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Module 7 properties of fourier series and complex fourier spectrum

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Module -7 Properties of Fourier Series and Complex Fourier Spectrum. Objective:To understand the change in Fourier series coefficients due to different signal operations and to plot complex Fourier spectrum. Introduction: The Continuous Time Fourier Series is a good analysis tool for systems with periodicexcitation. Understanding properties of Fourier series makes the work simple in calculating the Fourier series coefficients in the case when signals modified by some basic operations. Graphical representation of a periodic signal in frequency domain represents Complex Fourier Spectrum. Description: Properties of continuous- time Fourier series The Fourier series representation possesses a number of important properties that are useful for various purposes during the transformation of signals from one form to other . Some of the properties are listed below. [x1(t) and x2(t)] are two periodic signals with period T and with Fourier series coefficients Cn and Dn respectively. 1) Linearity property The linearity property states that, if 𝐹𝑠 x1(t) Cn and x2 (𝑑) then proof: 𝐹𝑠 𝐹𝑠 Dn Ax1(t)+Bx2(t) ACn+BDn From the definition of Fourier series, we have 1 𝑑 +𝑇 FS[Ax1(t)+Bx2(t)]=𝑇 𝑑 0 [𝐴π‘₯1 𝑑 + 𝐡π‘₯2 𝑑 ]𝑒 βˆ’π‘—π‘› πœ” 0 𝑑 𝑑𝑑 0 1 𝑑 +𝑇 1 =𝐴(𝑇 𝑑 0 π‘₯1 𝑑 𝑒 βˆ’π‘—π‘› πœ” 0 𝑑 𝑑𝑑)+ 𝐡(𝑇 0 =ACn+BDn 𝐹𝑠 or Ax1(t)+Bx2(t) ACn+BDn proved 2) Time shifting property The time shiftin ...
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