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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.

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Presentation on theme: "Linear Systems & Signals Signal Theory Zdzisław Papir Basic definitions Examples of signals and signal processing Classification of signal models Time-invariant."— Presentation transcript:

1 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

2 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

3 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

4 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

5 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

6 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

7 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

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

9 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

10 Time-invariant models Signal Theory Zdzisław Papir

11 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.

12 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

13 Linear models Signal Theory Zdzisław Papir

14 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.

15 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.

16 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

17 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

18 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.

19 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.

20 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 + –

21 Millers code Signal Theory Zdzisław Papir + –

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

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

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

25 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

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

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

28 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

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

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

31 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

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

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

34 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

35 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

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

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

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

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

40 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

41 Chebyshev filter n = 6 Signal Theory Zdzisław Papir Butterworth Chebyshev

42 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.

43 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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