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Unformatted text preview: 1/21 EECE 301 Signals & Systems Prof. Mark Fowler Note Set #13 • CT Signals: Fourier Series (for Periodic Signals) • Reading Assignment: Section 3.2 & 3.3 of Kamen and Heck 2/21 Ch. 1 Intro CT Signal Model Functions on Real Line DT Signal Model Functions on Integers System Properties LTI Causal Etc Ch. 2 Diff Eqs CT System Model Differential Equations DT Signal Model Difference Equations ZeroState Response ZeroInput Response Characteristic Eq. Ch. 2 Convolution CT System Model Convolution Integral DT System Model Convolution Sum Ch. 3: CT Fourier Signal Models Fourier Series Periodic Signals Fourier Transform (CTFT) NonPeriodic Signals New System Model New Signal Models Ch. 5: CT Fourier System Models Frequency Response Based on Fourier Transform New System Model Ch. 4: DT Fourier Signal Models DTFT (for “Hand” Analysis) DFT & FFT (for Computer Analysis) New Signal Model Powerful Analysis Tool Ch. 6 & 8: Laplace Models for CT Signals & Systems Transfer Function New System Model Ch. 7: Z Trans. Models for DT Signals & Systems Transfer Function New System Model Ch. 5: DT Fourier System Models Freq. Response for DT Based on DTFT New System Model Course Flow Diagram The arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (CT Freq. Analysis) and the blue blocks (DT Freq. Analysis). 3/21 3.2 & 3.3 Fourier Series In the last set of notes we looked at building signals using: ∑ − = = N N k t jk k e c t x ) ( ω N = finite integer We saw that these build periodic signals. Q: Can we get any periodic signal this way? A: No! There are some periodic signals that need an infinite number of terms: ∑ = + + = N k k k t k A A t x 1 ) cos( ) ( θ ω ∑ ∞ −∞ = = k t jk k e c t x ) ( ω Fourier Series (Complex Exp. Form) k are integers ∑ ∞ = + + = 1 ) cos( ) ( k k k t k A A t x θ ω k are integers Fourier Series (Trig. Form) These are two different forms of the same tool!! There is a 3 rd form that we’ll see later. Sect. 3.3 There is a related Form in Sect. 3.2 4/21 Q: Does this now let us get any periodic signal? A: No! Although Fourier thought so! Dirichlet showed that there are some that can’t be written in terms of a FS! But… those will never show up in practice! See top of p. 155 So we can write any practical periodic signal as a FS with infinite # of terms! So what??!! Here is what!! We can now break virtually any periodic signal into a sum of simple things… and we already understand how these simple things travel through an LTI system!...
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 Spring '08
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