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 \(\cos \omega_{0} t\) ?

Understanding the complex exponential function is crucial when diving into the world of Fourier transforms. Typically, this function is represented as \( e^{ix} \), where \( i \), is the imaginary unit, and \( x \) is a real number. The beauty of this function lies in its relation to trigonometry; it can be expressed using Euler's formula: \( e^{ix} = \text{cos}(x) + i\text{sin}(x) \).

Why is this relevant? The complex exponential function helps us describe oscillations. For instance, \( e^{-i \omega_{0} t} \) represents a sinusoidal oscillation with frequency \( \omega_{0}\) in the complex plane. In Fourier analysis, this function helps us capture both the amplitude and phase of signals, making it a powerful tool in signal processing.

When we compute the Fourier transform of a function modulated by a complex exponential, it results in a frequency shift represented in the Fourier domain—an aspect fundamental to our understanding of modulation and signal analysis.

The Modulation Theorem is a cornerstone of Fourier analysis, allowing us to comprehend how a signal's spectrum changes when it is modulated by another function. In simpler terms, when function \( f(t) \) is multiplied by a modulating signal like \( e^{-i \omega_{0} t} \), its Fourier transform experiences a frequency shift.

This principle is not just theoretical; it has real-world applications in communications where modulation is required to transmit signals over long distances. When we apply the modulation theorem to our rectangular pulse modulated by a complex exponential \( f(t)e^{-i \omega_{0} t} \), we observe that the Fourier transform \( F(\omega-\omega_{0}) \) represents a shift in the frequency domain by \( \omega_{0} \). This proves how modulation impacts the frequency content of signals and is vital for understanding concepts like frequency modulation and signal processing in modern technology.

Fourier transforms come with a set of properties that make analyzing signals more straightforward and intuitive. Among these, linearity lets us add and scale transforms just like the signals they represent. Convolution becomes simple multiplication in the Fourier domain, and similarly, modulation turns into a frequency shift.

Additionally, time-scaling affects the frequency domain's width, showing an inverse relationship. Notably, a time-domain function's duration inversely affects its frequency-domain bandwidth—longer time spans imply narrower bandwidths and vice versa.

The time-shifting property also tells us that shifting a function in time results in a phase shift in the frequency domain. These properties are not just mathematical curiosities; they provide an insightful language to discuss and manage signals in areas ranging from audio engineering to quantum mechanics.

The Fourier spectrum is a powerful way to visualize the frequency components of a sinusoidal oscillation. A pure sine or cosine function in the time domain translates to a single spike in the frequency domain at its oscillation frequency. However, real-world signals are rarely so neat; they may consist of multiple oscillations or be shaped differently.

For example, a rectangular pulse modulated by a sinusoidal wave doesn't have a single frequency but a band of frequencies, represented by a sinc function in the frequency domain. The 'sinc' shape arises due to the time-limited nature of the pulse—capturing the idea that constraining a signal in time broadens its frequency spectrum. This concept is pivotal in understanding the trade-offs between time and frequency in signal processing—increasing the duration of a sinusoidal pulse narrows its spectral width, and conversely, a shorter pulse has a wider spectral footprint.