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ELECTRONICS AND TELECOMMUNICATION
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ELECTRONICS AND TELECOMMUNICATION
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Electronics and Telecommunication
Wave modulation id the process by which an engineer changes the signal of the original
wave by either increasing or reducing its frequency or amplitude, especially when two waves
interact. The analysis of the configurations between waves can be used to explain the process of
wave modulation. When evaluating the modulation equations, the engineer should consider
various aspects like the existence of phase between two waves that are combined together to
form minimize or amplified wave. The description of both non-linear and linear modulations are
covered in this paper. The first types are modulation mechanisms which follow the superposition
theorem whereas the second type represents the devices which do not conform to the
superposition theorem. The discussion covers various examples including:
•
A linear modulation system illustrating the modulation of wave’s amplitude.
•
A linear modulation system that is involved in the suppressed carrier modulation due
to use of a ring modulator.
•
A non-linear modulation system that is involved in the modulation of the wave’s
frequency.
Linear modulation system 1: Amplitude modulation
Amplitude modulation is the modulation of a signal by either increasing or reducing its
amplitude by constructive interference of two waves. For this modulation to occur, the
modulating signal and the carrier should be at resonance. In this case, the frequencies of the two
waves that are corresponding to the primary wave or carrier & the secondary or modulating wave
must have a ratio that is a natural number i.e. 1, 2, 3…. This is because any fraction would result
in destructive interaction between the two waves, thereby making the waves to disappear. The
carrier and modulating signals in most cases don’t depict a similar pattern. Thus, some of the
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conventional techniques utilized in electronics entail the formulation of a triangular or scalar
modulating wave that has a feature of a sinusoidal electromagnetic wave. In the example
illustrated in figure 1 below, a sinusoidal wave with a lower frequency is combined with a
triangular modulating wave to produce a triangular modulated wave that has a considerably
lower amplitude as compared to the two waves that were combined during the modulation
process.
Figure 1: Combination of a sinusoidal wave with a triangular wave to produce a wave with
considerably lower amplitude
Question 1. Derivation of the equation for s(t)
In a situation where both the modulating and carrier signals conform to a sinusoidal
equation, the equation of the modulated signal can be represented as:
𝑠(𝑡) = 𝑚(𝑡) + 𝑐(𝑡) …………………………………...Eq.1
Where m(t) modulating wave signal and c(t) carrier wave signal. These two parameters can be
obtained separately as:
𝑚(𝑡) = cos(2𝜋𝑓𝑚 𝑡) ∗ 𝐴𝑚
𝑐(𝑡) = cos(2𝜋𝑓𝑐 𝑡) ∗ 𝐴𝑐
Substituting the values of m(t) and c(t) into equation 1 gives;
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𝑠(𝑡) = cos(2𝜋𝑓𝑚 𝑡) ∗ 𝐴𝑚 + cos(2𝜋𝑓𝑐 𝑡) ∗ 𝐴𝑐 ………………….Eq. 2
By considering;
𝑏+𝑑
𝑏−𝑑
𝑎 cos(𝑏𝑥) + 𝑐 cos(𝑑𝑥) = √𝑎2 + 𝑐 2 + 2𝑎𝑐 ∗ cos(∅) ∗ cos (
) ∗ cos (
)
2
2
Where ϕ is the phase existing between the carrier and modulating waves. By applying that
relationship, the resulting curve for the general equation of the modulated wave can be
represented as:
2𝜋(𝑓𝑐 − 𝑓𝑚 )
2𝜋(𝑓𝑐 + 𝑓𝑚 )
𝑠(𝑡) = √𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos (
∗ 𝑡) ∗ cos (
∗ 𝑡)
2
2
𝑠(𝑡) = √𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos(𝜋(𝑓𝑐 − 𝑓𝑚 ) ∗ 𝑡) ∗ cos(𝜋(𝑓𝑐 + 𝑓𝑚 ) ∗ 𝑡) …Eq. 3
Now assuming the most common situation where the two waves i.e. the modulating and
carrier waves are in phase, ϕ = 0, the initial equation of the modulated signal can be simplified
as:
𝑠(𝑡) = √𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos(𝜋(𝑓𝑐 − 𝑓𝑚 ) ∗ 𝑡) ∗ cos(𝜋(𝑓𝑐 + 𝑓𝑚 ) ∗ 𝑡)
𝑠(𝑡) = √(𝐴𝑐 + 𝐴𝑚 )2 ∗ cos(𝜋(𝑓𝑐 − 𝑓𝑚 ) ∗ 𝑡) ∗ cos(𝜋(𝑓𝑐 + 𝑓𝑚 ) ∗ 𝑡)
𝑠(𝑡) = (𝐴𝑐 + 𝐴𝑚 ) ∗ cos(𝜋(𝑓𝑐 − 𝑓𝑚 ) ∗ 𝑡) ∗ cos(𝜋(𝑓𝑐 + 𝑓𝑚 ) ∗ 𝑡) ……………………...Eq. 4
From the observation, the amplitude is the sum of the amplitudes of the modulating and
carrier waves due to constructive interference thereby producing the modulation effect. The
frequency of the modulated wave relies on the relationship between the frequencies of the
modulating and carrier waves. Based on the previous statement of the two waves being resonant
from each other, the resulting modulated wave will be a resonant wave whose frequency is the
same as that of the fastest vibrating wave. For instance, if the vibrating frequency of the carrier
wave is half that of the modulating wave, the equation of the modulated signal becomes;
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𝑠(𝑡) = (𝐴𝑐 + 𝐴𝑚 ) ∗ cos(𝜋𝑓𝑚 ∗ 𝑡) ∗ cos(𝜋3𝑓𝑚 ∗ 𝑡)…………….Eq. 5
Question 2. Derivation of the equation of the SNR
According to equation 5, the amplitude of the wave can be estimated as the sum of the
amplitudes of the carrier and modulated waves. Based on that equation, the general equation for
an acoustic signal combined with noise can be described as:
𝑠(𝑡) = (𝐴𝑆 + 𝐴𝑁 ) ∗ cos(𝜋(𝑓𝑆 − 𝑓𝑁 ) ∗ 𝑡) ∗ cos(𝜋(𝑓𝑆 + 𝑓𝑁 ) ∗ 𝑡) ………..Eq. 6
Whereas that of the noise would be:
𝑠(𝑛𝑜𝑖𝑠𝑒, 𝑡) = cos(2𝜋𝑓𝑁 𝑡) ∗ 𝐴𝑁 ……………………………………….Eq. 7
The relationship between the signal to noise ratio can be expressed as:
𝑠(𝑡)
𝑠(𝑛𝑜𝑖𝑠𝑒,𝑡)
=
(𝐴𝑆 +𝐴𝑁 )∗ cos(𝜋(𝑓𝑆 −𝑓𝑁 )∗𝑡)∗ cos(𝜋(𝑓𝑆 +𝑓𝑁 )∗𝑡)
𝐴𝑁 ∗ cos(2𝜋𝑓𝑁 𝑡)
……………………Eq. 8
Based on the information provided, a*fN = 10-11 WHz, fS = 5.5 kHz = 5,500 Hz, with the
attenuation factor a being of 50 dB = 105.
By considering these figures,
𝑓𝑁 =
10−11
= 10−16 𝐻𝑧
105
(𝐴𝑆 + 𝐴𝑁 ) ∗ cos(𝜋(5,500 − 10−16 ) ∗ 𝑡) ∗ cos(𝜋(5,500 + 10−16 ) ∗ 𝑡)
𝑠(𝑡)
=
𝑠(𝑛𝑜𝑖𝑠𝑒, 𝑡)
𝐴𝑁 ∗ cos(2𝜋10−16 𝑡)
The above equation can be simplified as:
(𝐴𝑆 + 𝐴𝑁 ) ∗ cos(5,500𝜋 ∗ 𝑡) ∗ cos(5,500𝜋 ∗ 𝑡)
𝑠(𝑡)
=
𝑠(𝑛𝑜𝑖𝑠𝑒, 𝑡)
𝐴𝑁
=
=
𝐴𝑆 + 𝐴𝑁
𝐴𝑁
∗ cos(5,500𝜋𝑡)
𝐴𝑆 + 𝐴𝑁
∗ cos(𝜋𝑓𝑆 𝑡)
𝐴𝑁
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By projecting how the intensity (amplitude) of the two signals will be higher than that of
the noise, the graph of the SNR against the time will also be similar with that of the original
acoustic signal as illustrated below.
𝑺𝑵𝑹 =
𝑨𝑺 + 𝑨𝑵
𝑨𝑵
𝑨
∗ 𝐜𝐨𝐬(𝝅 ∗ 𝒇𝑺 ∗ 𝒕) = 𝑨 𝑺 ∗ 𝐜𝐨𝐬(𝝅 ∗ 𝒇𝑺 ∗ 𝒕) = 𝒔𝒊𝒈𝒏𝒂𝒍………Eq. 9
𝑵
Question 3. Graphical representation of the SNR
For instance, consider the amplitude of 1 and the previous frequency of 5.5 kHz, the
graphical representation of SNR against time will be;
Linear modulation system 2: Suppressed carrier modulation
The process of suppressed carrier modulation is characterized by the presence of the
transmission of the carrier wave as occurred in the amplitude modulation and two extra
components of frequencies f1 = fc + fm and f2 = fc – fm. Despite the fact that these components
were present in the general equation obtained for the modulated wave when the amplitude
modulation process was considered, they were simplified through the resonance process as per
definition of the experimental constraints of the model. The suppressed carrier modulation
doesn’t emphasize on the resonance phenomenon like the amplitude modulation, but it focuses
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on the attenuation of the amplitude resulting from the destructive interactions that take place at
different points of the wave.
Question 1. Derivation of the equation for s(t)
The following two specific equations govern the function of the double sideband
suppressed carrier.
For the upper band:
𝑠(𝑢𝑝𝑝𝑒𝑟 𝑏𝑎𝑛𝑑) = cos(2𝜋𝑓𝑐 𝑡) ∗ 𝐴𝑐 (1 + 𝐴𝑚 𝑒𝑚 (𝑡)) ………………Eq. 10
For the lower band:
𝑠(𝑙𝑜𝑤𝑒𝑟 𝑏𝑎𝑛𝑑) = cos(2𝜋𝑓𝑐 𝑡) ∗ 𝐴𝑐 (1 − 𝐴𝑚 𝑒𝑚 (𝑡))………………..Eq. 11
Note that equations 10 and 11 resulted from the modulation that exists between the
modulating and carrier waves. The interaction between the modulating and carrier waves causes
the destructive interaction of the carrier wave due to the exact experimental setup of the
modulation model. The carrier wave will indicate an amplitude of 0 throughout the time. From
equation 1, the transmitted wave will be represented as;
𝑠(𝑡) = 𝑠𝑢𝑝𝑝𝑒𝑟 𝑏𝑎𝑛𝑑 − 𝑠𝑙𝑜𝑤𝑒𝑟 𝑏𝑎𝑛𝑑
𝑠(𝑡) = [𝐴𝑐 (1 + 𝐴𝑚 𝑒𝑚 (𝑡)) ∗ cos(2𝜋𝑓𝑐 𝑡)] − [𝐴𝑐 (1 − 𝐴𝑚 𝑒𝑚 (𝑡)) ∗ cos(2𝜋𝑓𝑐 𝑡)]
𝑠(𝑡) = cos(2𝜋𝑓𝑐 𝑡) ∗ 2 ∗ 𝐴𝑐 𝐴𝑚 𝑒𝑚 (𝑡)………………………………Eq. 12
Question 2. Derivation of the equation for the SNR
From equation 11 and 12, the SNR would be;
𝑠(𝑡) = 𝑠𝑢𝑝𝑝𝑒𝑟 𝑏𝑎𝑛𝑑 − 𝑠𝑙𝑜𝑤𝑒𝑟 𝑏𝑎𝑛𝑑
𝑠(𝑡) = cos(2𝜋𝑓𝑐 𝑡) ∗ 2 ∗ 𝐴𝑐 𝐴𝑚 𝑒𝑚 (𝑡)
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For the noise,
𝑠𝑁 (𝑡) = cos(2𝜋𝑓𝑚 𝑡) ∗ 2 ∗ 𝐴𝑁 𝐴𝑁 𝑒𝑚 (𝑡) …………………………Eq. 13
For the signal,
𝑠𝑆 (𝑡) = cos(2𝜋𝑓𝑐 𝑡) ∗ 2 ∗ 𝐴𝑆 𝐴𝑁 𝑒𝑚 (𝑡)…………………………….Eq. 14
The SNR equation is:
𝑆𝑁𝑅 =
𝑠𝑖𝑔𝑛𝑎𝑙
cos(2𝜋𝑓𝑐 𝑡) ∗ 2 ∗ 𝐴𝑆 𝐴𝑁 𝑒𝑚 (𝑡)
=
𝑛𝑜𝑖𝑠𝑒
cos(2𝜋𝑓𝑚 𝑡) ∗ 2 ∗ 𝐴𝑁 𝐴𝑁 𝑒𝑚 (𝑡)
Which can be simplified as;
cos(2𝜋𝑓 𝑡)∗𝐴
𝑆𝑁𝑅 = cos(2𝜋𝑓 𝑐 𝑡)∗𝐴𝑆 ………………………………………………Eq. 15
𝑚
𝑁
From the previous statement, fc = 5.5 kHz = 5,500 Hz & fm = 10-16 Hz,
Substituting these values to SNR equation gives;
𝑆𝑁𝑅 =
cos(2𝜋 ∗ 5,500 ∗ 𝑡) ∗ 𝐴𝑆
cos(2𝜋 ∗ 10−16 ∗ 𝑡) ∗ 𝐴𝑁
But cos (2π*10-16*t) = cos (0) = 1,
𝑆𝑁𝑅 =
𝐴𝑆 ∗ cos(2𝜋∗5,500∗𝑡)
𝐴𝑁
𝐴
= cos(𝜋𝑓𝑆 𝑡) ∗ 𝐴 𝑆 ……………………..Eq. 16
𝑁
Question 3. Graphical representation of the SNR
The graph illustrated below shows a representation of the SNR in the case of a double
sideband suppressed carrier modulation. This graph is a representation of the computed equation
in relation to amplitude modulation. For instance, consider the amplitude of 1 and the previous
frequency of 5.5 kHz, the graphical representation of SNR against time will be;
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Non-linear modulation systems: Frequency modulation
The linear modulation systems emphasized the resonance between the waves, that
resulted in the destructive or constructive interaction of the modulating wave and carrier signals.
In the non-linear modulation systems, there is no resonance phenomenon between the waves and
restrictions are not imposed on the available frequencies by the model.
Derivation of the equation for s(t)
Let m(t) and c(t) be:
𝑚(𝑡) = cos(2𝜋𝑓𝑚 𝑡) ∗ 𝐴𝑚
𝑐(𝑡) = cos(2𝜋𝑓𝑐 𝑡) ∗ 𝐴𝑐
From the previous case of amplitude modulation, the equation for s(t) becomes:
2𝜋(𝑓 −𝑓 )
2𝜋(𝑓 +𝑓 )
𝑠(𝑡) = √𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos ( 𝑐 𝑚 ∗ 𝑡) ∗ cos ( 𝑐 𝑚 ∗ 𝑡)…...Eq. 17
2
2
In the non-linear modulation system, there are no presumed simplifications of the s(t)
equation because it is not a must for the modulating and carrier waves to be in resonance. In this
regard, the physical phases existing between the two waves and the temporal one have a
significant impact on the phase and location of the wave (Consolo et al., 2010).
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Derivation of the SNR equation
The noise wave equation is:
2𝜋(𝑓𝑁 −𝑓𝑚 )
𝑠𝑁 (𝑡) = √𝐴𝑁 2 + 𝐴𝑚 2 + 2𝐴𝑁 𝐴𝑚 ∗ cos(∅′) ∗ cos (
2
2𝜋(𝑓𝑁 +𝑓𝑚 )
∗ 𝑡) ∗ cos (
2
∗ 𝑡)….Eq. 19
The signal wave equation is:
2𝜋(𝑓𝑐 −𝑓𝑚 )
𝑠(𝑡) = √𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos (
2
∗ 𝑡) ∗ cos (
2𝜋(𝑓𝑐 +𝑓𝑚 )
2
∗ 𝑡)……...Eq. 20
Additionally, the SNR equation will be:
𝑆𝑁𝑅 =
√𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos (2𝜋(𝑓𝑐 − 𝑓𝑚 ) ∗ 𝑡) ∗ cos (2𝜋(𝑓𝑐 + 𝑓𝑚 ) ∗ 𝑡)
2
2
√𝐴𝑁 2 + 𝐴𝑚 2 + 2𝐴𝑁 𝐴𝑚 ∗ cos(∅′) ∗ cos (2𝜋(𝑓𝑁 − 𝑓𝑚 ) ∗ 𝑡) ∗ cos (2𝜋(𝑓𝑁 + 𝑓𝑚 ) ∗ 𝑡)
2
2
Substituting fc = 5.5 kHz = 5,500 Hz and fN = 10-16 Hz gives:
𝑆𝑁𝑅
2𝜋(5500 − 𝑓𝑚 )
2𝜋(5500 + 𝑓𝑚 )
∗ 𝑡) ∗ cos (
∗ 𝑡) ∗
2
2
=
−16 − 𝑓 )
2𝜋(10−16 + 𝑓𝑚 )
𝑚
√𝐴𝑁 2 + 𝐴𝑚 2 + 2𝐴𝑁 𝐴𝑚 ∗ cos(∅′) ∗ cos (2𝜋(10
∗
𝑡)
∗
cos
(
∗ 𝑡)
2
2
√𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos (
But, cos (10-16) = cos (0) = 1,
𝑆𝑁𝑅
=
√𝐴𝑐 2 + 𝐴𝑚 2 + 2𝐴𝑐 𝐴𝑚 ∗ cos(∅) ∗ cos (2𝜋(5500 − 𝑓𝑚 ) ∗ 𝑡) ∗ cos (2𝜋(5500 + 𝑓𝑚 ) ∗ 𝑡) ∗
2
2
√𝐴𝑁 2 + 𝐴𝑚 2 + 2𝐴𝑁 𝐴𝑚 ∗ cos(∅′)
Graphical representation of the SNR
The figure below shows the graphical representation of the modulated signal obtained
from the non-linear modulation system. From the figure, it is observed that the pattern comprises
of a series of valleys and peaks that are distributed throughout a two-dimensional grid that
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accounts for both the time phase between the two waves and temporal (Fehenberger et al., 2016).
The peaks’ intensity reduces with an increase in the distance from the source.
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References
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