Aaron Reyes
ECE 3101 L
PRELAB
1 - Root Mean Square Voltage in dBV and dBm
Given x(t) = 1.25 cos(2π × 666.667t),
1. Find the rms voltage of x(t) and represent the rms voltage is dBV and dBm.
Vrms =
𝑉𝑝𝑒𝑎𝑘
√2
1.25
= √2 = 0.88𝑉
𝑑𝐵𝑣 = 20 log10 (0.88) = − 1.07𝑑𝐵
𝑑𝐵𝑚 = 20 ∗ log10 (0.88/(√50Ω ∗ 10−3 𝑊 = 0.22))=11.92 dBm
2 -, Vp=1.25V, T0=1.5ms, duty cycle=1/2
A rectangular pulse train x(t) with peak-to-peak amplitude 2.5V, period T0 = 1.5ms, duty cycle d = ½
has dc offset of 0V. The amplitude of the pulse train is +1.25V or −1.25V.
v(t)
1.25
t
T
-T
2T
-1.25
The Fourier series representation of x(t) is given by
𝑗𝑛𝑤0 𝑡
𝑥(𝑡) = ∑∞
= 𝑐0 + ∑∞
𝑛=−∞ 𝑋𝑛 𝑒
𝑛=1 𝑐𝑛 cos(𝑛𝑤0 𝑡 + 𝜃𝑛 ), 𝑤0 =
2𝜋
𝑇0
2.a. Plot the first five periods of x(t) as a function of time t using MATLAB.
2.b. Find the expression of the complex exponential Fourier coefficients Xn of x(t).
2.5 −𝑗(𝜋)𝑛
𝜋
𝑋𝑛 =
𝑒 2 [sin 𝑛]
𝑛𝜋
2
Aaron Reyes
ECE 3101 L
𝑋𝑛 =
𝜋
−𝑗( )𝑛
𝑒 2
𝜋𝑛
[sin 2 ]
𝑛𝜋
2.5
𝜋
𝑛
−𝑗( )𝑛
= 1.25 𝑒 2 sinc ( )
2
2.c. Find the expression of the magnitude of the trigonometric Fourier coefficients cn of x(t).
𝑛
𝐶𝑛 = 2|𝑋𝑛 | = 1.25 sinc ( ) , 𝑛 = 1,2,3, …
2
2.d. Find the expression of the phase coefficients θn of x(t).
𝜋
𝜃0 = 0 , 𝜃𝑛 = ∠(𝑋𝑛 ), 𝑛 = − 𝑛 , 𝑛 = 1,2,3, …
2
2.e. Evaluate cn for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and represent cn in dBV.
C0=0,C1=1.59, C2=0, C3=-0.53, C4=0, C5=0.31, C6=0, C7=-0.22, C8=0, C9=0.18, C10=0
2.f. Evaluate θn for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 in radians.
𝜋
𝜋
𝜋
C{1-10}=0,− 2 ,− 𝜋,− 2, − 𝜋, − 2,− 𝜋……..
2.g. Plot cn for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 as a function of frequency using MATLAB.
2.h. Plot θn for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 as a function of frequency using MATLAB.
_n_
____cn______
__theta___
___Cndbv____
0
1
2
0
0.79577
4.8727e-17
0
-90
-180
-Inf
-4.9945
-329.25
3
4
5
6
7
8
9
10
11
0.26526
4.8727e-17
0.15915
4.8727e-17
0.11368
4.8727e-17
0.088419
4.8727e-17
0.072343
-90
-180
-90
-180
-90
-180
-90
-180
-90
-14.537
-329.25
-18.974
-329.25
-21.896
-329.25
-24.079
-329.25
-25.822
Aaron Reyes
ECE 3101 L
3 - Fourier Coefficients for a rectangular pulse train, Vp/p=2.5V, Vp=1.25V, T0=1.5ms, duty
cycle=1/3
3. Repeat #2 with duty cycle d = 1/3.
4 - Fourier Coefficients for a rectangular pulse train, Vp/p=2.5V, Vp=1.25V, T0=1.5ms, duty
cycle=1/4
4. Repeat #2 with duty cycle d = 1/4.
5
5.
- Fourier Coefficients for a rectangular pulse train, Vp/p=2.5V, Vp=1.25V,
T0=1.5ms, duty cycle=1/5
Repeat #2 with duty cycle d = 1/5.
Aaron Reyes
ECE 3101 L