11 Lecture 2 Signals and Systems (II) Principles of Communications Fall 2008 NCTU EE Tzu-Hsien Sang

Outlines Signal Models & Classifications Signal Space & Orthogonal Basis Fourier Series &Transform Power Spectral Density & Correlation Signals & Linear Systems Sampling Theory DFT & FFT 2

Examples Symmetry Properties of x(t) and Its Fourier Function For real periodic x(t), For real aperiodic x(t), 3

Fourier Transform of Singular Functions is not an energy signal (hence doesn’t satisfy Dirichlet condition). However, its FT can be obtained by formal definition. Example: The FT of ? 4

Fourier Transform of Periodic Signals— Periodic signals are not energy signals (don’t satisfy Dirichlet’s conditions). But we are doing it anyway (at least formally)… Given a periodic signal  Example-1: Example-2: 5 (A pulse train! What good are they for?)

6 Note: This table uses “  ” instead of “f”. But it doesn’t hurt the fundamental facts.

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Transform Pairs (There is something nice to know in life…) 8

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Let FT of an aperiodic pulse signal p(t) be We can generate a periodic signal x(t) by duplicating p(t) at every interval T s, then From convolution theorem, 11

Taking inverse FT of the eq. on previous page. 12 Poisson sum formula

Power Spectral Density & Correlation Why should we care about the “frequency components” of a signal? For energy signals: The time-averaged autocorrelation function The squared magnitude of the FT represents the “energy” distributed on the frequency axis. 13

For power signals: For periodic power signals: 14 “Power spectral density function”

The functions  (  ) and R(  ) measure the similarity between the signal at time t and t+ . G(f) and S(f) represents the signal energy or power per unit frequency at freq. f., R(  ) is even for real x(t): If x(t) does not contain a periodic component: If x(t) is periodic with period T 0, then R(  ) is periodic in  with the same period. S(f) is non-negative. 15

Cross-correlation of two power signals: Cross-correlation of two energy signals: Remarks: 16

Signals & Linear Systems The standard input/output black box model for linear systems. Q: Why does it work? Linear: Satisfies superposition principle Time-invariant: Delayed input produces an output with the same delay. 17

Describing LTI Systems with Impulse Responses Let h(t) be the impulse response: 18 If time-invariant,

19 Note: This example is a linear, but not time-invariant system.

The convolution form holds iff LTI. Duality of signal x(t) & system h(t): The Convolution Theorem: Key application: generally is easier than … 20