Question

Let \(f: \mathbb{R} \rightarrow \mathbb{C}\) be a continuous, absolutely integrable function. Let \(F\) denote the Fourier transform of \(f\). Show that if \(F(\omega)=0\) for all \(\omega>\left|\omega_{0}\right|\) then for all \(a>\left|\omega_{0}\right|\) $$ f(x) * \frac{\sin a x}{\pi x}=f(x) . $$

Short answer

The convolution with the sinc function returns the original function: \( f(x) * \frac{\sin ax}{\pi x} = f(x) \).

Step-by-step solution

Understanding the Given Conditions

Given a function \( f: \mathbb{R} \rightarrow \mathbb{C} \) that is continuous and absolutely integrable, this means that \( \int_{-\infty}^{\infty} |f(x)| \, dx < \infty \). The Fourier transform \( F \) of \( f \) is defined, and we are told \( F(\omega) = 0 \) for all \( \omega > |\omega_0| \). This implies that \( F(\omega) \) is only non-zero within the interval \(-|\omega_0| \leq \omega \leq |\omega_0|\).

Formulating Convolution with Sinc Function

We are considering the convolution \( f(x) * \frac{\sin(ax)}{\pi x} \). The sinc function \( \frac{\sin(ax)}{\pi x} \) acts as a low-pass filter in the frequency domain, allowing frequencies up to \( a \). Given the condition that \( a > |\omega_0| \), it implies that this filter will pass all the frequencies of \( f \, \) since \( F(\omega) \) contains no frequencies beyond \( |\omega_0| \).

Applying Convolution Theorem

The Convolution Theorem states that the Fourier transform of a convolution is the product of the Fourier transforms of the individual functions. Let \( F_f(\omega) \) be the Fourier transform of \( f(x) \) and let \( S(\omega) \) be the Fourier transform of the \( \frac{\sin(ax)}{\pi x} \). The theorem gives us: \[ \text{Fourier Transform of } \left( f(x) * \frac{\sin(ax)}{\pi x} \right) = F_f(\omega) S(\omega) \].

Evaluating the Fourier Transforms

The Fourier Transform of \( \frac{\sin(ax)}{\pi x} \) is a gate function (or rectangular function) that equals 1 for \( -a \leq \omega \leq a \) and 0 otherwise. Because \( a > |\omega_0| \), the rectangular function \( S(\omega) \) is equal to 1 wherever \( F_f(\omega) \) is non-zero. Thus, \( F_f(\omega) S(\omega) = F_f(\omega) \).

Combining Results for the Final Solution

With the Fourier transform \( F_f(\omega) S(\omega) = F_f(\omega) \), applying the inverse Fourier transform gives us back \( f(x) \), showing: \[ f(x) * \frac{\sin(ax)}{\pi x} = f(x) \] for all \( a > |\omega_0| \). This confirms the statement in the exercise.

Key concepts

Convolution Theorem

The Convolution Theorem is an essential concept in signal processing and analysis. It states that the Fourier transform of a convolution of two functions is equal to the product of their individual Fourier transforms. This relationship is crucial when dealing with frequency domain analysis because convolution in the time domain corresponds to multiplication in the frequency domain.
Think of the Convolution Theorem as a bridge linking the time and frequency domains. This makes it easier to analyze the operation of filters and signals.

• In mathematical terms, for two functions, \(f(x)\) and \(g(x)\), the convolution \((f*g)(x)\) has a Fourier transform \(F_f(\omega)G(\omega)\), where \(F_f\) and \(G\) are the Fourier transforms of \(f\) and \(g\), respectively.
• The theorem simplifies the analysis by turning a complex integral operation (convolution) into a simpler algebraic product.

Understanding this theorem is vital for solving problems involving frequency response, such as filtering, modulation, and more.

Sinc Function

The sinc function, often encountered in mathematics and signal processing, is defined as \(\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}\). In signal processing applications, a modified version \(\frac{\sin(ax)}{\pi x}\) is also commonly used.
This function has defining properties that make it useful in various contexts, including:

• It acts as a "perfect" low-pass filter in the context of Fourier analysis, which means it allows frequencies below a certain threshold to pass through entirely and attenuates frequencies above this limit.
• The sinc function has a unique mathematical property known as the "reproducing property". When convolved with a signal, it can perfectly reconstruct the signal if the conditions are right.

This characteristic is particularly relevant in digital signal processing, such as in the reconstruction of a continuous-time signal from its samples.

Low-Pass Filter

A low-pass filter is a signal processing tool that allows signals with a frequency lower than a certain cutoff frequency to pass through and attenuates signals with frequencies higher than the cutoff. These filters are crucial in various applications, where they help remove unwanted high-frequency components such as noise.
In the context of the problem and the sinc function discussed, we see:

• The sinc function in the frequency domain acts precisely as a low-pass filter, allowing us to select desired frequency components while filtering out others.
• Applying such a filter helps isolate the essential frequency components needed for signal integrity, aligning with the Fourier transform's role.

In practical applications, low-pass filters are used in music equalization, image processing, and electrical circuits, providing smoother results by removing high-frequency elements.

Inverse Fourier Transform

The Inverse Fourier Transform is the process of transforming a function from the frequency domain back to the time domain. This is essential when moving between different domains to suit the analysis since it allows one to interpret how frequency components combine to shape the original signal.
It's incredibly useful because:

• It allows the reconstruction of the original continuous signal after manipulation has been performed in the frequency domain.
• In mathematical notation, if \(F(\omega)\) is the Fourier transform, the inverse Fourier transform is given by \(f(x) = \int_{-\infty}^{\infty} F(\omega) e^{i \omega x} \, d\omega\).

Applied correctly, it ensures no information is lost during transformation, such as in convolution operations highlighted previously in the exercise.