7 amplitude modulation

Amplitude Modulation

m(t)

s(t)

AM waveform with

( ) 1k ( ) 1ak m t <

s(t)

Over-modulated waveform

( ) 1k m t >( ) 1ak m t >

Amplitude Modulation 1


The sinusoidal carrier wave :

The AM waveform, DSB-LC :

( ) ( )cos 2c cc t A f tπ=

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 cos 2c a c

c a c

s t A k m t f t

A k AS f f f f f M f f M f f

π

δ δ

= +⎡ ⎤⎣ ⎦

⎡ ⎤+ + + + +⎡ ⎤⎣ ⎦

In AM, information pertaining to the message signal m(t) resides solely in the l hi h i d fi d th lit d f th d l t d ( )

( ) ( ) ( ) ( ) ( )2 2c c c cS f f f f f M f f M f fδ δ ⎡ ⎤= − + + + − + +⎡ ⎤⎣ ⎦ ⎣ ⎦

envelope, which is defined as the amplitude of the modulated wave s(t).

The carrier frequency fc is much greater than the highest frequency component Wof the message signal m(t).

Suppose the message signal m(t) is bandlimited to the interval −W ≤ f ≤ W, asshown in Fig. 3.2a.



Th i l hThe message signal has a bandwidth W Hz.

2W



Example 3.1 Single-Tone Modulation

( ) ( )cos 2m mm t A f tπ=

where fm is its frequency, and

Am is the amplitude of the sinusoidal modulating wave

( ) ( ) ( )f f⎡ ⎤( ) ( ) ( )[ ] [ ]

( )max min

1 cos 2 cos 2

where ; 1 ; 1

1

c m c

a m c c

s t A f t f t

k A A A A A

AA

μ π π

μ μ μ

= +⎡ ⎤⎣ ⎦= = + = −

+( )( )

maxmax max min min

min

max max min min max min max min

1and

1c

c c c cc

AA A A A A A A A AA A

A A A A A A A A

μμ μ

μμ μ μ μ

+= ⇒ − = +

−

− = + ⇒ − = +max max min min max min max min

max min PThe modulation factor, A AA A

μ μ μ μ

μ −= =

eak DSB-SC AmplitudeP k C i A lit d


max minA A+ Peak Carrier Amplitude


⎡ ⎤( ) ( ) ( )[ ] [ ]max min

1 cos 2 cos 2

where ; 1 ; 1c m c

a m c c

s t A f t f t

k A A A A A

μ π π

μ μ μ

= +⎡ ⎤⎣ ⎦= = + = −

max min

max min

Peak DSB-SC AmplitudeWith Peak Carrier Amplitude

m

c

A A AA A A

μ −= = =

+

[ ]

max min p

1 1

c

mAA A A A Aμ⎡ ⎤

= + = + = +⎢ ⎥[ ]

[ ]

max

min

1 1

1 1

c c c mc

mc c c

A A A A AA

AA A A A AA

μ

μ

+ + +⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

= − = − = −⎢ ⎥⎢ ⎥

m[ ]min c c ccA

μ ⎢ ⎥⎢ ⎥⎣ ⎦

( ) ( )( ) ( )

max min22

m

c m c m m mA A A A AA A A

A A A AA A A Aμ

+ − −−= = = =

+


( ) ( )max min 2 c cc m c mA A A AA A A A+ + + −

Amplitude Modulation, DSB-LCp ,

maxA

2

3

minA1

2

0

-1

0

0 0 5 1 1 5 2 2 5 3-3

-2

0 0.5 1 1.5 2 2.5 3

x 10-3

max min 2.8 1.2 1.6 0.42 8 1 2 4

A AA A

μ − −= = = =

+ +


max min 2.8 1.2 4A A+ +



Amplitude Modulation, DSB-LC

Example 3.1 Single-Tone Modulation (continued)

( ) ( ) ( )1 cos 2 cos 2c m cs t A f t f tμ π π= +⎡ ⎤⎣ ⎦( ) ( ) ( )

( ) ( ) ( )

cos 2 cos 2 cos 2

1 1cos 2 cos 2 cos 22 2

c c c m c

c c c c m c c m

A f t A f t f t

A f t A f f t A f f t

π μ π π

π μ π μ π

= +

= + + + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦( ) ( ) ( )

( ) ( ) ( )2 2

12

c c c c m c c m

c c cS f A f f f fδ δ

⎣ ⎦ ⎣ ⎦

⎡ ⎤= − + +⎣ ⎦

( ) ( ){ }( ) ( ){ }

141

c c m c mA f f f f f f

A f f f f f f

μ δ δ

μ δ δ

⎡ ⎤ ⎡ ⎤+ − + + + +⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤+ + +⎣ ⎦ ⎣ ⎦( ) ( ){ }4 c c m c mA f f f f f fμ δ δ⎡ ⎤ ⎡ ⎤+ − − + + −⎣ ⎦ ⎣ ⎦

Can you identify each component in Fig. 3.3c from the above expression?




( ) ( ) ( )1 cos 2 cos 2c m cs t A f t f tμ π π= +⎡ ⎤⎣ ⎦( ) ( ) ( )cos 2 cos 2 cos 2c c c m cA f t A f t f tπ μ π π= +

( ) ( ) ( ) ( )2 2 2 2 2 2 2

The mean square value or time averaged value:

( ) ( ) ( ) ( )

( )

2 2 2 2 2 2 2

2 2

cos 2 cos 2 cos 2

1 1 1 1 1 1Since cos 1 cos 2 cos cos 2 02 2 2 2 2 2

c c c m cs t A f t A f t f t

A A A A

π μ π π= +

= + ⇒ = + = + × =

( )2 2 2

2 2 2 2 2 21cos 2 Carrier Power2

1 1

c c cA f t Aπ = =

( ) ( )2 2 2 2 2 2 1 1cos 2 cos 22c m c cA f t f t Aμ π π μ= × ×

2 2

21 Sidebands Power=USB Power + LSB PowercAμ= =


4 cμ



21Sidebands power A=

2 2 2 2

Sidebands power2

1 1Upper sideband power ; Lower sideband power 8 8

c

c c

A

A Aμ μ

=

= =

2 2 2 2 22

22 2 2 2

1 1 1Total sidebands power 8 8 4

1 1 1 1Total power 2

c cA A

A A

μ μ μ μμ

+= = =

++ +2 2 2 2Total power 22 4 2 4c cA A μμ μ ++ +

Total power = ( )2 2 2 21 1s t A Aμ= +Total power ( )2 4c cs t A Aμ+




Envelope Detection


Envelope DetectionEnvelope Detection

An envelope detector is used to detect the envelope of the modulated waveform. Any circuit whose output follows the envelope of thecircuit whose output follows the envelope of the input signal waveform will serve as an envelope detector.

The simplest form of an envelope detector is a nonlinear charging circuit with a fast charge time and a slow discharge time.

On the positive half cycles of the input signal, the diode is forwarded biased and the capacitorthe diode is forwarded-biased and the capacitor C charges to the peak value of the input signal waveform. As the input signal falls below this value, the diode is turned off. The capacitor slowly discharges through the resistor R until theslowly discharges through the resistor R until the next positive half-cycle. When the input signal becomes greater than the capacitor voltage and the diode will conduct again and the process is repeated


repeated.

Double Sideband-Suppressed Carrier, DSB-SC

( )m t ( )s t

( ) ( )cos 2c cc t A f tπ=



Product modulator is used to generate DSB-SC modulated wave. Themodulated signal s(t) undergoes a phrase reversal whenever the message signalm(t) crosses zero, as indicated in Fig. 3-10b. The envelope of a DSB-SCm(t) crosses zero, as indicated in Fig. 3 10b. The envelope of a DSB SCmodulated signal is therefore different from the message signal m(t), whichmeans that envelope detection is not applicable to retrieve m(t).



( ) ( ) ( ) ( ) ( )

( ){ } ( ) ( ) ( )cos 2

1c cs t c t m t A f t m t

t S f A M f f M f f

π= =

⎡ ⎤ℑ

The message signal m(t) has a bandwidth of W. The modulation processtranslates the spectrum of the message signal by f to the right and by –f to the

( ){ } ( ) ( ) ( )2 c c cs t S f A M f f M f f⎡ ⎤ℑ = = − + +⎣ ⎦

translates the spectrum of the message signal by fc to the right and by fc to theleft. The transmission bandwidth required by DSB-SC modulation is the sameas that for amplitude modulation, 2W.

The only advantage for DSB-SC is saving transmitted power. The Amplitudemodulation is using DSB-LC. The demodulator is more complex for DSB-SCthan the envelope detector for DSB-LC.


Coherent Detection

The demodulation of a DSB-SC signal is under the assumption that the localoscillator of the receiver is exactly coherent or synchronized, in bothfrequency and phase, with the modulated signal.


Coherent Detection

If there is a small frequency error, ∆ f, and a phase error, φ, in the local oscillatorof the receiver.

( )( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

cos 2 where cos 2

cos 2 cos 2

c c c c

t

s t A f f t m t s t A f t

A A f t f f t m t

ν

π φ π

π π φ

′= + Δ + =⎡ ⎤⎣ ⎦′= + Δ +⎡ ⎤⎣ ⎦( ) ( ) ( )

( ) ( ) ( ) ( ){ }( ) ( ) ( ) ( ) ( ){ }

cos 2 cos 2

cos 2 cos 2 cos sin 2 sin

cos 2 cos 2 cos cos 2 sin 2 sin

c c c c

c c c c c

c c c c c c

A A f t f f t m t

A A m t f t f f t f f t

A A m t f t f f t f t f f t

π π φ

π π φ π φ

π π φ π π φ

= + Δ +⎡ ⎤⎣ ⎦′= + Δ − + Δ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

′= + Δ − + Δ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦( ) ( ) ( ) ( ) ( ){ }cA A

⎣ ⎦ ⎣ ⎦

= ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

cos 2 cos 2 cos 2 sin 2 sin 2 cos

cos 2 sin 2 cos 2 cos 2 sin 2 sin

c c c c

c c c c c

m t f t f t f t f t f t

A A m t f t f t f t f t f t

π π π π π φ

π π π π π φ

′ ⎡ ⎤Δ − Δ⎣ ⎦′ ⎡ ⎤− Δ − Δ⎣ ⎦

( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( ) ( ) ( )

2cos 2 cos 2 cos cos 2 sin 2 sin 2 cos

cos 2 sin 2 cos 2

c c c c c

c c c c

A A m t f t f t f t f t f t

A A m t f t f t f t

π π φ π π π φ

π π π

′= Δ − Δ

′− Δ ( ) ( ){ }2sin cos 2 sin 2 sincf t f tφ π π φ− Δ


Coherent Detection

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2cos 2 cos 2 cos sin 2 sin

cos 2 sin 2 sin 2 cos cos 2 sin

c c c

c c c c

t A A m t f t f t f t

A A m t f t f t f t f t

ν π π φ π φ

π π π φ π φ

′= Δ − Δ⎡ ⎤⎣ ⎦′− Δ + Δ⎡ ⎤⎣ ⎦

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2cos 2 cos 2 cos sin 2 sin

cos 2 sin 2 sin 2 cos cos 2 sin

c c c

c c c c

A A m t f t f t f t

A A m t f t f t f t f t

π π φ π φ

π π π φ π φ

′= Δ − Δ⎡ ⎤⎣ ⎦′− Δ + Δ⎡ ⎤⎣ ⎦

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 i 22 i 2A A f f A A ff fφ φ′ ′Δ Δ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( )

2cos 2 cos 2 sin 2

1With sin sin 2 2cos sin cos sin sin 2 , then2

cos 2 sin 2

sin 4

c c c c c c cA A m t f t f t A A m t f t

A A A A A A A A

f t f t

f t

π π φ π φπ π

π

′ ′= Δ + − Δ +

+ = = ⇒ =

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

2cos 2 cos 2 sin 2

cos 21 cos 42

sin 42cc c c c

c c

c

c

ft A A m t f t A A m t f t

A

t

A m t f

f t

tf t

ν π φ ππ

φ

π

π

π

′ ′= Δ + − Δ +

⎡ ⎤′= Δ⎢ ⎥+

⎣ ⎦( ) ( ) ( ) ( )sin 4 sin 2

2c c

c

A Am t f t f tφ π π φ′

+ − Δ +2⎢ ⎥⎣ ⎦

( ) ( ) ( ) ( ) ( )

2

cos 2 cos 4 cos 22 2

c c c cc

A A A Am t f t m t f t f tπ φ π π φ′ ′

= Δ + + Δ +

′

Amplitude Modulation 19( ) ( ) ( )sin 4 sin 2

2c c

c

A Am t f t f tπ π φ′

− Δ +

Coherent Detection

( ) ( ) ( ) ( ) ( )cos 22

cos 242

c cc cc

A AAt m t f t

Am t f t f tν π φπ π φ

′= + Δ +

′+ Δ +

The first term is centered at 2 and can be filtered out by the LP .Fcf f± + Δ

Unless both ∆ f and φ are zero, otherwise v(t) is not equal to ( ).2

c cA Am t′

( )2

If ∆ f = 0, then ( ) ( )cos .2

c cA At m tν φ

′=

This phase error in the local carrier causes an attenuation of the output signal proportional to the cosine of the phase error, φ.

If φ = 0 then ( ) ( ) 2c cA At t f t

′ΔIf φ = 0, then

If φ = 90º, the received signal will be wiped out, v(t) = 0.

( ) ( )cos 2 .2

c ct m t f tν π= Δ


Coherent Detection

( ) ( ) ( )0 cos 22

c cA At m t f tν π φ

′= Δ +

The second term can be expressed as:

( ) ( ) ( )2

1 cos 2 cos sin 2 sin2 c cA A m t f t f tπ φ π φ′= Δ − Δ⎡ ⎤⎣ ⎦

This phase error in the local carrier causes an attenuation of the output signalThis phase error in the local carrier causes an attenuation of the output signal proportional to the cosine of the phase error φ, and Δ f is usually within ± 2% of the carrier frequency.

If φ = 0 then ( ) ( ) 2c cA At t f t

′ΔIf φ = 0, then

If φ = 90º, the received signal will be wiped out, v0 (t) ≈ 0.

I l th h φ i t t t It i t h i th

( ) ( )0 cos 2 .2

t m t f tν π= Δ

In general, the phase error φ is not constant. It is necessary to synchronize thelocal oscillator in the receiver in both frequency and phase with the carrierwave to generate the DSB-SC modulated signal in the transmitter.


Coherent Detection

( ) ( ) 2c cA At t f t

′Δ( ) ( )0 cos 2 .

2t m t f tν π= Δ


Costas Receiver


Costas Receiver

This receiver consists of two coherent detectors supplied with the sameinput signal, Ac cos (2π fc t) m(t), but with two local oscillator signals thatare in phase quadrature with respect to each other.

Suppose the local oscillator drifts from its proper value by a small angle φradians. The I-channel output is proportional to cos φ and cos φ ≈ 1 forsmall φ. The Q-channel output will have the same polarity as the I-channeloutput for one direction of local oscillator phase driftφ and the oppositepolarity for the opposite direction of φ.

By combining the I- and Q-channel outputs in a phase discriminator (whichconsists of a multiplier followed by a time-averaging unit), a dc controlsignal proportional to the phase drift φ is generated. With negative feedbackacting around the Costas receiver, the control signal tends to automaticallycorrect for the local phase error φ in the voltage-controlled oscillator.correct for the local phase error φ in the voltage controlled oscillator.


Single-Sideband Modulation, SSB

( ) ( )( ) ( )

Modulating signal: cos 2

Carrier signal: cos 2m m

c c

m t A f t

c t A f t

π

π

=

=( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )

g

cos 2 cos 2

1 1cos 2 cos 22 2

c c

DSB SC c m c m

c m c m c m c m

f

S t c t m t A A f t f t

A A f f t A A f f t

π π

π π

− = =

= + + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

( ) ( )

( ) ( )

2 21 cos 221 1cos 2 cos 2 sin 2

SSB c m c mS t A A f f t

A A f t f t A A

π

π π π

+ = +⎡ ⎤⎣ ⎦

= − ( ) ( )sin 2f t f tπ( ) ( )cos 2 cos 2 sin 22 2c m c m c mA A f t f t A Aπ π π= − ( ) ( )

( ) ( )

sin 2

1 cos 221 1

c m

SSB c m c m

f t f t

S t A A f f t

π

π− = −⎡ ⎤⎣ ⎦

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1cos 2 cos 2 sin 2 sin 22 21 1cos 2 cos 2 sin 2 sin 22 2

c m c m c m c m

SSB c m c m c m c m

A A f t f t A A f t f t

S t A A f t f t A A f t f t

π π π π

π π π π±

= +

= ∓


2 2

Single-Sideband Modulation, SSB

( ) ( ) ( ) ( ) ( )1 1cos 2 cos 2 sin 2 sin 22 21 1

SSB c m c m c m c mS t A A f t f t A A f t f tπ π π π± = ∓

( ) ( ) ( ) ( )1 1cos 2 sin 22 2c c c cA m t f t A m t f tπ π= ∓

h ( ) b d i d f ( ) i l b hif i h h f hwhere m(t) can be derived from m(t) simply by shifting the phase of each frequency component by −90º.

The signal m(t) is the Hilbert transform of the signal m(t) .


Hilbert Transform

The transfer function of a Hilbert transform is defined by:

The signum function, sgn(f) is defined as:

( )1 0

sgn 0 0t

t t+ >⎧⎪= =⎨⎪

( )sgn t

1 0t⎪− <⎩

The Hilbert transformer is a wide-band phase-shifter, the frequency response is characterized in two parts:The magnitude response is unity for all frequencies, both positive and negative.g p y q p gThe phase response is +90º for negative frequencies -90º for positive frequencies.


Frequency Discrimination Method for SSB

The SSB modulator of Fig. 3.19 consists of two components: productmodulator followed by a band-pass filter (BPF). The product modulatorproduces a DSB-SC modulated wave with an upper sideband and a lowersideband. The BPF is designed to transmit one of these two sidebands,depending on whether the USB or LSB is the desired modulation.

( )M f

faf bf cf

( )S f

fc af f+ c bf f+

( )S f

fc af f−c bf f−

Figure 3.18 (a) Spectrum of a message signal m(t) with energy gay centered around zero frequency. Corresponding spectra of SSB-modulated waves using (b) upper sideband, and (c) lower sideband. In parts (b)


Corresponding spectra of SSB modulated waves using (b) upper sideband, and (c) lower sideband. In parts (b) and (c), the spectra are only shown for positive frequencies.

Frequency Discrimination Method for SSB

For the design of the BPF to be practically feasible, there must be a certain separation between the two sidebands that is wide enough to accommodate theseparation between the two sidebands that is wide enough to accommodate the transition band of the BPF. The separation is equal to 2fa, where fa is the lowest frequency component of the message signal, as shown in Fig. 3.18. This requirement limits the applicability of SSB modulation to speech signals for


which fa ≈ 100 Hz.

Coherent Detection of SSB

The coherent detection requires synchronization of a local oscillator in the receiver with the oscillator responsible for generating the carrier in the transmitter. The synchronization requirement has to be in both phase and frequency.


frequency.

Vestigial Sideband (VSB) Modulation

SSB modulation works well with speech signal that has an energy gap centeredaround zero frequency. For wideband signal like television transmission signalwill benefit by not using double sidebands, and SSB will make it impossible toimplement.

The television transmission, 525 lines of video information are sent in 1/30 (30frames per second) of a second (that is, 15,750 lines per second – thehorizontal trace frequency). Allowing time for retrace and synchronization, thisrequires a minimum video bandwidth of 4 MHz to transmit an array of pictureelements. The transfer function of a Hilbert transform is defined by:The signum function sgn(f) is defined as:The signum function, sgn(f) is defined as:

( )1, 0

sgn 0, 0t

t t+ >⎧

⎪= =⎨( )sgn 0, 01, 0

t tt

⎨⎪ − <⎩


Vestigial Sideband (VSB) Modulation

VSB modulation overcomes two of the difficulties present in SSB modulation.By allowing a portion of the unwanted sideband to appear at the output of anSSB modulator, the design of the sideband filter is simplified since the sharpcut-off at the carrier frequency is not required. The bandwidth of a VSBmodulated signal is defined by BT = fv + W where fv is the vestigialbandwidth and W is the message bandwidth. The VSB bandwidth BTcompromises between the SSB bandwidth W and DSB-SC bandwidth 2Wcompromises between the SSB bandwidth, W, and DSB SC bandwidth, 2W.


Sideband Shaping Filter

In VSB modulation, only a portion of one sideband is transmitted in such away that the demodulation process reproduces the original signal. The partialsuppression of one sideband reduces the required bandwidth from that requiredfor DSB but does not match the spectrum efficiency of SSB. The spectrumshaping is defined by the transfer function of the filter, which is denoted byH( f ). The only requirement that the sideband shaping performed by H( f )must satisfy is that the transmitted vestige compensates for the spectral portionmust satisfy is that the transmitted vestige compensates for the spectral portionmissing from the other sideband.

The sideband shaping filter must satisfy the following condition:

where

fc is the carrier frequency.

( ) ( ) 1c cH f f H f f for W f W+ + − = − ≤ ≤

fc is the carrier frequency.

The term H( f + fc ) is the positive-frequency part of the band-pass transferfunction H( f ) shifted to the left by fc, and H( f − fc ) is the negative-frequencypart of H( f ) shifted to the right by fc.


p ( f ) g y fc

Sideband Shaping Filter

The transfer function of the sideband shaping filter is anti-symmetric about thecarrier frequency, fc.

The transfer function is required to satisfy the odd symmetry only for theThe transfer function is required to satisfy the odd symmetry only for thefrequency interval, −W ≤ f ≤ W, where W is the message bandwidth.

fcf− cf

f

f


Demodulation of SSB signals (1 of 5)

For demodulation, the spectral density of the SSB signal must be translatedback to f = 0. Multiplication of the SSB signal by cos(2π fc t) translates half ofeach spectral density up in frequency by fc and half down by the same amount..

( )SSBS t±( )0e t

( )cos 2 cf tπ

f

f

mfmf−

( )SSBS t+

f

c mf f+cfcf( )c mf f− +

mfmf− 2 c mf f+2 cf2 cf( )2 c mf f− +



With the frequency and phase errors in the demodulation process:

( ) ( ) ( ) ( ) ( )cos 2 sin 2SSB c cS t f t f t f t f tπ π= ±∓

⎡ ⎤Let the locally generated carrier signal be where

Δ f is the frequency error and φ is the phase error.

( )cos 2 cf f tπ φ+ Δ +⎡ ⎤⎣ ⎦

( ) ( )⎡ ⎤( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

cos 2

cos 2 sin 2 cos 2

cos 2 cos 2 cos 2 sin 2

SSB c

c c c

S t f f t

f t f t f t f t f f t

f t f t f f t f t f f t f t

π φ

π π π φ

π π φ π φ π

+ Δ +⎡ ⎤⎣ ⎦= ± + Δ +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= + Δ + ± + Δ +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

∓

( ) ( ) ( ) ( ) ( ) ( )cos 2 cos 2 cos 2 sin 2c c c cf t f t f f t f t f f t f tπ π φ π φ π+ Δ + ± + Δ +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

( ) ( )( ) ( ) ( )

The first term can be expressed as: Note: 2cos cos cos cos

cos 2 cos 2c c

A B A B A B

f t f t f f tπ π φ

= + + −⎡ ⎤⎣ ⎦+ Δ +⎡ ⎤⎣ ⎦( ) ( ) ( )

( ) ( ) ( ) ( )

( ) [ ] ( ) [ ]

1 1cos 2 2 cos 2 22 21 1cos 4 2 cos 2

c c

c c c cf t f t f f t f t f t f f t

f t f t f t f t f t

π π φ π π φ

π π φ π φ

⎣ ⎦

= + + Δ + + − + Δ +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= + Δ + + Δ +


( ) [ ] ( ) [ ]cos 4 2 cos 22 2cf t f t f t f t f tπ π φ π φ= + Δ + + Δ +


( ) ( )( ) ( ) ( )

The seond term can be expressed as: Note: 2cos sin sin sin

cos 2 sin 2

1 1c c

A B A B A B

f t f f t f tπ φ π

= + − −⎡ ⎤⎣ ⎦+ Δ +⎡ ⎤⎣ ⎦

( ) ( ) ( ) ( )

( ) [ ] ( ) [ ]

1 1sin 2 2 sin 2 22 21 1sin 4 2 sin 22 2

c c c c

c

f t f f t f t f t f f t f t

f t f t f t f t f t

π φ π π φ π

π π φ π φ

= + Δ + + − + Δ + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= + Δ + − Δ +2 2

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

Now, cos 2

cos 2 cos 2 cos 2 sin 2SSB c

c c c c

S t f f t

f t f t f f t f t f f t f t

π φ

π π φ π φ π

+ Δ +⎡ ⎤⎣ ⎦= + Δ + ± + Δ +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

∓

( ) [ ] [ ]{ }

( ) [ ] [ ]{ }

1 cos 4 2 cos 22

1 sin 4 2 sin 22

c

c

f t f t f t f t

f t f t f t f t

π π φ π φ

π π φ π φ

= + Δ + + Δ +

± + Δ + − Δ +( ) [ ] [ ]{ }

( ) [ ] [ ]{ }

( ) [ ]

21 cos 4 2 cos 22

1 i 2 i 4

c

c

f f f f

f t f t f t f t

f f

φ φ

π π φ π φ

φ

= + Δ + + Δ +

Δ [ ]{ }2f f φΔ

Filtered out

Filtered out


( ) [ ]1 sin 2 sin 42

f t f tπ φ πΔ + −∓ [ ]{ }2cf t f tπ φ+ Δ +


The LPF in the SSB demodulator will filter out the higher frequenciescomponent. Thus the output

( ) ( ) [ ] ( ) [ ]1 12 i 2f f f fφ φΔ Δ

If Δ f = 0, and φ = 0, then ( ) ( )01 .2

e t f t=

( ) ( ) [ ] ( ) [ ]01 1cos 2 sin 22 2

e t f t f t f t f tπ φ π φ= Δ + Δ +∓

If Δ f = 0, and φ ≠ 0, then

2

( ) ( ) ( )01 1cos sin .2 2

e t f t f tφ φ= ∓

This give phase distortion in the receiver output. The SSB-SC demodulation isquite tolerate for voice communication.

If φ = 0 and Δ f ≠ 0 then ( ) ( ) ( ) ( ) ( )1 12 i 2t f t f t f t f tΔ Δ∓If φ = 0, and Δ f ≠ 0, then

This frequency errors give rise to spectral shifts as well as to phase distortionin the demodulation output. If Δ f is small, these spectral shifts can be tolerated

( ) ( ) ( ) ( ) ( )0 cos 2 sin 2 .2 2

e t f t f t f t f tπ π= Δ Δ∓


p f , pin voice communications.


If the frequency error, Δf = fm , then the spectrum is inverted. The highfrequency spectral components will become the low frequency components andvice versa.

This spectral inversion scrambles the speech quite unintelligible and this can beused as a low-level speech scramblers to ensure communication privacy

( )f t

( )cos2 c mf f tπ +cf c mf f+

fmfmf−

( )SSBS t+

cf

fc mf f+cfcf( )c mf f− +

c mf f+( )SSBS t−


f

c mf f+cfcf( )c mf f− +

Superheterodyne Receiver

Heterodyning means the translating or shifting in frequency. In the heterodynereceiver the incoming modulated signal is translated in frequency, thusoccupying an equal bandwidth centered about a new frequency. This newfrequency is known as an intermediate frequency (IF) which is fixed and is notdependent on the received signal frequency. The signal is amplified at the IFbefore demodulation. If the IF is lower than the received carrier frequency butabove the final output signal frequency it is called a superheterodyne receiverabove the final output signal frequency, it is called a superheterodyne receiver.



The intermediate frequency (IF) for most common AM broadcast receivers is455 KHz. AM band is from 540 KHz ~ 1600 KHz.

The required frequency translation to the IF is accomplished by mixing theThe required frequency translation to the IF is accomplished by mixing theincoming signal with a locally generated signal which differs from theincoming carrier by the IF (455 KHz). The received signal now translated to afixed IF, and it can be easily be amplified, filterer and demodulated. All theamplification and filtering is performed at a fixed frequency regardless ofstation selection.

The locally generated frequency is chosen to be 455 KHz higher than theincoming signal, because it is easier to build oscillators which are reasonablylinear within 1~2 MHz range than 0.1~1 MHz range (455 KHz lower than theincoming signal).

AM b d 540 KH 1600 KHAM band: 540 KHz ~ 1600 KHz

455 KHz higher than the incoming signal: ( 540 + 455) KHz ~ (1600 + 455) KHz

455 KHz lower than the incoming signal: (540 − 455) KHz ~ (1600 – 455) KHz



ff fff

cfcf−

IFf

cf

IFf

cf

2c IFf f+( )2c IFf f− +

IFfIFf

cf cf

fc IFf f+( )c IFf f− +

fIFfIFf− 2 c IFf f+( )2 c IFf f− +

If there is another station broadcasting at ( f +2f ) KHz the signal will also beIf there is another station broadcasting at ( fc+2fIF ) KHz, the signal will also be mapped into the IF band. This signal is called the image frequency which is not the desired signal. This is the only drawback in superheterodyne receiver.


Dual Conversion Receiver

A telemetry receiver is designed to receive satellite transmissions at 136 MHz.The receiver uses two heterodyne operations with intermediate frequencies of30 MHz and 10 MHz The first local oscillator is designed to operate below the30 MHz and 10 MHz. The first local oscillator is designed to operate below theincoming carrier frequency; the second one, above the first (30 MHz)intermediate frequency. What possible input frequencies could result in imagesfor both mixers if the filters were not ideal?


Dual Conversion Receiver

The first local oscillator is operating at 136 MHz − 30 MHz = 106 MHz.

The other frequency of transmission @ 106 MHz − 30 MHz = 76 MHz willThe other frequency of transmission @ 106 MHz 30 MHz 76 MHz will also map to IF1.

The second local oscillator is operating at 30 MHz +10 MHz = 40 MHz.

If 40 MHz +10 MHz = 50 MHz is not attenuated significantly after passingIf 40 MHz +10 MHz = 50 MHz is not attenuated significantly after passing IF1, then the 50 MHz will also map to IF2. Then the possible image frequencies are:

106 − 40 = 56 MHz ; 106 − 30 = 76 MHz ; 106 + 50 = 156 MHz106 − 40 = 56 MHz ; 106 − 30 = 76 MHz ; 106 + 50 = 156 MHz

Check:• LO1=106 MHz, 106 −56 = 50 MHz. If this 50 MHz appears at the output of

IF it will also map to IF 50 40 = 10 MHzIF1, it will also map to IF2. 50 −40 = 10 MHz.• LO1=106 MHz, 106 −76 = 30 MHz. (IF1=30 MHz)• LO1=106 MHz, 156 −106 = 50 MHz. If this 50 MHz appears at the output

of IF1, it will also map to IF2, for 50 −40 = 10 MHz.


Stereo Multiplex System

×( )l t +

−

l t r tb g b g−∑+

∑( )r t

+

19 KHz + x tbb gcos2 cf tπ2÷

+

+

38 KHz

+

l t r tb g b g+∑+

f0 15 KHz

19 KHz

23 KHz

38 KHz

53 KHz


Time-Averaged Noise Representations

Suppose n(t) is a noise voltage or current (assume a 1-ohm resistive load):

1. Mean value, :( )n t ( ) ( )1limn t n t dt= ∫,

This is often referred to as the dc or average, value of n(t).

( )n t ( ) ( )limT

T

n t n t dtT→∞ ∫

( )2 212. Mean-square value, :

The square root of is called the root-mean-square (rms) value of n(t).

( )2n t ( ) ( ) 22 1limT

T

n t n t dtT→∞

= ∫( )2n t

3. AC component, σ(t) :( ) ( ) ( ) ( ) ( ) ( )t n t n t n t n t tσ σ≈ − ⇒ = +

( ) ( ) ( ) ( )

( ) ( )

222

2 2

1 1lim lim

1 1lim lim

T TT T

n t n t dt n t t dtT T

n t dt t dt

σ

σ

→∞ →∞= = +

= +

∫ ∫

∫ ∫Amplitude Modulation 46

( ) ( )lim limT T

T T

n t dt t dtT T

σ→∞ →∞

= +∫ ∫


The ac, or fluctuation, component of n(t) is that component which remains after the mean value, , had been taken out.( )n t

( ) ( ) ( ) ( )

( ) ( )

222

2 2

1 1lim lim

1 1lim lim

T TT T

n t n t dt n t t dtT T

n t dt t dt

σ

σ

→∞ →∞= = +

= +

∫ ∫

∫ ∫

The term on the left-hand side of the above equation is the time-averagedpower in n(t) across a 1 ohm resistor The first term on the right hand side is

( ) ( )lim limT T

T T

n t dt t dtT T

σ→∞ →∞

= +∫ ∫

power in n(t) across a 1-ohm resistor. The first term on the right-hand side isthe dc power and the second term, the ac power in n(t). The rms value of n(t)is equal to the rms value of σ(t) only if the mean value is zero.( )n t



Example:

Compute the (a) average value, (b) ac power, and © rms value of the periodic f ( ) 1 2t f twaveform .

a)

( ) 01 cos 2v t f tπ= +

( ) ( )01 1 cos 2 1

T

v t f t dtT

π= + =∫

b) ( ) ( ) ( )220 0

1 1 1 1cos 2 1 cos 42 2T T

t f t dt f t dtT T

σ π π= = + =∫ ∫

c) ( ) ( ) ( )22 20 0 0

1 1 1 31 cos 2 1 cos 4 cos 2 12 2T T

v t f t dt f t f t dtT T

π π π= + = + + = + =∫ ∫

( )2 3 true rms value2rmsv v t= = =


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7 amplitude modulation