VLSI homework 10, ECOMMS homework 3

7th-Semester-Fall-2024/ECOMMS/homework/homework-3/homework-3.md

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+# ECOMMS Homework 3 - Aidan Sharpe
+
+## Problem 1
+A sinusoidal signal $m(t) = \cos(2\pi f_m t)$ is the input to an angle-modulated transmitter, $A_c = 1$, and the carrier frequency is $f_c = 1$ Hz and $f_m = f_c/4$. Plot $m(t)$, and the corresponding phase and frequency modulated signals, $S_p(t)$ and $S_f(t)$ respecively. $D_p = \pi$ and $D_f = \pi$.
+
+```python
+f_c = 1
+f_m = f_c/4
+
+f_s = 8E3
+T_s = 1/f_s
+
+A_c = 1
+D_p = np.pi
+D_f = np.pi
+
+omega_m = 2*np.pi*f_m
+omega_c = 2*np.pi*f_c
+
+# Time samples from to 10 seconds with a sampling frequency of 8kHz.
+t = np.arange(0,10,T_s)
+
+# The message signal
+m = np.cos(omega_m * t)
+
+# The phase modulated signal
+S_p = np.cos(omega_c*t + D_p*m)
+
+# The time integral of the mesage signal
+M = np.sin(omega_m * t) / omega_m
+
+# The frequency modulated signal
+S_f = np.cos(omega_c*t + D_f*M)
+```
+
+![](1.png)
+
+## Problem 2
+An FM signal has sinusoidal modulation with a frequency of $f_m = 15$kHz and a modulation index of $\beta = 2.0$. Find the transmission bandwidth by using Carson's rule, and find the percentage of total FM signal power that lies within the Carson rule bandwidth.
+
+```python
+# Modulation index
+beta = 2.0
+
+# Message frequency
+f_m = 15E+3
+
+# Transmission bandwidth
+B_T = 2*(beta+1)*f_m
+```
+
+$\boxed{B_T = 90\text{kHz}}$
+
+```python
+A_c = 1
+n = np.arange(-3,4,1)
+
+# Evaluate the Bessel function at values in Carson rule bandwidth
+bessel_values = np.abs(sp.special.jv(n,beta))
+
+P_C = 0.5 * A_c**2 * np.sum(bessel_values**2))
+P = 0.5 * A_c**2
+```
+
+\boxed{P_c = 0.9976 P}
+
+## Problem 3
+A modulated RF waveform is given by $500\cos(\omega_c t + 20\cos(\omega_1 t))$, where $\omega_1 = 2\pi f_1$, $f_1 = 1$kHz, $\omega_c = 2\pi f_c$, and $f_c = 100$MHz.
+
+### 3a
+If the phase sensitivity $D_p = 100$ rad/V, find the mathematical expression for the corresponding phase modulation voltage $m(t)$. What is its peak value and frequency?
+
+$m(t) = \frac{20\cos(\omega_1 t)}{D_p} = 0.2\cos(\omega_1 t)$
+
+Peak value: $2 \times 10^{-1}$
+
+Frequency: 1kHz
+
+### 3b
+If the frequency deviation constant $D_f = 10^6$rad/Vs, find the mathematical expression for the corresponding FM voltage $m(t)$. What is its peak value and its frequency?
+
+$\theta(t) = 20\cos(\omega_1 t)$
+
+$m(t) = \frac{1}{D_f} \frac{d \theta(t)}{dt} = -2 \times 10^{-5} \omega_1 \sin(\omega_1 t)$
+
+Peak value: $2 \times 10^{-5}$
+
+Frequency: 1kHz
+
+### 3c
+If the RF waveform appears across a 50$\Omega$ load, determine the average power and the PEP.
+
+The average power of the signal is:
+$$\langle s^2(t) \rangle = \frac{1}{T} \int\limits_{-T/2}^{T/2} 500\cos(\omega_c t + 20\cos(\omega_1 t)) dt$$
+