ECE454_HW08_SP21_solution SUNY Oswego ECE454: Communications Systems Spring 2021 Homework 8- Solution Problem 1: [10 points] An AM signal ??(??) with a carrier frequency ???? = 100?????? has a complex...

ECE454_HW08_SP21_solution
SUNY Oswego
ECE454: Communications Systems
Spring 2021


Homework 8- Solution
Problem 1: [10 points]
An AM signal ??(??) with a carrier frequency ???? = 100?????? has a complex envelope ??(??) =
????[1 + ??(??)], where ???? = 20??, and the modulation ??(??) is sinusoidal test tone ??(??) =
0.5 sin(2????????), with frequency ???? = 20??????.
a) Determine the in-phase and quadrature components ??(??) and ??(??) of this signal.
??(??) = ℜ{??(??)} = ????[1 + ??(??)]
??(??) = ℑ{??(??)} = 0
b) Determine the baseband amplitude and phase ??(??) and ??(??) for this signal.
??(??) = |??(??)| = ????[1 + ??(??)]
??(??) = ∠??(??) = 0
c) Determine the expression of the bandpass signal ??(??).
??(??) = ??(??) cos(?????? + ??(??)) = ????[1 + ??(??)] cos(??????)
= 20[ XXXXXXXXXXsin(2????????)] cos(??????) = [ XXXXXXXXXXsin(2????????)] cos(??????)
d) Compute the average normalized power ???? of the AM signal ??(??).
The average power of the bandpass signal is ???? =
1
2
????
The baseband power is ???? = 〈|??(??)|2〉 = 〈|????[1 + ??(??)]|2〉 = ????2〈1 + 2??(??) + ??2(??)〉
Assuming 〈??(??)〉 = 0, then:
???? = ????2 + ????2〈??2(??)〉 = ????2 + ????2〈0.52 sin2(2????????)〉 = ????2 +
????2
4

1 − cos(4????????))
2

= ????2 +
????2
8 =
9
8????
2 =
9
8 × 400 = 450??
Therefore, ???? =
1
2
???? = 225??
e) Using MatLab, plot both the baseband amplitude ??(??) and the bandpass signal v(t). Plot
both signals in the same figure over a ??0 = 0.2???? time duration with time resolution
(time-step) of ???? = 0.1????.
XXXXXXXXXX
t 10 -4
-30
-20
-10
0
10
20
30
R(t): baseband envelope
v(t): bndpass AM signal
Problem 2: [4 points]
Using the same AM signal of problem 1, determine the following frequency-domain
characteristics:
a) Derive the expression of the complex baseband envelope spectrum ??(??).
??(??) = 0.5 sin(2????????) then ??(??) = 0.5 ×
??
2
[??(?? + ????) − ??(?? − ????)]
??(??) = ????[??(??) + ??(??)] = ???? �??(??) +
??
4
[??(?? + ????) − ??(?? − ????)]�
b) Derive the expression of the bandpass spectrum ??(??) using the following equation:
??(??) =
1
2
[??(?? − ????) + ??∗(−?? − ????)]
??(??) =
????
2
�??(?? − ????) +
??
4
[??(?? − ???? + ????) − ??(?? − ???? − ????)]�
+
????
2
�??(−?? − ????) +
??
4
[??(−?? − ???? + ????) − ??(−?? − ???? − ????)]�

=
????
2
�??(?? − ????) +
??
4
[??(?? − ???? + ????) − ??(?? − ???? − ????)]�
+
????
2
�??(?? + ????) −
??
4
[??(?? + ???? − ????) − ??(?? + ???? + ????)]�
Problem 3: [6 points]
Given a pulse-modulated signal (bandpass) of the form:
??(??) = ??−???? cos[(???? + Δ??)??]??(??)
where ??(??) is the unit-step function, a, ωc, and Δω are positive constants and the carrier angular
frequency ωc>> Δω.
a) Find the corresponding in-phase (I) and quadrature (Q) components x(t) and y(t).
You may use the following trigonometric identity: cos(?? + ??) = cos?? ???????? − sin ?? sin ??
??(??) = ??−???? cos[?????? + Δ????]??(??)
??(??) = ??−???? cos (Δ????)cos(??????)??(??) − ??−???? sin (Δ????)sin(??????)??(??)
Or
??(??) = ??−???? cos (Δ????)??(??)cos(??????) − ??−???? sin(Δ????)??(??)sin(??????)
= ??(??) cos(??????) − ??(??)sin (??????)
where ??(??) = ??−???? cos(Δ????)??(??) and ??(??) = ??−???? sin(Δ????)??(??) are the in-phase (I) and
quadrature (Q) components, respectively.
b) Determine the magnitude R(t) and phase θ(t) of the complex baseband envelope ??(??).
??(??) = |??(??)| = �??2(??) + ??2(??) = ??−??????(??)
??(??) = arctan �
??(??)
??(??)� = arctan
(tan (Δ?? ??)) = Δ?? ??
c) Find the spectrum G(f) of the complex baseband envelope.
??(??) = ??(??)??????(??) = ??(??)????Δ????
Using the Fourier transform properties:
??(??) = ??−??????(??) ↔
1
?? + ??2????
??(??)????Δ???? ↔ 1
??+??2??�??−Δ??2??�
(using frequency-shift)
Therefore, ??(??) = 1
??+??2??�??−Δ??2??�

Problem 4: [4 points]
An amplifier is tested for total harmonic distortion (THD) by using a single-tone test. The output
is observed on a spectrum analyzer. It is found that the peak values of the measured harmonics
are ???? = ????????2 �
??
5
�, where ?? = 1, … ,4. These amplitude are obtained from a single-sided
spectrum (?? > 0).
- Compute the percentage total harmonic distortion (THD).
?????? =
�??22 + ??32 + ??42
??1
=
�????????4 �25�+ ????????
4 �35�+ ????????
4 �45�
????????2 �15�
= 71.9%
n=[1:4];
V=sinc(n/5).^2
THD=sqrt(sum(V(2:4).^2))/V(1)

V =
XXXXXXXXXX XXXXXXXXXX
THD =
XXXXXXXXXX
May 11, 2021

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