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**Linear Systems & Signals**

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 „Signal Theory” Zdzisław Papir

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**Basic definitions SYSTEM Signal Theory is related to modeling of both:**

input signals output 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 In analog models input and output signals**

are continuous functions of time. Seismogram recorded on an analog device Electrocardiogram recorded on an analog device „Signal Theory” Zdzisław Papir

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**Discret models Buffer Transmission channel t**

In discret models signals are changing stepwise. Buffer Transmission channel 3 t 1 2 4 5 6 7 Packet count is one of the possible teletraffic models. „Signal Theory” Zdzisław Papir

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**Static models Buffer Channel Static models do not depend on time.**

Packet buffering leads to multiplexing of traffic streams over a channel. „Signal Theory” Zdzisław Papir

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**Dynamic models Buffer Channel Diffusion approximation**

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

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**Time – invariant models**

In time-invariant models both signal parameters and system characteristics do not depend on time. IN OUT LOGISTIC ITERATION FEEDBACK „Teoria sygnałów” Zdzisław Papir

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**Time-invariant models**

100 200 300 400 500 600 700 800 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 „Signal Theory” Zdzisław Papir

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**Time-variant models Frequency Modulation FM**

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

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**Linear models R C r x1(t) y1(t) x2(t) y2(t) Preemphasis filter**

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

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**Linear models H(f) [dB] f [dec] Preemphasis filter f2/f1 = 100**

1 2 3 4 -2 -1 f [dec] H(f) [dB] Preemphasis filter f2/f1 = 100 log-log amplitude response „Signal Theory” Zdzisław Papir

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**Nonlinear models Weber-Fechner Law**

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

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**Nonlinear models -compression y x**

The aim of a nonlinear compression is to emphasize weak signals while leaving strong signals almost unchanged. -compression 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y „Signal Theory” Zdzisław Papir

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**Nonlinear models Kompresja **

0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 Kompresja Signal before compression Signal after compression -compression law is used in Northern America; European digital telephony exploits the A-compression concept. „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**

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

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**Deterministic models In deterministic models signal fluctuations**

are described by functions or equations. The exact formula modeling the signal makes future signal values known. Double-sideband Amplitude modulation AM „Signal Theory” Zdzisław Papir

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**Stochastic models + – Transition graph for the Miller’s code**

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 Miller’s code „Signal Theory” Zdzisław Papir

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Miller’s code + – 1 „Signal Theory” Zdzisław Papir

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**Spectral density function**

Miller’s code + – S() Spectral density function Bipolar code Miller’s code Biphase code „Teoria sygnałów” Zdzisław Papir

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**Time-invariant Linear Systems (TILS)**

„Signal Theory” Zdzisław Papir

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**Time-invariant System**

Exponential input TILS Linear System 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 TILS Let’s assume that an extra solution does exist:**

Let’s substracte the identity side by side: „Signal Theory” Zdzisław Papir

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**Exponential input We state that: The conclusion is:**

We do not receive a new solution: „Signal Theory” Zdzisław Papir

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**TILS transfer function**

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 R C L**

TILS impedance (voltage/current transfer function): „Signal Theory” Zdzisław Papir

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**TILS (R, L, C) admittance**

ULS R C L Admittance (current/voltage transfer function): „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, Kirchoff’s current law, Kirchoff’s 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) y(t) „Signal Theory” Zdzisław Papir

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**TILS response to a sinusoidal input**

TILS response to a sinusoidal (harmonic) input: TILS „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) y(t) „Signal Theory” Zdzisław Papir

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**Preemphasis filter H(f) [dB] f [dek] Preemphasis filter f2/f1 = 100**

1 2 3 4 -2 -1 f [dek] H(f) [dB] Preemphasis filter f2/f1 = 100 Log-log amplitude response „Signal Theory” Zdzisław Papir

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**Butterworth filter A-f function n = 2, fg = 1 kHz**

10 -2 2 4 6 -4 -200 -150 -100 -50 A-f function n = 2, fg = 1 kHz P-f function n = 2, fg = 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 A2() 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 Chebyshev Butterworth n = 6**

„Signal Theory” Zdzisław Papir

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**Summary Signal Theory is related to modeling of both:**

signal properties, signal processing in systems. 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” Zdzisław Papir

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Summary The transfer function of the TILS = (R, L, C) can be derived from a differential equation or using theorems of the circuit theory. 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. „Signal Theory” Zdzisław Papir

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