3 Fourier Series Representation of Periodic Signals
3 Fourier Series Representation of Periodic Signals
3 Fourier Series Representation of Periodic Signals
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3.1 The Response <strong>of</strong> LTI Systems to Complex Exponentials<br />
<strong>Fourier</strong> series = representation <strong>of</strong> periodic signals as weighted superposition<br />
<strong>of</strong> complex exponentials<br />
– continuous-time case: e jωt<br />
– discrete-time case: e jωn<br />
Why useful ?<br />
Because e jωt and e jωn are eigenfunctions <strong>of</strong> LTI systems<br />
Remark:<br />
– Continuous–time LTI system<br />
– Discrete–time LTI system<br />
e jωt −→ H(jω) e jωt<br />
e jωn −→ H(e jω ) e jωn<br />
Complex constants H(jω) and H(e jω ) are the eigenvalues associated<br />
with e jωt and e jωn , respectively.<br />
The statements above can be generalized to complex exponentials<br />
e st and z n , respectively, with s, z ∈ C. These generalizations are<br />
discussed in Chapters 9 (The Laplace Transform) and 10 (The z-<br />
Transform) <strong>of</strong> our text book and are treated in the complementary<br />
course EECE 360.<br />
Lampe, Schober: <strong>Signals</strong> and Communications<br />
67
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