Chapter 4Amplitude 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 information-

bearing 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, suppressed-carrier (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 low-pass.

• 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 trade-off.

• 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 ≤1µ = (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,