Fourier Series Quiz: Representing Periodic Signals Quiz

Challenge your understanding of Fourier series, coefficients, and the mathematical representation of periodic signals in this focused quiz. Explore fundamental concepts, key formulas, and common applications in signal analysis using Fourier series.

1. Basic Concept of Fourier Series

   Which statement best describes the main idea behind representing a periodic signal using a Fourier series?

   1. Periodic signals are represented only by even harmonics in Fourier series.
   2. Fourier series requires the signal to be non-periodic.
   3. Fourier series expresses periodic signals with a single exponential function.
   4. A periodic signal can be expressed as a sum of sinusoids with different frequencies, amplitudes, and phases.

   Explanation: The core concept of a Fourier series is that any periodic signal can be decomposed into an infinite sum of sinusoids (sines and cosines), each with their own frequency, amplitude, and phase. The statement about a single exponential function is incorrect, as Fourier series involve multiple terms. Limiting representation to only even harmonics is false; all harmonics may be present depending on the signal. Additionally, Fourier series specifically applies to periodic, not non-periodic, signals.

2. Fourier Series Coefficients

   In the context of Fourier series, what do the coefficients (such as a_n and b_n in the trigonometric form) represent for a given periodic function?

   1. They determine the frequencies of each component.
   2. They indicate the period of the signal.
   3. They specify the amplitudes of the sine and cosine terms.
   4. They identify the signal's sampling rate.

   Explanation: In the trigonometric form of the Fourier series, coefficients like a_n and b_n determine the amplitudes of the respective cosine and sine components. These coefficients do not set the frequencies, as frequencies are determined by the harmonic order and fundamental frequency. The coefficients are unrelated to the signal’s sampling rate or its period, which are defined separately.

3. Application Example

   If a square wave with period T is represented by its Fourier series, which type of harmonics will be absent if the wave is perfectly symmetric about the horizontal midline?

   1. All even harmonics
   2. All fundamental frequencies
   3. All odd harmonics
   4. All harmonics

   Explanation: A square wave that is symmetric around the midline (an odd function) contains only odd harmonics in its Fourier series decomposition, meaning all even harmonics are absent. Saying 'all odd harmonics' or 'all harmonics' are absent is incorrect because odd harmonics actually persist in the series. The fundamental frequency is always present as it corresponds to the first harmonic.

4. Mathematical Representation

   What mathematical operation is typically used to calculate the value of a specific Fourier coefficient for a continuous periodic function?

   1. Division by the signal’s frequency
   2. Differentiation of the function
   3. Integration of the function over one period
   4. Multiplication by the signal's amplitude

   Explanation: Calculating a Fourier coefficient involves integrating the function, multiplied by sine or cosine terms, over one complete period. Multiplying by the amplitude or dividing by the frequency does not yield the coefficient. Differentiation is not used for finding Fourier coefficients; it might instead be used to analyze frequency responses or signal changes.

5. Complex Exponential Form

   How does the complex exponential form of the Fourier series differ from the standard trigonometric form?

   1. It uses only cosine functions instead of both sine and cosine.
   2. It does not allow for negative frequency components.
   3. It expresses the periodic function as a sum of complex exponentials with complex coefficients.
   4. It represents the series as a sum of real-valued polynomials.

   Explanation: The complex exponential form rewrites the Fourier series in terms of exponentials with complex coefficients, making analysis of frequency content and phase shifts more convenient. It is not represented as real-valued polynomials nor limited to cosine functions. In fact, the complex exponential form inherently includes both positive and negative frequency components, making the last option incorrect.