amplitude modulation the amplitude of the sinusoidal carrier with frequency 𝑓 varies over time, in
proportion to the amplitude of the message waveform 𝑚 𝑡
• The information in the message is impressed in the amplitude of the carrier, i.e., the envelope of the
modulated carrier has the same shape as the message waveform 𝑚 𝑡
• If 𝐴 denotes the unmodulated carrier amplitude, modulation by 𝑚 𝑡 produces the modulated envelope

𝐴 𝑡 =𝐴 1+𝜇×𝑚 𝑡 Parameter 𝜇 is a positive constant named modulation index

• Carrier’s frequency 𝑓 and initial phase 𝜃 remain fixed over time

• For analytical simplification the initial phase is typically assumed as 𝜃 = 0

Complete AM signal waveform in time domain

𝑥 𝑡 = 𝐴 1 + 𝜇 × 𝑚 𝑡 cos 𝜔 𝑡 = 𝐴 cos 𝜔 𝑡 + 𝐴 𝜇 × 𝑚 𝑡 cos 𝜔 𝑡

• With reference to the representation of the bandpass signal in quadrature-carrier description we

observe that, as the AM signal has no time-varying phase its in-phase and quadrature components are
In-phase component 𝑥 𝑡 =𝐴 1+𝜇×𝑚 𝑡 Quadrature component 𝑥 𝑡 =0

• i.e., information is only conveyed in the in-phase component

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Amplitude or Linear Modulation (3/8)

• From the waveform expression 𝐴 + 𝜇 × 𝑚 𝑡 cos 𝜔 𝑡 of the AM signal it can be seen that that the
carrier amplitude 𝐴 is sent together with the message modulated in the amplitude of the carrier
𝑚 𝑡 𝜇>1
𝑥 𝑡
Phase inversion
𝐴
𝑡
𝑡
𝜇<1 −𝐴
𝑥 𝑡
𝐴 =𝐴 1+𝜇
𝐴
𝐴 =𝐴 1−𝜇 Over-modulation: 𝝁 > 𝟏
• Over-modulation causes phase reversals and
𝑡
−𝐴 envelope distortion

Under-modulation: 𝝁 ≤ 𝟏 • The rectifying diode rectifies the signal and,

• Under-modulation ensures that 𝐴 + 𝜇 × 𝑚 𝑡 does not therefore, the envelope detector is not able

go negative and that the envelope detector recovers the to recover the negative amplitudes

carrier’s envelope • The waveform rectification changes the

• Condition 𝑓 ≫ 𝐵 must be accomplished, i.e., the carrier’s format of the envelope relative to the

oscillation has to be significantly faster than the variation message waveform 𝑚 𝑡 to be recovered

of the information signal with time, for the envelope 𝐴 𝑡

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Amplitude or Linear Modulation (4/8)

Bandwidth and spectral efficiency
• Applying Fourier transform to the AM signal, and bearing in mind that the AM signal is real, i.e. that
it has Hermitian symmetry, it is enough to consider the positive portion of the spectrum
1 𝜇
𝑋 𝑓 = 𝐴 𝛿 𝑓−𝑓 + 𝐴 𝑀 𝑓−𝑓 ; 𝑓 >0
2 2
𝑋 𝑓 carrier

Bandwidth: 2𝐵 Lower Upper

sideband sideband

−𝑓 0 𝑓 −𝐵 𝑓 𝑓 +𝐵
𝑓
2𝐵

• The two impulses at the carrier frequency ±𝑓 are the lines of the unmodulated sinusoidal carrier
which is sent together with the carrier modulated in amplitude
• The bandwidth of the AM signal is twice the bandwidth of the information signal
• We observe the existence of the two sidebands, a lower one (LSB) and an upper one (USB), as the AM
signal is real, and thus has symmetry around the carrier frequency
• For this reason the AM signal is also named Double Sideband Amplitude Modulation (DSB)

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Amplitude or Linear Modulation (5/8)

Power and energy efficiency
• A parameter of fundamental importance in the evaluation of the performance of any modulation
scheme is the average power of the modulated signal
1 1
𝑃 = 𝑥 𝑡 = 𝐴 1 + 2𝜇 × 𝑚 𝑡 + 𝜇 × 𝑚 𝑡 + 𝐴 1+𝜇×𝑚 𝑡 cos 2𝜔 𝑡
2 2

with 𝑚 𝑡 = 0 and 𝑃 𝑡 = 𝑚 𝑡 with 𝑓 ≫ 𝐵 ⟹ cos 2𝜔 𝑡 = 0

1
• It results 𝑃 = 𝐴 1+𝜇 ×𝑃
2
1
Average power of the unmodulated carrier 𝑃 = 𝐴
2
𝑃 = 𝑃 + 2𝑃
Average power in each sideband 𝑃 = 𝐴 ×𝜇 ×𝑃 = 𝜇 𝑃 ×𝑃

• From the condition imposed upon the modulation index 𝜇 × 𝑚 𝑡 ≤ 1 it results 𝜇 × 𝑃 ≤ 1

1 1 1
𝑃 ≤ 𝑃 𝑃 = 𝑃 − 2𝑃 ≥ 𝑃 𝑃 ≤ 𝑃
2 2 4

o In AM at least 50% of the power available in the emitter is used in the transmission of the
unmodulated carrier
o The unmodulated carrier does not convey information, and this reflects in the poor energy
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Amplitude or Linear Modulation (6/8)

Tone-modulation: the modulating waveform is a single sinusoid
• The message or information signal 𝑚 𝑡 is a single sinusoid with amplitude 𝐴 and frequency 𝑓 Hz

𝑚 𝑡 = 𝐴 cos 𝜔 𝑡 𝑚 =𝐴 e 𝑚 = −𝐴

• Modulation index
𝐴 − −𝐴 𝐴
𝜇= = →𝐴 = 𝜇𝐴 𝑚 𝑡 = 𝜇𝐴 cos 𝜔 𝑡
2𝐴 + 𝐴 + −𝐴 𝐴

• AM signal 𝑥 𝑡 = 𝐴 + 𝐴 cos 𝜔 𝑡 cos 𝜔 𝑡 = 𝐴 1 +

𝐴
cos 𝜔 𝑡 cos 𝜔 𝑡
𝐴

𝑥 𝑡 = 𝐴 1 + 𝜇 × cos 𝜔 𝑡 cos 𝜔 𝑡

𝜇 = 0,5 𝜇=1
1 + cos 𝜔 𝑡
1 + 0,5 cos 𝜔 𝑡
𝐴 /2 𝐴
𝐴 𝐴 /2 𝐴

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Amplitude or Linear Modulation (7/8)

Tone-modulation: the modulating waveform is a single sinusoid (continuation)

1
𝑚 𝑡 = 𝜇𝐴 cos 𝜔 𝑡 𝑚 𝑡 = 𝜇 𝐴
2
Energy efficiency 1
𝜂=
𝑚 𝑡
= 2𝜇 𝐴 =
𝜇
%
𝐴 + 𝑚 𝑡 1 2+𝜇
𝐴 + 𝜇 𝐴
2
• We note that we need to verify the condition 0 ≤ 𝜇 ≤ 1, and therefore we can observe that:
o The efficiency 𝜂 increases continuously with the modulation index 𝜇
o Its peak value is achieved for 𝜇 = 1 (100% modulation)
𝜇 1
𝜂 = = ≈ 33%
2+𝜇 3
• Under the best conditions of full modulation (𝜇 = 1), roughly 2/3 of the power available in the emitter
for the diffusion of the AM signal is spent is the transmission of the unmodulated carrier alone
o In reality, efficiency can be even worse, namely when 𝜇 < 1
• All information in the modulator output is contained in the sidebands
o Unmodulated carrier is wasted power as far as information transfer is concerned
o This fact can be of considerable importance in an environment where power is limited
and can completely preclude the use of AM as a modulation technique in power-limited
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Amplitude or Linear Modulation (8/8)

Tone-modulation: the modulating waveform is a single sinusoid (continuation)
• Phasor diagram representation of the AM signal modulated by a single sinusoid with 𝜇𝐴 = 2/3

1 • The two vectors representing the two sidebands rotate with equal
𝐴
2 angular frequencies 𝜔 ± 𝜔 , or with angular frequencies ±𝜔
𝜔
relative to the carrier frequency 𝜔
• As expected, the sum of the vectors of both sidebands (upper and
𝐴 𝐴 𝑡
lower sides) result in a single vector, collinear with the real axis
𝜔
1
• This is due to the symmetry inherent to the AM signal (a real
𝐴
2 function of time)
• What happens if one of the sidebands is eliminated by the channel?
o The signal has no symmetry anymore
o The phasor of the AM signal makes an angle 𝜃 𝑡 with the real axis (the length of the envelope 𝐴 𝑡 is
smaller) and both A 𝑡 and 𝜃 𝑡 change with time (which is something characteristic of the distortion
resulting from the elimination of the lower sideband by the channel)
𝐴 𝑡
1
𝐴 sin 𝜔 𝑡
𝜔 𝑡 3
𝜃 𝑡

𝐴
1
𝐴 cos 𝜔 𝑡
3 SC - 2019/2020 - Alberto Nascimento 23

Double Side Band Modulation with Suppressed Carrier (1/4)

• The “squander” of power incurred by the transmission of the unmodulated carrier over the channel
with AM can be overcome if:
1. The modulation index is programmed to its highest admissible level (𝜇 = 1)
2. The unmodulated carrier is supressed
• This particular form of amplitude modulation is designated “Double Side Band modulation with
Suppressed Carrier” (DSB)

𝑥 𝑡 = 𝐴 𝑚 𝑡 cos 2𝜋𝑓 𝑡 𝑚 𝑡

• Envelope and phase of the DSB carrier 𝑡

Envelope 𝐴 𝑡 =𝐴 𝑚 𝑡 𝑥 𝑡 𝐴 =𝐴
𝐴

0; 𝑚 𝑡 > 0 𝑡
Phase 𝜃 𝑡 =
±𝜋; 𝑚 𝑡 < 0
−𝐴
Inversion of the phase

• In DSB modulation the envelope has the shape of the magnitude of the message waveform 𝑚 𝑡
and, therefore, the phase of the DSB signal is inverted whenever the amplitude becomes negative
• For this reason it is not possible to recover the information 𝑚 𝑡 with the envelope detector

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Double Side Band Modulation with Suppressed Carrier (2/4)

1
Bandwidth and spectral efficiency 𝑋 𝑓 = 𝐴 𝑀 𝑓−𝑓 ; 𝑓 >0
2
𝑋 𝑓

Bandwidth: 2𝐵

Average Power
⁄ −𝑓 0 𝑓 −𝐵 𝑓 𝑓 +𝐵
1 𝑓
𝑃 = 𝑥 𝑡 = lim 𝐴 𝑚 𝑡 cos 𝜔 𝑡 𝑑𝑡 2𝐵
→ 𝑇 ⁄

Average Power of the information signal 𝑚 𝑡

1
𝑃 = 𝑥 𝑡 = 𝐴 𝑚 𝑡 = 2𝑃
2 𝑃 = 𝑚 𝑡

• As expected, DSB modulation has better power efficiency than AM, as it makes better use of the
available power in the emitter by not transmitting the carrier unmodulated over the channel
• This result is valid even when the information waveform 𝑚 𝑡 has a DC component (we shall
remember that, previously in the AM modulation, it was assumed that 𝑚 𝑡 had no DC value
• Spectral optimization and regulation, interference mitigation, and operational and safety margins of
radio equipment (power amplifier’s, etc. ) impose limits to the peak of power achieved by the
carrier’s envelope, 𝐴
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Double Side Band Modulation with Suppressed Carrier (3/4)

Power efficiency
• We want to compare performances of AM and DSB with reference to the power efficiency of both
schemes, using as a common reference the peak of the carrier’s envelope power (𝐴 ) and with
reference to the same value of the emitter’s average power (𝑃 )
• The comparison is carried out assuming that the carrier is tone modulated by a sinusoid with
amplitude 𝐴 , and using the maximum allowable value of the carrier’s amplitude (𝐴 =𝐴 )
• For the DSB signal we replace this value in the expression relating average power in both sidebands
with average power
1 𝑃 𝑃
𝑃 = 𝐴 𝑃 = 2𝑃 =
2 𝐴 4

• Under the same conditions for the AM signal and after making the necessary modifications (𝐴 =
2𝐴 and 𝜇 = 1) in the analytic expression of the average power per sideband, it results
1 𝑃 𝑃
𝑃 = 𝐴 ×𝑃 =
16 𝐴 16

• Under these assumptions we note that the average power per sideband used in the diffusion of the
DSB signal is four times higher than the power used in the diffusion of AM
• The poor efficiency of AM is the price we have to pay for the use of a much simpler receiver (the
envelope detector) in demodulating the AM signal, than with DSB (use of a coherent receiver)
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Double Side Band Modulation with Suppressed Carrier (4/4)

Special case: tone modulation 1
𝑀 𝑓 = 𝛿 𝑓−𝑓 +𝛿 𝑓+𝑓
2
𝑚 𝑡 = cos 𝜔 𝑡 = cos 2𝜋𝑓 𝑡

1
𝑥 𝑡 = 𝑚 𝑡 cos 2𝜋𝑓 𝑡 = cos 𝜔 + 𝜔 𝑡 + cos 𝜔 − 𝜔 𝑡
2

Spectrum of DSB-SC signal

Suppressed by the lowpass filter Suppressed by the lowpass filter

1
𝑟 𝑡 = cos 𝜔 𝑡 cos 𝜔 𝑡 = cos 𝜔 𝑡 1 + cos 2𝜔 𝑡
2
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AM and DSB Modulators (1/7)

• Both sidebands of AM and DSB contain lines in spectra which did not exist (both in the carrier and in
the message waveforms) before modulation
• the circuit which physically implement the linear modulation must be a nonlinear device varying
with time Block diagram of the AM modulator
Product modulator 𝜇𝑚 𝑡
x +
Analog multiplier 𝑥 𝑡 = 𝐴 cos 𝜔 𝑡 + 𝜇𝑚 𝑡 𝐴 cos 𝜔 𝑡
~ 𝐴 cos 𝜔 𝑡

Analog multiplication can be carried out electronically with the variable

transconductance multiplier
o Input voltage v1 is applied to a differential amplifier whose gain
depends on the transconductance of the transistors which, in turn,
varies with the total emitter current
o Input voltage v2 controls the emitter current by means of a voltage to
current converter
o The differential output 𝑣 equals 𝐾𝑣 𝑣

• Analog multipliers have a very important limitation: their application is to signals with low power
levels and relatively low frequencies only
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AM and DSB Modulators (2/7)

Square law modulator
• Signal multiplication at higher frequencies can be accomplished by the square law modulator, in
which a nonlinear element approximates the square-law transfer function
RLC bandpass filter
square law device: Transfer function NLD cos 𝜔 𝑡
𝑣 =𝑎 𝑣 +𝑎 𝑣
𝑚 𝑡
nonlinear 𝑥 𝑡
+ Filter
device 𝑚 𝑡
𝑣 𝑡 𝑣 𝑡 𝑉

~ cos 𝜔 𝑡
FET transistor polarized in the saturation
region (nonlinear mode)
Signal at the input 𝑣 𝑡 = 𝑚 𝑡 + cos 𝜔 𝑡
2𝑎
Signal at the output 𝑣 𝑡 = 𝑎 𝑚 𝑡 + 𝑎 𝑚 𝑡 + 𝑎 cos 𝜔 𝑡 + 𝑎 1 +
𝑎
𝑚 𝑡 cos 𝜔 𝑡

Supressed by the RLC filter

• Making the substitution 𝐴 = 𝑎 and 𝜇 = 2𝑎 /𝑎 and assuming that the bandpass RLC filter is
selective enough to filter out the undesirable components, it results at the output the AM signal with
the following analytical waveform
𝑣 𝑡 = 𝐴 1 + 𝜇𝑚 𝑡 cos 𝜔 𝑡

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AM and DSB Modulators (3/7)

Square law modulator (continuation) Note: 𝑉 𝑓 =ℱ 𝑣 𝑡
1 1 1
𝑎 𝛿 𝑓 𝑎 𝛿 𝑓−𝑓 𝑎 𝛿 𝑓 − 2𝑓
2 2 4 ℱ 𝑚 𝑡 =𝑀 𝑓 ∗𝑀 𝑓

𝑎 𝑀 𝑓 o 𝑓 > 3𝐵 to avoid spectral overlapping

𝑎 𝑀 𝑓−𝑓
𝑎 𝑀 𝑓 ∗𝑀 𝑓 o The RLC bandpass filter has bandwidth
𝐵 = 2𝐵, with centre frequency at 𝑓 to
𝑓
0 𝐵 2𝐵 𝑓 −𝐵 𝑓 𝑓 +𝐵 2𝑓 filter out the undesirable spectral lines

Square law modulator for DSB

• If 𝑎 = 0 we have the square law device for the DSB Balanced Modulator

signal at the output: 𝑣 =𝑎 𝑣

1
𝐴 1 + 𝑚 𝑡 cos 𝜔 𝑡
• It filters out the carrier and generate the DSB signal 1
𝑚 𝑡
2
2
Balanced modulator
• There exist no two perfect square-law devices in reality 𝑚 𝑡 𝐴 cos 𝜔 𝑡

• The alternative is two use two AM-modulators arranged

𝐴 cos 𝜔 𝑡
in a balanced configuration to cancel out the carrier 1
− 𝑚 𝑡
2
• We assume the AM modulators are identical, save for 1
𝐴 1 − 𝑚 𝑡 cos 𝜔 𝑡
2
the reversed sign of one input
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AM and DSB Modulators (4/7)

Balanced modulator 𝑦 𝑡 = 𝑎𝑥 𝑡 + 𝑏𝑥 𝑡

NL – non-linear device
𝑚 𝑡 𝑥 𝑡 𝑦 𝑡

𝑧 𝑡
±𝜔
4𝑏𝑚 𝑡 cos 𝜔 𝑡

cos 𝜔 𝑡 𝑥 𝑡 𝑦 𝑡

𝑦 𝑡 = 𝑎 × 𝑚 𝑡 + cos 𝜔 𝑡 + 𝑏 × 𝑚 𝑡 + cos 𝜔 𝑡 𝑧 𝑡 = 𝑦 𝑡 − 𝑦 𝑡 = 2𝑎 × 𝑚 𝑡 + 4𝑏 × 𝑚 𝑡 × cos 𝜔 𝑡

𝑦 𝑡 = 𝑎 × −𝑚 𝑡 + cos 𝜔 𝑡 + 𝑏 × −𝑚 𝑡 + cos 𝜔 𝑡
Attenuated by the bandpass filter It goes through the
bandpass filter

• Simple balanced modulator: at the input we have the carrier (to compose the modulated signal) but it
does not show up at the output (the adder rejects it)
• Complete balanced modulator: at the input we have both waveforms: the carrier and the message; at
the output we have none (in isolation of course)
• Note: in the absence of the carrier, and with the assumption that both transformers are well balanced
and the diodes are perfectly identical, there is no signal at the output of the bandpass filter
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AM and DSB Modulators (5/7)

Ring modulator for DSB modulation
• The 4 diodes, D , D , D e D , which compose the ring are controlled by the square wave 𝑐 𝑡 with
𝐷
frequency 𝑓

1. When 𝑐 𝑡 > 0 ⟹ D1 and D3 are switched on 𝐷

𝑚 𝑡 𝑣 𝑡 𝑣 𝑡
and D2 and D4 are switched off 𝑣 𝑡 = 𝑚(𝑡) 𝐷

2. When 𝑐 𝑡 > 0 ⟹ D2 and D4 are switched on

𝐷
and D1 and D3 are switched off 𝑣 𝑡 = −𝑚(𝑡)
𝑣 𝑡 =𝑚 𝑡 ×𝑐 𝑡
• The ring modulator behaves as an analog 𝑚 𝑡 𝑐 𝑡

multiplier for signals 𝑚(𝑡) and 𝑐 𝑡 𝑡

𝑡

• 𝑐 𝑡 is a periodic wave (period 1/𝑓 ) and as such can be expressed in Fourier series
• We express 𝑐 𝑡 in Fourier trigonometric series and multiply if by the message waveform 𝑚 𝑡 , and
thus, at the output of the transformer results the balanced signal
4 4 4
𝑣 𝑡 = 𝑚 𝑡 ×𝑐 𝑡 = 𝑚 𝑡 cos 𝜔 𝑡 − 𝑚 𝑡 cos 3𝜔 𝑡 + 𝑚 𝑡 cos 5𝜔 𝑡 − ⋯
𝜋 3𝜋 5𝜋
• The DSB signal can be obtained by passing 𝑣 𝑡
𝑥 𝑡 = 𝑘𝑚 𝑡 cos 𝜔 𝑡 with 𝑘 = 4/𝜋
through a bandpass filter with bandwidth 2𝐵
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AM and DSB Modulators (6/7)

• An inherent limitation to the operations principle of the square law modulator is the selectivity
imposed to the bandpass filter
• We shall remember that the bandpass of this filter is centered at the carrier’s frequency 𝑓 , the
bandwidth is 2𝐵 (for AM or DSB signals), and at the same time the intermodulation products which
result from the nonlinearity of the square law device are filtered out (these are not relevant to the
final result)
• Therefore, the application of this kind of modulator is restricted to the generation of signals of low
power, which do not require the use of filters with too much selectivity; otherwise, with a filter with
“normal” selectivity” the use of high power signals would not avoid the spectral leakage to nearby
channels, and would result in adjacent channel interference antenna
• Signal average power 𝑃 must be appropriate for the
signal diffusion over the air and therefore we should
message
have at our disposal at the output of the square law
audio
modulator
modulator, and before the antenna for signal amplifier

amplification, linear power amplifiers, 𝑓 power

• However power amplifiers used in RF typically ~ amplifier
crystal oscillator
operate in the saturation region, as this is the best
mode of operation regarding power efficiency SC - 2019/2020 - Alberto Nascimento 33

AM and DSB Modulators (7/7)

• The best approach to this setback is to opt for a modulator which generate the AM or DSB signal with the
average power already appropriate for the diffusion over the air
• The principle of this modulator avoids the use of nonlinear devices which create intermodulation products
inherent to the nonlinear characteristic, and, as a consequence avoid the use of the bandpass filters that must
used to filter out these nonlinearities • The active device is a transistor operating as a switch driven

Switched modulator at the carrier frequency, closing briefly every 1/𝑓 , so the
𝑚 𝑡 switching action causes the tank circuit to “ring” sinusoidally
active element “tank circuit”
• The tank circuit is a resonant oscillator with resonant
1: 𝑁
𝑓 frequency 1/𝑓
𝑉 𝑣 𝑡
• The steady-state load voltage in absence of modulation is
𝑣 𝑡 = 𝑉 cos 𝜔 𝑡

𝑚 𝑡
• The message 𝑚 𝑡 is amplified by the transformer with turns ratio 𝑁 𝑉 + 𝑁𝑚 𝑡

• Signal 𝑁𝑚 𝑡 at the transformer output is added to the signal from the DC

voltage source, with voltage 𝑉, resulting 𝑉 + 𝑁𝑚 𝑡 at the input of the switch 𝑡

• If voltage 𝑉 and turns ratio 𝑁 are correctly proportioned the desired

modulation is accomplished without appreciable generation of undesired
𝑣 𝑡 = 𝑉 + 𝑁𝑚 𝑡 cos 𝜔 𝑡
components (intermodulation products)

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Frequency Conversion (1/2)

• Amplitude modulation in the emitter is basically an upward translation of the message spectrum
because the higher portions of the channel are better suited for the transmission of the information
• Demodulation in the receiver is the inverse operation, i.e., downward frequency translation to the
lower portion of the spectrum, in order to recover the message from the modulated wave
• Demodulators that perform this operation fall into the two broad categories of synchronous detectors
and envelope detectors
• Translation of frequency bands is carried out by the frequency converter or mixer
• Downconversion
o The converter changes the signal from bandpass to baseband format
o The mixer changes the frequency of the carrier from radio frequency (RF) 𝑓 to intermediate
frequency (IF) 𝑓
𝑚 𝑡 × cos 𝜔 𝑡 𝑥 𝑡 𝑚 𝑡 × cos 𝜔 𝑡
BPF tuned at 𝜔
Signal at the input of the filter

𝑥 𝑡 = 2𝑚 𝑡 × cos 𝜔 𝑡 × cos 𝜔 𝑡 2 cos 𝜔 ± 𝜔 𝑡

Frequency of the local oscillator of the mixer Signal at the output of the filter

𝜔 =𝜔 ±𝜔 𝑥 𝑡 = 𝑚 𝑡 × cos 𝜔 − 𝜔 𝑡 + cos 𝜔 + 𝜔 𝑡

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Frequency Conversion (2/2)

Carrier frequency 𝑓 is translated in the spectrum to the intermediate frequency 𝑓

Up conversion: modulation operation

Frequency of the local oscillator

𝜔 =𝜔 +𝜔 𝑥 𝑡 = 𝑚 𝑡 × cos 𝜔 𝑡 + 𝑚 𝑡 × cos 2𝜔 + 𝜔 𝑡

Down conversion: demodulation operation

Frequency of the local oscillator

𝜔 =𝜔 −𝜔 𝑥 𝑡 = 𝑥 𝑡 = 𝑚 𝑡 × cos 𝜔 𝑡 + 𝑚 𝑡 × cos 2𝜔 − 𝜔 𝑡

bandpass filter centered at intermediate frequency 𝜔 passes the component 𝑚 𝑡 cos 𝜔 𝑡

downconversion upconversion

0 𝜔
𝜔 2𝜔 − 𝜔 2𝜔 2𝜔 + 𝜔
• The following condition must be achieved in order for the sidebands not to overlap

2𝜔 + 𝜔 ≥ 2𝜋𝐵 or 2𝑓 + 𝑓 ≥ 𝐵
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Synchronous or Coherent Demodulation (1/4)

• The demodulation operation consists in the multiplication of the modulated signal at the output of
the channel by a replica of the carrier waveform, generated locally at the receiver which is to perform
the demodulation
• Therefore the demodulation is identical to the modulation operation: the only difference being the
relation between the signal at the input, and the filter used at the output of the demodulator
o In the demodulator the filter at the output is a lowpass one (not bandpass as in the modulator)
o This filter is used to remove the high-frequency components which result from the
multiplication and to recover the information waveform (baseband signal)
• With the exception of the product modulator the operation of any of the modulators presented
previously can be reversed in order to perform the demodulation operation

𝑚 𝑡 × cos 𝜔 𝑡 Ideally there should be a perfect

𝑥 𝑡 𝑚 𝑡
LPF
X synchronism between the carrier
LB = 𝐵
synchronism of the local oscillator and the
~ cos 𝜔 𝑡 carrier of the modulated signal

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Synchronous or Coherent Demodulation (2/4)

Double Side Band with Supressed Carrier (DSB) demodulation

Signal at the channel local oscillator output 𝑟 𝑡 = 𝑚 𝑡 cos 𝜔 𝑡 × 2 cos 𝜔 𝑡

1
= 𝑚 𝑡 × 2 cos 𝜔 𝑡 = 𝑚 𝑡 × 2 × 1 + cos 2𝜔 𝑡 = 𝑚 𝑡 + 𝑚 𝑡 × cos 2𝜔 𝑡
2

• This component is eliminated by the lowpass

𝑚 𝑡 × cos 𝜔 𝑡 𝑥 𝑡
LPF
𝑚 𝑡 filter at the receiver
X
LB = 𝐵 • Assuming an ideal filter, with amplitude gain of
synchronism 1 over it s bandpass, results the information
~ 2cos 𝜔 𝑡
𝑚 𝑡 at the output

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Synchronous or Coherent Demodulation (3/4)

Double Side Band with Supressed Carrier (DSB) demodulation (continuation)
• In the derivation of previous result it was assumed that the local oscillator is coherent with the carrier
waveform at the output of the channel
• By “coherent” we mean that both carriers (original one at the emitter conveying the information we
want to recover, and the local one at the receiver, which is a replica used in the demodulation) are
synchronized, both in phase and in frequency
• Nevertheless, the performance of the synchronous detector can be seriously affected, depending on
the amount of the error we make in the recovery of the phase and/or frequency of the carrier
generated locally
Imperfect phase synchronism in the demodulation of the DSB-SC signal
𝑚 𝑡 × cos 𝜔 𝑡 𝑥 𝑡 𝑟 𝑡
𝑟 𝑡 = 𝑚 𝑡 cos 𝜔 𝑡 × 𝑘 cos 𝜔 𝑡 + 𝜙 LPF
X
𝑘 𝑘 LB = 𝐵
= 𝑚 𝑡 cos 𝜙 + 𝑚 𝑡 cos 2𝜔 𝑡 + 𝜙
2 2
synchronism
~ 𝑘 cos 𝜔 𝑡 + 𝜙
with 𝜙 = 𝜋/2 the message 𝑚 𝑡 is lost
• As long as the phase 𝜙 remains fixed the output of the demodulator is a distorted version of the
original signal
• In reality the phase 𝜙 varies randomly over time and thus the receiver must track it over time
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Synchronous or Coherent Demodulation (4/4)

Double Side Band with Supressed Carrier (DSB) demodulation (continuation)
Imperfect frequency synchronism in the demodulation of the DSB-SC signal
• There is another problem when the frequencies are not exactly the same

𝑟 𝑡 = 𝑚 𝑡 cos 𝜔 𝑡 × 𝑘 cos 𝜔 + Δ𝜔 𝑡
𝑘 𝑘
= 𝑚 𝑡 cos Δ𝜔𝑡 + 𝑚 𝑡 cos 2𝜔 𝑡 + Δ𝜔𝑡
2 2

Now the original signal is affected by a cosine waveform with period equal to 1/Δ𝑓

• If it happens that the frequency deviation Δ𝑓 is or the same order of the frequency of the baseband
signal it will raise oscillations in the amplitude of the original signal
o In telephone systems the limit Δ𝑓 ≤30 Hz is acceptable

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Non coherent or Non synchronous Demodulation (1/4)

Rectification and filtering
𝐴 1 + 𝜇 × 𝑚 𝑡 cos 𝜔 𝑡 𝑣 𝑡 𝐴 1+𝜇×𝑚 𝑡
1 1
𝐴 1+𝜇×𝑚 𝑡 𝐴 1+𝜇×𝑚 𝑡
𝜋 𝜋

The capacitor filters the DC component

(the carrier’s amplitude 𝐴 )
𝑣 𝑡 1
𝐴 ×𝜇×𝑚 𝑡
𝜋
𝐴 1 + 𝜇 × 𝑚 𝑡 cos 𝜔 𝑡 𝐶
Lowpass 𝑅 1
filter 𝑚 𝑡
𝜋

• The diode is ON at the positive alternations of the carrier waveform (half wave rectifier)
o It is functionally equivalent to the multiplication operation between the AM signal and the
waveform 𝑝 𝑡 which gives rise to the signal 𝑣 𝑡
𝑐 𝑡

𝜔𝑡
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Non coherent or Non synchronous Demodulation (2/4)

Rectification and filtering (continuation)
• The rectifier (formed by the diode and the resistance) generate the half-wave rectified signal at the
output
𝑣 𝑡 = 𝐴 1 + 𝜇 × 𝑚 𝑡 cos 𝜔 𝑡 × 𝑐 𝑡

1 2 1 1
=𝐴 1+𝜇×𝑚 𝑡 × cos 𝜔 𝑡 + cos 𝜔 𝑡 − cos 3𝜔 𝑡 + cos 5𝜔 𝑡 − ⋯
2 𝜋 3 5
1
1 Note: cos 𝜔 𝑡 = 1 + cos 𝜔 𝑡
= 𝐴 1+𝜇×𝑚 𝑡 + high frequency terms … 2
𝜋

• The lowpass filter filters out the high-frequency components of the signal 𝑣 𝑡 and the coupling
capacitor blocks the DC component: it generates the signal 𝑚 𝑡 /𝜋 at the output
• It is worth mentioning that:
o The multiplication of the AM signal by the square wave 𝑐 𝑡 in the wave rectifying demodulator
is functionally equivalent to a synchronous or coherent detector, without the local generation of
the carrier (as it happens in the synchronous detectors)
o It is possible to raise the signal amplitude if a full wave rectifier is used in place of the half wave
one

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Non coherent or Non synchronous Demodulation (3/4)

Envelope detector

The signal at the output of the envelope

AM signal
𝐶 𝑅 𝑣 𝑡 detector follows the envelope of the
Capacitor modulated waveform which is 𝐴 + 𝑚 𝑡
discharge

If we suppose the diode is ideal:

• When the voltage at the input is positive the diode is ON and the capacitor 𝐶 charges until the
voltage reaches the peak of the voltage signal
• When the voltage of the signal at the input decrease in relation to the peak of the voltage at the
capacitor, the diode “opens” and the capacitor 𝐶 discharges over the load resistance 𝑅 (with time
constant 𝑅𝐶) until the voltage level at the input becomes higher than the voltage level at the capacitor
• It is important to note that the capacitor time constant of the discharge must be much slower than the
time constant of charge, in order to give enough time for the envelope detector to recover the
envelope

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Non coherent or Non synchronous Demodulation (4/4)

Envelope detector (continuation)
𝑅 - source resistance
1
Time constant of charge 𝑅 +𝑟 𝐶 ≪
𝑓 𝑟 - direct resistance of the diode

• The capacitor must charge rapidly for the voltage 𝑣 𝑡 at the output to follow the envelope
• For this to happen the time constant of charge of the RC filter must be much lower than the period
1/𝑓 of the carrier
1
Time constant of discharge 1/𝜔 ≪ 𝑅𝐶 < 1/ 2𝜋𝐵 or 2𝜋𝐵 < ≪𝜔
𝑅𝐶

• The discharge of the capacitor through the load resistance 𝑅 must be slow between peaks in the
amplitude of the voltage signal, but not too slow in order to avoid the capacitor to discharge at the
maximum rate of variation (1/𝐵) of the signal (𝐵 is the signal bandwidth)

• 𝑣 𝑡 is equal to 𝐴 + 𝑚 𝑡 , with the

addition of a ripple (sawtooth
waveform) at frequency 𝜔
• The ripple is removed partially by
filtering the signal with an appropriate
lowpass filter

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Spectral Efficiency (1/2)

• One of the most relevant characteristics of amplitude modulation (AM or DSB-SC) is its reduced
spectral efficiency, due to the use of both sidebands in the spectrum to convey information
o Both bands to each side of the carrier are used
o It results in under-utilization of the available spectrum

message with bandwidth B

DSC signal with bandwidth 2B

Redundancy
• DSB signal transmits two identical replicas of the same information 𝑚 𝑡
o One replica is associated to the Upper Side Band (USB)
o Another replica is associated to the Lower Side Band (LSB)
• However, there exist alternative forms of modulation, which result in a better use of the spectra
available in the channel for the transmission of information, i.e., with a better spectral efficiency
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Spectral Efficiency (2/2)

Quadrature Amplitude Modulation (QAM)
• Although the bandwidth of the QAM signal is twice the bandwidth of the information signal, it
optimizes the use of the spectrum in the same way as SSB, since it transmits two messages
simultaneously, modulating in phase-quadrature the same carrier 𝑓
• The in-phase and quadrature versions of the carrier are orthogonal, and although they occupy the
same band in the spectrum they do not interference with each other
But there is a price to pay for this gain in spectral efficiency, which is the increased complexity in the
emitter and the receiver resulting from the implementation of either of these two schemes
Single Side Band modulation (SSB)
• Either the upper side band (USB) or the lower side band (LSB) is removed before the signal is sent
through the channel
o USSB (Upper Single Side Band): only the upper sideband remains and is transmitted
o LSSB (Lower Single Side Band): only the lower sideband remains and is transmitted
• In any case, the signal bandwidth (L/U)SSB is equal to the bandwidth of the baseband signal 𝑚 𝑡
(𝐵 = 𝐵)

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Quadrature Amplitude Modulation (QAM) (1/2)

• Quadrature amplitude modulation (QAM) is an alternative to the modulation in single side band
(SSB), since (as we will see) SSB is not appropriate in those applications in which the baseband
spectrum imposes highly selective filters
• QAM utilizes carrier phase shifting and synchronous detection to permit two DSB signals to occupy
the same frequency band
Modulation
• The transmitted signal has the form 𝑥 𝑡 = 𝐴 𝑚 𝑡 cos 𝜔 𝑡 ± 𝑚 𝑡 sin 𝜔 𝑡

Demodulation
• The demodulation is coherent and recovers the information as long as the local oscillators generate
carriers multiplexed in phase and in quadrature In order to recover 𝑚 𝑡 we need to filter
out the QAM signal with a lowpass filter
𝑥 𝑡 = 2𝑥 𝑡 × cos 𝜔 𝑡 = 2 𝑚 𝑡 cos 𝜔 𝑡 + 𝑚 𝑡 sin 𝜔 𝑡 cos 𝜔 𝑡
lowpass
𝑥 𝑡 filter 𝑚 𝑡
= 2𝑚 𝑡 × cos 𝜔 𝑡 + 2𝑚 𝑡 sin 𝜔 𝑡 × cos 𝜔 𝑡
2 cos 𝜔 𝑡
𝑚 𝑡 + 𝑚 𝑡 cos 2𝜔 𝑡 𝑚 𝑡 sin 2𝜔 𝑡
−𝜋/2
• Proceeding in the same way for 𝑥 𝑡 2 sin 𝜔 𝑡
lowpass
𝑥 𝑡 = 𝑚 𝑡 − 𝑚 𝑡 cos 2𝜔 𝑡 + 𝑚 𝑡 sin 2𝜔 𝑡 𝑥 𝑡 filter 𝑚 𝑡

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Quadrature Amplitude Modulation (QAM) (2/2)

• QAM modulation requires a very precise phase synchronism
• What happens then if there is a small phase deviation?

Assume that 2 cos 2𝜋𝑓 𝑡 + 𝜃 is the carrier generated locally with a phase
deviation 𝜃

Thus: 𝑥 𝑡 = 2𝑥 𝑡 × cos 𝜔 𝑡 + 𝜃

= 2𝑚 𝑡 cos 𝜔 𝑡 × cos 𝜔 𝑡 + 𝜃 + 2𝑚 𝑡 sin 𝜔 𝑡 × cos 𝜔 𝑡 + 𝜃

= 𝑚 𝑡 cos 𝜃 − 𝑚 𝑡 sin 𝜃 + high − frequency components

Co-channel interference (or cross-talk) between the two branches

• We have interference between both branches (cross-talk) namely when the attenuations caused by the
phase deviation in both branches are not the same

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Single Sideband Modulation (SSB) (1/5)

𝑀 𝑓 USSB spectrum: the upper

baseband spectrum (𝑚 𝑡 ) baseband sideband of DSB signal
goes through the ideal

upper sideband lower sideband upper sideband bandpass filter

lower sideband
(bandwidth B) without
DSB
distortion

LSSB spectrum: the lower

USB
sideband of DSB signal
goes through the ideal
LSB bandpass filter
(bandwidth B) without
spectrum bandpass signal (DSB-SC)
distortion

SSB bandwidth: 𝐵 Hz (better efficiency)

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Single Sideband Modulation (SSB) (2/5)

The signal modulated in single sideband is generated in two stages
1. The information signal 𝑚 𝑡 is the input to a product modulator used in the generation of a DSB signal
with carrier frequency 𝑓 and amplitude 𝐴

𝑥 𝑡 = 𝑚 𝑡 × 𝐴 cos 𝜔 𝑡 𝑚 𝑡 𝑥 𝑥
+ balanced quadrature
𝐴 ℱ 𝐴 modulator filter
𝑥 𝑡 = 𝑚 𝑡 ↔𝑋 𝑓 = 𝑀 𝑓
2 2
(in the lowpass equivalent representation) ~ cos 𝜔 𝑡

2. The quadrature filter (or phase splitter) filters out the 𝐻 𝑓 𝐻 𝑓

negative (positive) band from the spectrum of the
1 - 1
DSB signal in order to generate the USSB (LSSB)
signal at the output 𝑓
−𝑓 − 𝐵 −𝑓 0 𝑓 𝑓 +𝐵 𝑓 0 𝐵
Filter used in the generation of USSB
𝐻 𝑓 𝐻 𝑓

𝐻 𝑓 =𝐻 𝑓+𝑓 𝑢 𝑓+𝑓 = 𝑢 𝑓 −𝑢 𝑓−𝐵 1 - 1

Filter used in the generation of LSSB

−𝑓 −𝑓 + 𝐵 0 𝑓 −𝐵 𝑓 𝑓 −𝐵 0 𝑓
𝐻 𝑓 =𝐻 𝑓−𝑓 𝑢 𝑓−𝑓 =𝑢 𝑓+𝐵 −𝑢 𝑓

1
Joining both results 𝐻 𝑓 = 1 ± sgn 𝑓 ; 𝑓 ≤𝐵
2
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Single Sideband Modulation (SSB) (3/5)

• The DSB signal 𝑋 𝑓 is filtered by 𝐻 𝑓 in order to produce at the output the spectrum of the
SSB signal
• 𝐻 𝑓 is the lowpass representation of the transfer function of the filter used in the generation of the
LSSB signal

1 1 Note: −j sgn 𝑓 𝑀 𝑓 = ℱ 𝑚 𝑡
𝑋 𝑓 = 𝐴 1 ± sgn 𝑓 𝑀 𝑓 = 𝐴 𝑀 𝑓 ± sgn 𝑓 × 𝑀 𝑓
4 4 with 𝑚 𝑡 the Hilbert transform
of 𝑚 𝑡
ℱ sgn 𝑓 × 𝑀 𝑓 = j𝑚 𝑡

• We first apply the inverse Fourier transform to obtain the waveform of the (complex) SSB signal in
time domain 1
𝑥 𝑡 = 𝐴 𝑚 𝑡 ± j𝑚 𝑡
4

• Then we apply the transformation 𝑥 𝑡 = 2Re 𝑥 𝑡 e to generate the signal in bandpass

format
𝑥 𝑡 = 𝐴 𝑚 𝑡 cos 𝜔 𝑡 ∓ 𝐴 𝑚 𝑡 sin 𝜔 𝑡 = 𝐴 𝑚 𝑡 cos 𝜔 𝑡 ∓ 𝑚 𝑡 sin 𝜔 𝑡

1 1 (envelope)
𝑥 , 𝑡 = 𝐴 𝑚 𝑡 𝑥 , 𝑡 =± 𝐴 𝑚 𝑡
2 2  
𝐴 𝑡 = 𝐴 𝑚 𝑡 +𝑚 𝑡
(in-phase component) (quadrature component)

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Single Sideband Modulation (SSB) (4/5)

1
1
𝑀 𝑓 𝐻 𝑓 = 1 ± sgn 𝑓 ; 𝑓 ≤𝐵
2 2
ℱ
𝑚 𝑡 ↔𝑀 𝑓
𝑚 𝑡 =𝑚 𝑡 +𝑚 𝑡 with ℱ
𝑚 𝑡 ↔𝑀 𝑓
−𝐵 𝐵
1 1
𝑚 𝑡 =ℱ 𝑀 𝑓 +ℱ sgn 𝑓 𝑀 𝑓
𝑀 𝑓 2 2

1 1 1
= 𝑚 𝑡 + j ∗𝑚 𝑡
𝐵 2 2 𝜋𝑡
𝑀 . 𝑓
1
𝑚 𝑡 = 𝑚 𝑡 + j𝑚 𝑡
2
−𝐵
𝑀 𝑓+𝑓 𝑀 𝑓−𝑓
1 1
𝑚 𝑡 =ℱ 𝑀 𝑓 −ℱ sgn 𝑓 𝑀 𝑓
2 2

1 1 1
−𝑓 𝑓 = 𝑚 𝑡 − j ∗𝑚 𝑡
𝑀 𝑓+𝑓 2 2 𝜋𝑡
𝑀 𝑓−𝑓 1
𝑚 𝑡 = 𝑚 𝑡 − j𝑚 𝑡
2
−𝑓 𝑓
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Single Sideband Modulation (SSB) (5/5)

• Continuing with the analysis from the figure

1 1 1 1
𝑥 𝑡 = 𝐴 𝑚 𝑡 e + 𝐴 𝑚 𝑡 e = 𝐴 𝑚 𝑡 + j𝑚 𝑡 e + 𝐴 𝑚 𝑡 − j𝑚 𝑡 e
2 2 4 4

1 1 1 1 1 1
= 𝐴 𝑚 𝑡 e + e − 𝐴 𝑚 𝑡 𝑒 − e
2 2 2 2 2j 2j

1 1
𝑥 𝑡 = 𝐴 𝑚 𝑡 cos 𝜔 𝑡 − 𝐴 𝑚 𝑡 sin 𝜔 𝑡
2 2

• We proceed in the same way to generate the LSSB signal

1 1
𝑥 𝑡 = 𝐴 𝑚 𝑡 cos 𝜔 𝑡 + 𝐴 𝑚 𝑡 sin 𝜔 𝑡
2 2

Bandwidth and power efficiency

• The SSB signal has only one sideband (the upper or lower) and therefore it results:

Bandwidth: 𝐵 =𝐵

1
Average power: 𝑃 =𝑃 = 𝐴 𝑃
4
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SSB Modulators (1/4)

Phase-shift method for SSB generation

• SSB signal consists of two DSB waveforms with quadrature carriers and modulating signals 𝑚 𝑡 and 𝑚 𝑡

1 1
𝑥 𝑡 = 𝐴 𝑚 𝑡 cos 𝜔 𝑡 ± 𝐴 𝑚 𝑡 cos 𝜔 𝑡 − 90º
2 2

• This system implements the phase shift method for SSB generation
• It produces either USSB or LSSB, depending upon the sign at the summer
• The quadrature phase shifter 𝐻 𝑓 implements the Hilbert transform
1
𝐴 𝑚 𝑡 cos 𝜔 𝑡
• The Hilbert transform computation imposes an DSB 2

abrupt phase change of 𝜋 rads at DC frequency, modulator

𝐴
cos 𝜔 𝑡
but the phase shift filter cannot implement this 2

behaviour in reality + 𝑥 𝑡
𝑚 𝑡 𝜋
−
• Nevertheless, the Hilbert transform can be 2
∓
approximated by a real filter if the spectral 𝐴
sin 𝜔 𝑡
2
content of the information 𝑚 𝑡 is null at DC,
DSB
with very little energy content at lower 𝐻 𝑓
modulator
1
frequencies (around DC) 𝑚 𝑡 𝐴 𝑚 𝑡 sin 𝜔 𝑡
2
• Therefore, the practical phase-shift filter will not Wideband phase shifter (Hilbert transform
affect too much the quality of the signal 𝐻 𝑓 = −j sgn 𝑓
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SSB Modulators (2/4)

Two step SSB generation with bandpass filters
• The straightforward way to generate the SSB is indirectly: (1) generation of a DSB signal and then (2)
using a bandpass filter to erase one of the sidebands of the DSB signal to produce the SSB signal
𝑀 𝑓
• There is an inherent difficulty with this method as it
energy gap
demands filters with steep transition bands (even ideal
ones with no transition band), namely when the sidebands
have frequencies adjacent to each other
• However, this method has application for audio signals 𝑓
𝑓ℓ 𝑓ℓ + 𝐵
(voice and music) due to their little low-frequency content
(spectra with “holes” at DC extending at, roughly, 300 Hz 𝑋 𝑓 2𝛽 𝐻 𝑓

(spectral hole of roughly 600 Hz around DC frequency)

• BPF: the width of the transition band should be less than
twice the lowest frequency with significant content in the 𝑓
spectrum, and attenuation around 40 dB inside the hole
𝑓 − 𝑓ℓ 𝑓 + 𝑓ℓ
• Empirical rule: 𝑓 < 200𝛽
o 𝑓 : filter’s cutoff frequency (set equal to carrier’s frequency 𝑓
o 2𝛽: transition bandwidth (equal to the width of the spectral hole)
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SSB Modulators (3/4)

Two step SSB generation with bandpass filters (continuation)
• The synthesis of an analog filter with an attenuation of 40 dB over a transition band of width 600 Hz
is a very complex problem, even unrealizable in reality
• For carriers in the very high portions of the electromagnetic spectrum it can happen that the ratio
between the width of the spectral hole (roughly 600 Hz for the audio signal for example) and the
carrier frequency 𝑓 is not enough to guarantee the desired attenuation for the filter’s transition band
to suppress one of the sidebands from the spectrum of the DSB signal, and then create the SSB signal
• The “modified method of filters” works around this problem by generating the SSB in two stages

o The first modulator filters out one of

𝑚 𝑡 𝑥 𝑡
the sidebands of the DSB signal at X BPF1 X BPF2

frequency 𝑓 , which is lower than

the carrier’s frequency: the spectral ~ 𝐴 cos 𝜔 𝑡 ~ 𝐴 cos 𝜔 𝑡

hole at the output is roughly 2𝑓

o The second modulator filters out one of the sidebands of the signal at the output of the second
modulator, at the frequency of the carrier of the SSB signal: the spectral hole is still 2𝑓 but it is
now centred at frequency 𝑓

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SSB Modulators (4/4)

Example: telephonic channel (300 - 3000 Hz) modulating a carrier 10MHz
(1) Using the filter’s method
• The bandpass filter would have to attenuate the signal in approximately 40 dB inside the transition
band of 600 Hz
• The change in frequency (in percent) over the transition
band would be 600/107=0,006%: it is very difficult to Normalized PDS of the voice signal in
the telephonic channel (baseband)
obtain an analog filter with such strict requirements
o Note: 10 log 0,00006 = −42 dB)

(2) Using the modified filter’s method

• Based in a two stage design: 𝑓 = 100 KHz and
𝑓 = 10 MHz
• Stage 1 (𝑓 = 100 KHz): the change in
Normalized PDS of the voice signal in the
frequency in percent for the first filter is now
telephonic channel (baseband) for the Weaver
600/105=0,6% modulator

• Stage 2 (𝑓 = 10 MHz): the change in frequency

in percent for the second filter is 2105/107 =2%

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Interference in Linear Modulation (1/4)

• Interference in real-physical systems arises from various sources, such as RF emissions from
transmitters having carrier frequencies close to that of the carrier being demodulated
• It is advantageous to to study interference influence in linear systems because it provides us with
important insights into the behaviour of systems operating in the presence of noise
• As a simple scenario and performance of a linear modulation system in the presence of interference
we consider the received signal having the spectrum (only positive frequencies shown)

Scenario of analysis
• The received signal consists of three components:
i. The carrier component 𝑓
ii. A pair of sidebands 𝑓 ± 𝑓 representing a sinusoidal
message signal
iii. An undesired interfering tone of frequency 𝑓 + 𝑓

• The input to the demodulator is therefore

𝑥 𝑡 = 𝐴 cos 2𝜋𝑓 𝑡 + 𝐴 cos 2𝜋 𝑓 + 𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡 cos 2𝜋𝑓 𝑡 1

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Interference in Linear Modulation (2/4)

Coherent Demodulation
• Multiplying 𝑥 𝑡 by 2 cos 2𝜋𝑓 𝑡 and lowpass filtering yields

The interference component is passed by the filter and the DC

𝑦 𝑡 = 𝐴 cos 2𝜋𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡
terms resulting from the carrier is blocked

• Due to the linear nature of amplitude modulation, the signal and interference are additive at the
output of the coherent demodulator, since the interference is additive at the receiver input
Envelope Detector
• The effect of interference with envelope detection is quite different because of the non-linear nature of
the envelope detector, and therefore the analysis is more complex
• We gain some insight by writing 𝑥 𝑡 in a form that leads to the phasor diagram
1 1
𝑥 𝑡 = Re 𝐴 +𝐴 e + 𝐴 e + 𝐴 e e
2 2
Phasor diagram: scenario without interference Phasor diagram: scenario with interference

The output of an ideal envelope detector is 𝑅 𝑡

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Interference in Linear Modulation (3/4)

Envelope Detector (continuation)
• The phasor diagrams illustrate that interference induces both an amplitude distortion and a phase
deviation
• The effect of interference with envelope detection is determined by writing (1) as

𝑥 𝑡 = 𝐴 cos 2𝜋𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡 cos 2𝜋𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡 cos 2𝜋𝑓 𝑡 − sin 2𝜋𝑓 𝑡 sin 2𝜋𝑓 𝑡

= 𝐴 + 𝐴 cos 2𝜋𝑓 𝑡 cos 2𝜋𝑓 𝑡 − 𝐴 sin 2𝜋𝑓 𝑡 sin 2𝜋𝑓 𝑡 (2)

• We now consider two different scenarios regarding the ratio between the amplitudes of the carrier
and the interference
(1) 𝐴 ≫ 𝐴 : the last term in (2) is negligible compared to the first one: the output of the ED is

𝑦 𝑡 ≅ 𝐴 cos 2𝜋𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡

• For the small interference case envelope detection and coherent demodulation are essentially
equivalent
(2) 𝐴 ≪ 𝐴 : the last term of (2) must be considered now and we perform a small trick by writing (1) in
the following way

𝑥 𝑡 = 𝐴 cos 2𝜋 𝑓 + 𝑓 − 𝑓 𝑡 + 𝐴 cos 2𝜋 𝑓 + 𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡 cos 2𝜋 𝑓 + 𝑓 − 𝑓 𝑡

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Interference in Linear Modulation (4/4)

Envelope Detector (continuation)
• Using trigonometric identities we write previous equation in the appropriate format

𝑥 𝑡 = 𝐴 + 𝐴 cos 2𝜋𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡 cos 2𝜋𝑓í 𝑡 cos 2𝜋 𝑓 + 𝑓 𝑡

+ 𝐴 sin 2𝜋𝑓 𝑡 + 𝐴 cos 2𝜋𝑓 𝑡 sin 2𝜋𝑓í 𝑡 cos 2𝜋 𝑓 + 𝑓 𝑡

• We see that with envelope detection the largest high-frequency component is treated as the carrier
o If 𝐴 ≫ 𝐴 the effective demodulation carrier has a frequency 𝑓
o if 𝐴 ≫ 𝐴 the effective carrier becomes the interference frequency 𝑓 + 𝑓

𝑨 𝒄 ≫ 𝑨𝒊 The interfering tone appears as a sinusoidal component with frequency 𝑓 at the

output of the envelope which performs as a linear or coherent demodulator

• The sinusoidal message signal with frequency 𝑓 modulates the

interference tone
• The output of the envelope detector has a spectrum that reminds us of the
𝑨 𝒄 ≪ 𝑨𝒊 spectrum of an AM signal with carrier frequency 𝑓 and sideband
components at 𝑓 + 𝑓 and 𝑓 − 𝑓
• Message signal is effectively lost (threshold effect) and is a consequence
of the nonlinear nature of the envelope detector
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Carrier Acquisition (1/5)

Sensitivity to frequency and phase deviations
• The wireless channel is affected by propagation delay and frequency deviation (resulting from
Doppler effect), and these two effects combined result in the phase and frequency deviation of the
carrier signal
𝑟 𝑡 𝑒 𝑡 𝑒 𝑡
LPF
Example: SSB signal X
LB = 𝐵

synchronism Phase deviation: 𝛿

~ 2 cos 𝜔 𝑡
Signal at the output of the channel: 𝑟 𝑡 Frequency deviation: Δ𝜔

𝑟 𝑡 = 𝑚 𝑡 cos 𝜔 + Δ𝜔 𝑡 + 𝛿 − 𝑚 𝑡 sin 𝜔 + Δ𝜔 𝑡 + 𝛿

Portadora local: 2 cos 𝜔 𝑡

𝑒 𝑡 = 2 cos 𝜔 𝑡 𝑚 𝑡 cos 𝜔 𝑡 + Δ𝜔𝑡 + 𝛿 − 𝑚 𝑡 sin 𝜔 𝑡 + Δ𝜔𝑡 + 𝛿

= 𝑚 𝑡 cos Δ𝜔𝑡 + 𝛿 − 𝑚 𝑡 sin Δ𝜔𝑡 + 𝛿

+𝑚 𝑡 cos 2𝜔 𝑡 + Δ𝜔𝑡 + 𝛿 − 𝑚 𝑡 sin 2𝜔 𝑡 + Δ𝜔𝑡 + 𝛿

Sinal SSB passa-banda centrado em 2𝜔 + Δ𝜔

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Carrier Acquisition (2/5)

Sensitivity to frequency and phase deviations (continuation)
• At the output of the low-pass filter:

𝑒 𝑡 = 𝑚 𝑡 cos Δ𝜔𝑡 + 𝛿 − 𝑚 𝑡 sin Δ𝜔𝑡 + 𝛿

1. Ideal scenario: Δ𝜔 = 0 and 𝛿 = 0 (there is no phase or frequency deviation)

𝑒 𝑡 =𝑚 𝑡

2. Phase deviation only: 𝛿 = ±𝜋/2 (e Δ𝜔 = 0 ) and thus the baseband signal is completely erased

3. Frequency deviation only: Δ𝜔 ≠ 0 (e 𝛿 = 0 ) (the signal is affected by both attenuation and distortion
at the same)
𝑚 𝑡 cos Δ𝜔𝑡

• The addition of a pilot carrier to the sidebands with the information facilitates the acquisition of the
carrier in the coherent detector (note that this is not the envelope detector)
• The pilot carrier is an unmodulated sinusoid at the carrier frequency 𝑓 , with amplitude -20 dB
relative to the maximum amplitude level of the useful signal

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Carrier Acquisition (3/5)

Square circuit
BPF: narrowband bandpass filter
𝑐 cos 2𝜔 𝑡 𝑘 cos 𝜔 𝑡
𝑚 𝑡 cos 𝜔 𝑡 𝑥 𝑡
𝟐 BPF Frequency
±2𝜔 PLL
divider (2:1)

1 1 signal (1/2) 𝑚 𝑡 has average DC

𝑥 𝑡 = 𝑚 𝑡 cos 𝜔 𝑡 = 𝑚 𝑡 + 𝑚 𝑡 cos 2𝜔 𝑡
2 2 value different from zero
Making the association
1 𝑘 is the average DC value (DC component)
𝑚 𝑡 =𝑘+𝜙 𝑡
2 𝜙 𝑡 is equal to the signal (1/2) 𝑚 𝑡 without its DC component

1
𝑥 𝑡 = 𝑚 𝑡 + 𝑘 cos 2𝜔 𝑡 + 𝜙 𝑡 cos 2𝜔 𝑡 BPF: highly selective bandpass filter (high
2
𝑄 factor) centred in 2𝜔

This term in eliminated This term is mostly eliminated by the BPF: signal 𝜙 𝑡 has no DC component, i.e.,
by the BPF the power of the signal at the frequency component 2𝜔 is (essentially) zero

The signal at the input of the PLL is 𝑘 cos 2𝜔 𝑡 plus the residue remaining at the output of the
bandpass filter which filters the signal 𝜙 𝑡 cos 2𝜔 𝑡
The PLL captures the signal cos 2𝜔 𝑡 and the frequency divider 2:1 recovers the carrier 𝜔
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Carrier Acquisition (4/5)

Costas Receiver
• The Costas receiver is another means of implementation of a device for carrier acquisition, typically
used in coherent demodulation of linear modulated signals
• The Costas receiver encompasses the devices:
o Two coherent detectors with the same signal at the input
o Two local oscillators in quadrature and with local frequency of oscillation adjusted to the
frequency 𝑓 of the carrier
• Branch in phase (I): upper
𝑚 𝑡 cos 𝜃 output detector which is in-phase
lowpass
filter • Branch in quadrature (Q):
2 cos 𝜔 𝑡 + 𝜃
lower detector which is in
𝑚 𝑡 cos 𝜔 𝑡 + 𝜃 narrowband quadrature
VCO lowpass filter
1 The two detectors compose a
𝑚 𝑡 sin 2𝜃
2 feedback system which keeps
−𝜋/2 𝑅 sin 2𝜃
the local oscillator in
2 sin 𝜔 𝑡 + 𝜃
synchronism with the carrier
lowpass
filter 𝑚 𝑡 sin 𝜃

• The outputs of the two branches (in-phase and in-quadrature) are combined in the phase
discriminator
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Carrier Acquisition (5/5)

Costas Receiver (continuation)
Phase discriminator
• It is a multiplier followed by a lowpass filter
• Signal at the output of the phase discriminator

1 1
𝑚 𝑡 sin 𝜃 cos 𝜃 = 𝑚 𝑡 sin 𝜃 − 𝜃 + sin 𝜃 + 𝜃 = 𝑚 𝑡 sin 2𝜃
2 2

Lowpass filter of the phase discriminator

• This filter passes the DC component of the signal at the input which is affected by the phase
deviation
1
Output of the filter 𝑅 sin 2𝜃 𝑅 is the DC component of the signal 𝑚 𝑡
2

VCO (Voltage Controlled Oscillator)

• The VCO generates the local carrier which reflects this local phase 𝜃 =𝜃 −𝜃
deviation (error) of amplitude 𝜃

Output of the VCO 2cos 𝜔 𝑡 + 𝜃

The frequency increases in order to reduce the error (and vice-versa)

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Pulse Amplitude Modulation (PAM) (1/4)

• Pulse Amplitude Modulation (PAM) is a kind of linear modulation in which the periodic samples
(with sampling period 𝑇 ) of the message waveform 𝑚 𝑡 can be transmitted over the channel using
pulse modulation, wherein the sample values modulate the amplitude of a pulse train 𝑝 𝑡
• The sampling is performed by flat top rectangular pulses of duration 𝜏

o The amplitude of the pulse is the value of the sample of the 𝑚 𝑡

information signal 𝑚 𝑡 at the beginning of the sampling 𝑇
𝑡

period PAM

o The amplitude of the pulse remains fixed over the duration 𝜏 of

𝐴
the pulse 𝑡

o The pulses repeat with period equal to the sampling interval 𝑇

Flat-Top Sampling and PAM sampling discharge

switch switch
• PAM is typically generated with a sample and hold (S/H) circuit
which produces the flat-top pulses 𝐺2
𝑚 𝑡 𝐶 𝑥 𝑡
o A gate pulse at G1 briefly closes the sampling switch and the
𝐺1
capacitor holds the sample voltage
o The sample is hold until the capacitor is discharged by a pulse
applied to G2
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Modulação da Amplitude do Impulso (PAM) (2/4)

• Periodic gating of the sample and hold circuit 𝑚 𝑡 𝑚 𝑛𝑇

generates the sampled wave 𝜏

 
𝑛𝑇 𝑡
𝑥 𝑡 = 𝑚 𝑘𝑇 𝑝 𝑡 − 𝑘𝑇
Each output pulse of duration 𝜏 represents
a single instantaneous sample value
• According to signal analysis theory we have the relation
 

𝑝 𝑡 − 𝑘𝑇 = 𝑝 𝑡 ∗ 𝛿 𝑡 − 𝑘𝑇 𝑥 𝑡 =𝑝 𝑡 ∗ 𝑚 𝑛𝑇 𝛿 𝑡 − 𝑘𝑇 =𝑝 𝑡 ∗𝑥 𝑡

• From signal analysis we know that the signal 𝑥 𝑡 is the resulting of ideal sampling of the analog
signal 𝑚 𝑡 by a train of Dirac pulses
  𝑋 𝑓
𝑥 𝑡 = 𝑚 𝑛𝑇 𝛿 𝑡 − 𝑘𝑇

• Applying Fourier transform to 𝑥 𝑡 𝑓 𝑓

0 𝑓
  𝑋 𝑓
𝑃 𝑓
𝑋 𝑓 =𝑃 𝑓 𝑓 𝑀 𝑓 − 𝑛𝑓 =𝑃 𝑓 𝑋 𝑓
Ideal
sampling
• Flat-top sampling is equivalent to passing an ideally sampled 𝑓 0 𝑓 𝑓

wave through a network having transfer function 𝑃 𝑓 = ℱ 𝑝 𝑡

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Modulação da Amplitude do Impulso (PAM) (3/4)

Aperture effect
• The high frequency roll off characteristics of a typical 𝑃 𝑓 acts like a lowpass filter and attenuates the
upper portion of the message spectrum
• The larger the pulse duration or aperture 𝜏 the larger the effect
• Aperture effect can be corrected when the analog signal 𝑚 𝑡 is to be reconstructed by including an
equalizer with frequency response
𝐾𝑒
𝐻 𝑓 = Little equalization is needed when 𝜏/𝑇 ≪ 1 𝑚 𝑡
𝑃 𝑓
𝑡
𝑇
Unipolar flat-top PAM signal
  PAM
𝑥 𝑡 = 𝐴 1 + 𝜇𝑚 𝑘𝑇 𝑝 𝑡 − 𝑘𝑇
𝐴
• 𝐴 : unmodulated pulse amplitude 𝑡

• 𝜇: modulation index (controls the amount of amplitude variation)

o The condition 1 + 𝜇𝑚 𝑡 > 0 ensures a unipolar (single polarity) waveform with no missing
pulses
• 𝑓 : pulse rate (particularly important for synchronization in time-division multiplexing)

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Modulação da Amplitude do Impulso (PAM) (4/4)

PAM spectrum
• The PAM signal can be obtained from a sample and hold circuit with input 𝐴 1 + 𝜇𝑚 𝑡 and
therefore the spectrum results from replacing the spectra of the information signal 𝑚 𝑡

o The spectrum of the PAM signal resulting from unipolar flat

top sampling contains impulses at all harmonics of 𝑓 and
ℱ 𝐴 1 + 𝜇𝑚 𝑡 = 𝐴 𝛿 𝑓 + 𝜇𝑀 𝑓
𝑓=0
o Therefore, reconstruction of 𝑚 𝑡 from 𝑥 𝑡 requires a DC
block, low pass filtering and equalization

• The PAM spectrum extends form DC up through several harmonics of 𝑓

o The estimate of required transmission bandwidth 𝐵 must be based on time domain
considerations
1
o Assuming a small pulse duration 𝜏 compared to the time between pulses 𝜏≪𝑇 ≤
2𝐵
1
o Adequate pulse resolution then requires 𝐵 ≥ ≫𝐵
2𝜏

• Practical applications of PAM are limited to those situations in which the advantages of a pulse
waveform outweigh the disadvantages of large bandwidth, as for example in Time Division
Multiplexing (TDM)
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Time Division Multiplexing (TDM) (1/3)

• The information modulated in the train of pulses occupy only a small time interval from the time
available for the transmission of the information
• There is enough “temporal space” for the simultaneous transmission of different and independent
signals over the same bandwidth of the channel (time multiplexing the signals)

𝑚 𝑡 In the figure we have:

• PAM modulation plus time division multiplexing
𝑚 𝑡
of two independent information signals
𝑡 • The pulses with the samples of the signals 𝑚 𝑡 e
𝑚 𝑡 are interlaced in time
• They are transmitted over the same channel
bandwidth

This scheme can be generalized for the simultaneous transmission of 𝑛 signals

• It can happen that the duration of each pulse is decreased to accommodate the signals, and therefore
the bandwidth demanded to the channel is increased

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Time Division Multiplexing (TDM) (2/3)

Implementation of time division multiplexing using PAM modulation
comutator decommutator PAM signals
𝑇
𝑚 𝑡 channel1
𝑚 𝑡
𝑚 𝑡 channel 2
𝑚 𝑡 𝑇
𝑚 𝑡
𝑚 𝑡 𝑚 (𝑡)
communication channel
𝑚 (𝑡) one sample of 𝑚 (𝑡)
one sample of 𝑚 (𝑡)
Emitter: sampling points

• The information signals at the input of the commutator

(or rotating switch) have the same bandwidth Receiver:

• Lowpass filters limit the spectrum of each signal before • The commutator is time-synchronized to

the transmission to mitigate the cross talk, i.e., the one in the emitter to process the

interference with the samples in nearby channels samples outputted from the channel and

• In each turn of the commutator a sample is obtained from to send them to the proper destination

each one of the 𝑁 signals being time multiplexed • Each train of impulses is processes by a

• If 𝐵 is the bandwidth of each signal, from the sampling low pass filter in order to recover the

theorem the commutator must perform (at least) 2𝐵 turns signal

in each second
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Time Division Multiplexing (TDM) (3/3)

𝑁 baseband signals 𝑚 𝑡 , … , 𝑚 (𝑡) (each one with maximum frequency 𝐵 i.e., the same bandwidth)
• Determine the bandwidth demanded for the simultaneous transmission of 𝑁 signals in the PAM-
TDM format
• According to the Nyquist criteria each signal 𝑚 𝑡 , 𝑖 = 1, … , 𝑁 is sampled with a sampling interval
equal or smaller then 𝑇 = 1/2𝐵

o Samples of the remaining 𝑁 − 1 signals are 𝑚 (𝑡)

inserted between adjacent samples of the same 𝑚 (𝑡)
𝑚 (𝑡)
signal
𝑇 1 𝑇 t
o Consecutive samples as separated by = 𝑇/𝑁 = 1/ 2𝐵𝑁
𝑁 2𝐵𝑁

• The bandwidth of the channel must be large enough for the waveform of the received signal to be
approximately of the same format of the waveform at the emitter: it simplifies the demultilexing
• Since the channel is typically bandlimited the channel frequency response changes the format of the
pulses which result in overlapping and interference among symbols
o “Crosstalk”: in each sampling instant the signal is affected by contributions from nearby
channels
o This overlapping of signals from different sources must be avoided
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