Signal A funcion of ime Sysem Processes inpu signal (exciaion) and produces oupu signal (response) Exciaion Inpu Sysem Oupu Response
1. Types of signals 2. Going from analog o digial world 3. An example of a sysem 4. Mahemaical represenaion of signals
Types of Signals Time Value 1 Coninuous Coninuous 2 Coninuous Discree Analog Digial 3 Discree Coninuous 5 Discree Discree
Types of Signals Time Value 1 Coninuous Coninuous 2 Coninuous Discree 3 Discree Coninuous Analog Digial 5 Discree Discree Role of Noise!
Advanage of Digial World
Going from Analog o Digial World Three Sep Process Sampling Quanizaion Encoding
Example of Sysem A simple sysem example Sound Recording Sysem Wha consiues a sound recording sysem?
Recorded Sound as a Signal Example s i gn al
Represenaion of Signals Time Domain Frequency Domain
Anoher Example of Sysem A very complex sysem example Human Brain
Sampling a CT Signal o Creae a DT Signal Sampling is acquiring he values of a CT signal a discree poins in ime x() is a CT signal --- x[n] is a DT signal xn xnt s where T s is he ime beween samples
Mahemaical Represenaion of Signals Coninuous Time x ( ) A sin( 0 ) A sin( 2 f 0 ) Discree Time x[ nts ] Asin[ onts ] A sin[ 2 f nt o s ]
Coninuous Time Signals Sinusoidal Signal: g( ) Asin( ) o Asin( 2f o 2 Asin( ) T o ) General Form: g( ) Asin(2 f ) o
Review of Euler s Ideniy Complex valued sinusoidal signals ) sin(2 ) cos(2 2 f j f e F F f j F Euler s Ideniy ) sin(2 ) cos(2 2 f j f e F F f j F 2 ) cos(2 2 2 f j f j F F F e e f j e e f f j f j F F F 2 ) sin(2 2 2 and Also
Review of Euler s Ideniy Complex valued sinusoidal signals ) sin(2 ) cos(2 ) (2 f jc f C Ce F F f j F Euler s Ideniy 2 ) cos(2 ) (2 ) 2 ( f j f j F F F Ce Ce f C and Also ) sin(2 ) cos(2 ) (2 f jc f C Ce F F f j F f j j f j j F F F e Ce e Ce f C 2 2 2 2 ) cos(2
Exponenial Funcions g( ) Ae g( ) Ae Complex valued exponenial signal: g( ) Ae ( j) Ae [cos j sin ] Where do hese funcions occur in real life?
Disconinuiy of a funcion Definiion: lim 0 g( ) lim 0 g( ) Simple words: If he value of funcion is differen a ime 0 when approached a 0 by decreasing and increasing ime, hen he funcion is disconinuous a ime 0 Examples:
Uni Sep Funcion Definiion: u( ) 1 1/ 0 2 0 0 0 Real Physical Phenomenon: Swiching
Signum Funcion Definiion: sgn( ) 1 0 1 0 0 0 2u( ) 1
Ramp Funcion Definiion: ramp( ) 0 0 0 u() u ( x) dx Can you generae his funcion?
Uni Impulse Funcion 1 a 1 a 1 a a lim a 0 Area 1 a ( a) 1
Definiion: ( ) 0 ( ) d 1 0 Can you represen u() in erms of uni impulse funcion? u ( ) ( x) dx
Anoher Imporan Fac abou Uni Impulse Funcion! ( ) d 1 g ( ) ( ) d g(0) Isn i Sampling?
Uni Comb n n comb ( ) ( n) comb() 3 2 1 0 1 2 3
Recangular Funcion rec ( ) 1 1/ 0 2 1/ 2 1/ 2 1/ 2 1 1 2 1 2
Triangular Funcion ri( ) 1 0 1 1 1 1 1
Uni Sinc Funcion sin c( ) sin( ) 1 5 4 3 2 1 1 2 3 4 5
Combinaions of Funcions g( ) sin c( )cos(20 ) g( ) Ae cos 20 g( ) u( ) ramp( ) g( ) sgn( )sin(2 )
Some More Examples
Ampliude Transformaions of Funcions Ampliude Shifing g( ) A g( ) Ampliude Scaling g( ) Ag( )
Time Transformaions of Funcions Time Shifing g( ) g( a) Time Scaling g( ) g( a )
Muliple Transformaions Case 1 g( ) Ag( a o ) g() Ampliude Scaling Time Time Scaling Shifing o Ag () Ag ( ) Ag( ) a a Case 2 g ( ) Ag ( b o) Ampliude Scaling Time Shifing g () Ag () Ag ) Ag b ) ( o Time Scaling ( o
Example of Case 1
Example of Case 2
Some More Examples
Differeniaion and Inegraion of Funcions Differeniaion: Slope of he funcion a ime dg ( ) d Inegraion: Accumulaive area under he curve g ( x) dx
Differeniaion A kind of Transformaion of a Signal
Inegraion A kind of Transformaion of a Signal
Even and Odd Funcions Funcion is Even if g( ) g( ) Example: cos( ) Funcion is Odd if g( ) g( ) Example: sin( )
If funcion is neiher even nor odd, hen ) ( ) ( ) ( g g g o e Even and Odd Componens of a Funcion Where 2 ) ( ) ( ) ( g g g e 2 ) ( ) ( ) ( g g g o
Producs of Even and Odd CT Funcions Even x Even = Even
Producs of Even and Odd CT Funcions Even x Odd = Odd
Producs of Even and Odd CT Funcions Even x Odd = Odd
Producs of Even and Odd CT Funcions Odd x Odd = Even
Inegrals of Even and Odd CT Funcions a a g d a 0 2 g d a g d 0 a
Coninuous Time Periodic Funcions Funcion is periodic wih period T, if g( ) g( nt) Wha is he effec on periodic funcion of ime shifing by nt?
Examples of Periodic Signals g( ) 3sin(400 ) g( ) 2 2 g( ) sin(12 ) sin(6 ) g( ) sin( ) sin(6 )
Discree Time Signals Coninuous Time x ( ) A sin(2 f 0 ) Discree Time x[ nts ] Asin[2 fonts ] Asin[ 2f T o s n] 2f Asin[ f s o n]
2f o x[ nts ] Asin[ n] f fo x[ n] Asin[2 n] f Ts x[ n] Asin[2 n] T s s o x[ n] Asin[2 Kn] p x[ n] Asin[2 n] q To be periodic, Kn has o be an ineger for some n => K has o be a raio of inegers Period = q
Discree-Time Sinusoids Periodic Sinusoids
How Many CT periodic Cycles are Presen in One DT Periodic Cycle p x[ n] Asin[2 n] q Period = q x[ n] Asin[2p n q ] x[ n] Asin[2 p] When n = q one DT period => There are p cycles of CT periodic sinusoidal funcion per one cycle of DT periodic sinusoidal funcion
Examples g [ n] 2 cos[ 5 1 n ] Period = 5 g [ n] 12 cos[ 5 2 n ] Period = 5
Two Discree Sinusoids could be similar? Two differen-looking DT sinusoids, g 1 n Acos 2K 1 n g 2 n and Acos 2K 2 n may acually be he same. If K 2 K 1 m, where m is an ineger hen (because n is discree ime and herefore an ineger), Acos2K 1 n Acos2K 2 n (Example on nex slide)
Examples g [ n] 2 cos[ 5 1 n ] Period = 5 g [ n] 12 cos[ 5 2 n ] Period = 5 g [ n] 16 cos[ 5 3 n ] Period = 5
2 cos[ n] 5 12 cos[ n] 5 16 cos[ n] 5
1 2 cos[ n] 5 0-1 0 1 2 3 4 5 6 7 8 9 10 1 12 cos[ n] 5 16 cos[ n] 5 0-1 0 1 2 3 4 5 6 7 8 9 10 1 0-1 0 1 2 3 4 5 6 7 8 9 10
Oher Discree Funcions Uni Impulse Funcion [ n] 1 0 n n 0 0 Please noe: [ n] [ an] n x[ n] [ n] x[0] Discree Sampling n x[ n] [ n n0 ] x[ n0 ]
Uni Sep Funcion u[ n] 1 0 n 0 n 0 Uni Ramp Funcion ramp[ n] n 0 n n 0 0
w w N N n N n n rec w 0 1 ] [ Recangular Funcion 1] [ ] [ ] [ w w N N n u N n u n rec w Please noe ha
Transformaions on Discree Time Funcions Ampliude Shifing g[ n] A g[ n] Ampliude Scaling g[ n] Ag[ n]
Time Shifing g[ n] g[ n a] Same as coninuous ime Time Scaling g[ n] n g[ a ] Tricky! Isn i?
Example of Time Shifing
Example of Time Compression
Discree Time Even and Odd Funcions ] [ ] [ n g n g ] [ ] [ n g n g Funcion is Even if Funcion is Odd if If funcion is neiher even nor odd, hen ] [ ] [ ] [ n g n g n g o e Where 2 ] [ ] [ ] [ n g n g n g e 2 ] [ ] [ ] [ n g n g n g o
Differencing and Accumulaion g[ n] g[ n 1] g[ n] n g [ n]
Energy of a Signal For Coninuous Time Signals 2 E x x( ) d For Discree Time Signals E x n x[ n] 2
Visual Example of Energy of a Signal CT Signal
Visual Example of Energy of a Signal DT Signal
Power of a Signal Some signals have infinie signal energy. In ha case I is more convenien o deal wih average signal power. For Coninuous Time Signals P x lim T 1 T T / 2 T / 2 x( ) 2 d For Discree Time Signals P x lim N 1 2N N 1 nn x[ n] 2
Power of a Periodic Signal For a periodic CT signal, x(), he average signal power is P x 1 T T x 2 d where T is any period of he signal. For a periodic DT signal, x[n], he average signal power is P x 1 N n N xn 2 where N is any period of he signal.
Energy and Power Signals A signal wih finie signal energy is called an energy signal. A signal wih infinie signal energy and finie average signal energy is called a power signal.