Question

Let \(f, g: \mathbf{R} \rightarrow \mathbf{C}\) be two \(2 \pi\)-periodic signals. If \(f, g\) are sufficiently smooth, then the convolution $$ (f * g)(t)=\int_{0}^{2 \pi} f(s) g(t-s) d s $$ exists for all \(t \in R\). Prove that \(f * g\) is again \(2 \pi\)-periodic, and that the convolution property \(\widehat{f * g}=\widehat{f} \cdot \hat{g}\) holds, so that $$ (\widehat{f * g})(k)=\hat{f}(k) \cdot \bar{g}(k) \text { for all } k \in Z \text {. } $$ (Thus the Fourier Transform converts convolution into pointwise multiplication.) You may assume that all occurring integrals exist.

Short answer

The convolution of two periodic signals is periodic, and its Fourier transform equals the product of the original Fourier transforms.

Step-by-step solution

Show that the Convolution is Periodic

To prove that \((f*g)(t)\) is \(2\pi\)-periodic, we need to show that it satisfies \((f*g)(t + 2\pi) = (f*g)(t)\). We have:\[(f*g)(t + 2\pi) = \int_{0}^{2\pi} f(s) g(t + 2\pi - s) \, ds\]Since both \(f\) and \(g\) are \(2\pi\)-periodic, it follows that \(g(t+2\pi-s) = g(t-s)\). Therefore:\[(f*g)(t + 2\pi) = \int_{0}^{2\pi} f(s) g(t-s) \, ds = (f*g)(t)\]This proves that \((f*g)(t)\) is \(2\pi\)-periodic.

Understand the Fourier Transform of Convolution

The problem requires us to show \( \widehat{f*g} = \hat{f} \cdot \hat{g} \). Given the Fourier coefficient representation:\[\hat{f}(k) = \frac{1}{2\pi} \int_{0}^{2\pi} f(s) e^{-iks} \, ds\]and similarly for \(\hat{g}(k)\), we calculate the Fourier transform of the convolution \((f*g)(t)\):\[\widehat{(f*g)}(k) = \frac{1}{2\pi} \int_{0}^{2\pi} \left( \int_{0}^{2\pi} f(s) g(t-s) \, ds \right) e^{-ikt} \, dt\]This requires us to interchange the order of integration and use the periodicity properties of \(f\) and \(g\).

Compute and Simplify the Fourier Transform of Convolution

Interchanging the order of integration and making the substitution \( u = t-s \), we get:\[\widehat{(f*g)}(k) = \frac{1}{2\pi} \int_{0}^{2\pi} f(s) \left( \int_{0}^{2\pi} g(u) e^{-ik(u+s)} \, du \right) \, ds\]Using periodicity and substituting back \(u = t - s\), the inner integral becomes:\[\int_{0}^{2\pi} g(u) e^{-iku} e^{-iks} \, du = e^{-iks} \hat{g}(k)\]Therefore:\[\widehat{(f*g)}(k) = \frac{1}{2\pi} \int_{0}^{2\pi} f(s) e^{-iks} \, ds \cdot \hat{g}(k) = \hat{f}(k) \cdot \hat{g}(k)\]

Conclusion

We have shown that the convolution \((f*g)(t)\) is \(2\pi\)-periodic and that the Fourier transform of the convolution is the product of the individual Fourier transforms of \(f\) and \(g\). Therefore, \((\widehat{f * g})(k) = \hat{f}(k) \cdot \hat{g}(k)\) for all integers \(k\), confirming that convolution in the time domain becomes pointwise multiplication in the frequency domain.

Key concepts

Fourier Transform

The Fourier Transform is a fundamental concept in signal processing that transforms a signal from its original domain, often time or space, into a frequency domain. It's essential for analyzing the frequencies present in a signal. In practical terms, it helps us understand how much of each frequency exists in a complex signal.

• The Fourier Transform can handle both continuous and discrete signals, depending on the context. For continuous signals, it's called the Continuous Fourier Transform (CFT); for discrete signals, it's the Discrete Fourier Transform (DFT).
• By converting signals into the frequency domain, we can easily analyze and understand their behavior under various conditions, including filtering and modulation.

Mathematically, the Fourier Transform of a function \(f\) is represented by \(\hat{f}(k) = \int_{-\infty}^{\infty} f(t)e^{-ikt} \, dt\). However, in periodic cases, we use the Fourier coefficients, which are a discrete set of values that represent the original signal in the frequency domain.

Understanding the Fourier Transform and its applications like convolution is key to mastering signal processing. Convolution in the time domain can be transformed into multiplication in the frequency domain, making complex operations simpler.

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals or periods. In signal processing, periodicity is a crucial property, as it allows us to understand and predict the behavior of signals over time.

• A function \(f(t)\) is said to be periodic if there exists a positive number \(T\), called the period, such that \(f(t + T) = f(t)\) for all \(t\) in its domain.
• In the given exercise, functions \(f\) and \(g\) are \(2\pi\)-periodic, which means they repeat every \(2\pi\) units. This periodicity plays a significant role in simplifying the analysis of these signals using techniques like Fourier series or convolution.
• The periodicity of a convolution, such as \( (f*g)(t) \), helps to maintain the periodic nature of composed signals, crucial in applications like signal filtering and system analysis.

Knowing how to handle periodic functions efficiently can greatly simplify many problems in signal processing, especially when working with time-domain convolutions and transformations.

Fourier Coefficients

Fourier coefficients are the building blocks for representing a periodic function in terms of sines and cosines. These coefficients offer a way to deconstruct a periodic signal into its component frequencies, providing deep insights into its structure.

• Each Fourier coefficient corresponds to a specific frequency and indicates the amplitude of that frequency in the signal. They are essential in Fourier series, which express periodic signals as the sum of sinusoidal functions.
• Given a function \(f(t)\), the nth Fourier coefficient, represented as \(\hat{f}(n)\), is calculated using: \[ \hat{f}(n) = \frac{1}{T} \int_{0}^{T} f(t) e^{-i n \frac{2\pi}{T} t} \, dt \]
• In practice, knowing the Fourier coefficients of a signal lets us reconstruct the original signal or transform it into different domains, such as frequency, which aids in analysis and filter design.

Understanding Fourier coefficients is crucial because they enable transformations like the Fourier Transform and help in deriving properties such as convolution theorems, which are vital for signal manipulation and understanding in both theoretical and practical applications.