Question

A function \(f(\alpha)\) is nonzero only in the region of width \(2 \delta\) centered at \(\alpha=0\) $$ f(\alpha)=\left\\{\begin{array}{ll} C & |\alpha| \leq \delta \\ 0 & |\alpha|>\delta \end{array}\right. $$ where \(C\) is a constant. (a) Find and plot versus \(\beta\) the Fourier transform \(A(\beta)\) of this function. (b) The function \(f \alpha\) ) might represent a pulse occupying either finite distance \((\alpha=\) position) or finite time \((\alpha\) = time). Comment on the wave number spectrum if \(\alpha\) is position and on the frequency spectrum if \(\alpha\) is time. Specifically address the dependence of the width of the spectrum on \(\delta\).

Short answer

The Fourier transform of function \(f(\alpha)\) is \(A(\beta) = 2C \frac{sin(\beta \delta)}{\beta}\). This function, when plotted, gives a sinc function. If \(\alpha\) is the position, the function describes the wave number spectrum, and if \(\alpha\) is time, it describes the frequency spectrum. The localisation of the function and the width of the sinc is controlled by \(\delta\) and this can be attributed to the Uncertainty Principle.

Step-by-step solution

Calculate the Fourier transform for the given function

The Fourier transform of a function \(f(\alpha)= C\) for \(|\alpha| \leq \delta \) and \(f(\alpha)=0\) otherwise, can be found using the definition of the Fourier transformation. It is calculated as \(A(\beta) = \int_{-\infty}^{\infty} f(\alpha) e^{-i \beta \alpha} d\alpha \). Since \(f(\alpha) = 0\) outside the interval \(-\delta \leq \alpha \leq \delta\), we can restrict the integration to this interval. So, \(A(\beta) = \int_{-\delta}^{\delta} C e^{-i \beta \alpha} d\alpha\).This can be further worked upon as \(A(\beta) = \frac{C}{-i\beta}[e^{-i\beta\alpha}]_{-\delta}^{\delta} = \frac{C}{-i\beta}(e^{-i\beta\delta} - e^{i\beta\delta}) = 2C \frac{sin(\beta \delta)}{\beta}\).

Plot the Fourier transform function

We can plot the function obtained in step 1 which is \(A(\beta) = 2C \frac{sin(\beta \delta)}{\beta}\) against \(\beta\) which will give a sinc function

Interpret the wave number and frequency spectrum

If \(\alpha\) represents position, the Fourier transform gives a wave number spectrum. The width of the peak in the wave number spectrum is related to the spatial extension of the function in position space i.e., \(2\delta\). If the function is more localized (smaller \(\delta\)), the width of the peak in wave number space will be larger. This is due to Uncertainty Principle. If \(\alpha\) represents time instead, the Fourier transform gives a frequency spectrum, and the result is similar, with the Uncertainty Principle applying between time and frequency.

Key concepts

Wave Number Spectrum

The concept of the wave number spectrum is rooted in the study of waves and spatial phenomena. The wave number, denoted usually by the symbol \(\beta\), is a measure of the number of wave cycles per unit length. It's akin to frequency, but whereas frequency measures cycles per unit time, wave number measures them per unit space.

When we analyze the Fourier transform of a spatially bounded function, like the one given in our textbook problem, we essentially decompose the function into waves of different numbers. The Fourier transform \(A(\beta)\) reveals how much of each wave number is present in the initial function. In this exercise, a function with sharp boundaries, only nonzero within a width of \(2\delta\), yields a sinc-shaped wave number spectrum when Fourier transformed.

The width of the sinc function in the wave number spectrum (the sharpness of its peak) is inversely proportional to \(\delta\); this means a narrower spatial pulse (smaller \(\delta\)) results in a broader wave number spectrum. These insights connect to Heisenberg's Uncertainty Principle, which we will delve into in a later section, dictating an intrinsic trade-off between the precision of a wave's position and its wave number.

Frequency Spectrum

Switching our focus to temporal analysis, the frequency spectrum is a powerful tool for understanding signals in time. Just as with the wave number spectrum's spatial analysis, the frequency spectrum analyzes the composition of a signal in terms of its frequency constituents.

If the variable \(\alpha\) is considered time, then the Fourier transform of our given function becomes an expression of how different frequencies contribute to the formation of a temporal pulse. In this instance, the sinc function obtained from our Fourier transform \(A(\beta)\) depicts the frequency spectrum of a pulse that occurs for a limited duration of \(2\delta\) in time.

Similarly to the wave number spectrum, the width of the frequency spectrum is inversely related to the duration \(\delta\). A shorter pulse in time (with smaller \(\delta\)) corresponds to a broader frequency spectrum. Once again, the Uncertainty Principle provides a foundational explanation for this behavior, highlighting the inherent limitations in precisely knowing both the time a signal occupies and its frequency content.

Uncertainty Principle

The Uncertainty Principle, formulated by Werner Heisenberg, is a fundamental concept in quantum mechanics, but its implications reach into the realms of signal processing and wave analysis. It states that there is a limit to how precisely we can know certain pairs of physical properties simultaneously. The classic pair often discussed is position and momentum, but for our purposes, the relevant pairs are wave number and position, or frequency and time.

In the context of our exercise, the principle can be expressed as \(\Delta \alpha \Delta \beta \geq \frac{1}{2}\), where \(\Delta \alpha\) represents the uncertainty in position or time, and \(\Delta \beta\) represents the uncertainty in wave number or frequency. The smaller the region \(\delta\) in which the function is non-zero, the larger the spread in the corresponding spectrum. Hence, as we confine our function in position (or time), we necessarily broaden its wave number (or frequency) spectrum.

This principle underpins the relationship described in the earlier sections, exemplifying why a pulse that is sharply defined in time or space must have a broad wave number or frequency spread, respectively. It's a profound reflection of the intrinsic limitations present in the analysis of waves and signals, and understanding this principle is crucial for students to grasp the fundamental trade-offs in signal processing and wave mechanics.