Telecommunication Principles Assignment I

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TELECOMMUNICATION PRINCIPLES ASSIGMENT I
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QUESTION 1 (40 MARKS)
Student ID: 0061072809. The waveform chosen is WaveParamsAssign9.pdf.
The waveform contains a reference sinewave and an unknown wave.
Using the waveform below,
Determining the unknown waveform parameters from the figure above,
x (t )= A sin (wt + φ)
The following parameters were obtained from the unknown wave versus the reference wave
Parameter Reference wave Unknown wave
Amplitude 1 2
Phase angle (Time shift) 00 100.80
Angular Velocity 40 π rads/sec 40 π rads /sec
The angular velocity is given by,
ω=2 πf ... angular velocity
f = 1
T frequency∧period relationship
1

f = 1
0.05 =20 Hertz
ω=2∗π∗20
ω=40 π rads /sec
When the sine wave fails to pass through zero at the entry point, it is considered to have a phase
difference. The phase difference can be expressed as a time shift based on the period in seconds.
The common or standard mode of expressing it is done in angular measurements.
The time shift is given as,
Time shift=0.012 seconds
1 T =0.04 seconds ( full cycle )
full cycle=3600
¿ 0.05 s →3600 ; 0.014 s → ?
3600 x 0.014
0.05 =90.80 ... . phase angle
The phase offset or the phase difference of the unknown wave is given as 100.80.
The reference signal is given as,
x (t )=sin ( 40 π ) volts
The resulting equation of the unknown waveform is,
x ( t ) =−2sin ( 40 π −90.80 ) volts
QUESTION 2 (40 MARKS)
Amplitude modulated waveform
The waveform used is AMassign9.pdf.
The amplitude modulation is used in the transfer of voice information from one place to another.
The voice frequencies are said to range from 50Hz to 3000Hz. Amplitude modulation receives
voice frequencies and mixes them with some radio frequency signals. In this form, they can
easily be converted to radio frequencies that will radiate or propagate through free space.
The modulated signal is a simple sinusoid of the form,
2

m ( t ) =Am sin ( ωm t )
The AM waveform is then described by
x AM ( t ) = Ac sin ωm t + Am sin ( ωm t ) sin ( ωc t )
x AM ( t )= Ac ( 1+ μ sin ωm t ) sin ωc t
The modulation index is given by,
μ= Am
Ac
From a quick view of the waveform, one can settle on roughly 50 percent modulation. The
modulation index is the measure based on the ratio of the modulation excursions of the radio
frequency signal to the extent of the unmodulated carrier. It is given based on the message wave
envelope and the carrier wave form signal,
μ=50 %
Using time shift,
T =1.7 seconds ( period )
To get the angular velocity of the message envelope,
f = 1
T = 1
1.7 =0.588235 Hz
ω=2∗π∗f
ω=2∗0.588235∗π
ω=1.176 π rad /sec
Parameter Message signal Carrier signal
Amplitude Am =7 volts Ac=3.5 volts
3