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Chapter 12: Problem 3

Find the frequency spectrum of a wave phase-modulated simultaneously at two frequencies \(\omega_{1}\) and \(\omega_{2}\) with modulation indices \(m_{1}\) and \(m_{2}\).

Short Answer

Expert verified
Answer: The frequency spectrum of a wave with simultaneous phase modulation at two frequencies can be expressed as: $$ X(\omega) = \pi A\sum_{n_1=-\infty}^{\infty} \sum_{n_2=-\infty}^{\infty} J_{n_1}(m_{1})J_{n_2}(m_{2})\Big[\delta(\omega - (\omega_c + n_1\omega_{1} + n_2\omega_{2}))\Big] $$ where \(A\) is the amplitude, \(\omega_c\) is the carrier frequency, \(\omega_{1}\) and \(\omega_{2}\) are the given modulation frequencies, \(m_{1}\) and \(m_{2}\) are the modulation indices, \(J_{n_1}(m_{1})\) and \(J_{n_2}(m_{2})\) are the Bessel functions of the first kind of orders \(n_1\) and \(n_2\), respectively, and \(\delta()\) is the Dirac delta function. In this spectrum, there are multiple frequency components defined by the central carrier frequency and frequencies centered around the sum and difference of the modulation frequencies, each multiplied by integer values \(n_1\) and \(n_2\).

Step by step solution

01

Represent the modulated wave equation

The wave with simultaneous phase modulation at two frequencies can be written as: $$ x(t) = A \cos(\omega_ct + m_{1}\cos(\omega_{1}t) + m_{2}\cos(\omega_{2}t)) $$ Where \(A\) is the amplitude, \(\omega_c\) is the carrier frequency, and \(\omega_{1}\), \(\omega_{2}\), \(m_{1}\), and \(m_{2}\) are given modulation frequencies and indices.
02

Apply the Bessel function approximation to the modulated wave

In order to find the frequency spectrum of the modulated wave, we need to express the phase-modulated signal in terms of the Bessel functions of the first kind. The Bessel function expansion of the wave above can be written as: $$ x(t) = A\sum_{n_1=-\infty}^{\infty}\sum_{n_2=-\infty}^{\infty} J_{n_1}(m_{1})J_{n_2}(m_{2})\cos((\omega_c + n_1\omega_{1} + n_2\omega_{2})t) $$ Where \(J_{n_1}(m_{1})\) and \(J_{n_2}(m_{2})\) represent the Bessel functions of the first kind of orders \(n_1\) and \(n_2\), respectively.
03

Apply the Fourier Transform to find the frequency spectrum

Now we can apply the Fourier Transform to the expanded modulated wave equation to obtain its frequency spectrum. The Fourier Transform of \(x(t)\) is defined as: $$ X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt $$ By calculating the Fourier Transform of \(x(t)\), we obtain the frequency spectrum: $$ X(\omega) = \pi A\sum_{n_1=-\infty}^{\infty} \sum_{n_2=-\infty}^{\infty} J_{n_1}(m_{1})J_{n_2}(m_{2})\Big[\delta(\omega - (\omega_c + n_1\omega_{1} + n_2\omega_{2}))\Big] $$
04

Identify the frequency components

The obtained frequency spectrum indicates that there are multiple frequency components in the modulated wave. They are defined by the central carrier frequency \(\omega_c\) and frequencies centered around the sum and difference of the modulation frequencies, \(\omega_{1}\) and \(\omega_{2}\), each multiplied by integer values \(n_1\) and \(n_2\). The Bessel functions \(J_{n_1}(m_{1})\) and \(J_{n_2}(m_{2})\) characterize the amplitude of these frequency components. Thus, the frequency spectrum of the phase-modulated wave simultaneously modulated at two frequencies \(\omega_{1}\) and \(\omega_{2}\) with modulation indices \(m_{1}\) and \(m_{2}\) has been found.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bessel Functions
Bessel functions are a series of solutions to a specific type of differential equation called Bessel's equation. They frequently appear in physics and engineering contexts, particularly when dealing with problems that have cylindrical or spherical symmetry. For instance, when a drum head vibrates, the resulting patterns can be described using Bessel functions.

The Bessel functions often show up in the analysis of systems with circular or cylindrical geometry, perhaps in the radial part of the wave functions in quantum mechanics, or in the study of electromagnetic waves in cylindrical structures.

In the context of phase modulation, Bessel functions of the first kind, denoted as \(J_n(x)\), where \(n\) is the order of the function, become crucial. They describe the amplitude of various frequency components when a carrier signal is phase-modulated. Each order of the Bessel function indicates a frequency component in the spectrum resulting from modulation, with the modulation index defining the intensity or contribution of that frequency component. In the step-by-step solution, Bessel functions help in expressing the phase-modulated signal in a series form that can be further analyzed to find the frequency spectrum.
Fourier Transform
The Fourier Transform is a mathematical technique that transforms a time-domain signal into its constituent frequencies, referred to as the frequency domain. It provides a way to analyze the frequencies present within a complex signal and is widely used in signal processing, physics, and engineering.

The essence of the Fourier Transform is that it can decompose any periodic or non-periodic signal into a spectrum of frequencies, revealing information that might not be easily observed in the time domain. This is akin to understanding a musical chord not just as one sound, but as a combination of individual notes.

In solving for the frequency spectrum of a phase-modulated wave, applying the Fourier Transform allows us to express the signal in terms of delta functions, \(\delta(\omega)\), centered at each frequency component. This mathematical operation essentially filters out the different frequencies that make up the signal, showing how much of each frequency is present. In the provided solution, the Fourier Transform is used to convert the complex time-domain expression into a frequency-domain representation, uncovering the spectrum of frequencies generated by the phase modulation.
Phase Modulation
Phase Modulation (PM) is a modulation technique used in signal processing where the phase of the carrier wave is varied in accordance with the instantaneous amplitude of the modulating signal. This is different than amplitude modulation, which changes the amplitude of the carrier wave, and frequency modulation, which changes its frequency.

Phase modulation is particularly useful in digital transmission systems, such as Wi-Fi and mobile phone networks, due to its resiliency against signal degradation and ability to carry more data in a fixed bandwidth. In the context of the original exercise, we're looking at phase modulation with two modulating signals, which means the carrier wave's phase is being influenced by two different signals at the same time.

Understanding the effect of these modulating signals on the carrier is critical for designing communication systems. The ability to calculate the frequency spectrum of a phase-modulated wave, as demonstrated in the textbook solution, is a foundational skill that helps engineers and scientists predict how a signal will propagate, how much bandwidth it will occupy, and how it might interact with other signals or noise in a transmission system.

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Most popular questions from this chapter

The voltage gain of a multistage low-pass a mplifier is accurately approximated by the gaussian function $$ |\breve{G}(\omega)|=G_{0} e^{-(\omega / 2 \pi \Delta \omega)^{2}} . $$ Show that \(\Delta \omega\) is the rms bandwidth of the amplifier. Show also that the so-called \(u p p e r\) halfpower frequency \(\nu_{0}\) [the frequency at which \(\left(|\breve{G}| / G_{0}\right)^{2}=\frac{1}{2}\) ] is related to \(\Delta \omega\) by $$ \nu_{0} \approx 0.6 \Delta \omega, $$ where \(\nu_{0}\) is an ordinary (cyclic) frequency. Show then that the uncertainty relation (12.2.7) implies that $$ \nu_{0} \Delta t \approx \frac{1}{3} $$ when \(\Delta t\), the duration of a narrow pulse whose frequency spectrum is limited by the amplifier, has its minimum permitted value. This form of the uncertainty relation expresses reasonably well the relation between the rise- time of an amplifier (see Prob. 11.4.3 for a definition of risetime) and its bandwidth expressed by its upper half-power frequency.

Apply the Kramers-Kronig relations (12.7.15) and (12.7.16) to the two parts of the (delay) transfer function of Prob. 12.6.1, T) \((\omega)=\cos \omega t_{2}-j \sin \omega t_{2}\), to verify that one part implies the other, and vice versa.

Show that the amplitude spectrum of a rectangular pulse of the sinusoidal oscillation \(e^{-i \omega_{0} t}\) lasting from \(-\tau\) to \(+\tau\) is given by $$ F(\omega)=2 a \frac{\sin \left(\omega-\omega_{0}\right) \tau}{\left(\omega-\omega_{0}\right) \tau} $$ Relate this spectrum to that of the rectangular pulse discussed in Sec. 12.2. Generalize this result into a modulation theorem: if \(F(\omega)\) is the transiorm of \(f(t)\), then \(F\left(\omega-\omega_{0}\right)\) is the transform of \(f(t) e^{-i \omega_{0} t}\). What form does the theorem take if \(e^{-i \omega_{0} t}\) is replaced by \(\cos \omega_{0} t\) ?

Establish the integral $$ \int_{0}^{\infty} \sin a t \cos b t d t=\frac{a^{2}}{a^{2}-b^{2}} $$ by first multiplying the integrand by \(e^{-s t}\) and allowing \(s \rightarrow 0\) after integration. This, or a similar artifice, is of necessary in evaluating Fourier transform integrals. A Fourier transform so evaluated is then described as a limiting form.

Show that a necessary and sufficient condition for \(p(t)\) to be a real function of time is that its Fourier transform \(\breve{P}(\omega)\) have the symmetry property \(\breve{P}(-\omega)=\breve{P} *(\omega)\), which shows that the real part of \(\breve{P}(\omega)\) is an even function of \(\omega\) but that its imaginary part is an odd function of \(\omega\) Then show that a necessary and sufficient condition for \(q(t)\), as given by (12 7.2) also to be real is for \(\breve{P}(\omega) \breve{T}(\omega)\), and hence \(\widetilde{T}(\omega)\), to obey the same symmetry property, so that \(\breve{T}(-\omega)={T}^{*}(\omega) .\) Note that this argument establishes the evenness and oddness of the real and imaginary parts of a transfer function without using the delta function.
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