Download Chapter 4 Amplitude Modulations and Demodulations

Chapter 4
Amplitude Modulations and Demodulations
Instructor – Oluwayomi Adamo
Introduction
• Modulation is a process that moves the
message signal into a specific frequency band
that is dictated by the physical channel
• Advantages of modulation is Ease of RF
transmission and frequency division
multiplexing
• Modulation can be digital or analog
• Traditional communication systems e.g
AM/FM radios and NTSC television signals are
based on analog modulations
• Second and third generation cellular phone
systems, HDTV and DSL are all digital
Introduction (continued)
• We will study classic analog modulations:
amplitude modulation and angle modulation
• Communication systems that does not use
modulation – baseband communications
• Communication systems that use modulation
– carrier communications
Baseband Versus Carrier Communications
• Baseband is used to designate frequency band of
original message signal from input transducer or
source
• In telephony-baseband is the audio band (0 -3.5kHz)
• NTSC – video baseband is video band (0-4.3 MHz)
• Digital data or PCM that uses bipolar signaling at a
rate of Rb pulses per second (0 – Rb Hz)
• In baseband communication, message signals are
directly transmitted without modification
Baseband Communication
• Baseband signals such as audio and video contain
significant low-frequency content,
• They cannot be effectively transmitted over radio
(wireless) link
• As a result, dedicated user channel such as
twisted pairs of copper wires and coaxial cables
are assigned to each user for long distance
communications
• Baseband signals will interfere with one another
severely since their band overlaps
• FDM allow utilization of one channel by signals
through modulation and shifting of spectra to
nonoverlapping bands.
Carrier Modulations
• Communication that uses modulation to shift the
frequency spectrum of a signal is known carrier
communication
• In terms analog modulation, amplitude, frequency,
or phase of a sinusoidal carrier of high frequency fc
Hz (or ωc = 2πfc rad/s) is varied linearly with the
baseband signal m(t)
• This results in amplitude modulation(AM), frequency
modulation(FM),or phase modulation (PM),
respectively.
• Amplitude modulation is linear while the latter two
types of carrier modulation are similar and nonlinear
( called angle modulation)
Carrier Modulations
• Pulse modulated signal [pulse amplitude
modulation(PAM), pulse width modulation
(PWM), pulse position modulation, pulse code
modulation (PCM) and delta modulation (DM)
will be studied.
• In these cases, analog message is modulating
parameters of a digital pulse train. These
signals can still modulate a carrier in order to
shift the spectra.
Amplitude Modulation and Angle Modulation
• If m(t) is the source message to be transmitted
by the sender to its receivers and
• If M(f) is its Fourier transform
• To move the frequency response of m(t) to a
new frequency band centered at fc Hz –
frequency shifting property will allow us.
• To achieve that, m(t) is multiplied by a
sinusoid of frequency fc:
• Signal frequency content is centered at ±fc
Amplitude modulations
• This allow changes in the amplitude of the sinusoid s1(t) to be
proportional to the message signal (Amplitude Modulation)
• Consider a sinusoidal signal
• There are 3 variables in a sinusoidal: amplitude,
(instantaneous) frequency, and phase.
• The message can be used to modulate any of the 3
parameters to allow s(t) to carry the information from
transmitter to the receiver
• Amplitude A(t) is proportional to m(t) (amplitude modulation)
• Frequency is proportional to m(t) (frequency modulation)
• Phase is proportional to m(t) (phase modulation)
Double-Sideband Amplitude Modulation
• Amplitude modulation is characterize by an informationbearing carrier amplitude A(t) that is a linear function of
the baseband (message) signal m(t)
• At the same time, angular frequency ωc and the phase θc
remains constant ( assume phase θc = 0)
• If carrier amplitude A is made directly proportional to the
modulating signal m(t), then modulated signal is:
m(t)cos ωct (shifts spectrum of m(t) to carrier frequency
• If m(t) M(f) then
• M(f-fc) is M(f) shifted to what direction?
• M(f+fc) is M(f) shifted to what direction?
Double-Sideband Amplitude Modulation
• If the bandwidth of m(t) is B Hz, then the modulated
signal has a bandwidth of 2B Hz.
• The modulated signal spectrum centered at ±fc (or ωc
rad/s) consists of two parts: a portion that lies outside
±fc and is know as upper sideband (USB)
• A portion that lies inside ±fc is known as Lower Sideband
(LSB)
• The modulated signal does not contain a discrete
component of the carrier frequency fc.
• The modulation process does not introduce sinusoid at fc
and as a result, it is called Double-sideband, suppressedcarrier (DSB-SC modulation)
Double-Sideband Amplitude Modulation
• The relationship of B to fc is of interest
• From fig c, if fc >= B thus avoiding overlap, of modulated spectra centered
at ±fc
• if fc < B, the two message spectra overlap and m(t) is distorted during
modulation
• This will make it to recover m(t) from m(t)cos ωct
• Practical factors may impose additional restrictions on fc
•E.g in broadcast applications, a transmit antenna can radiate only a narrow
band without distortion.
Demodulation
• DSB-SC modulation shifts spectrum to right and left by fc
• To recover original signal m(t) from the modulated signal,
it is necessary to retranslate the spectrum to its original
position (Demodulation)
• If modulated signal spectrum in fig c (previous figure) is
shifted to the left and to the right by fc and multiplied by
half, we obtain:
• The figure contains the desired baseband spectrum plus
and unwanted spectrum at ±2fc.
• The unwanted spectrum can be suppressed by a lowpass.
• Demodulation is similar to modulatioin
Demodulation
• Demodulation consists of multiplication of the incoming
modulated signal m(t)cos ωct by a carrier cos ωct
followed by a low pass filter
• This can be verified in the time domain by observing e(t)
as follows:
• Finding the Fourier transform of the signal e(t)
• Signal e(t) consists of two components (1/2)m(t) and
(1/2)m(t)cos2ωct, with their nonoverlapping spectra
Demodulation
• The spectrum of the second component, being a
modulated signal with carrier frequency 2fc, is centered
at ±2fc
• This component is suppressed by low-pass filter
• On the other hand, the desired component (1/2)M(f),
being a low-pass spectrum (centered at f = 0) passes
through the filter unharmed, resulting in (½)m(t)
• You can get rid of the inconvenient fraction ½ in the
output by using a carrier 2cosωct instead of cosωct
• This method of recovering the baseband signal is called
synchronous detection or coherent detection where we
use a carrier of exactly the same frequency(same phase)
as the carrier used for modulation
Example
• For a baseband signal
m(t) = cos ωmt = cos 2πfmt
Find the DSB-SC signal, and sketch its spectrum. Identify
the upper and lower sidebands (USB and LSB). Verify
that the DSB-SC modulated signal can be demodulated
by the demodulator shown previously (synchronous
detection or coherent detection)
• This case is called tone modulation because the
modulating signal is a pure sinusoid or tone, cos ωmt
• Working in frequency domain, the spectrum of the
baseband signal m(t) = cos ωmt is given by
Example (contd)
• The message spectrum consists of two impulses located
at ±fm as shown below
• The DSB-SC modulated spectrum is the baseband
spectrum above shifted left and right by fc as shown
below:
• This spectrum consists of impulses at angular frequencies
±(fc-fm) and ±(fc+fm).
• The spectrum beyond fc is the USB, and the one below fc
is LSB.
• The DSB-SC spectrum does not have the component of
the carrier frequency fc (suppressed carrier)
Example (Contd)
• In the time domain, for the baseband signal m(t), the
DSB-SC signal
• When the baseband is a single sinusoid of frequency fm,
the modulated signal consists of two sinusoids; the
component of frequency fc+fm (USB) and the component
of frequency fc-fm (LSB)
• Each component of frequency fm in the modulating
signal turns into two components of frequencies fc+fm
and fc –fm in the modulated signal.
Example (Contd)
• When the modulated signal is applied to the input of
demodulator (coherent detector)
• The spectrum of the term cos ωmt cos 2ωct is centered at
2fc and will be suppressed by the low-pass filter, yielding
1/2cos ωmt as the output
• This results in the spectrum shown below:
Modulators
• Multiplier Modulators: Modulation is achieved directly
by using an analog multiplier whose output is
proportional to the product of two signals m(t) and cos
ωct.
• Typically, the multiplier is obtained from a variable-gain
amplifier in which the gain parameter is controlled by
one of the signals e.g m(t).
• When cos ωct is applied to the input of the amplifier, the
output is proportional to m(t)cos ωct
• Non-Linear Modulator: Modulation is achieved through
nonlinear devices such as a semiconductor diode or a
transistor.
Non-linear Modulator
• Let the input-output characteristics of either of the
nonlinear elements be approximated by a power series
y(t) = ax(t) + bx2(t)
Where x(t) and y(t) are the input and output of the
nonlinear element.
• The summer output z(t) is given by
• Substituting the two inputs x1(t) and x2(t)
Nonlinear Modulators
• The spectrum of m(t) is centered at the origin and
M(t)coswc(t) is centered at ±ωc
• Passing z(t) through a bandpass filter tuned to ωc, the
signal am(t) is suppressed and the desired modulated
signal 4bm(t)cos ωct can pass through the system
without distortion
• Because the cos ωct does not appear at the z(t), this
setup is called balanced circuit
• The nonlinear modulator is an example of a class of
modulators known as balanced modulator.
• Because m(t) appear in z(t), it is called single balance
modulator, however, m(t) is removed through bandpass
filter.
Switching Modulators
• Switching Modulator: multiplication operation for
modulation is replaced by a simple switching operation
• This is because the sinusoid can be replaced by any
periodic signal φ(t) with fundamental radian frequency
ωc.
• The periodic signal can be expressed as:
• Hence
• This shows that the spectr4um of the product m(t)φ(t) is
the spectrum M(f) shifted to ±fc, ±2fc, ……… ±nfc….
• Passing the signal through bandpass filter of bandwidth
2B Hz and tuned to fc will result c1m(t)cos(ωct+θ1)
Switching Modulator
• The square pulse in b is a periodic signal whose Fourier
series is:
• The signal m(t)w(t) is given by
• The signal m(t)w(t) consists of m(t) and an infinite
number of modulated signals with angular frequency ωc,
3ωc, 5ωc…..
• Spectrum of m(t)w(t) consists of m(t) shifted by ±fc, ±3fc
….. (with decreasing relative weight)
Switching Modulator
• We are only interested in m(t)cosωct, hence the signal
m(t)w(t) is passed through a bandpass filter of
bandwidth 2B Hz centered at ±fc
• This will suppress all spectra components not centered at
±fc to yield the desired modulated signal (2/π)m(t)cos
ωct as shown in fig (d)
Nonlinear Modulator – Switching Modulator
• The advantage of this scheme is that multiplication of a
signal by a square pulse train is in reality a switching
operation.
• It involves switching the signal m(t) on and off
periodically and can be implemented using simple
switching element controlled by w(t)
• Example is the diode bridge modulator driven by a
sinusoid Acos ωct to produce the switching action.
• Diode D1, D2 and D3 and D4 are matched pairs.
• When the signal cos ωct is of a polarity that will make
terminal c positive with respect to d, all diodes conduct
Switching Modulators
• Because diodes D1 and D2 are matched, terminals a and
b have the same potential and are effectively shorted
• During the next half-cycle, terminal d is positive with
respect to c and all four diodes open, thus opening
terminal a and b.
• This therefore acts as a switch.
• Terminals a and b open and close periodically with carrier
frequency fc when a sinusoid A cos ωct is applied across c
and d.
Switching Modulator
• To obtain m(t)cos wct, terminals a and b are connected
in series or across (parallel) to m(t) as shown below:
• This is called series bridge diode modulator and the
shunt bridge diode modulator.
• The switching on and off periodically with fc results in
switched signal m(t)w(t) which when bandpassed yields
modulated signal (2/π)m(t)cosωct
Ring Modulator
• This another switching modulator.
• During the positive half-cycles of the carrier, diodes D1
and D3 conduct and D2 and D4 are open.
• Terminal a is therefore connected to c and terminal b to
d.
• During negative half-cycles of the carrier, D1 and D3 are
open, D2 and D4 are conducting
• Terminal a and d are connected and so is b and c.
• Output is proportional to m(t) during positive half-cycle
and to –m(t) during the negative half-cycle.
Ring Modulator
• m(t) is multiplied by a square pulse w0(t) shown below:
• The fourier series of w0(t) is given below:
• The signal m(t)w0(t) is shown below:
• When m(t)w0(t) is passed through a bandpass filter tuned
to ωc , the filter output will be (4/π)m(t)cos ωct
Ring Modulator
• The Ring modulator circuit has two: m(t) and cos ωct
• The input to the bandpass filter does not contain either
of these inputs
• As a result, this circuit is an example of a double
balanced modulator
Example (Frequency Mixer or converter)
• Frequency mixer or converter: is used to change the
carrier angular frequency of a modulated signal
m(t)cosωct from ωc to ωI
• This is achieved by multiplying m(t)cos ωct by 2cosωmixt,
where ωmix = ωc + ωI or ωc - ωI and bandpass filtering the
product
• Product x(t) is
• If ωmix = ωc - ωI then
Example (Frequency Mixer or converter)
• If ωmix = ωc + ωI then
• The spectra in the figure below will not overlap as long as
ωc + ωI >= 2πB and ωI >= 2πB.
• When a bandpass filter tuned to ωI is applied at the
output, m(t) cosωIt will be passed and the other spectra
will be suppressed.
• As a result, carrier frequency ωc has been translated to ωI
Example (Frequency Mixer or converter)
• The operation of frequency mixing/conversion is known
as heterodyning.
• This is basically a shifting of spectra by an additional ωmix
• This is also equivalent to the operation of modulation
with modulating carrier frequency (mixer oscillator
frequency ωmix ) that differs from incoming carrier
frequency by ωI .
• Already discussed modulator can be used for the
frequency mixing
• When local carrier frequency is ωmix = ωc + ωI , the
operation is called super-heterodyning and when we
select ωmix = ωc - ωI , it is called sub-heterodyning.
Demodulation of DSB-SC
• Demodulation of DSB-SC signal is essentially
multiplication with the carrier signal and is identical to
modulation
• At the receiver, the modulated signal is multiplied with a
local carrier of frequency and phase in synchronism with
the incoming carrier.
• The product is passed through a lowpass filter.
• The only difference between modulator and
demodulator lies in the input signal and output filter?
• In modulator, input is m(t) and output is passed through
a bandpass filter, while in demodulator, DSB-SC signal is
the input and the multiplier output is passed through a
lowpass filter.
• All previous modulator can be used.
Demodulation of DSB-SC
• Demodulators whereby a carrier is synchronous in phase
and in frequency with the incoming carrier are called
synchronous or coherent (homodyne) demodulators
Example:
Analyze the switching demodulator that uses the electronic switch
(diode bridge) as a switch
• The input signal is m(t)cos ωct
• The carrier causes periodic switching on and off of the
input signal, therefore output is m(t)cos ωct x w(t).
• Using cos x cos y = 0.5[cos (x+y) + cos (x-y) gives:
• Spectra of the terms of the form m(t)cos ωct are centered
at nωc rad/s and are filtered out using lowpass filter
Amplitude Modulation (AM)
• DSB-SC amplitude modulation is easy to understand in both
time and frequency domains, but does not have equivalent
simplicity in practical implementation
• The coherent modulation of DSB-SC requires the receiver to
possess a carrier signal that is synchronized with incoming
carrier. (Not easy to achieve in practice)
• The modulated signal may have traveled hundreds of miles
and could have suffered from unknown frequency shift.
• The bandpass received signal has the form:
r(t) = Acm(t-t0)cos[(ωc +Δω)(t-t0)] = Acm(t-t0)cos[(ωc +Δω)t-θd)]
Δω represents the Doppler effect and comes from unknow delay t0 :
θd =(ωc +Δω)td
• To utilize the coherent demodulator, the receiver most be
sophisticated enough to generate a local oscillator cos[(ωc
+Δω)t-θd)] from r(t) -This will be difficult and costly
Amplitude Modulation (AM)
• Alternative to coherent demodulator is for the
transmitter to sent A cos ωct [along with the modulated
signal m(t) cos ωct.
• As a result, there will be no need to generate a carrier at
the receiver
• However, transmitter will need to transmitt at a much
higher power level which makes it more costly as a tradeoff.
• This option is obvious choice in broadcasting because of
desirable trade-offs.
• In point to point communication, there is one transmitter
for every receiver, hence substantial complexity in the
receiver system can be justified if the cost is offset by less
expensive transmitter.
Amplitude Modulation (AM)
• Transmitting the carrier with the modulated signal leads
to so-called AM in which the transmitted signal is given
as:
• The spectrum is the same as the DSB-SC (m(t) cos ωct)
except for two additional impulses ±fc
• Comparing φAM(t) and φDSB-SC (t) = m(t) cos ωct, Am
signal is identical to the DSB-SC signal with A+m(t) as the
modulating signal [instead of m(t)]
• To sketch φAM(t), we sketch the envelope |A+m(t)| and
its mirror image -|A+m(t)| and fill in between with the
sinusoid of carrier frequency fc.
• The size of A affects the time domain envelope of the
modulated signal
Amplitude Modulation (AM)
• Example cases for A:
• In fig b (first case), A is large enough to ensure that
A+m(t)≥0 is always nonnegative
• In fig c (second case), A is not large enough to satisfy the
previous condition.
• In the first case, the envelope has the same shape as m(t)
(although riding on a direct current of magnitude A)
• In the second case, envelope shape differs from the shape
of m(t) because the negative part of A+m(t) is rectified
Amplitude Modulation (AM)
• The desired signal m(t) can be detected by detecting the
envelope in the first case when A+m(t)>0
• However, it is not possible in the second case.
• Envelope detection is a simple and inexpensive operation
that does not require the generation of local carrier at the
receiver and will be studied further later
• Envelope of AM has the information about m(t) only if AM
signal [A+m(t)]cos ωct satisfies the condition A+m(t)>0 for
all t
• Consider a signal E(t)cos ωct . If E(t) varies slowly in
comparison with the sinusoidal carrier cos ωct , then the
envelope of E(t) cos ωct is |E(t)| Hence (if A+m(t)>0 ):
Envelope of φAM(t) = |A+m(t)|=A+m(t)
Amplitude Modulation (AM)
• For envelope detection to properly detect m(t), these
condition needs to be met:
• Fc>> bandwidth of m(t)
• A+m(t) ≥ 0
• The conclusion is verified from:
• In d) A+m(t) is the envelope and m(t) can be recovered
• In e) A+m(t) is not always positive, The envelope |A+m(t)|
is rectified from A+m(t), hence m(t) cannot be recovered.
• Demodulation therefore amounts to simple envelope
detection.
Amplitude Modulation (AM)
• The condition for envelope detection of an AM signal is
A+m(t) ≥ 0 for all t
• If m(t) ≥0 for all t, then A=0 already satisfies the condition
above and as a result, you don’t need to add any carrier
because the envelope of the DSB-SC signal is m(t),
• DSB-SC signal can be detected by envelope detection.
• If we assume that m(t) is not greater than or equal 0
• m(t) with zero offset
• Let ±mp be the maximum and the minimum values of m(t)
respectively.
• Then m(t) ≥ -mp hence the condition of envelope detection
is :
A≥-mmin=mp
• The minimum carrier amplitude required for the viability of
envelope detection is mp.
Amplitude Modulation (AM)
• Modulation index: µ = mp/A
• For envelope detection to be distortionless, the condition
is
0 ≤ µ ≤1
• Hence the same condition holds for distortionless
demodulation of AM by an envelope detector
• If A < mp, µ >1 (overmodulation)
• Envelope detection will not be viable, synchronous
demodulation is used instead
Message signals m(t) with nonzero offset:
• On some occasions, the message signal m(t) will have a
nonzero offset such that maximum mmax and its
minimum mmin are not symmetric
• Envelope detection still remains distortionless if 0 ≤ µ ≤1
µ = (mmax – mmin)/(2A+mmax+mmin)
Example
• Sketch φAM(t) for modulation indices of µ = 0.5 and µ = 1,
when m(t) = bcosωmt (tone modulation because
modulating signal is a pure sinusoid or tone)
• Mmax=b and mmin = -b, hence
µ = b/A
B = µA and
Therefore:
Sideband and Carrier Power
• The price of envelope detection is that the carrier term
does not carry any information and hence the carrier
power is wasteful
• The carrier power is the mean square value of Acos ωct,
which is A2/2 .
• The sideband power Ps is the power of m(t) cos ωct, which
is 0.5 m2(t) hence:
• Useful message resides in the sideband power and the
carrier power is used for convenience in modulation and
demodulation.
• Total power is the sum of carrier (wasted) and sideband
(useful) power.
Sideband and Carrier Power
• The power efficiency:
• Considering tone modulation:
• Hence
• Efficiency increases monotonically with µ and ηmax occurs
at µ=1 for which ηmax =33%
• For tone modulation, under the best condition (µ=1), only
one-third of the transmitted power is used for carrying
messages.
Example
• Determine η and the percentage of the total power
carried by the sidebands of the AM wave for tone
modulation when a) µ = 0.5 b) µ = 0.3
• For µ = 0.5
• Hence, only 11% of the total power is in the sidebands
• Hence, only 4.3% of the total power is the useful
information power (in sidebands)
Generation of AM Signals
• Generation is similar to DSB-SC modulation. What is the
difference?
Demodulation of AM signal
• Demodulation can be carried out coherently like th DSB-SC
where local carrier is generated.
• However, coherent or synchronous demodulation defeats
the purpose of AM. Does not take advantage of the
additional carrier
• We saw that the envelope of AM signal follows the
message signal m(t) for µ ≤ 1.
• Hence, two noncoherent methods of AM demodulation
will be consider for 0< µ ≤ 1 a) Rectifier detection and b)
envelope detection
Rectifier
• If an AM signal is applied to a diode and a resistor circuit,
the negative part of the AM wave will be removed.
• The output across the resistor is half-wave rectified
version of the AM signal.
• The diode acts like a pair of scissors by cutting off any
negative half-cycle of the modulated sinusoid. At the
rectifier output, the AM signal is multiplied by w(t). Half
wave rectified output VR(t) is
Rectifier
• When VR(t) is applied to a low pass filter of cutoff B Hz,
the output [A+m(t)]/π and all the other terms of
frequencies higher than B Hz are suppressed.
• The dc term A/π may be blocked by a capacitor to give the
desired output m(t)/π
• It should be noted that even though it is a w(t) that is used
for multiplication, it is still a synchronous detection that is
performed with using a local carrier.
Envelope Detection
• The output of the detector follows the envelope of the
modulated signal.
Envelope Detector
• On positive cycle of the input signal, the input grows and
may exceed the charged voltage on the capacity vc(t)
• This turns on the diode and allowing the capacitor C to
charge up to the peak voltage of the input signal cycle.
• As the input signal falls below this peak value, it falls
quickly below the capacitor voltage (near peak voltage)
the diode therefore opens.
• Capacitor then discharges through the resistor R at a slow
rate with time constant (RC). The same scenario during
the next positive cycle.
•
Envelope Detector
• During each positive cycle, the capacitor charges up to the
peak voltage of the input signal and then decays slowly
until the next positive cycle.
• As a result, the output voltage Vc(t) closely follows the
rising envelop of the input AM signal.
• Slow capacity discharge via the resistor R allows the
capacity voltage to follow a declining envelope
• Capacitor discharge between positive peaks causes a ripple
signal of frequency ωc in the output.
• Ripple can be reduced by choosing a larger RC (drawback?)
• Design Criteria (RC < 1/2πB)
•
Bandwidth-Efficient Amplitude Modulations
• The DSB spectrum( including AM) has two sidebands: upper
sideband (USB) and lower Sideband (LSB) both containing
information about m(t)
• As a result, for a baseband signal m(t) with bandwidth B Hz,
DSB modulation require twice the radio frequency
bandwidth to transmit.
Bandwidth-Efficient Amplitude Modulations
• To improve the spectral efficiency of the amplitude
modulation, two schemes can be used to either utilize or
remove the 100% spectra redundancy.
• Single-sideband (SSB) modulation, which removes either
the LSB or USB so that for one message signal m(t), there is
only a bandwidth of B Hz
• Quadrature amplitude modulation (QAM), which utilizes
spectral redundancy by sending two messages over the
same bandwidth of 2B Hz
Amplitude Modulation: Single Sideband (SSB)
• Either the LSB or USB can be suppressed from the DSB
signal via a bandpass filtering.
• A scheme in which only one sideband is transmitted is
known as Single sideband (SSB) transmission and requires
only half the bandwidth of the DSB signal
• An SSB signal can be coherently (synchronously)
demodulated just like DSB –SC signals.
• Multiplication of a USB signal in fig c by cos ωct shifts its
spectrum to the left and right by fc yielding the spectrum as
shown in fig e.
• Low pass filtering of this signal in e yields the baseband
signal. Since no additional carrier accompany the
modulated signal it is called SSB-SC
Hilbert Transform
• New tool called Hilbert transform
• The Hilbert transform of a signal x(t)
• The right handside of equation is the form of convolution
• Applying duality to pair 12 of the transform table:
• Applying time convolution property to topmost equation:
• Looks as if m(t) is passed through a transfer function H(f) = jsgn(f), then output is mh(t), Hilbert transform of m(t)
Hilbert Transform
• Hence H(f) = 1 and θh(f) = -π/2 for f>0 and π/2 for f <0 as
shown below:
• Hence if we change the phase of every component m(t) by
π/2 (without changing the amplitude) mh(t) results which is
the Hilbert transform of m(t).
• Hilbert transformer is an ideal phase shifter that shifts the
phase of every positive spectra component by - π/2
Time domain representation of SSB signals
• The building blocks of SSB signal are sidebands, to obtain
time domain expression:
• Expressing the SSB signal in terms of m(t) and mh(t).
• Finding inverse transform using frequency shifting property
Time domain representation of SSB signals
• We can also show for LSB
• Hence general SSB can be shown:
• Minus sign is for USB and plus for LSB
• To check whether SSB can be coherently demodulated:
• The product
yields the baseband signal.
What type of filter will be applicable for this output
Example
• Find
for the simple case of a tone modulation that is a
modulating signal that is a sinusoid m(t) = cos ωmt. Also
demonstrate the coherent demodulation of the SSB signal
• Remember that Hilbert transform delays the phase of each
spectral component by π/2.
• There is only one spectral component of frequency ωm.
Delaying the phase by π/2 gives:
• From previous equation:
Example (contd)
• Consider the spectrum m(t) in fig a and the DSB-SC in fig b,
USB in fig c and LSB in fig d.
• Coherent demodulation of the SSB tone modulation can be
achieved by
• This can be sent to a lowpass filter to retrieve the message
tone cos ωmt
SSB Modulation Systems
• Three methods used in generating SSB signals:
• Phase shifting, Selective filtering and the Weaver method
• None of them are precise and they all require that the
baseband signal spectrum have little power around the
origin
• Phase shift method uses directly
• The figure below shows its implementation:
• π/2 designates a ±π/2 phase shifter, which delays the phase
of every positive spectral component by π/2 (Hilbert
Transformer). There is no ideal phase shifter.
SSB Modulation Systems
• Selective-filtering method is the most commonly used
method of generating SSB signals.
• A DSB-SC signal is passed through a sharp cutoff filter to
eliminate the undesired sideband.
• To obtain the USB, the filter should pass all components
above frequency fc unattenuated and completely suppress
all components below fc.
• This will require ideal filter which is unrealizable
• It is realizable if there is a separation between the passband
and stopband.
• Weavers Method utilizes two stages of SSB amplitude
modulation.
• The modulation is carried out using smaller carrier fc1
Detection of SSB Signals with a Carrier (SSB+C)
• Consider SSB signals with additional carrier (SSB+C)
• M(t) can be recovered by synchronous detection
(multiplying by cos ωct) if the carrier amplitude A is large
enough,