Question

Let \(\operatorname{sinc} x=(\sin \pi x) / \pi x(\operatorname{sinc} 0=1)\). a. If \(a>0, \widehat{\chi}_{|-a, a|}(x)=\chi_{[a, a]}^{V}(x)=2 a \operatorname{sinc} 2 a x\). b. Let \(\mathcal{H}_{a}=\left\\{f \in L^{2}: \hat{f}(\xi)=0\right.\) (a.c.) for \(\left.|\xi|>a\right\\}\). Then \(\mathcal{T}\) is a Hilbert space and \(\\{\sqrt{2 a}\) sinc \((2 a x-k): k \in Z\\}\) is an orthonormal basis for \(\mathcal{H}\). c. (The Sampling Theorem) If \(f \in \mathcal{T}_{a}\), then \(f \in C_{0}\) (after modification on a null set), and \(f(x)=\sum_{-\infty}^{\infty} f(k / 2 a) \operatorname{sinc}(2 a x-k)\), where the series converges both uniformly and in \(L^{2}\). (In the terminology of signal analysis, a signal of bandwidth \(2 a\) is completely determined by sampling its values at a sequence of points \(\\{k / 2 a\\}\) whose spacing is the reciprocal of the bandwidth.)

Short answer

The sinc function helps express a signal's Fourier transform and the Sampling Theorem states that a bandlimited signal can be determined from discrete samples.

Step-by-step solution

Understanding the Sinc Function

The sinc function is defined as \( \operatorname{sinc} x = \frac{\sin \pi x}{\pi x} \) with \( \operatorname{sinc} 0 = 1 \). It is a mathematical function often used in signal processing and Fourier analysis.

Examine Part a

For part (a), we have \( \widehat{\chi}_{|-a, a|}(x) = \chi_{[a, a]}^{V}(x) = 2 a \operatorname{sinc} (2 a x) \). This shows how the Fourier transform of the window function \( \chi_{[-a, a]} \) is represented as a scaled sinc function, conveying that the bandwidth is limited to \([ -a, a ]\).

Understanding Hilbert Space in Part b

In part (b), the space \( \mathcal{H}_{a} = \{f \in L^{2}: \hat{f}(\xi) = 0 \) (a.c.) for \(|\xi|>a\}\) is introduced. This is a space of functions whose Fourier transform is zero outside \([-a, a]\). The question states that the set \{\sqrt{2 a}\operatorname{sinc} (2 a x - k) : k \in \mathbb{Z}\} is an orthonormal basis, meaning any function in this space can be represented uniquely by these sinc functions.

Apply the Sampling Theorem in Part c

Part (c) uses the Sampling Theorem, which applies to bandlimited signals (functions in \(\mathcal{T}_a\)). If \( f \in \mathcal{T}_a \), then we can express \(f(x)\) exactly by using a series: \( f(x) = \sum_{k=-\infty}^{\infty} f\left(\frac{k}{2a}\right) \operatorname{sinc}(2 a x - k) \). This series describes how the continuous signal can be completely reconstructed from its samples at intervals of \(\frac{1}{2a}\), the reciprocal of its bandwidth.

Key concepts

Sinc Function

The sinc function is a mathematical function that is very useful in signal processing and Fourier analysis. It is defined as \[\text{\( sinc \)} x = \frac{\sin \pi x}{\pi x},\]where the special case at zero is defined as \( \operatorname{sinc}(0) = 1 \). This definition avoids the division by zero problem by assigning the limiting value at zero.
The shape of the sinc function makes it ideal for representing band-limited functions. It oscillates, forming ripples on either side of the peak at \( x = 0 \), and these ripples decay gradually as \( x \) moves away from zero. The sinc function plays a crucial role in the process of signal reconstruction, as seen in the Sampling Theorem. Understanding this function is key to grasping how continuous signals can be fully reconstructed from discrete sample points.

Fourier Transform

The Fourier Transform is a mathematical operation that transforms a function of time into a function of frequency. It is a powerful tool for analyzing the frequencies present in a signal. When you perform a Fourier Transform on a time-domain function, like a signal, you convert it into its frequency components.
In the context of part (a) of the exercise, the Fourier Transform of the window function \[\chi_{[-a, a]}(x)\]is shown to be represented as a scaled sinc function. This representation indicates that the function's bandwidth is constrained to the interval \([-a, a]\). This scaling effect is vital for understanding how functions behave in the frequency domain and how different frequencies contribute to the overall signal.
By examining the Fourier Transform, we can understand how a signal decomposes into its constituent frequencies, which is essential for applications like signal processing and communications.

Hilbert Space

Hilbert spaces are a generalization of Euclidean spaces and contain functions that possess an inner product. They are foundational in mathematical analysis, particularly in the context of quantum mechanics and signal processing. In this scenario, we consider the Hilbert space \[\mathcal{H}_{a} = \{ f \in L^2 : \hat{f}(\xi) = 0 \text{ a.c. for } |\xi| > a \}\].
This means that the Fourier transform of any function in this space is zero outside the interval \([-a, a]\),representing a band-limited function. Functions in this space can be completely characterized by their behavior within this frequency band.
The orthonormal basis is particularly important here, as any function in this Hilbert space can be represented uniquely as a linear combination of the basis functions. These basis functions in our exercise are the sinc functions, which are normalized appropriately to maintain orthonormality. Understanding how Hilbert spaces function allows us to perform various mathematical and physical analyses with a sound theoretical basis.

Orthonormal Basis

An orthonormal basis is a set of vectors in a Hilbert space that are both orthogonal and normalized. This concept is essential for the decomposition and reconstruction of functions. In the given exercise scenario, the set of functions \[\{\sqrt{2a} \operatorname{sinc}(2ax - k) : k \in \mathbb{Z}\}\]forms an orthonormal basis for the Hilbert space \(\mathcal{H}_a\).This basis allows any function in the space to be expressed as a sum of these sinc functions.
Orthonormal means each pair of different functions in the set is orthogonal with respect to the inner product in the space, and each function in the set is of unit length. This property ensures that the representation of any function as a series of basis functions is unique and stable. Such a basis simplifies numerous mathematical processes, such as finding approximations and decompositions, by providing a framework to work within the orthogonal dimensions laid out by these functions.