AM: Math and Spectrum

Hello! Welcome back to our course on radioelectronics.

In our last lesson, we established why modulation is a fundamental requirement for radio communication, focusing on the need for practical antenna sizes and the ability to share the radio spectrum through Frequency Division Multiplexing (FDM).

Today, we move from the "why" to the "how." Our learning outcome is to understand Amplitude Modulation (AM): its mathematical representation and frequency spectrum. This is one of the oldest and simplest forms of modulation, and understanding it provides the foundation for more complex techniques. We will derive the mathematical expression for an AM signal and analyze what it looks like in the frequency domain, a critical skill in radio engineering.

1. The Concept of Amplitude Modulation

At its core, Amplitude Modulation is a straightforward process: the amplitude of a high-frequency carrier wave is made to vary in direct proportion to the instantaneous amplitude of a low-frequency message signal.

Let's define the signals involved:

• Message Signal, $m(t)$: This is the baseband information you want to transmit (e.g., an audio signal). It has a relatively low frequency.
• Carrier Signal, $c(t)$: This is a high-frequency sinusoidal wave. Its purpose is to "carry" the message signal.
• AM Signal, $s(t)$: This is the final modulated signal that is transmitted.

The process works as follows: The message signal $m(t)$ is added to the DC amplitude of the carrier, creating a time-varying amplitude. This combined amplitude then modulates the high-frequency carrier wave. The resulting AM signal has an "envelope" that traces the shape of the original message signal.

This image illustrates the concept of Amplitude Modulation. The top waveform is the low-frequency message signal, $m(t)$. The middle waveform is the high-frequency carrier, $c(t)$. The bottom waveform is the resulting AM signal, $s(t)$, where the envelope (the dashed line tracing the peaks) has the same shape as the message signal.

2. Mathematical Representation of an AM Signal

To analyze AM properly, we need to describe it mathematically. Let's start by defining our signals as simple sinusoids, which is a standard approach for analysis.

1. The Carrier Signal: We represent the high-frequency carrier as a cosine wave:
   $c(t)=A_c\cos (2\pi f_{c}t)$
   where:

   • $A_c$ is the peak amplitude of the carrier.
   • $f_c$ is the carrier frequency.

2. The Message Signal: For simplicity, let's represent the low-frequency message signal as a single-tone cosine wave:
   $m(t)=A_m\cos (2\pi f_{m}t)$
   where:

   • $A_m$ is the peak amplitude of the message.
   • $f_m$ is the message frequency.
     We assume $f_c\gg f_m$.

The instantaneous amplitude of the AM signal's envelope is the sum of the carrier's peak amplitude and the message signal: $A_{env}(t)=A_c+m(t)$.

The complete AM signal, $s(t)$, is this time-varying amplitude multiplied by the carrier wave:
$s(t)=[A_c+m(t)]\cos (2\pi f_{c}t)$
Substituting our sinusoidal message signal gives us the standard equation for a single-tone AM signal:
$s(t)=[A_c+A_m\cos (2\pi f_{m}t)]\cos (2\pi f_{c}t)$

The Modulation Index

To make this equation more useful, we introduce a critical parameter called the modulation index (often denoted by $\mu$ or $k_a$). It is the ratio of the message amplitude to the carrier amplitude:
$\mu =\frac{A_m}{A_c}$
The modulation index quantifies the extent to which the carrier amplitude is varied. We can factor $A_c$ out of the main equation to express it in terms of $\mu$:
$s(t)=A_c[1+\frac{A_m}{A_c}\cos (2\pi f_{m}t)]\cos (2\pi f_{c}t)$
$s(t)=A_c[1+\mu \cos (2\pi f_{m}t)]\cos (2\pi f_{c}t)$

The value of $\mu$ is very important for the quality of the signal:

• $\mu <1$ (Undermodulation): The message amplitude is less than the carrier amplitude. The envelope never reaches zero. This is the desired state for standard AM, as the envelope shape perfectly preserves the message signal.
• $\mu =1$ (100% Modulation): The message amplitude equals the carrier amplitude. The envelope just touches zero at its minimum points. This provides the maximum signal variation without distortion.
• $\mu >1$ (Overmodulation): The message amplitude is greater than the carrier amplitude. The envelope attempts to go below zero, causing it to be clipped at zero and introducing a 180° phase reversal in the carrier. This severely distorts the signal and makes it impossible to recover the original message with a simple detector. This condition must be avoided.

3. The Frequency Spectrum of an AM Signal

The time-domain equation tells us what the signal looks like on an oscilloscope, but to understand its bandwidth and how it fits into the radio spectrum, we need to analyze its frequency components. We can do this by applying a trigonometric identity to our AM equation.

Let's expand the equation for $s(t)$:
$s(t)=A_c\cos (2\pi f_{c}t)+A_c\mu \cos (2\pi f_{m}t)\cos (2\pi f_{c}t)$
The first term is simply the carrier. For the second term, we use the product-to-sum identity:
$\cos (A)\cos (B)=\frac{1}{2}[\cos (A-B)+\cos (A+B)]$
Applying this to the second term, with $A=2\pi f_{c}t$ and $B=2\pi f_{m}t$, we get:
$\mu A_c\cos (2\pi f_{c}t)\cos (2\pi f_{m}t)=\frac{\mu A_c}{2}[\cos (2\pi (f_c-f_m)t)+\cos (2\pi (f_c+f_m)t)]$
Now, we substitute this back into the full expression for $s(t)$:
$s(t)=\underset{\text{Carrier}}{\underbrace{A_c\cos (2\pi f_{c}t)}}+\underset{\text{Lower Sideband (LSB)}}{\underbrace{\frac{\mu A_c}{2}\cos (2\pi (f_c-f_m)t)}}+\underset{\text{Upper Sideband (USB)}}{\underbrace{\frac{\mu A_c}{2}\cos (2\pi (f_c+f_m)t)}}$

This equation is fundamental. It reveals that a simple AM signal is not a single frequency but a composite of three distinct sinusoidal components:

1. Carrier Component: A powerful component at the original carrier frequency $f_c$, with amplitude $A_c$.
2. Upper Sideband (USB): A new component at frequency $f_c+f_m$, with amplitude $\frac{\mu A_c}{2}$.
3. Lower Sideband (LSB): Another new component at frequency $f_c-f_m$, with amplitude $\frac{\mu A_c}{2}$.

The two sidebands are mirror images of each other around the carrier, and they contain the identical message information.

Visualizing the Spectrum and Calculating Bandwidth

We can visualize this as a frequency spectrum plot:

This plot shows the frequency components of a single-tone AM signal. A large spectral line appears at the carrier frequency $f_c$. Two smaller lines of equal amplitude appear symmetrically around the carrier at $f_c-f_m$ (the Lower Sideband) and $f_c+f_m$ (the Upper Sideband).

From this spectrum, we can determine the bandwidth (BW) of the AM signal. Bandwidth is the range of frequencies occupied by the signal, calculated as the difference between the highest and lowest frequencies.

$BW=(f_c+f_m)-(f_c-f_m)=2f_m$

This is a crucial result: The bandwidth of a standard AM signal is twice the highest frequency in the message signal.

For a more complex message signal like voice or music, which contains a range of frequencies, the sidebands are not single lines but "side-bands" of frequencies. If the message signal has a maximum frequency of $f_{m(\text{max})}$, the total bandwidth will be $2f_{m(\text{max})}$. For example, AM radio broadcast channels limit the audio message frequency to about 5 kHz, resulting in a signal bandwidth of 10 kHz, which is why broadcast stations in the AM band are spaced 10 kHz apart in North America.

4. Power Distribution in AM

A final important consideration is how the transmitter power is distributed among the three frequency components. Since the power of a sinusoidal signal delivered to a load is proportional to the square of its amplitude, we can analyze the power distribution.

• Carrier Power: $P_c\propto A_c^2$
• Total Sideband Power: $P_{sb}=P_{LSB}+P_{USB}$. Since both sidebands have amplitude $\frac{\mu A_c}{2}$, their combined power is:
  $P_{sb}\propto {\left(\frac{\mu A_c}{2}\right)}^2+{\left(\frac{\mu A_c}{2}\right)}^2=2\frac{{\mu }^2{A}_c^2}{4}=\frac{{\mu }^2{A}_c^2}{2}$
  We can express this in terms of the carrier power: $P_{sb}=P_c\frac{{\mu }^2}{2}$.
• Total Power: $P_t=P_c+P_{sb}=P_c+P_c\frac{{\mu }^2}{2}=P_c\left(1+\frac{{\mu }^2}{2}\right)$

The efficiency of AM, defined as the ratio of useful power (in the sidebands) to total power, is:
$\eta =\frac{P_{sb}}{P_t}=\frac{P_c({\mu }^2/2)}{P_c(1+{\mu }^2/2)}=\frac{{\mu }^2}{2+{\mu }^2}$
Even at 100% modulation ($\mu =1$), the maximum efficiency is $\eta =\frac{1^2}{2+1^2}=\frac{1}{3}$, or 33.3%. This means at least two-thirds of the transmitted power is concentrated in the carrier, which carries no information. This inefficiency is a major drawback of standard AM and motivates other modulation schemes like Single Sideband (SSB), which we may discuss later.

Conclusion

In this lesson, we have taken a detailed look at Amplitude Modulation. You should now have a solid grasp of its mathematical underpinnings and spectral characteristics.

Key Takeaways:

• AM involves varying a carrier's amplitude in proportion to a message signal.
• The standard equation is $s(t)=A_c[1+\mu \cos (2\pi f_{m}t)]\cos (2\pi f_{c}t)$, where $\mu =A_m/A_c$ is the modulation index.
• Overmodulation ($\mu >1$) must be avoided as it causes distortion.
• The frequency spectrum of an AM signal consists of a carrier frequency ($f_c$) and two sidebands at $f_c\pm f_m$.
• The bandwidth required for an AM signal is twice the highest frequency of the message signal ($BW=2f_{m(\text{max})}$).
• Standard AM is power-inefficient, with a maximum of only 33.3% of the power residing in the information-carrying sidebands.

Now that we know how to create an AM signal, the next logical step is to understand how to get the information back. In our next lesson, we will explore AM Demodulation using an Envelope Detector, a simple and effective circuit for recovering the original message signal.