Question

For a function \(f(t)\) that is non-zero only in the range \(|t|

Short answer

The Fourier series coefficients relate directly to the Fourier transform as \[c_{n} = \frac{\sqrt{2 \pi}}{T} \hat{f}\left(\frac{2 \pi n}{T}\right)\]. Using this, the full frequency spectrum is shown to be \[\hat{f}(\omega)=\sum_{n=-\infty}^{\infty} \hat{f}\left(\frac{2 \pi n}{T}\right) \operatorname{sinc}\left(n \pi-\frac{\omega T}{2}\right)\].

Step-by-step solution

Define Fourier Series Coefficients

In a Fourier series representation, a function with period T can be written as a sum of harmonics. The coefficients of the complex Fourier series for a function f(t) with period T are given by \[c_{n} = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-i \frac{2\pi n}{T} t} \, dt\].

Express Using Fourier Transform

The Fourier transform of f(t) is defined as \[\hat{f}(\omega)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt\]. Because f(t) is non-zero only for \ |t| < T/2, the integral limits can be replaced by T/2: \[\hat{f}(\omega)= \frac{1}{\sqrt{2 \pi}} \int_{-T/2}^{T/2} f(t) e^{-i \omega t} \, dt\].

Relate Fourier Series Coefficients to Fourier Transform

Compare the formulas for the Fourier series coefficients and the Fourier transform. Notice that if \omega = \frac{2 \pi n}{T}, the transform integrates over exactly one period of the Fourier series, so \[c_{n} = \frac{\sqrt{2 \pi}}{T} \hat{f}\left(\frac{2 \pi n}{T}\right)\]. This shows the required expression.

Construct Infinite Sum

From the inverse Fourier transform, f(t) can be represented as \[f(t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i \omega t} \, d\omega\]. Since the Fourier transform \hat{f}(\omega) can be reconstructed from discrete points, substitute the series expression for \hat{f}(\omega): \[\hat{f}(\omega) = \sum_{n=-\infty}^{\infty} \hat{f}\left(\frac{2 \pi}{T}n\right) \delta \left(\omega - \frac{2 \pi n}{T}\right)\].

Utilize Sinc Function

The sinc function is defined as sinc(x) = \frac{\sin x}{x}. When represented in terms of the delta function and Fourier transform: \[\hat{f}(\omega) = \sum_{n=-\infty}^{\infty} \hat{f}\left(\frac{2 \pi n}{T}\right) \operatorname{sinc}\left(n \pi-\frac{\omega T}{2}\right)\]. This expression shows the full frequency spectrum can be constructed from the discrete points \hat{f}\left(\frac{2 \pi n}{T}\right), thus completing the proof.

Key concepts

Fourier Series

The Fourier Series is a way to represent a periodic function as a sum of simple sine and cosine waves. The idea is that any function can be broken down into a collection of oscillating functions. This approach is highly beneficial for analyzing periodic functions in various engineering and physics problems. The complex Fourier series for a function \(f(t)\) with period \(T\) is written as:\[f(t) = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-i \frac{2\pi n}{T} t} \, dt\].Each term in the series corresponds to a particular frequency, and the coefficients \(c_n\) determine the amplitude and phase of these frequencies. These coefficients are crucial in understanding how the function behaves over time.
By decomposing a signal into its Fourier series, we can analyze its frequency components, which is particularly useful in signal processing.

Frequency Spectrum

The Frequency Spectrum represents the range of frequencies present in a signal. It shows how much of the signal lies within each given frequency band over a range of frequencies. For example, in a musical signal, different notes correspond to different frequencies.
The frequency spectrum is typically obtained through the Fourier Transform, which converts a signal from its time domain representation to its frequency domain representation. This helps in identifying the dominant frequencies and analyzing the signal's behavior.
In the given problem, we express \(f(t)\) as a sum to demonstrate how the full frequency spectrum \(\hat{f}(\omega)\) can be constructed from discrete sample points. This highlights how we can understand and manipulate signals based on their frequency components rather than just their time-based representations.

Sinc Function

The Sinc Function, written as \(\operatorname{sinc}(x) = \frac{\sin x}{x}\), is an essential function in signal processing. It commonly appears in the analysis and reconstruction of signals. One of the key properties of the sinc function is its role in interpolation and signal reconstruction.
In this exercise, we use the sinc function to show that \(\hat{f}(\omega)\) can be obtained as a sum of sampled values of the Fourier transform times the sinc function. This is mathematically represented as:\[\hat{f}(\omega) = \sum_{n=-\infty}^{\infty} \hat{f}\left(\frac{2 \pi n}{T}\right) \operatorname{sinc}\left(n \pi-\frac{\omega T}{2}\right)\].
This expression indicates how the sinc function aids in the continuous representation of the frequency spectrum based on discrete points.

Complex Fourier Coefficients

Complex Fourier Coefficients \(c_n\) are the terms in the Fourier series that describe the amplitude and phase of each frequency component in the function. They are obtained by integrating the function multiplied by a complex exponential function over a period:\[c_{n} = \frac{1}{T} \int_{-T/2}^{T/2} f(t) e^{-i \frac{2\pi n}{T} t} \, dt\].
These coefficients are crucial because they capture the essence of the function's frequency content. In effect, the Fourier series converts the original function into a sum of sine and cosine terms, with each term weighted by its corresponding Fourier coefficient.
From the derived relationship in the problem:\[c_{n} = \frac{\sqrt{2 \pi}}{T} \hat{f}\left(\frac{2 \pi n}{T}\right)\].This links the Fourier coefficients directly to the Fourier transform values at specific frequencies. As a result, it provides a bridge between time-domain and frequency-domain representations of the function.