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Chapter 4 Frequency Analysis of Signals and Systems Contents Motivation: complex exponentials are eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Frequency Analysis of Continuous-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Fourier series for continuous-time signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Fourier transform for continuous-time aperiodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Frequency Analysis of Discrete-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The Fourier series for discrete-time periodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Relationship of DTFS to z -transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Preview of 4.4.3 , analysis of LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 The Fourier transform of discrete-time aperiodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Relationship of the Fourier transform to the z -transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Gibb’s phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Energy density spectrum of aperiodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Properties of the DTFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 The sampling theorem revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 Proofs of Spectral Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.23 Frequency-domain characteristics of LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.30 A first attempt at filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.32 Relationship between system function and frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.32 Computing the frequency response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Steady-state and transient response for sinusoidal inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.35 LTI systems as frequency-selective filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.37 Digital sinusoidal oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.42 Inverse systems and deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.44 Minimum-phase, maximum-phase, and mixed-phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.47 Linear phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.47 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.48 4.1
4.2 c J. Fessler, May 27, 2004, 13:11 (student version) Motivation: complex exponentials are eigenfunctions Why frequency analysis? Complex exponential signals, which are described by a frequency value, are eigenfunctions or eigensignals of LTI systems. Period signals, which are important in signal processing, are sums of complex exponential signals. Eigenfunctions of LTI Systems Complex exponential signals play an important and unique role in the analysis of LTI systems both in continuous and discrete time. Complex exponential signals are the eigenfunctions of LTI systems. The eigenvalue corresponding to the complex exponential signal with frequency ω 0 is H ( ω 0 ) , where H ( ω ) is the Fourier transform of the impulse response h ( · ) . This statement is true in both CT and DT and in both 1D and 2D (and higher). The only difference is the notation for frequency and the definition of complex exponential signal and Fourier transform. Continuous-Time x ( t ) = e 2 πF 0 t LTI, h ( t ) y ( t ) = h ( t ) * e 2 πF 0 t = H a ( F 0 ) e 2 πF 0 t . Proof: ( skip ) y ( t ) = h ( t ) * x ( t ) = h ( t ) * e 2 πF 0 t = Z -∞ h ( τ ) e 2 πF 0 ( t - τ ) d τ = e 2 πF 0 t Z -∞ h ( τ ) e - 2 πF 0 τ d τ = H a ( F 0 ) e 2 πF 0 t , where H a ( F ) = Z -∞ h ( τ ) e - 2 πF τ d τ = Z -∞ h ( t ) e - 2 πF t d t . Discrete-Time x [ n ] = e ω 0 n LTI, h [ n ] y [ n ] = h [ n ] * e ω 0 n = H ( ω 0 ) e ω 0 n = |H ( ω 0 ) | e ( ω 0 n + H ( ω 0 )) Could you show this using the z -transform? No, because z -transform of e ω 0 n does not exist! Proof of eigenfunction property: y [ n ] = h [ n ] * x [ n ] = h [ n ] * e ω 0 n = X k = -∞ h [ k ] e ω 0 ( n - k ) = " X k = -∞ h [ k ] e - ω 0 k # e ω 0 n = H ( ω 0 ) e ω 0 n , where for any ω R : H ( ω ) = X k = -∞ h [ k ] e - ωk = X n = -∞ h [ n ] e - ωn .

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