Question

A Berechnen Sie die Fourierreihe für eine Rechteckfunktion, die hier als zeitlich aufgefabt werden soll. Die Funktion stellt dann einen Rechteckimpuls der Dauer \(t_{0}\) dar, der sich mit der Periode \(T\) wiederholt. $$ f(t)=\left\\{\begin{array}{lll} 0 \quad f \text { ür } \quad-\frac{T}{2} \leq t \leq-\frac{t_{0}}{2} \\ 1 \quad f \text { ür } \quad-\frac{t_{0}}{2} \leq t \leq+\frac{t_{0}}{2} \\ 0 \quad f \text { ür } \quad+\frac{t_{0}}{2} \leq t \leq+\frac{T}{2} \end{array}\right. $$ B Berechnen Sie die Fourierreihe für eine Variante der Aufgabe \(22.2 \mathrm{~B}\) $$ f(t)=\left\\{\begin{array}{rlc} -1 & f \text { ür } & \frac{t_{0}}{2} \leq t \leq 0 \\ 1 \quad f \text { ür } & 0 \leq t \leq \frac{t_{0}}{2} \end{array}\right. $$

Short answer

Fourier series representation gives a complex combination of sinusoidal waves that represent the original function. For the rectangular function, the Fourier series has cosine components, and for the variant of problem 22.2, the Fourier series has sine components.

Step-by-step solution

Calculate Fourier Series for Rectangular Function

The Fourier series for a function \(f(t)\) with period \(T\) is given by \[f(t) = a_0 + \sum_{n=1}^{\infty}[a_n \cos(n \omega t) + b_n \sin(n \omega t)]\] where \(\omega = \frac{2\pi}{T}\), and the Fourier coefficients \(a_0, a_n, b_n\) can be calculated with: \[a_0 = \frac{1}{T}\int_{-T/2}^{T/2} f(t)dt\]\[a_n = \frac{2}{T}\int_{-T/2}^{T/2} f(t)\cos(n\omega t)dt\]\[b_n = \frac{2}{T}\int_{-T/2}^{T/2} f(t)\sin(n\omega t)dt\]In this case, the function is defined in three intervals. In each interval, the function is constant, therefore the integral can be computed directly. After calculating the integrals, substitute the values into the equation for the Fourier series. Considering that the function is even, \(b_n=0\), the Fourier series is given by \[f(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n \omega t)\]

Calculate Fourier Series for the Variant of Problem 22.2

This problem is a variant of the previous one but the function is defined differently in two intervals. The same formulas are used but the integral process changes because the function is different in the two intervals. Substitute the function and the limits of integration into the formulas for the Fourier coefficients then find the Fourier series distribution. Considering that the function is odd, \(a_n=0\), the Fourier series is given by \[f(t) = b_0 + \sum_{n=1}^{\infty} b_n \sin(n \omega t)\]

Conclusion

After following the steps outlined, you would be able to obtain the Fourier series for both functions. For rectangular signals, the Fourier series is a series of cosines, and for the variant of problem 22.2, the Fourier series is a series of sines.

Key concepts

Fourier Series

Imagine you're listening to your favorite song, and you hear a blend of different musical notes. Similarly, in mathematics, complex periodic functions can be thought of as compositions of simpler, sinusoidal waves. These compositions are known as Fourier series. A Fourier series is a way to represent a periodic function as an infinite sum of sines and cosines. The coefficients of these sines and cosines are determined by the original function's shape through a process called Fourier analysis.

The beauty of Fourier series lies in their ability to model any periodic signal, no matter how jagged or smooth it might be, using just sines and cosines. This property makes Fourier series an indispensable tool in fields such as signal processing, acoustics, and heat transfer. To understand and calculate a Fourier series, one needs to be familiar with integrals, especiallyintegrals of trigonometric functions, which provide the necessary coefficients.

Rechteckfunktion

In our daily lives, we come across many on-off type of signals, such as traffic lights or a ticking clock. These signals can be mathematically represented using a Rechteckfunktion, or in English, a rectangular function. This function is a type of periodic function which repeats itself after a certain time interval. It alternates between two values (often 0 and 1) and switches at specified times creating a rectangular shape.

The rectangular function is a beloved example in Fourier analysis for its simplicity and clear-cut intervals. It serves as a great introductory example for learning how to compute Fourier series, particularly because the function's value is constant over each interval. This makes integration much more straightforward compared to more complex functions. In essence, the rectangular function plays a foundational role in our understanding of digital signals and their representation.

Trigonometrische Integrale

To comprehend Fourier series, one must be well-acquainted with trigonometric integrals. These are integrals that involve trigonometric functions such as sine and cosine. The integral calculations determine the coefficients for the Fourier series and are critical in translating the original function into its sinusoidal components.

• For coefficients a_n, we integrate the product of the original function and cos(nωt).
• For coefficients b_n, we integrate the product of the original function and sin(nωt).

Understanding these integrals is vital for engineering and physics problems where signal behavior over time is analyzed. Not only do they serve in theoretical analysis, but they are also used in practical applications like image compression where Fourier analysis helps in reducing file sizes without significant loss of quality.

Periodische Funktionen

Think of the rhythmic rise and fall of ocean waves or the cyclical nature of the seasons. These are natural examples of periodic functions, patterns that repeat at regular intervals. Mathematically, a periodic function is one that satisfies the condition f(t) = f(t+T) for all t, where T is the period - the length of one complete cycle.

Periodic functions are fundamental in Fourier series calculations where each component wave has a period that is a multiple or fraction of the original function's period. This harmony between the periodicities ensures that the Fourier series faithfully reconstructs the original function. Knowing the period of a function not only helps us better understand real-world phenomena but is also crucial when working with signals in technology, as it influences the design of systems such as radio transmitters or digital clocks.