Solve in two step and add explanation box in each stepQuestion 4
The message signal is:
m(t) = 4cos(400mt + 7/3)
itis modulated with a carrier:
c(t) = 10sin(2 - 10°7t)
using a modulator with sens

Question

Solve in two step and add explanation box in each stepQuestion 4
The message signal is:
m(t) = 4cos(400mt + 7/3)
itis modulated with a carrier:
c(t) = 10sin(2 - 10°7t)
using a modulator with sensitivity
ks = 200 [rad/ Volt].

You are also provided with the following Bessel function coefficient table Jj(8):

B k=0 k=1 k=2 k=3 k=4 k=5 k=6
0.0 1.00

05 0.94 024 0.03

1.0 0.77 0.44 011 0.02

15 051 0.56 0.23 0.06 0.01

20 0.22 0.58 0.35 0.13 0.03

24 0.00 0.52 0.43 0.20 0.06 0.02
3.0 -0.26 034 0.49 031 0.13 0.04 0.01
4A.
Write the expression of the modulated signal.
4B.

Express the modulated signal as a harmonic series based on its frequency components.
4c.

Calculate the phase and frequency modulation indices, and determine the total transmitted power (assume
normalized resistance).
4D.
Write the amplitude and phase values only for the 3rd harmonic.

Ensure the amplitudes are positive, and adjust the phases to lie within the range

—m< $

VIPSolutions · Accepted Answer

Let's solve **Question 4A** and **4B** step by step, with an explanation box for each:

---

## **Step 1: Write the expression of the modulated signal (4A)**

### **Given:**

- Message:

VIPSolutions · Answer

Let's carefully solve the problem step by step, explaining each part clearly.

---

### **Given Data:**

- Message signal:  
  $$
  m(t) = 4 \cos(400 \pi t + \frac{7}{3})
  $$
  
- Carrier signal:  
  $$
  c(t) = 10 \sin(2 \times 10^7 t)
  $$

- Modulation sensitivity:  
  $$
  k_s = 200 \, \text{rad/V}
  $$

- Bessel coefficients table $ J_j(8) $:

| $j$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---------|-----|-----|-----|-----|-----|-----|-----|
| $J_j(8)$ | 1.00 | 0.94 | 0.24 | 0.03 | 0.05 | 0.02 | 0.00 |
| $J_j(15)$ | 0.77 | 0.44 | 0.11 | 0.02 | 0.15 | 0.23 | 0.06 |
| $J_j(20)$ | 0.22 | 0.58 | 0.35 | 0.13 | 0.03 | 0.06 | 0.02 |
| $J_j(24)$ | 0.00 | 0.52 | 0.43 | 0.20 | 0.06 | 0.02 | - |

---

## **4A. Write the expression of the modulated signal**

### **Step 1: Find the modulating angular frequency $\omega_m$ and carrier angular frequency $\omega_c$.**

- Message: $ m(t) = 4 \cos(400 \pi t + 7/3) $

$\Rightarrow$  
  $\omega_m = 400 \pi \, \text{rad/sec}$

- Carrier: $ c(t) = 10 \sin(2 \times 10^7 t) $

$\Rightarrow$  
  $\omega_c = 2 \times 10^7 \, \text{rad/sec}$

---

### **Step 2: Write the phase deviation $\Delta \phi(t)$ for FM**

In frequency modulation, the instantaneous phase deviation is proportional to the message:

$$
\Delta \phi(t) = k_s \times m(t) = 200 \times 4 \cos(400 \pi t + 7/3) = 800 \cos(400 \pi t + 7/3)
$$

The FM signal is:

$$
s(t) = A_c \sin\left[\omega_c t + \Delta \phi(t)\right]
$$

where $A_c = 10$.

### **Step 3: Express the FM modulated signal**

Using the general FM expression:

$$
\boxed{
s(t) = 10 \sin \left[ 2 \times 10^7 t + 800 \cos(400 \pi t + 7/3) \right]
}
$$

---

## **4B. Express the modulated signal as a harmonic series**

### **Step 1: Recognize the form of FM signal**

The FM signal can be expanded using Bessel functions:

$$
s(t) = A_c \sum_{j=-\infty}^{\infty} J_j(\beta) \sin\left[\omega_c t + j \omega_m t + \phi_0\right]
$$

where:

- $\beta = $ frequency modulation index for FM, which is:

$$
\boxed{
\beta = \frac{\Delta f}{f_m}
}
$$

But, in terms of phase deviation:

$$
\beta = \frac{\text{Peak phase deviation}}{\text{modulating frequency}} = \frac{800}{400 \pi} \approx \frac{800}{1256.637} \approx 0.6366
$$

Alternatively, since the phase deviation amplitude is $800$, and the message frequency is $\omega_m/2\pi = 200 \pi$, the **phase modulation index** is:

$$
\beta = \frac{\text{Peak phase deviation}}{\omega_m} = \frac{800}{400 \pi} \approx 0.6366
$$

---

### **Step 2: Write the harmonic series**

Using the Bessel expansion, the modulated signal becomes:

$$
\boxed{
s(t) = 10 \sum_{j=-6}^{6} J_j(8) \sin \left( \omega_c t + j \omega_m t + \phi_j \right)
}
$$

with $\omega_m = 400 \pi$, and phases:

$$
\phi_j = \frac{7}{3}
$$

**Note:** Since $J_j(8)$ are symmetric with respect to $j$, and the sum runs over $|j| \leq 6$ (based on the table), the harmonic components are at frequencies:

$$
f_j = \frac{\omega_c}{2\pi} + j \frac{\omega_m}{2\pi} = 10^7 + j \times 200\, \text{Hz}
$$

---

## **Summary of answers:**

### **4A: Expression of the modulated FM signal**

$$
\boxed{
s(t) = 10 \sin\left( 2 \times 10^7 t + 800 \cos(400 \pi t + \frac{7}{3}) \right)
}
$$

### **4B: Harmonic series representation**

$$
\boxed{
s(t) = 10 \sum_{j=-6}^{6} J_j(8) \sin \left( 2 \times 10^7 t + j \times 200 \pi t + \frac{7}{3} \right)
}
$$

---

Would you like to proceed with parts **4C** (calculating indices and power) and **4D** (amplitude and phase for the 3rd harmonic)?

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