CSE 314 S YSTEMS A NALYSIS Lecture 06

CSE 314: S YSTEMS A NALYSIS Lecture 06: Filtering Dr. Ahmed Mahmoud, 29/11/2020

What Is Filtering? • LTI systems can alter scales or phases of sinusoidal components of input signals. • LTI systems cannot add new frequency components. • LTI systems can filter out frequency components of the input, and also enhance other components, but it cannot introduce components that are not already present in the input. • It merely changes the relative amplitudes and phases of the frequency components that are present in the inputs. • LTI can be either a model of a physical system or an ideal (Behavior specification) of an engineered system. 29/11/2020 CSE 314: Systems Analysis, Lec 06 2

Filtering Examples: Acoustics 29/11/2020 CSE 314: Systems Analysis, Lec 06 3

Filtering Examples: Communications • A channel may be a physical medium, such as a pair of wires, that carries an electrical signal. • That physical medium distorts the signal. • This distortion is often reasonably well approximated as linear and time invariant. • An equalizer in the receiver compensates for this distortion. • It should consider both magnitude and phase. Why? 29/11/2020 CSE 314: Systems Analysis, Lec 06 4

Filtering Examples: Image Processing • The top image is the original, undistorted image. The lower left image is blurred, as might result for example from unfocused optics. • Image processing systems are LSI systems. • The blurred image is constructed from the original by an LSI system that eliminates high (spatial) frequencies, passing unaltered the low frequencies. Low pass. • The lower right image is constructed from the original by an LSI system that eliminates low frequencies, passing unaltered the high frequencies. High pass. 29/11/2020 CSE 314: Systems Analysis, Lec 06 5

Imperative Description of Frequency Response • The frequency response of a system is a declarative description of the system. • It tells us what it is, not how it works. • For example, it tells us that it is a lowpass filter, but it does not tell us whether it is a defocused lens or a computer program, much less telling us how the computer program works. • In this lecture, we explore imperative descriptions of systems, and build up to detailed descriptions of software that can implement certain kinds of LTI (or LSI) systems. • These imperative descriptions are based on convolution. 29/11/2020 CSE 314: Systems Analysis, Lec 06 6

Convolution Sum 29/11/2020 CSE 314: Systems Analysis, Lec 06 7

Example: Convolution Sum 29/11/2020 CSE 314: Systems Analysis, Lec 06 8

Convolution Integral 29/11/2020 CSE 314: Systems Analysis, Lec 06 9

Example: Convolution Integral 29/11/2020 CSE 314: Systems Analysis, Lec 06 10

Impulses • An impulse is a signal that is zero everywhere except at time zero. Discrete-time case Continuous-time case • The Kronecker delta function defined by: is • The Dirac delta function is defined as where and to satisfy the following property: for any 29/11/2020 CSE 314: Systems Analysis, Lec 06 11

Signals as Sums of Weighted Delta Functions 29/11/2020 CSE 314: Systems Analysis, Lec 06 12

Impulse Response and Convolution (1) 29/11/2020 CSE 314: Systems Analysis, Lec 06 13

Impulse Response and Convolution (2) 29/11/2020 CSE 314: Systems Analysis, Lec 06 14

Frequency Response and Impulse Response (1) 29/11/2020 CSE 314: Systems Analysis, Lec 06 15

Frequency Response and Impulse Response (2) 29/11/2020 CSE 314: Systems Analysis, Lec 06 16

Causality 29/11/2020 CSE 314: Systems Analysis, Lec 06 17

Finite Impulse Response (FIR) Filters 29/11/2020 CSE 314: Systems Analysis, Lec 06 18

Finite Impulse Response (FIR) Filters 29/11/2020 CSE 314: Systems Analysis, Lec 06 19

Example: FIR Filter … (1) 29/11/2020 CSE 314: Systems Analysis, Lec 06 20

Example: FIR Filter … (2) 29/11/2020 CSE 314: Systems Analysis, Lec 06 21

Infinite Impulse Response (IIR) Filters 29/11/2020 CSE 314: Systems Analysis, Lec 06 22

Example: IIR Filter … (1) • Consider an LTI filter defined by the following difference equation: • The impulse response: 29/11/2020 CSE 314: Systems Analysis, Lec 06 23

Example: IIR Filter … (2) 29/11/2020 CSE 314: Systems Analysis, Lec 06 24

Filters Implementation • An LTI system (a filter) can be described by a state-space, a frequency response, an impulse response, or a difference equation. • The state-space description and the difference equations prove the most useful when constructing the filter. • A realization of a filter in hardware or software is called an implementation. • Filter design refer to the choice of frequency response, impulse response, or coefficients for a difference equation. 29/11/2020 CSE 314: Systems Analysis, Lec 06 25

Signal Flow Graphs 29/11/2020 CSE 314: Systems Analysis, Lec 06 26

IIR Direct Form 1 29/11/2020 CSE 314: Systems Analysis, Lec 06 27

IIR Direct Form 1 29/11/2020 CSE 314: Systems Analysis, Lec 06 28

IIR Direct Form 2 29/11/2020 CSE 314: Systems Analysis, Lec 06 29

Reading • Read carefully Chapter 9: Filtering. 29/11/2020 CSE 314: Systems Analysis, Lec 06 30