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 = πππππ √2 1.25 = √2 = 0.88π ππ΅π£ = 20 log10 (0.88) = − 1.07ππ΅ ππ΅π = 20 ∗ log10 (0.88/(√50β¦ ∗ 10−3 π = 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 πππ€0 π‘ π₯(π‘) = ∑∞ = π0 + ∑∞ π=−∞ ππ π π=1 ππ cos(ππ€0 π‘ + ππ ), π€0 = 2π π0 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 −π(π)π π ππ = π 2 [sin π] ππ 2 Aaron Reyes ECE 3101 L ππ = π −π( )π π 2 ππ [sin 2 ] ππ 2.5 π π −π( )π = 1.25 π 2 sinc ( ) 2 2.c. Find the expression of the magnitude of the trigonometric Fourier coefficients cn of x(t). π πΆπ = 2|ππ | = 1.25 sinc ( ) , π = 1,2,3, … 2 2.d. Find the expression of the phase coefficients θn of x(t). π π0 = 0 , ππ = ∠(ππ ), π = − π , π = 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