3 Fourier Series Representation of Periodic Signals

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Proofs: 1. Continuous–time LTI system with input x(t) = ejωt ∞ ∞ y(t) = h(τ)x(t − τ) dτ = h(τ)e jω(t−τ) dτ −∞ = e jωt ∞ −∞ = H(jω)e jωt h(τ)e −jωτ dτ −∞ with the definition of the amplitude factor ∞ H(jω) = h(τ)e −jωτ dτ −∞ 2. Discrete–time LTI system with input x[n] = ejωn y[n] = ∞ h[k]x[n − k] = ∞ k=−∞ = e jωn ∞ k=−∞ = H(e jω )e jωn h[k]e −jωk with the definition of the amplitude factor H(e jω ∞ ) = Lampe, Schober: Signals and Communications k=−∞ k=−∞ h[k]e −jωk h[k]e jω(n−k) 68
Usefulness – Continuous–time LTI system Represent input signal as x(t) = to obtain output signal as y(t) = – Discrete–time LTI system Represent input signal as k k x[n] = to obtain output signal as y[n] = k ake jω kt akH(jωk)e jω kt k ake jω kn akH(e jω k )e jω kn Convolution with impulse response h(t) and h[n] is replaced by multiplication with eigenvalues H(jωk) and H(e jω k), respectively Lampe, Schober: Signals and Communications 69
Page 1 and 2: 3 Fourier Series Representation of
Page 3: 3.1 The Response of LTI Systems to
Page 7 and 8: Short-hand notation Denominations:
Page 9 and 10: 2. Periodic time function −2t/T
Page 11 and 12: 3.2.1 Convergence of the Fourier Se
Page 13 and 14: Gibbs phenomenon Discontinuity of u
Page 15 and 16: Time Reversal - Let then - Observe:
Page 17 and 18: Conjugation and Conjugate Symmetry
Page 19 and 20: Short-hand notation Denomination:
Page 21 and 22: Properties of Discrete-Time Fourier
Page 23 and 24: Process of x(t) −→ y(t) ak −
Page 25 and 26: −π −π 3. Ideal Lowpass Filter
Page 27 and 28: - Discrete-time case 1 H(e jω )
Page 29 and 30: - Observe ∗ |H(jω)| ≈ 1 for |
Page 31: Lampe, Schober: Signals and Communi

3 Fourier Series Representation of Periodic Signals

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