---
tags: Communication System Design, ss, ncu
author: N0-Ball
title: Modulation
GA: UA-208228992-1
---

# Intro

- first form of modulation used in radio transmission
- MF band (medium frequency, 300 kHz - 3 MHz)
- Basic Form:
- DSB-LC (Double Side Band)
- DSB-SC
- offers narrow bandwidth transmission
- $B = 2 \times f_{max}$ (DSB-LC)
- Very inefficient in use of transmit power
- Message signal is $k_a m(t)$

# DSB-LC

## Modulate
**With the modulated signal**
$$
\begin{split}
v_{AM}(t) &= A_c \cos \left( \omega_c t \right) \times \left[ 1 + k_am(t) \right] \\[1em]
&= A_c \cos \left( \omega_c t \right) + k_aA_cm(t) \cos \left( \omega_c t\right)
\end{split}
$$
**Considering** $k_am(t) = k_a \cos \left( \omega_m t \right)$
$$
\begin{split}
&v_{AM}(t) = A_c \cos \left( \omega_c t \right) + \frac{1}{2}k_aA_c \cos \left( \omega_c t + \omega_m t \right) + \frac{1}{2}k_aA_c \cos \left( \omega_c t - \omega_m t \right)
\end{split}
$$


:::info
- Two copies of m(t) appear on either side of $f_c$
- Power of each term:
- Carrier: $\frac{1}{2} A_c^2$
- Upper sideband: $\frac{\left( \frac{1}{2}k_aA_c \right)^2}{2}$
- Lower sideband: $\frac{\left( \frac{1}{2}k_aA_c \right)^2}{2}$
- if $k_a = 1$(maximum value), ratio of message power to carrier power = 0.5
:::
:::danger
**Typical mean value for $k_a = 0.3$**
$$
\frac{P_{msg}}{P_c} = \frac{1}{2}k_a^2 = 0.045
$$
**Less than 5% of transmitted power is in the message**
:::

## Demodulate

- Apply large AM signal to a diode $v_{AM}(t) = A_c \cos \left( \omega_c t \right) \times \left[ 1 + k_am(t) \right]$
- Sole action of carrier is to switch the demoduulator diode in the AM receiver

1. Generates switching function s(t) at carrier frequency
- $s(t) = \frac{1}{2} + \frac{2}{\pi}\left( \cos \left( \omega_ct \right) - \frac{1}{3} \cos \left( 3 \omega_ct \right) + \cdots \right)$
2. Multiply s(t) by AM signal
- $x(t) = v_{AM}(t) \times \left(\frac{1}{2} + \frac{2}{\pi}\cos \left( \omega_ct \right) - \frac{2}{3\pi} \cos \left( 3 \omega_ct \right) + \cdots \right)$
3. for convenience, set k~a~ = 1
4. $\frac{2}{\pi} A_c \left( \cos \left( \omega_c t \right) \right)^2 = \frac{2}{\pi}A_c\left( 1 + \frac{1}{2} \cos \left(2 \omega_c t \right) \right) = \frac{2}{\pi} A_c + \frac{1}{\pi} A_c \cos \left( A_c \cos \left( 2 \omega_c t \right) \right)$
5. Place a low pass filter at the output of the diode to get the following lower-frequency terms.
- $y(t) = \frac{2}{\pi}A_c + \frac{1}{\pi}A_cm(t)$
6. Output contains a DC term and the message signal

- Wanted signal has amplitude proportional to received carrier voltage A~c~
- Control gain of AM receiver to make output signal of demodulator constant
- requires Automatic Gain Control (AGC)

# Diode Detector

- Since mesage abides as the amplitude of the received signal, detecting amplitude change = obtaining message!

### Adventage

- verfy simple circuit
- Very low cost

### Disadventage

- Poor linearity - distorion
- Poor selectivity
- Affected by selective fading