Question

A periodic amplitude modulated (AM) radio signal has the form $$ y=(A+B \sin 2 \pi f t) \sin 2 \pi f_{c}\left(t-\frac{x}{v}\right) $$ The factor \(\sin 2 \pi f_{c}(t-x / v)\) is called the carrier wave; it has a very high frequency (called radio frequency; \(f_{\epsilon}\) is of the order of \(10^{6}\) cycles per second). The amplitude of the carrier wave is \((A+B \sin 2 \pi f t)\). This amplitude varies with time-hence the term "amplitude modulation"..-with the much smaller frequency of the sound being transmitted (called audio frequency; \(f\) is of the order of \(10^{2}\) cycles per second). In order to see the general appearance of such a wave, use the following simple but unrealistic data to sketch a graph of \(y\) as a function of \(t\) for \(x=0\) over one period of the amplitude function: \(A=3, B=1, f=1\), \(f_{c}=20 .\) Using trigonometric formulas, show that \(y\) can be written as a sum of three waves of frequencies \(f_{c}, f_{c}+f\), and \(f_{c}-f ;\) the first of these is the carrier wave and the other two are called side bands.

Short answer

The amplitudes are combined into waves with frequencies \(f_c = 20\), \(f_c + f = 21\), and \(f_c - f = 19\).

Step-by-step solution

Write down the given information

Given the equation: \(y = (A + B \, \sin 2 \pi f t) \sin 2 \pi f_{c}\left(t - \frac{x}{v}\right)\) Data: \(A = 3\) \(B = 1\) \(f = 1\) \(f_{c} = 20\) Given \(x = 0\), the equation simplifies to \(y = (A + B \, \sin 2 \pi f t) \sin 2 \pi f_{c} t\)

Substitute the given values

Substitute \(A = 3\), \(B = 1\), \(f = 1\), and \(f_{c} = 20\) into the equation: \[ y = (3 + \sin 2 \pi \cdot 1 \cdot t) \sin 2 \pi \cdot 20 \cdot t \] Simplify: \[ y = (3 + \sin 2 \pi t) \sin 40 \pi t \]

Apply trigonometric identity

Use the trigonometric identity \( \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \).With \(A = 2\pi f t\) and \(B = 2\pi f_{c} t\), we have:\[ \sin 2 \pi t \sin 40 \pi t = \frac{1}{2} [\cos(40\pi t - 2\pi t) - \cos(40\pi t + 2\pi t)] \]Simplify the expression:\[ \sin 2 \pi t \sin 40 \pi t = \frac{1}{2} [\cos 38 \pi t - \cos 42 \pi t] \]

Expand the expression for y

Now expand the expression for \(y\):\[ y = 3 \sin 40\pi t + \sin 2\pi t \sin 40\pi t \]Using the identity from Step 3, \(\sin 2\pi t \sin 40\pi t\) becomes:\[ y = 3 \sin 40\pi t + \frac{1}{2} [\cos 38\pi t - \cos 42 \pi t] \]Combine terms:\[ y = 3 \sin 40\pi t + \frac{1}{2} \cos 38\pi t - \frac{1}{2} \cos 42 \pi t \]

Identify the frequencies

The simplified form can be written as the sum of three waves: \[ y = 3 \sin 40\pi t + \frac{1}{2} \cos 38\pi t - \frac{1}{2} \cos 42 \pi t \] The frequencies are: 1. Carrier wave: Frequency \(40 \pi\) (or \(f_{c} = 20\)) 2. Side band (lower): Frequency \(38 \pi\) (or \(f_{c} - f = 19\)) 3. Side band (upper): Frequency \(42 \pi\) (or \(f_{c} + f = 21\))

Key concepts

Carrier Wave

In the realm of Amplitude Modulation (AM), the carrier wave is incredibly important. It is essentially the primary signal that carries the information you want to send. For our problem, the carrier wave is represented by the function \( \sin 2 \pi f_c \left(t - \frac{x}{v}\right) \). This wave has a very high frequency, typically in the range of millions of cycles per second. This frequency is much higher than that of the audio information being transmitted. Because of this high frequency, the carrier wave can effectively travel long distances, making it ideal for radio transmissions.

Side Bands

Side bands are essential when dealing with AM signals. When you modulate the amplitude of a carrier wave with an audio signal, new frequencies appear on either side of the original carrier frequency. These are called side bands. In the exercise provided, two side bands emerge when you modulate the amplitude. These side frequencies are mathematically expressed as \( f_c + f \) and \( f_c - f \).
These side bands contain the actual information from the audio signal. The carrier frequency delivers the signal, but the side bands make sure that the information is accurately represented.

Trigonometric Identities

Trigonometric identities are powerful tools for simplifying and understanding AM signals. In this problem, we use one specific identity: \(\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]\). This identity helps break down the modulation process by converting the product of two sine functions into a sum of cosine functions.
This makes it easier to see the components of the modulated signal \(y\). When we apply this identity, the complicated multiplication of sine waves at different frequencies becomes more straightforward, turning into a sum of waves at different, distinct frequencies.

Frequency Modulation

While this specific problem focuses on amplitude modulation, it's useful to briefly understand frequency modulation (FM) for context. Unlike AM, where the amplitude of the carrier wave varies, in FM, the carrier wave's frequency is changed according to the information signal.
FM provides better noise resistance compared to AM. However, AM is simpler to implement, making it widely used for different broadcast purposes. Understanding both concepts enriches your grasp of how complex signals are structured and transmitted in real-world scenarios.