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Linear Systems & Signals Signal Theory Zdzisław Papir Basic definitions Examples of signals and signal processing Classification of signal models Time-invariant & Linear Systems (TILSs) TILS transfer function Components of a TILS response TILS response to a harmonic input Summary

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Basic definitions SYSTEM input signals output signals Signal – variation of some physical quantity in (t;x,y,z). Input signals – signals driving the system. Output signals – response of the system to input signals. Signal Theory is related to modeling of both: signal properties, signal processing in systems. Signal/system model – description of signal/system using functions or differential/integral equations Signal Theory Zdzisław Papir

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Examples of signals & signal processing INFORMATION TRANSMISSION: radio and television signals, mobile and fixed telephony data transmission (data networks) OBJECT IDENTIFICATION SIGNALS: ultrasound scanning, X-ray scanning, radar techniques, stock analysis, demographic trends. Signal Theory Zdzisław Papir

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Types of models of signals & signal processing Analog models Discret models Time-invariant models Time-variant models Linear models Nonlinear models Lumped models Distributed models Deterministic models Stochastic models Static models Dynamic models Signal Theory Zdzisław Papir

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Analog models Signal Theory Zdzisław Papir In analog models input and output signals are continuous functions of time. Seismogram recorded on an analog device Electrocardiogram recorded on an analog device

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Discret models In discret models signals are changing stepwise. Buffer Transmission channel 3 t Packet count is one of the possible teletraffic models. Signal Theory Zdzisław Papir

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Static models Static models do not depend on time. Packet buffering leads to multiplexing of traffic streams over a channel. Buffer Channel Signal Theory Zdzisław Papir

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Dynamic models Buffer Channel Diffusion approximation Signal Theory Zdzisław Papir Dynamic models do depend on time.

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Time – invariant models In time-invariant models both signal parameters and system characteristics do not depend on time. INOUT LOGISTIC ITERATION FEEDBACK Teoria sygnałów Zdzisław Papir

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Time-invariant models Signal Theory Zdzisław Papir

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Time-variant models Frequency Modulation FM Instantaneous frequency of the FM signal depends on the modulating signal. Signal Theory Zdzisław Papir In time-variant models both signal parameters and system characteristics do depend on time.

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Linear models In linear models the system response to a composite input signal is combination of system responses to component signals. Preemphasis filter R C r x1(t)x1(t)y1(t)y1(t) x2(t)x2(t)y2(t)y2(t) Signal Theory Zdzisław Papir

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Linear models Signal Theory Zdzisław Papir

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Nonlinear models Weber-Fechner Law The sensation change depends linearly on a relative stimulus change. Signal Theory Zdzisław Papir In nonlinear models the system response to a composite input signal is not combination of system responses to component signals.

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Nonlinear models - compression x y Signal Theory Zdzisław Papir The aim of a nonlinear compression is to emphasize weak signals while leaving strong signals almost unchanged.

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Kompresja Signal before compression Signal after compression -compression law is used in Northern America; European digital telephony exploits the A-compression concept. Nonlinear models Teoria sygnałów Zdzisław Papir

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Lumped models In lumped models energy is accumulated/disspated in isolated system points. Signals are transferred within the system without any delay. R C r Signal Theory Zdzisław Papir

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Distributed parameter models power networks CATV coaxial network Digital Subscriber Lines Printed Circuit Boards (> 100 MHz) Signal Theory Zdzisław Papir In distributed models energy is accumulated/disspated in all system points. Signals are transferred within the system with some delay.

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Deterministic models Double-sideband Amplitude modulation AM Signal Theory Zdzisław Papir In deterministic models signal fluctuations are described by functions or equations. The exact formula modeling the signal makes future signal values known.

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Stochastic models Stochastic models allow for a signal description exact to a probability distribution. The future signal values can be predicted with some accuracy only. Transition graph for the Millers code Signal Theory Zdzisław Papir + –

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Millers code Signal Theory Zdzisław Papir + –

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Millers code + – S( ) Spectral density function Bipolar code Millers code Biphase code Teoria sygnałów Zdzisław Papir

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Time-invariant Linear Systems (TILS) TILS Linear System Time-invariant System Signal Theory Zdzisław Papir TILS

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Exponential input Linear System TILS Time-invariant System Signal Theory Zdzisław Papir

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Exponential input TILS The single and nontrivial solution to an equation: is an exponential signal: The amplitude H(s) depends on some constant s C. The exponential signal is an invariant to Linear Time-invariant Systems (TILS). Signal Theory Zdzisław Papir

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Exponential input Lets assume that an extra solution does exist: Lets substracte the identity side by side: Signal Theory Zdzisław Papir TILS

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The conclusion is: We state that: We do not receive a new solution: Signal Theory Zdzisław Papir Exponential input

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TILS transfer function TILS The transfer function of any TILS: is defined as a ratio of the system response to the exponential driving function. The transfer function can be interpreted as a TILS amplification. Signal Theory Zdzisław Papir

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TILS (R, L, C) impedance TILS impedance (voltage/current transfer function): TILS R C L Signal Theory Zdzisław Papir

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TILS (R, L, C) admittance Admittance (current/voltage transfer function): ULS R C L Signal Theory Zdzisław Papir

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TILS (R, L, C) transfer function Derivation of the TILS (R, L, C) transfer function is supported by various theorems: serial/parallel combination of impedances, Kirchoffs current law, Kirchoffs voltage law, Thevenin/Norton theorems, transformation of current/voltage sources. Signal Theory Zdzisław Papir

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Preemphasis filter R 1/Cs r x(t)x(t)y(t)y(t) Signal Theory Zdzisław Papir

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TILS response to a sinusoidal input TILS TILS response to a sinusoidal (harmonic) input: Signal Theory Zdzisław Papir

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Harmonic excitation TILS The transfer function H(j ) is a rational function so it follows the Hermite symmetry: Using the exponential representation we get: Signal Theory Zdzisław Papir

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Harmonic excitation TILS TILS response to the harmonic excitation: A( )- amplitude-frequency characteristic ( )- phase -frequency characteristic A-f function A( ) is an even function, A( ) = A(- ) P-f function ( ) is an odd function, ( ) = - (- ) Signal Theory Zdzisław Papir

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Preemphasis filter R 1/Cs r x(t)x(t)y(t)y(t) Signal Theory Zdzisław Papir

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Preemphasis filter Signal Theory Zdzisław Papir Preemphasis filter f2/f1 = 100 Log-log amplitude response

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Butterworth filter A-f function n = 2, f g = 1 kHz P-f function n = 2, f g = 1 kHz Signal Theory Zdzisław Papir

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Butterworth filter Butterworth filters have a maximaly flat a-f function in both passband and stopband. Signal Theory Zdzisław Papir

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Chebyshev filter Chebyshev polynomials: Oscillation level of A 2 ( ) in the passband: The Chebyshev a-f function decreases faster than the Butterworth a-f function (for the same order). Signal Theory Zdzisław Papir

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Chebyshev filter n = 6 Signal Theory Zdzisław Papir Butterworth Chebyshev

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Summary Signal Theory Zdzisław Papir In time-invariant models both signal parameters and system characteristics do not depend on time. In linear models the system response to a composite input signal is combination of system responses to component signals. The exponential signal is an invariant to Linear Time-invariant Systems (TILS). The transfer function of any TILS is defined as a ratio of the system response to the exponential driving function. Signal Theory is related to modeling of both: signal properties, signal processing in systems.

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Summary Signal Theory Zdzisław Papir The TILS response to a harmonic excitation is a harmonic signal as well. The frequency remains unchanged. Amplitude and phase can be derived from amplitude and phase functions. The transfer function of the TILS = (R, L, C) can be derived from a differential equation or using theorems of the circuit theory.

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