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SIGNALS AND SIGNAL SPACE
C H A P T E R 2 SIGNALS AND SIGNAL SPACE
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Description of a Signal
Amplitude Radian Frequency Phase Angle Period
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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. ELCT Fall 2011
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Example ELCT Fall 2011
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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. ELCT Fall 2011
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Analog and continuous time Digital and continuous time
Analog and discrete time Digital and discrete time ELCT Fall 2011
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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 ELCT Fall 2011
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Signal Operations Time shifting a signal. ELCT Fall 2011
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Time scaling a signal. ELCT Fall 2011
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? ? Examples of time compression and time expansion of signals.
ELCT Fall 2011
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Time inversion (reflection) of a signal.
g(-t)? Time inversion (reflection) of a signal. ELCT Fall 2011
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Example of time inversion. g(-t)?
ELCT Fall 2011
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Multiplication of a Function by an Impulse
(a) Unit impulse and (b) its approximation. Multiplication of a Function by an Impulse ELCT Fall 2011
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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? ELCT Fall 2011
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Signals Versus Vectors
Magnitude and Direction <x,x>=? Component (projection) of a vector along another vector. ELCT Fall 2011
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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 ELCT Fall 2011
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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: ELCT Fall 2011
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Correlation of Signals
Correlation coefficient b:1, c:1, d:-1,e: 0.961,f:0.628, f:0 Signals for Example 2.6. ELCT Fall 2011
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Application to Signal Detection
Physical explanation of the auto-correlation function. ELCT Fall 2011
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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 ELCT Fall 2011
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Compact Trigonometric Fourier Series
ELCT Fall 2011
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Amplitude spectrum Phase spectrum
(a, b) Periodic signal and (c, d) its Fourier spectra. ELCT Fall 2011
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Figure 2.20 (a) Square pulse periodic signal and (b) its Fourier spectrum.
ELCT Fall 2011
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Bipolar square pulse periodic signal.
ELCT Fall 2011
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(a) Impulse train and (b) its Fourier spectrum.
ELCT Fall 2011
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Exponential Fourier spectra for the signal
Exponential Fourier Series Exponential Fourier spectra for the signal ELCT Fall 2011
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