ELEG 3124 SYSTEMS AND SIGNALS Ch. 1 Continuous-Time Signals

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1 Deparmen of Elecrical Engineering Universiy of Arkansas ELEG 3124 SYSTEMS AND SIGNALS Ch. 1 Coninuous-Time Signals Dr. Jingxian Wu

2 OUTLINE 2 Inroducion: wha are signals and sysems? Signals Classificaions Basic Signal Operaions Elemenary Signals

3 INTRODUCTION Examples of signals and sysems (Elecrical Sysems) Volage divider Inpu signal: x = 5V Oupu signal: y = Vou The sysem oupu is a fracion of he inpu (y = R 2 R 1 +R 2 x) Mulimeer Inpu: he volage across he baery Oupu: he volage reading on he LCD display The sysem measures he volage across wo poins Radio or cell phone Inpu: elecromagneic signals Oupu: audio signals The sysem receives elecromagneic signals and conver hem o audio signal

4 INTRODUCTION Examples of signals and sysems (Biomedical Sysems) Cenral nervous sysem (CNS) Reina Inpu signal: a nerve a he finger ip senses he high emperaure, and sends a neural signal o he CNS Oupu signal: he CNS generaes several oupu signals o various muscles in he hand The sysem processes inpu neural signals, and generae oupu neural signals based on he inpu Inpu signal: ligh Oupu signal: neural signals Phoosensiive cells called rods and cones in he reina conver inciden ligh energy ino signals ha are carried o he brain by he opic nerve.

5 INTRODUCTION Examples of signals and sysems (Biomedical Insrumen EEG (Elecroencephalography) Sensors Inpu: brain signals Oupu: elecrical signals Convers brain signal ino elecrical signals Magneic Resonance Imaging (MRI) Inpu: when apply an oscillaing magneic field a a cerain frequency, he hydrogen aoms in he body will emi radio frequency signal, which will be capured by he MRI machine Oupu: images of a cerain par of he body Use srong magneic fields and radio waves o form images of he body.

6 INTRODUCTION Signals and Sysems Even hough he various signals and sysems could be quie differen, hey share some common properies. In his course, we will sudy: How o represen signal and sysem? Wha are he properies of signals? Wha are he properies of sysems? How o process signals wih sysem? The heories can be applied o any general signals and sysems, be i elecrical, biomedical, mechanical, or economical, ec.

7 OUTLINE 7 Inroducion: wha are signals and sysems? Signals Classificaions Basic Signal Operaions Elemenary Signals

8 SIGNALS AND CLASSIFICATIONS 8 Wha is signal? Physical quaniies ha carry informaion and changes wih respec o ime. E.g. voice, elevision picure, elegraph. Elecrical signal Carry informaion wih elecrical parameers (e.g. volage, curren All signals can be convered o elecrical signals Speech Microphone Elecrical Signal Speaker Speech Signals changes wih respec o ime

9 SIGNALS AND CLASSIFICATIONS 9 Mahemaical represenaion of signal: Signals can be represened as a funcion of ime Suppor of signal: E.g. E.g. s 1 2 1( sin(2 and are wo differen signals! The mahemaical represenaion of signal conains wo componens: The expression: The suppor: The suppor can be skipped if E.g. s(, 1 2 s2( sin(2 0 s ( ) ) 1 s ( 2 s( 1 2 s ( sin(2 ) 1

10 SIGNALS AND CLASSIFICATIONS 10 Classificaion of signals: signals can be classified as Coninuous-ime signal v.s. discree-ime signal Analog signal v.s. digial signal Finie suppor v.s. infinie suppor Even signal v.s. odd signal Periodic signal v.s. Aperiodic signal Power signal v.s. Energy signal

11 OUTLINE 11 Inroducion: wha are signals and sysems? Signals Classificaions Basic Signal Operaions Elemenary Signals

12 SIGNALS: CONTINUOUS-TIME V.S. DISCRETE-TIME 12 Coninuous-ime signal If he signal is defined over coninuous-ime, hen he signal is a coninuous-ime signal E.g. sinusoidal signal E.g. voice signal E.g. Recangular pulse funcion s( sin(4 p( p( A, 0, 0 1 oherwise A 0 1

13 SIGNALS: CONTINUOUS-TIME V.S. DISCRETE-TIME 13 Discree-ime signal If he ime can only ake discree values, such as, kt s k 0, 1, 2, hen he signal s( s( kts ) is a discree-ime signal E.g. he monhly average precipiaion a Fayeeville, AR (weaher.com) T s 1monh k 1, 2,,12 Wha is he value of s( a ( k 1) Ts kt s? Discree-ime signals are undefined a kt s!!! Usually represened as s(k)

14 SIGNALS: ANALOG V.S. DIGITAL 14 Analog v.s. digial Coninuous-ime signal coninuous-ime, coninuous ampliude analog signal Example: speech signal Coninuous-ime, discree ampliude Example: raffic ligh Discree-ime signal Discree-ime, discree-ampliude digial signal Example: Telegraph, ex, roll a dice Discree-ime, coninuous-ampliude Example: samples of analog signal, average monhly emperaure

15 SIGNALS: EVEN V.S. ODD 15 Even v.s. odd x( is an even signal if: E.g. x( is an odd signal if: E.g. Some signals are neiher even, nor odd E.g. x( cos(2 ) x( sin(2 x( e x( x( x( x( x( cos(2 ), 0 Any signal can be decomposed as he sum of an even signal and odd signal y( y ( y ( e even odd o proof

16 SIGNALS: EVEN V.S. ODD Example Find he even and odd decomposiion of he following signal x( e

17 SIGNALS: EVEN V.S. ODD 17 Example Find he even and odd decomposiion of he following signal 2sin(4, 0 x( 0 oherwise

18 SIGNALS: PERIODIC V.S. APERIODIC 18 Periodic signal v.s. aperiodic signal An analog signal is periodic if There is a posiive real value T such ha I is defined for all possible values of, s( s( nt) (why?) Fundamenal period : he smalles posiive ineger ha saisfies T 0 s( s( nt0 ) T 0 E.g. T1 2T 0 s( nt1 ) s( 2nT0 ) s( So is a period of s(, bu i is no he fundamenal period of s( T 1

19 SIGNALS: PERIODIC V.S. APERIODIC 19 Example Find he period of s( Acos( 0 ) Ampliude: A Angular frequency: Iniial phase: Period: T 0 Linear frequency: f 0 0

20 SIGNALS: PERIODIC V.S. APERIODIC 20 Complex exponenial signal Euler formula: e jx cos( x) Complex exponenial signal j sin( x) e j cos( 0 jsin( 0 0 ) Complex exponenial signal is periodic wih period Proof: T Complex exponenial signal has same period as sinusoidal signal!

21 SIGNALS: PERIODIC V.S. APERIODIC 21 The sum of wo periodic signals x( has a period y( has a period Define z( = a x( + b y( Is z( periodic? T 1 T 2 z( T) ax( T) by( T) In order o have x(=x(+t), T mus saisfy In order o have y(=y(+t), T mus saisfy Therefore, if T kt 1 lt 2 z T) ax( kt) by( lt The sum of wo periodic signals is periodic if and only if he raio of he wo periods can be expressed as a raional number. T1 l T2 k The period of he sum signal is T kt 1 T lt 2 ( 1 2) ax( by( z( T kt 1 lt 2 )

22 SIGNALS: PERIODIC V.S. APERIODIC 22 Example 2 2 x( sin( y( exp( j z( exp( j Find he period of x(, y(, z( Is 2x( 3y( periodic? If periodic, wha is he period? Is x( z( periodic? If periodic, wha is he period? Is y( z( periodic? If periodic, wha is he period? Aperiodic signal: any signal ha is no periodic.

23 SINGALS: ENERGY V.S. POWER 23 Signal energy Assume x( represens volage across a resisor wih resisance R. Curren (Ohm s law): i( = x(/r 2 Insananeous power: p ( x ( / R Signal power: he power of signal measured a R = 1 Ohm: p( x 2 ( Signal energy a: [, ] n n p( E n p( ) n p( n ) Toal energy E lim p( n) 0 n E x( 2 d p ( d n Review: inegraion over a signal gives he area under he signal.

24 SINGALS: ENERGY V.S. POWER 24 Energy of signal x( over [, ] E x( If 0 E, hen x( is called an energy signal. 2 d Average power of signal x( P 1 lim 2T T T T x( 2 d If 0 P, hen x( is called a power signal. A signal can be an energy signal, or a power signal, or neiher, bu no boh.

25 SINGALS: ENERGY V.S. POWER 25 Example 1: x( Aexp( 0 Example 2: x( Asin( 0 ) x( Example 3: j (1 j) e 0 10 All periodic signals are power signal wih average power: P 1 T T 0 x( 2 d

26 OUTLINE 26 Inroducion: wha are signals and sysems? Signals Classificaions Basic Signal Operaions Elemenary Signals

27 OPERATIONS: SHIFTING 27 Shifing operaion x( ) 0 : shif he signal x( o he righ by Why righ? x(0) A y ( ) x ( 0) y( ) x( ) x(0) A x ( 0) y ( 0)

28 OPERATIONS: SHIFTING Example o.w ) ( x Find ) ( 3 x 28

29 OPERATIONS: REFLECTION 29 Reflecion operaion x( is obained by reflecing x( w.r.. he y-axis ( = 0)

30 OPERATIONS: REFLECTION 30 Example: Find x( x( o.w. x ( y( x( z( y( 3) x( ( 3)) The operaions are always performed w.r.. he ime variable direcly!

31 OPERATIONS: TIME-SCALING 31 Time-scaling operaion x(a a a is obained by scaling he signal x( in ime. 1 1, signal shrinks in ime domain, signal expands in ime domain x( a 3 a 1/ 2

32 OPERATIONS: TIME-SCALING 32 Example: Find x( 3 6) x( o.w. x( a b) 1. scale he signal by a: y( = x(a 2. lef shif he signal by b/a: z( = y(+b/a) = x(a(+b/a))=x(a+b) The operaions are always performed w.r.. he ime variable direcly (be careful abou or a!

33 OUTLINE 33 Signals Classificaions Basic Signal Operaions Elemenary Signals

34 ELEMENTARY SIGNALS: UNIT STEP FUNCTION 34 Uni sep funcion u( 1, 0, 0 0 Example: recangular pulse Express 1, p( 0, p ( as a funcion of u( 2 2 oherwise

35 ELEMENTARY SIGNALS: RAMP FUNCTION 35 The Ramp funcion r( r( u( 0 The Ramp funcion is obained by inegraing he uni sep funcion u( u( d

36 ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION 36 Uni impulse funcion (Dirac dela funcion) (0) ( 0, 0 1, 0, 0 ( d 0 ( dela funcion can be viewed as he limi of he recangular pulse ( lim 0 p Δ ( Relaionship beween and u( ( 0 ( d u( ( du( d

37 ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION 37 Sampling propery x( ( 0) x( 0) ( 0) Shifing propery x( ( 0) d x( 0) Proof:

38 ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION 38 Scaling propery ( a b) 1 a b a Proof:

39 ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION 39 Examples 4 2 ( 2 ) ( 3) d 1 2 ( 2 ) ( 3) d 3 2 exp( 1) (2 4) d

40 ELEMENTARY SIGNALS: SAMPLING FUNCTION 40 Sampling funcion Sa( x) sin x x Sampling funcion can be viewed as scaled version of sinc(x) Sinc ( x) sin x x sa( x)

41 ELEMENTARY SIGNALS: COMPLEX EXPONENTIAL 41 Complex exponenial x( e ( r j0 ) Is i periodic? Example: Use Malab o plo he real par of x( e ( 1 j2 ) [ u( 2) u( 4)]

42 SUMMARY 42 Signals and Classificaions Mahemaical represenaion Coninuous-ime v.s. discree-ime Analog v.s. digial Odd v.s. even Periodic v.s. aperiodic Power v.s. energy Basic Signal Operaions Time shifing reflecion Time scaling Elemenary Signals s(, 1 2 Uni sep, uni impulse, ramp, sampling funcion, complex exponenial

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