Download presentation

1
**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

2
**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

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 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

6
**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

7
**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

8
**Dynamic models Buffer Channel Diffusion approximation**

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

9
**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

10
**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

11
**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

12
**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

13
**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

14
**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

15
**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

16
**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

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**

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

19
**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

20
**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

21
Miller’s code + – 1 „Signal Theory” Zdzisław Papir

22
**Spectral density function**

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

23
**Time-invariant Linear Systems (TILS)**

„Signal Theory” Zdzisław Papir

24
**Time-invariant System**

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

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

27
**Exponential input We state that: The conclusion is:**

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

28
**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

29
**TILS (R, L, C) impedance TILS R C L**

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

30
**TILS (R, L, C) admittance**

ULS R C L Admittance (current/voltage transfer function): „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, Kirchoff’s current law, Kirchoff’s voltage law, Thevenin/Norton theorems, transformation of current/voltage sources. „Signal Theory” Zdzisław Papir

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

33
**TILS response to a sinusoidal input**

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

37
**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

38
**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

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 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

41
**Chebyshev filter Chebyshev Butterworth n = 6**

„Signal Theory” Zdzisław Papir

42
**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

43
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

Similar presentations

OK

The Physical Layer Lowest layer in Network Hierarchy. Physical transmission of data. –Various flavors Copper wire, fiber optic, etc... –Physical limits.

The Physical Layer Lowest layer in Network Hierarchy. Physical transmission of data. –Various flavors Copper wire, fiber optic, etc... –Physical limits.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on group development theories Ppt on 9/11 conspiracy essay Ppt on brand marketing companies Ppt on time management for engineering students Ppt on indian administrative services Ppt on organic chemistry-some basic principles and techniques Ppt on any topic in hindi Ppt on obstructive sleep apnea Ppt on power diode testing Download ppt on oxidation and reduction chemistry