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

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

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

Deterministic signals: completely specified functions of time 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.

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

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: ;

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

Energy Signals & Power Signals Energy signals: iff Power signals: iff Examples:

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.

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:

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

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.

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

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

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

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

Some Examples

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.

“The wisdom is to tell the minute differences between similar-looking things and to find the common features of seemingly-unrelated ones…” Fourier Series Fourier Transform Good orthogonal basis functions for a periodic function: Intuitively, basis functions should be also periodic. Intuitively, periods of the basis functions should be equal to the period or integer fractions of the target signal. Fourier found that sinusoidal functions are good and smooth functions to expand a periodic function.   Good orthogonal basis functions for a aperiodic function: Already know sinusoidal functions are good choice. Sinusoidal components should not be in a “fundamental & harmonic” relationship. Aperiodic signals are mostly finite duration. Consider the aperiodic function as a special case of periodic function with infinite period

Synthesis & analysis: (reconstruction & projection) Given periodic with period , and , it can be synthesized as : Spectra coefficient, spectra amplitude response   Before synthesizing it, we must first analyze it first and find out . By orthogonality Given aperiodic with period , and , we can synthesize it as Hence,

Frequency components: Decompose a periodic signal into countable frequency components. Has a fundamental freq. and many other harmonics. Discrete line spectra   Power Spectral Density: and (by Parseval’s equality) Decompose an aperiodic signal into uncountable frequency components No fundamental freq. and contain all possible freq. Continuous spectral density Energy Spectral Density: and

In real basis functions:   note that for real x(t). Exercises!

Conditions of Existence Expansion by orthogonal basis functions can be shown is equivalent to finding using the LSE (or MSE) cost function: Would as ? This requires square integrable condition (for the power signal): and not necessarily Dirichlet’s conditions: finite no. of finite discontinuities; finite no. of finite max & min.; absolute integrable: Dirichlet’s condition implies convergence almost everywhere, except at some discontinuities. This requires square integrable condition (for the energy signal):