Cosine waves are functions of the form x ↦A cos(v x+ϕ) where A is the amplitude of the wave, v t

step 1 So we're still looking into cosine waves. We know their use is very frequent in a lot of differe

Cosine waves are functions of the form x ↦A cos(v x+ϕ) where...

Cosine waves are functions of the form $x \mapsto A \cos (v x+\phi)$ where $A$ is the amplitude of the wave, $v$ the frequency, and $\phi$ the phase shift. The physical superposition of two waves is obtained mathematically by adding the two wave functions. Concern the superposition of cosine waves. The identities from Exercises 49 and 50 are particularly useful in this context. An amplitude modulation (AM) radio transmitter broadcasts an audio tone (or baseband signal) $\cos (\nu \cdot t)$ by modulating the amplitude of a very high-frequency carrier signal $A \cdot \cos (\omega \cdot t)$ by a positive factor $(1+m \cdot \cos (\nu \cdot t)),$ where the modulation index $m$ is less than $1 .$ Thus the emitted signal has the form $$S(t)=A \cdot(1+m \cdot \cos (\nu \cdot t)) \cos (\omega \cdot t)$$ a. Use the identity from Exercise 49 to express $S(t)$ as a superposition of three cosine waves. The largest and smallest frequencies of these three waves are called sidebands. b. Set $A=1, m=1 / 2, \nu=2,$ and $\omega=8 .$ Graph the signal $S(t)$ and envelopes $\pm(1+m \cdot \cos (\nu \cdot t))$ for $0 \leq t \leq 2 \pi$ Explain how the sidebands can be determined visually from your graph. c. Suppose that $v_{\max }<\omega$ is the highest baseband frequency. The largest and smallest frequencies broadcast are called the upper and lower sidebands. What are they in terms of $\omega$ and $v_{\max } ?$ Their difference is called the bandwidth. d. Carrier frequencies for AM radio range from $550 \mathrm{kHz}$ (kilohertz), or $550 \times 10^{3}$ cycles per second, to $1610 \mathrm{kHz}$. Typically, $\nu_{\max }$ can be as high as $15 \mathrm{kHz}$. An AM receiver works by recovering the audio source $\cos (\nu \cdot t)$ from the received signal $S(t),$ a process known as demodulation. Demodulation requires that the bandwidths of different radio stations that broadcast to the same location not overlap. What must the minimum frequency separation between the carrier signals of two such stations be?

Cosine waves are functions of the form $x \mapsto A \cos (v x+\phi)$ where $A$ is the amplitude of the wave, $v$ the frequency, and $\phi$ the phase shift. The physical superposition of two waves is obtained mathematically by adding the two wave functions. Concern the superposition of cosine waves. The identities from Exercises 49 and 50 are particularly useful in this context.
An amplitude modulation (AM) radio transmitter broadcasts an audio tone (or baseband signal) $\cos (\nu \cdot t)$ by modulating the amplitude of a very high-frequency carrier signal $A \cdot \cos (\omega \cdot t)$ by a positive factor $(1+m \cdot \cos (\nu \cdot t)),$ where the modulation index $m$ is less than $1 .$ Thus the emitted signal has the form
$$S(t)=A \cdot(1+m \cdot \cos (\nu \cdot t)) \cos (\omega \cdot t)$$
a. Use the identity from Exercise 49 to express $S(t)$ as a superposition of three cosine waves. The largest and smallest frequencies of these three waves are called sidebands.
b. Set $A=1, m=1 / 2, \nu=2,$ and $\omega=8 .$ Graph the signal $S(t)$ and envelopes $\pm(1+m \cdot \cos (\nu \cdot t))$ for $0 \leq t \leq 2 \pi$ Explain how the sidebands can be determined visually from your graph.
c. Suppose that $v_{\max }<\omega$ is the highest baseband frequency. The largest and smallest frequencies broadcast are called the upper and lower sidebands. What are they in terms of $\omega$ and $v_{\max } ?$ Their difference is called the bandwidth.
d. Carrier frequencies for AM radio range from $550 \mathrm{kHz}$ (kilohertz), or $550 \times 10^{3}$ cycles per second, to $1610 \mathrm{kHz}$. Typically, $\nu_{\max }$ can be as high as $15 \mathrm{kHz}$. An AM receiver works by recovering the audio source $\cos (\nu \cdot t)$ from the received signal $S(t),$ a process known as demodulation. Demodulation requires that the bandwidths of different radio stations that broadcast to the same location not overlap. What must the minimum frequency separation between the carrier signals of two such stations be?

Key Concepts

Demodulation

Demodulation is the process of extracting the original baseband signal from the received modulated carrier. It involves techniques that reverse the modulation process, allowing a receiver to recover the transmitted information efficiently and accurately without interference from other signals.

Bandwidth

Bandwidth, in the context of amplitude modulation, is defined as the difference between the highest and lowest frequencies present in the modulated signal. It is determined by the frequency range of the baseband signal and is critical for ensuring that different radio stations do not have overlapping frequency components.

Sidebands

Sidebands are the frequency components generated on either side of the carrier frequency as a result of the modulation process. They arise due to the frequency mixing of the carrier and the baseband signal, and they carry the actual information of the baseband signal in the transmission.

Trigonometric Identities for Cosine Functions

Trigonometric identities, such as the product-to-sum formulas for cosine, are essential mathematical tools used to decompose a modulated signal into its constituent frequency components. By rewriting products of cosine terms as sums, these identities facilitate the analysis and understanding of the signal's spectral content.

Baseband Signal

The baseband signal, often representing the original audio or data information, is the low-frequency waveform used to modulate the amplitude of a carrier. It contains the essential content of the transmission and determines the envelope of the modulated waveform.

Carrier Signal

The carrier signal is a high-frequency waveform that serves as the basis for modulation in communication systems. In amplitude modulation, the carrier's amplitude is adjusted according to the information signal, and its main role is to transport the modulated signal over long distances through various media.

Amplitude Modulation (AM)

Amplitude modulation is a technique in which the amplitude of a high-frequency carrier wave is varied in accordance with the instantaneous amplitude of a lower-frequency information signal. This method allows the transmission of audio or other data by imprinting the baseband information onto the carrier wave, making it suitable for radio broadcasting.

Modulation Index

The modulation index is a parameter that measures the extent of amplitude variation in the carrier wave relative to its nominal amplitude in an amplitude modulation system. A modulation index less than one ensures that the modulated signal remains within non-overmodulation bounds, preventing various forms of signal distortion.

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