Lecture 2 Signals and Systems (I)

1. Lecture 2Signals and Systems (I) Principles of Communications Fall 2008 NCTU EE Tzu-Hsien Sang 1

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

3. Signal Models and Classifications • The first step to knowledge: classify things. • What is a signal? • Usually we think of one-dimensional signals; can our scheme extend to higher dimensions? • How about representing something uncertain, say, a noise? • Random variables/processes – mathematical models for signals

4. Deterministic signals: completely specified functions of time. Predictable, no uncertainty, e.g. , with A and are fixed. • Random signals (stochastic signals):take on random values at any given time instant and characterized by pdf (probability density function).“Not completely predictable”, “with uncertainty”, e.g. x(n) = dice value at the n-th toss.

6. Periodic vs. Aperiodic signals • Phasors and why are we obsessed with sinusoids?

7. Singularity functions (they are not functions at all!!!) • Unit impulse function : • Defined by • It defines a precise sample point of x(t) at the incidence t0: • Basic function for linearly constructing a time signal • Properties: ;

8. What is precisely? some intuitive ways of imaging it: • Unit step funcgtion:

9. Energy Signals & Power Signals • Energy: • Power: • Energy signals: iff • Power signals: iff • Examples:

10. If x(t) is periodic, then it is meaningless to find its energy; we only need to check its power. • Noise is often persistent and is often a power signal. • Deterministic and aperiodic signals are often energy signals. • A realizable LTI system can be represented by a signal and mostly is a energy signal. • Power measure is useful for signal and noise analysis. • The energy and power classifications of signals are mutually exclusive (cannot be both at the same time). But a signal can be neither energy nor power signal.

11. Signal Spaces & Orthogonal Basis • The consequence of linearity: N-dimensional basis vectors: • Degree of freedom and independence: For example, in geometry, any 2-D vector can be decomposed into components along two orthogonal basis vectors, (or expanded by these two vectors) • Meaning of “linear” in linear algebra:

12. A general function can also be expanded by a set of basis functions (in an approximation sense) or more feasibly • Define the inner product as (“arbitrarily”) and the basis is orthogonal then

13. Examples: cosine waves What good are they? • Taylor’s expansion: orthogonal basis? • Using calculus can show that function approximation expansion by orthogonal basis functions is an optimal LSE approximation. • Is there a very good set of orthogonal basis functions? • Concept and relationship of spectrum, bandwidth and infinite continuous basis functions.

14. Fourier Series & Fourier Transform • Fourier Series: • Sinusoids (when?): • If x(t) is real, Notice the integral bounds.

15. Or, use both cosine and sine: with Yet another formulation:

16. Some Properties • Linearity If x(t) akand y(t) bk then Ax(t)+By(t) Aak + Bbk • Time Reversal If x(t) akthen x(-t) a-k • Time Shifting • Time Scaling x(at) ak But the fundamental frequency changes • Multiplicationx(t)y(t)  • Conjugation and Conjugate Symmetry x(t) akand x*(t) a*-k If x(t) is real a-k = ak*

17. Parseval’s Theorem Power in time domain = power in frequency domain

18. Some Examples

19. Extension to Aperiodic Signals • Aperiodic signals can be viewed as having periods that are “infinitely” long. • Rigorous treatments are way beyond our abilities. Let’s use our “intuition.” • If the period is infinitely long. What can we say about the “fundamental frequency.” • The number of basis functions would leap from countably infinite to uncountably infinite. • The synthesis is now an integration.. • Remember, both cases are purely mathematical construction.

20. “The wisdom is to tell the minute differences between similar-looking things and to find the common features of seemingly-unrelated ones…”

24. Conditions of Existence