SIGNALS AND SIGNAL SPACE C H A P T E R 2 SIGNALS AND SIGNAL SPACE

Description of a Signal Amplitude Radian Frequency Phase Angle Period

Size of a Signal Examples of signals: (a) signal with finite energy; Signal Energy Signal Power: Examples of signals: (a) signal with finite energy; (b) signal with finite power. ELCT332 Fall 2011

Example ELCT332 Fall 2011

Classification of Signals Continuous time and discrete time signals Analog and digital signals Periodic and aperiodic signals Energy and power signals Deterministic and radaom signals Physical description is known completely in either mathematical or graphical form Probabilistic description such as mean value, mean squared value and distributions a) Continuous time and (b) discrete time signals. ELCT332 Fall 2011

Analog and continuous time Digital and continuous time Analog and discrete time Digital and discrete time ELCT332 Fall 2011

Energy signal: a signal with finite energy g(t)=g(t+T0) for all t Periodic signal of period T0. Energy signal: a signal with finite energy Power signal: a signal with finite power ELCT332 Fall 2011

Signal Operations Time shifting a signal. ELCT332 Fall 2011

Time scaling a signal. ELCT332 Fall 2011

? ? Examples of time compression and time expansion of signals. ELCT332 Fall 2011

Time inversion (reflection) of a signal. g(-t)? Time inversion (reflection) of a signal. ELCT332 Fall 2011

Example of time inversion. g(-t)? ELCT332 Fall 2011

Multiplication of a Function by an Impulse (a) Unit impulse and (b) its approximation. Multiplication of a Function by an Impulse ELCT332 Fall 2011

a) Unit step function u(t). (b) Causal exponential function e−atu(t). Causal signal: a signal starts after t=0 Question: how to convert any signal to a causal signal? ELCT332 Fall 2011

Signals Versus Vectors Magnitude and Direction <x,x>=? Component (projection) of a vector along another vector. ELCT332 Fall 2011

g=cx+e=c1x+e1=c2x+e2 Component of a Vector along Another Vector Approximations of a vector in terms of another vector. g=cx+e=c1x+e1=c2x+e2 ELCT332 Fall 2011

Decomposition of a Signal and Signal Components Approximation of square signal in terms of a single sinusoid. Find the component in g(t) of the form sin(t) to make the energy of the error signal is minimum Hint: ELCT332 Fall 2011

Correlation of Signals Correlation coefficient b:1, c:1, d:-1,e: 0.961,f:0.628, f:0 Signals for Example 2.6. ELCT332 Fall 2011

Application to Signal Detection Physical explanation of the auto-correlation function. ELCT332 Fall 2011

Representation of a vector in three-dimensional space. Parseval’s Theorem The energy of the sum of orthogonal signal is equal to the sum of their energies. Orthogonal Signal Space Generalized Fourier Series ELCT332 Fall 2011

Compact Trigonometric Fourier Series ELCT332 Fall 2011

Amplitude spectrum Phase spectrum (a, b) Periodic signal and (c, d) its Fourier spectra. ELCT332 Fall 2011

Figure 2.20 (a) Square pulse periodic signal and (b) its Fourier spectrum. ELCT332 Fall 2011

Bipolar square pulse periodic signal. ELCT332 Fall 2011

(a) Impulse train and (b) its Fourier spectrum. ELCT332 Fall 2011

Exponential Fourier spectra for the signal Exponential Fourier Series Exponential Fourier spectra for the signal ELCT332 Fall 2011