7 amplitude modulation

Amplitude Modulation
m(t)

s(t)
AM waveform with
ka m ( t ) < 1

s(t)
Over-modulated waveform
ka m ( t ) > 1

Amplitude Modulation 1


The sinusoidal carrier wave : c ( t ) = Ac cos ( 2π f c t )

The AM waveform, DSB-LC :

s ( t ) = Ac ⎡1 + ka m ( t ) ⎤ cos ( 2π f c t )
⎣ ⎦
Ac ka Ac
S( f )= ⎡δ ( f − f c ) + δ ( f + f c ) ⎤ +
⎣ ⎦ ⎡ M ( f − f c ) + M ( f + f c )⎤
2 2 ⎣ ⎦

In AM, information pertaining to the message signal m(t) resides solely in the
envelope, which is defined as th amplitude of the modulated wave s(t).
l hi h i d fi d the lit d f th d l t d ()
The carrier frequency fc is much greater than the highest frequency component W
of the message signal m(t).

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



The
Th message signal has a
i lh
bandwidth W Hz.

2W



Example 3.1 Single-Tone Modulation

m ( t ) = Am cos ( 2π f m t )

where fm is its frequency, and
Am is the amplitude of the sinusoidal modulating wave
s ( t ) = Ac ⎡1 + μ cos ( 2π f m t ) ⎤ cos ( 2π f c t )
⎣ ⎦
where μ = ka Am ; Amax = Ac [1 + μ ] ; Amin = Ac [1 − μ ]
Amax Ac (1 + μ )
and = ⇒ Amax Ac − μ Amax Ac = Amin Ac + μ Amin Ac
Amin Ac (1 − μ )
Amax − μ Amax = Amin + μ Amin ⇒ Amax − Amin = μ Amax + μ Amin

Amax − Amin Peak DSB-SC Amplitude
The modulation factor, μ = =
Amax + Amin Peak Carrier A lit d
P k C i Amplitude


⎣ ⎦
where μ = ka Am ; Amax = Ac [1 + μ ] ; Amin = Ac [1 − μ ]

Amax − Amin Peak DSB-SC Amplitude Am
With μ = = =
Amax + Amin p
Peak Carrier Amplitude Ac

⎡ Am ⎤
Amax = Ac [1 + μ ] = Ac ⎢1 + ⎥ = Ac + Am
⎢ Ac ⎥
⎣ ⎦
⎡ Am ⎤
Amin = Ac [1 − μ ] = Ac ⎢1 − ⎥ = Ac − Am
⎢ Ac ⎥
⎣ ⎦
Amax − Amin ( Ac + Am ) − ( Ac − Am ) 2 Am Am
μ= = = =
Amax + Amin ( Ac + Am ) + ( Ac − Am ) 2 Ac Ac

Amplitude Modulation, DSB-LC
p ,
3
Amax

2

Amin
1

0 0

-1

-2

-3
0 0.5
05 1 1.5
15 2 2.5
25 3
-3
x 10

Amax − Amin 2.8 − 1.2 1.6
μ= = = = 0.4
Amax + Amin 2 8 + 1.2 4
2.8 1 2



Example 3.1 Single-Tone Modulation (continued)

⎣ ⎦
= Ac cos ( 2π f c t ) + μ Ac cos ( 2π f m t ) cos ( 2π f c t )
1 1
= Ac cos ( 2π f c t ) + μ Ac cos ⎡ 2π ( f c + f m )t ⎤ + μ Ac cos ⎡ 2π ( f c − f m )t ⎤
⎣ ⎦ 2 ⎣ ⎦
2
1
S ( f ) = Ac ⎡δ ( f − f c ) + δ ( f + f c ) ⎤
2 ⎣ ⎦
1
4
{
+ μ Ac δ ⎡ f − ( f c + f m ) ⎤ + δ ⎡ f + ( f c + f m ) ⎤
⎣ ⎦ ⎣ ⎦ }
1
4
{
+ μ Ac δ ⎡ f − ( f c − f m ) ⎤ + δ ⎡ f + ( f c − f m ) ⎤
⎣ ⎦ ⎣ ⎦ }
Can you identify each component in Fig. 3.3c from the above expression?




⎣ ⎦
= Ac cos ( 2π f c t ) + μ Ac cos ( 2π f m t ) cos ( 2π f c t )
The mean square value or time averaged value:
s 2 ( t ) = Ac 2 cos 2 ( 2π f c t ) + μ 2 Ac 2 cos 2 ( 2π f m t ) cos 2 ( 2π f c t )
1 1 1 1 1 1
Since cos 2 A = (1 + cos 2 A ) ⇒ cos 2 A = + cos 2 A = + × 0 =
2 2 2 2 2 2
1
Ac 2 cos 2 ( 2π f c t ) = Ac 2 = Carrier Power
2
1 1
μ 2 Ac 2 cos 2 ( 2π f m t ) cos 2 ( 2π f c t ) = μ 2 Ac 2 × ×
2 2
1
= μ 2 Ac 2 = Sidebands Power=USB Power + LSB Power
4



1 2
Sidebands power = Ac
2
1 1
Upper sideband power = μ 2 Ac 2 ; Lower sideband power = μ 2 Ac 2
8 8
1 2 2 1 2 2 1 2
Total sidebands power 8
μ Ac + μ Ac μ μ2
= 8 = 4 =
Total power 1 2 1 2 2 1 1 2 2 + μ2
Ac + μ Ac + μ
2 4 2 4
1 2 1 2 2
Total power = s ( t ) = Ac + μ Ac
2

2 4


Envelope Detection


Envelope Detection
An envelope detector is used to detect the
envelope of the modulated waveform. Any
circuit 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 capacitor
forwarded biased
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 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 )

c ( t ) = A c cos ( 2π f c t )



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



s ( t ) = c ( t ) m ( t ) = A c cos ( 2π f c t ) m ( t )
1
ℑ{s ( t )} = S ( f ) = A c ⎡ M ( f − f c ) + M ( f + f c )⎦
⎣
⎤
2
The message signal m(t) has a bandwidth of W. The modulation process
translates the spectrum of the message signal by fc to the right and by –fc to the
f
left. The transmission bandwidth required by DSB-SC modulation is the same
as that for amplitude modulation, 2W.

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


Coherent Detection
The demodulation of a DSB-SC signal is under the assumption that the local
oscillator of the receiver is exactly coherent or synchronized, in both
frequency and phase, with the modulated signal.


Coherent Detection

If there is a small frequency error, ∆ f, and a phase error, φ, in the local oscillator
of the receiver.
ν (t )
= s ( t ) Ac′ cos ⎡ 2π ( f c + Δ f ) t + φ ⎤ m ( t )
⎣ ⎦ where s ( t ) = Ac cos ( 2π f c t )

= Ac Ac′ cos ( 2π f c t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤ m ( t )
⎣ ⎦
{
= Ac Ac′m ( t ) cos ( 2π f c t ) cos ⎡ 2π ( f c + Δ f ) t ⎤ cos φ − sin ⎡ 2π ( f c + Δ f )t ⎤ sin φ
⎣ ⎦ ⎣ ⎦ }
= A A ′m ( t ) cos ( 2π f t ) {cos ⎣ 2π ( f + Δ f ) t ⎦ cos φ − cos ( 2π f t ) sin ⎣ 2π ( f + Δ f ) t ⎦ sin φ }
c c c ⎡ c ⎤ c ⎡ c ⎤

= A A ′m ( t ) cos ( 2π f t ) ⎡cos ( 2π f t ) cos ( 2π Δ f t ) − sin ( 2π f t ) sin ( 2π Δ f t ) ⎤ cos φ
c c c⎣ c c ⎦
− A A ′m ( t ) cos ( 2π f t ) ⎡sin ( 2π f t ) cos ( 2π Δ f t ) − cos ( 2π f t ) sin ( 2π Δ f t ) ⎤ sin φ
c c ⎣c c c ⎦
= A A ′m ( t ) {cos ( 2π f t ) cos ( 2π Δ f t ) cos φ − cos ( 2π f t ) sin ( 2π f t ) sin ( 2π Δ f t ) cos φ }
c c
2
c c c

− A A ′m ( t ) {cos ( 2π f t ) sin ( 2π f t ) cos ( 2π Δ f t ) sin φ − cos ( 2π f t ) sin ( 2π Δ f t ) sin φ }
c c c c
2
c


Coherent Detection
ν ( t ) = Ac Ac′m ( t ) cos 2 ( 2π f c t ) ⎡cos ( 2π Δ f t ) cos φ − sin ( 2π Δ f t ) sin φ ⎤
⎣ ⎦
− Ac Ac′m ( t ) cos ( 2π f c t ) sin ( 2π f c t ) ⎡sin ( 2π Δ f t ) cos φ + cos ( 2π Δ f t ) sin φ ⎤
⎣ ⎦
= Ac Ac′m ( t ) cos 2 ( 2π f c t ) ⎡cos ( 2π Δ f t ) cos φ − sin ( 2π Δ f t ) sin φ ⎤
⎣ ⎦
− Ac Ac′m ( t ) cos ( 2π f c t ) sin ( 2π f c t ) ⎡sin ( 2π Δ f t ) cos φ + cos ( 2π Δ f t ) sin φ ⎤
⎣ ⎦
= Ac Ac′m ( t ) cos 2 ( 2π f c t ) cos ( 2π Δ f t+ φ ) − Ac Ac′m ( t ) cos ( 2π f c t ) sin ( 2π f c t ) sin ( 2π Δ f t+ φ )
i i
1
With sin ( A + A ) = sin 2 A = 2 cos A sin A ⇒ cos A sin A = sin 2 A, then
2
sin ( 4π f c t )
ν ( t ) = Ac Ac′m ( t ) cos 2 ( 2π f c t ) cos ( 2π Δ f t+ φ ) − Ac Ac′m ( t ) sin ( 2π Δ f t+ φ )
2
A A′
= Ac Ac ′m ( t ) ⎡1 + cos 4π f c t ⎤ cos ( 2π Δ f t + φ ) − c c m ( t ) sin ( 4π f c t ) sin ( 2π Δ f t + φ )
⎢ 2 ⎥ 2
⎣ ⎦
Ac Ac′ Ac Ac′
= m ( t ) cos ( 2π Δ f t + φ ) + m ( t ) cos ( 4π f c t ) cos ( 2π Δ f t + φ )
2 2
Ac Ac′
− m ( t ) sin ( 4π f c t ) sin ( 2π Δ f t + φ )
2

Coherent Detection

Ac Ac′ Ac Ac′
ν (t ) = m ( t ) cos ( 4π f c t + 2π Δ f t + φ ) + m ( t ) cos ( 2π Δ f t + φ )
2 2
The first term is centered at ± 2 f c + Δ f and can be filtered out by the LPF.

Ac Ac′
Unless both ∆ f and φ are zero, otherwise v(t) is not equal to m (t ).
2
Ac Ac′
If ∆ f = 0, then ν ( t ) = m ( t ) cos φ .
2

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

Ac Ac′
If φ = 0, then ν ( t ) =
0 m ( t ) cos 2π Δ f t.
2

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


Coherent Detection
The second term can be expressed as:
Ac Ac′
ν 0 (t ) = m ( t ) cos ( 2π Δ f t + φ )
2
1
= Ac Ac′m ( t ) ⎡ cos ( 2π Δ f t ) cos φ − sin ( 2π Δ f t ) sin φ ⎤
⎣ ⎦
2

This 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.
Ac Ac′
If φ = 0, then ν 0 ( t ) =
0 m ( t ) cos 2π Δ f t.
2
If φ = 90º, the received signal will be wiped out, v0 (t) ≈ 0.
I general, th phase error φ i not constant. It i necessary t synchronize th
In l the h is t t t is to h i the
local oscillator in the receiver in both frequency and phase with the carrier
wave to generate the DSB-SC modulated signal in the transmitter.


Coherent Detection

Ac Ac′
ν 0 (t ) = m ( t ) cos 2π Δ f t.
2


Costas Receiver


Costas Receiver
This receiver consists of two coherent detectors supplied with the same
input signal, Ac cos (2π fc t) m(t), but with two local oscillator signals that
are in phase quadrature with respect to each other.
Suppose the local oscillator drifts from its proper value by a small angle φ
radians. The I-channel output is proportional to cos φ and cos φ ≈ 1 for
small φ. The Q-channel output will have the same polarity as the I-channel
output for one direction of local oscillator phase driftφ and the opposite
polarity for the opposite direction of φ.
By combining the I- and Q-channel outputs in a phase discriminator (which
consists of a multiplier followed by a time-averaging unit), a dc control
signal proportional to the phase drift φ is generated. With negative feedback
acting around the Costas receiver, the control signal tends to automatically
correct for the local phase error φ in the voltage-controlled oscillator.
voltage controlled


Single-Sideband Modulation, SSB

Modulating signal: m ( t ) = Am cos ( 2π f m t )
Carrier signal:
g c ( t ) = Ac cos ( 2π f c t )
S DSB − SC ( t ) = c ( t ) m ( t ) = Ac Am cos ( 2π f c t ) cos ( 2π f m t )
1 1
= Ac Am cos ⎡ 2π ( f c + f m ) t ⎤ + Ac Am cos ⎡ 2π ( f c − f m ) t ⎤
⎣ ⎦ 2 ⎣ ⎦
2
1
S SSB + ( t ) = Ac Am cos ⎡ 2π ( f c + f m ) t ⎤
⎣ ⎦
2
1 1
= Ac Am cos ( 2π f c t ) cos ( 2π f m t ) − Ac Am sin ( 2π f c t ) sin ( 2π f m t )
2 2
1
S SSB − ( t ) = Ac Am cos ⎡ 2π ( f c − f m ) t ⎤
⎣ ⎦
2
1 1
= Ac Am cos ( 2π f c t ) cos ( 2π f m t ) + Ac Am sin ( 2π f c t ) sin ( 2π f m t )
2 2
1 1
S SSB ± ( t ) = Ac Am cos ( 2π f c t ) cos ( 2π f m t ) ∓ Ac Am sin ( 2π f c t ) sin ( 2π f m t )
2 2


Single-Sideband Modulation, SSB

1 1
S SSB ± ( t ) = Ac Am cos ( 2π f c t ) cos ( 2π f m t ) ∓ Ac Am sin ( 2π f c t ) sin ( 2π f m t )
2 2
1 1
= Ac m ( t ) cos ( 2π f c t ) ∓ Ac m ( t ) sin ( 2π f c t )
2 2

where m(t) can b d i d from m(t) simply by shifting the phase of each
h () be derived f ( ) i l b hif i h h f h
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:
sgn ( t )

⎧+1 t > 0
⎪
sgn ( t ) = ⎨ 0 t = 0
⎪
⎩−1 t < 0

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 g
The 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: product
modulator followed by a band-pass filter (BPF). The product modulator
produces a DSB-SC modulated wave with an upper sideband and a lower
sideband. The BPF is designed to transmit one of these two sidebands,
depending on whether the USB or LSB is the desired modulation.
M(f)

f
fa fb fc
S( f )

f
fc + f a f c + fb
S( f )

f
fc − fb fc − fa

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)
SSB-modulated
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 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.


Vestigial Sideband (VSB) Modulation
SSB modulation works well with speech signal that has an energy gap centered
around zero frequency. For wideband signal like television transmission signal
will benefit by not using double sidebands, and SSB will make it impossible to
implement.

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

⎧ + 1, t >0
⎪
sgn ( t ) = ⎨ 0, t =0
⎪ −1, t<0
⎩


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 an
SSB modulator, the design of the sideband filter is simplified since the sharp
cut-off at the carrier frequency is not required. The bandwidth of a VSB
modulated signal is defined by BT = fv + W where fv is the vestigial
bandwidth and W is the message bandwidth. The VSB bandwidth BT
compromises between the SSB bandwidth W and DSB-SC bandwidth 2W
bandwidth, W, DSB SC bandwidth, 2W.


Sideband Shaping Filter
In VSB modulation, only a portion of one sideband is transmitted in such a
way that the demodulation process reproduces the original signal. The partial
suppression of one sideband reduces the required bandwidth from that required
for DSB but does not match the spectrum efficiency of SSB. The spectrum
shaping is defined by the transfer function of the filter, which is denoted by
H( f ). The only requirement that the sideband shaping performed by H( f )
must satisfy is that the transmitted vestige compensates for the spectral portion
missing from the other sideband.
The sideband shaping filter must satisfy the following condition:

H ( f + fc ) + H ( f − fc ) = 1 for − W ≤ f ≤ W
where
fc is the carrier frequency.
The term H( f + fc ) is the positive-frequency part of the band-pass transfer
function H( f ) shifted to the left by fc, and H( f − fc ) is the negative-frequency
p
part of H( f ) shifted to the right by fc.
( g y


Sideband Shaping Filter
The transfer function of the sideband shaping filter is anti-symmetric about the
carrier frequency, fc.
The transfer function is required to satisfy the odd symmetry only for the
frequency interval, −W ≤ f ≤ W, where W is the message bandwidth.

f
− fc fc

f

f


Demodulation of SSB signals (1 of 5)
For demodulation, the spectral density of the SSB signal must be translated
back to f = 0. Multiplication of the SSB signal by cos(2π fc t) translates half of
each spectral density up in frequency by fc and half down by the same amount..

e0 ( t )
S SSB ± ( t )

cos ( 2π f c t )

f
− fm fm

S SSB + ( t )

f
− ( fc + fm ) fc fc fc + fm

f
− ( 2 fc + f m ) 2 fc − fm fm 2 fc 2 fc + f m


With the frequency and phase errors in the demodulation process:
S SSB ∓ ( t ) = f ( t ) cos ( 2π f c t ) ± f ( t ) sin ( 2π f c t )

Let the locally generated carrier signal be cos ⎡ 2π ( f c + Δ f ) t + φ ⎦
⎣ ⎤ where
Δ f is the frequency error and φ is the phase error.

S SSB ∓ ( t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤
⎣ ⎦
= ⎡ f ( t ) cos ( 2π f c t ) ± f ( t ) sin ( 2π f c t ) ⎤ cos ⎡ 2π ( f c + Δ f ) t + φ ⎤
⎣ ⎦ ⎣ ⎦
= f ( t ) cos ( 2π f c t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤ ± f ( t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤ sin ( 2π f c t )
⎣ ⎦ ⎣ ⎦

The first term can be expressed as: ⎡ Note: 2 cos A cos B = cos ( A + B ) + cos ( A − B ) ⎤
⎣ ⎦
f ( t ) cos ( 2π f c t ) cos ⎣ 2π ( f c + Δ f ) t + φ ⎦
⎡ ⎤
1 1
= f ( t ) cos ⎡ 2π f c t + 2π ( f c + Δ f ) t + φ ⎤ + f ( t ) cos ⎡ 2π f c t − 2π ( f c + Δ f ) t + φ ⎤
⎣ ⎦ 2 ⎣ ⎦
2
1 1
= f ( t ) cos [ 4π f c t + 2π Δ f t + φ ] + f ( t ) cos [ 2π Δ f t + φ ]
2 2

The seond term can be expressed as: ⎡ Note: 2 cos A sin B = sin ( A + B ) − sin ( A − B ) ⎤
⎣ ⎦
f ( t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤ sin ( 2π f c t )
⎣ ⎦
1 1
= f ( t ) sin ⎡ 2π ( f c + Δ f ) t + φ + 2π f c t ⎤ − f ( t ) sin ⎡ 2π ( f c + Δ f ) t + φ − 2π f c t ⎤
⎣ ⎦ 2 ⎣ ⎦
2
1 1
= f ( t ) sin [ 4π f c t + 2π Δf t + φ ] − f ( t ) sin [ 2π Δ f t + φ ]
2 2
Now, S SSB ∓ ( t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤
⎣ ⎦
= f ( t ) cos ( 2π f c t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤ ± f ( t ) cos ⎡ 2π ( f c + Δ f ) t + φ ⎤ sin ( 2π f c t )
⎣ ⎦ ⎣ ⎦
1
= f ( t ) {cos [ 4π f c t + 2π Δ f t + φ ] + cos [ 2π Δ f t + φ ]}
2
1
± f ( t ) {sin [ 4π f c t + 2π Δf t + φ ] − sin [ 2π Δ f t + φ ]}
2 Filtered out
1
= f ( t ) {cos [ 4π f c t + 2π Δ f t + φ ] + cos [ 2π Δ f t + φ ]}
2 Filtered out
1
∓ f ( t ) {sin [ 2π Δ f t + φ ] − sin [ 4π f c t + 2π Δf t + φ ]}
i i
2

The LPF in the SSB demodulator will filter out the higher frequencies
component. Thus the output
1 1
e0 ( t ) = f ( t ) cos [ 2π Δ f t + φ ] ∓ f ( t ) sin [ 2π Δ f t + φ ]
i
2 2
1
If Δ f = 0, and φ = 0, then e0 ( t ) = f (t ).
2
1 1
If Δ f = 0, and φ ≠ 0, then e0 ( t ) = f ( t ) cos φ ∓ f ( t ) sin φ .
2 2

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

1 1
If φ = 0 and Δ f ≠ 0 then e0 ( t ) =
0, 0, f ( t ) cos ( 2π Δf t ) ∓ f ( t ) sin ( 2π Δf t ) .
i
2 2
This frequency errors give rise to spectral shifts as well as to phase distortion
in the demodulation output. If Δ f is small, these spectral shifts can be tolerated
p , p
in voice communications.


If the frequency error, Δf = fm , then the spectrum is inverted. The high
frequency spectral components will become the low frequency components and
vice versa.
This spectral inversion scrambles the speech quite unintelligible and this can be
used as a low-level speech scramblers to ensure communication privacy

f (t )

fc fc + fm
cos2π ( fc + fm ) t

f
− fm fm
fc
S SSB + ( t )

f
S SSB − ( t ) fc + fm

f

Superheterodyne Receiver
Heterodyning means the translating or shifting in frequency. In the heterodyne
receiver the incoming modulated signal is translated in frequency, thus
occupying an equal bandwidth centered about a new frequency. This new
frequency is known as an intermediate frequency (IF) which is fixed and is not
dependent on the received signal frequency. The signal is amplified at the IF
before demodulation. If the IF is lower than the received carrier frequency but
above the final output signal frequency it is called a superheterodyne receiver
frequency, receiver.


The intermediate frequency (IF) for most common AM broadcast receivers is
455 KHz. AM band is from 540 KHz ~ 1600 KHz.
The required frequency translation to the IF is accomplished by mixing the
incoming signal with a locally generated signal which differs from the
incoming carrier by the IF (455 KHz). The received signal now translated to a
fixed IF, and it can be easily be amplified, filterer and demodulated. All the
amplification and filtering is performed at a fixed frequency regardless of
station selection.
The locally generated frequency is chosen to be 455 KHz higher than the
incoming signal, because it is easier to build oscillators which are reasonably
linear within 1~2 MHz range than 0.1~1 MHz range (455 KHz lower than the
incoming signal).
AM band: 540 KH ~ 1600 KH
b d KHz 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



f IF f IF f IF f IF
f
− ( f c + 2 f IF ) − fc fc f c + 2 f IF

fc fc fc fc

f
− ( f c + f IF ) f c + f IF

f
− ( 2 f c + f IF ) − f IF f IF 2 f c + f IF

If there is another station broadcasting at ( fc+2fIF ) KHz the signal will also be
KHz,
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 of
30 MHz and 10 MHz The first local oscillator is designed to operate below the
MHz.
incoming carrier frequency; the second one, above the first (30 MHz)
intermediate frequency. What possible input frequencies could result in images
for 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 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 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 MHz
Check:
• LO1=106 MHz, 106 −56 = 50 MHz. If this 50 MHz appears at the output of
IF1, it will also map to IF2. 50 −40 = 10 MHz.
40 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 ) +
∑
bg bg ×
l t −r t

−
+
r (t )
cos2π fc t
38 KHz 19 KHz +
∑
bg
xb t
÷2
+
+

+ ∑ l bt g + r bt g

0 15 KHz 23 KHz 53 KHz
f
19 KHz 38 KHz


Time-Averaged Noise Representations

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

1
1. Mean value, n ( t ) : n ( t ) = lim n ( t ) dt
T →∞ T ∫
,
T

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

2. Mean-square value, n 2( t ): n 2( t ) = lim
1
n ( t ) dt
2

T →∞ T ∫
T

The square root of n 2( t ) is called the root-mean-square (rms) value of n(t).

3. AC component, σ(t) :
σ (t ) ≈ n (t ) − n (t ) ⇒ n (t ) = n (t ) + σ (t )
1 1 2
n ( t ) = lim ∫ n ( t ) dt = lim ∫ n ( t ) + σ ( t ) dt
2 2

T →∞ T T →∞ T
T T

1 2 1
= lim ∫ n ( t ) dt + lim ∫ σ ( t ) dt
2

T →∞ T T →∞ T
T T



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

1 1 2
n ( t ) = lim ∫ n ( t ) dt = lim ∫ n ( t ) + σ ( t ) dt
2 2

T →∞ T T →∞ T
T T

1 2 1
= lim ∫ n ( t ) dt + lim ∫ σ ( t ) dt
2

T →∞ T T →∞ T
T T

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



Example:
Compute the (a) average value, (b) ac power, and © rms value of the periodic
waveform v ( t ) = 1 + cos 2π f 0 t .
f

1
a) v (t ) = ∫ (1 + cos 2π f0 t )dt = 1
TT

1 1 1 1
b) σ 2( t ) = ∫ ( cos 2π f0 t ) dt = ∫ (1 + cos 4π f 0 t ) dt =
2

TT TT2 2

c) v 2( t ) = 1 ∫ (1 + cos 2π f 0 t )2 dt = 1 ∫ (1 + cos 4π f 0 t + cos 2 2π f 0 t ) dt = 1 + 1 = 3
T T
T T
2 2

3
vrms = v 2( t ) = = true rms value
2


7 amplitude modulation

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