Amplitude Modulation (AM) Cheatsheet

Cheatsheet Content

Amplitude Modulation (AM) Basics Definition: A modulation technique where the amplitude of the carrier wave is varied in proportion to the instantaneous amplitude of the message signal. Message Signal: $m(t) = A_m \cos(2\pi f_m t)$ Carrier Signal: $c(t) = A_c \cos(2\pi f_c t)$ Bandwidth: $BW = 2 f_m$ Methods for Standard AM (DSB-LC) Generation 1. Square-Law Modulator Principle: Uses a non-linear device (e.g., diode, transistor operating in non-linear region) to generate products of signals. Input: $v_{in}(t) = c(t) + m(t) = A_c \cos(2\pi f_c t) + A_m \cos(2\pi f_m t)$ Square-Law Device Output: $v_{out}(t) = a v_{in}(t) + b v_{in}^2(t)$ (where $a, b$ are constants) Expanded Output: Contains terms at $f_c$, $f_m$, $2f_c$, $2f_m$, $f_c \pm f_m$, etc. Filtering: A bandpass filter centered at $f_c$ is used to select the desired AM signal components. Resulting AM signal: $A_c [1 + k_a m(t)] \cos(2\pi f_c t)$ Derivation: $v_{in}^2(t) = (A_c \cos(2\pi f_c t) + A_m \cos(2\pi f_m t))^2$ $ = A_c^2 \cos^2(2\pi f_c t) + A_m^2 \cos^2(2\pi f_m t) + 2 A_c A_m \cos(2\pi f_c t) \cos(2\pi f_m t)$ Terms like $2 A_c A_m \cos(2\pi f_c t) \cos(2\pi f_m t) = A_c A_m [\cos(2\pi(f_c-f_m)t) + \cos(2\pi(f_c+f_m)t)]$ After filtering and combining with $a v_{in}(t)$ terms, we get the AM wave. 2. Switching Modulator Principle: Uses a switch (e.g., diode, transistor) controlled by the carrier signal to periodically switch the message signal. Input: Message signal $m(t)$ is applied to a switch controlled by the carrier $c(t)$. Switching: The carrier effectively multiplies $m(t)$ by a square wave $s(t)$ which has fundamental frequency $f_c$. $s(t) = \frac{1}{2} + \frac{2}{\pi} \cos(2\pi f_c t) - \frac{2}{3\pi} \cos(6\pi f_c t) + \dots$ Output: $v_{out}(t) = m(t) s(t) = m(t) [\frac{1}{2} + \frac{2}{\pi} \cos(2\pi f_c t) + \dots]$ Filtering: A bandpass filter extracts the AM signal. $v_{out}(t) = \frac{1}{2} m(t) + \frac{2}{\pi} m(t) \cos(2\pi f_c t) + \dots$ The filter selects $\frac{2}{\pi} m(t) \cos(2\pi f_c t)$, and with an added carrier, forms AM. This method is more commonly used for DSB-SC, but can be adapted for AM by adding a DC offset to $m(t)$ before modulation or adding a carrier after. Methods for DSB-SC Generation 1. Balanced Modulator Principle: Uses two identical AM modulators in a balanced configuration (e.g., using two square-law devices or two switching modulators). Configuration: Modulator 1: Input $m(t)$ and $c(t)$, output $s_1(t) = [A_c + k_a m(t)] \cos(2\pi f_c t)$ Modulator 2: Input $m(t)$ and $180^\circ$ phase-shifted carrier $-c(t)$, output $s_2(t) = [A_c + k_a m(t)] \cos(2\pi f_c t + \pi) = -[A_c + k_a m(t)] \cos(2\pi f_c t)$ Alternatively, Modulator 1: Input $m(t)$ and $c(t)$, output $s_1(t) = [A_c + k_a m(t)] \cos(2\pi f_c t)$. Modulator 2: Input $-m(t)$ and $c(t)$, output $s_2(t) = [A_c - k_a m(t)] \cos(2\pi f_c t)$. Output: The sum or difference of the outputs cancels the carrier. Using the second configuration: $s_{DSB}(t) = s_1(t) - s_2(t) = ([A_c + k_a m(t)] - [A_c - k_a m(t)]) \cos(2\pi f_c t) = 2 k_a m(t) \cos(2\pi f_c t)$. This is a DSB-SC signal. Advantage: Suppresses the carrier more effectively than single modulators. 2. Ring Modulator (Diode Bridge Modulator) Principle: A type of balanced modulator using four diodes in a ring or bridge configuration. The carrier signal acts as a switch. Operation: When $c(t)$ is positive, two diodes conduct, connecting $m(t)$ (with a polarity inversion) to the output. When $c(t)$ is negative, the other two diodes conduct, connecting $m(t)$ (without inversion) to the output. Effective Multiplication: The message signal $m(t)$ is effectively multiplied by a square wave $s(t)$ which is proportional to the carrier with amplitude $\pm 1$. $s(t) = \frac{4}{\pi} [\cos(2\pi f_c t) - \frac{1}{3} \cos(6\pi f_c t) + \dots]$ Output: $v_{out}(t) = m(t) s(t) = \frac{4}{\pi} m(t) \cos(2\pi f_c t) - \frac{4}{3\pi} m(t) \cos(6\pi f_c t) + \dots$ Filtering: A bandpass filter centered at $f_c$ extracts the DSB-SC signal: $\frac{4}{\pi} m(t) \cos(2\pi f_c t)$. Methods for SSB Generation 1. Filter Method Principle: Generate a DSB-SC signal, then use a highly selective bandpass filter to remove one of the sidebands. Steps: Generate DSB-SC signal $s_{DSB}(t) = A_m A_c \cos(2\pi f_m t) \cos(2\pi f_c t)$. Apply $s_{DSB}(t)$ to a sharp cutoff bandpass filter. To obtain USB: Filter centered at $(f_c + f_m/2)$ with bandwidth $f_m$. To obtain LSB: Filter centered at $(f_c - f_m/2)$ with bandwidth $f_m$. Challenge: Requires extremely sharp and stable filters, especially for voice signals where $f_m$ can be low (e.g., $f_m \approx 300$ Hz, requiring a filter to separate $f_c \pm 300$ Hz). This is difficult for high carrier frequencies. 2. Phase-Shift Method (Weaver's Method) Principle: Uses phase-shifting networks (Hilbert transformers) to cancel one sideband. Steps: Input $m(t)$ and apply to a $90^\circ$ phase shifter to get $\hat{m}(t)$ (Hilbert transform). Input $c(t)$ and apply to a $90^\circ$ phase shifter to get $\hat{c}(t) = A_c \sin(2\pi f_c t)$. Multiply $m(t)$ with $c(t)$ to get $P_1 = m(t) A_c \cos(2\pi f_c t)$. Multiply $\hat{m}(t)$ with $\hat{c}(t)$ to get $P_2 = \hat{m}(t) A_c \sin(2\pi f_c t)$. Sum/Subtract the products: For USB: $s_{USB}(t) = P_1 - P_2 = A_c [m(t) \cos(2\pi f_c t) - \hat{m}(t) \sin(2\pi f_c t)]$ For LSB: $s_{LSB}(t) = P_1 + P_2 = A_c [m(t) \cos(2\pi f_c t) + \hat{m}(t) \sin(2\pi f_c t)]$ Advantage: Does not require sharp filters, making it suitable for high carrier frequencies and wideband signals. Challenge: Designing ideal $90^\circ$ phase shifters for a wide range of message frequencies can be complex. Methods for VSB Generation 1. Filter Method (from DSB-SC) Principle: Generate a DSB-SC signal, then pass it through a VSB filter. Steps: Generate a DSB-SC signal $s_{DSB}(t) = m(t) \cos(2\pi f_c t)$. Pass $s_{DSB}(t)$ through a VSB filter $H_{VSB}(f)$. The filter characteristics ensure that one sideband is fully passed, and a vestige of the other sideband is also passed, with a smooth transition around $f_c$. Filter Characteristics: The VSB filter's magnitude response $|H_{VSB}(f)|$ is designed such that $|H_{VSB}(f_c - f)| + |H_{VSB}(f_c + f)| = \text{constant}$ for $0 Advantages: Simpler filter design than SSB, allows transmission of low-frequency components (including DC), crucial for video signals.