Fourier Series Analysis of Continuous-Time Periodic Signals
The chapter delves into Fourier Series Analysis, covering the decomposition of continuous-time periodic signals into harmonically related sinusoidal components. It introduces concepts of orthogonality, establishes the mathematical foundation for calculating Fourier coefficients, and discusses properties of these series. Additionally, key applications including filtering, circuit analysis, and the implications of the Gibbs phenomenon are also highlighted.
Sections
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What we have learnt
- Fourier Series allows the representation of periodic signals as sums of sinusoidal functions, providing insights into their frequency components.
- Orthogonality is fundamental for calculating unique Fourier coefficients, ensuring each term in the series contributes independently.
- The Gibbs phenomenon illustrates the overshoot behavior of Fourier series approximations at discontinuities, affecting applications in signal processing.
Key Concepts
- -- Orthogonality
- A property indicating that two functions are orthogonal if their inner product over a given interval is zero, facilitating unique representation in Fourier series.
- -- Fourier Series
- A mathematical representation of periodic functions as a sum of sine and cosine functions, providing a way to analyze frequency components.
- -- Gibbs Phenomenon
- The overshoot and ringing behavior observed in Fourier series approximations near discontinuities, where the approximated series does not converge to the actual function exactly at points of discontinuity.
- -- Parseval's Theorem
- A relationship that equates the average power of a signal in the time domain to the sum of the squares of its Fourier coefficients in the frequency domain.
- -- Linear TimeInvariant (LTI) Systems
- Systems through which the Fourier series analysis simplifies the understanding of system behavior when processing periodic inputs.
Additional Learning Materials
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