Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1

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2 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits i time. x t discrete-time sigal. x = x T s ( ) is a cotiuous-time sigal, x ( ) where T s is the time betwee samples is a Samplig Uiform Samplig x() t x x t [ ] () x[ ] ω s or f s 9/10/16 M. J. Roberts - All Rights Reserved 2

3 Samplig ad Discrete Time

4 Siusoids Ulike a cotiuous-time siusoid, a discrete-time siusoid is ot ecessarily periodic. If it is periodic, its period must be a iteger. If a siusoid has the form g [ ] = Acos( 2π F 0 +θ ) the [ ] to be F 0 must be a ratio of itegers (a ratioal umber) for g periodic. If F 0 is ratioal i the form q / N 0 ( q ad N 0 itegers)i which all commo factors i q ad N 0 have already bee cacelled, the the fudametal period of the siusoid is N 0, ot N 0 / q (uless q = 1). Therefore, the geeral form of a periodic siusoid with fudametal period N 0 is g ( ). [ ] = Acos 2πq / N 0 +θ 9/10/16 M. J. Roberts - All Rights Reserved 4

7 Siusoids Two siusoids whose aalytical expressios look differet, [ ] = Acos( 2π F 01 +θ ) ad g 2 g 1 may actually be the same. If F 02 = F 01 + m, where m is a iteger [ ] = Acos( 2π F 02 +θ ) the (because is discrete time ad therefore a iteger), Acos( 2π F 02 +θ ) = Acos 2π ( F 01 + m) +θ Acos 2π F 02 +θ ( ) +θ Iteger Multiple of 2π ( ) = Acos 2π F π m (Example o ext slide) = Acos( 2π F 01 +θ ) 9/10/16 M. J. Roberts - All Rights Reserved 7

9 Siusoids x[] = cos(2 F) Dashed lie is x(t) = cos(2 Ft) x[] 1 F = 0 x[] 1 F = 0.25 x[] 1 F = 0.5 x[] 1 F = ,t 4,t 4,t 4,t x[] 1 F = 1 x[] 1 F = 1.25 x[] 1 F = 1.5 x[] 1 F = ,t 4,t 4,t 4,t /10/16 M. J. Roberts - All Rights Reserved 9

10 Expoetials The form of the expoetial is [ ] = Az x!# "# $ or x Preferred [ ] = Ae β where z = e β 0 < z < 1 Real z -1 < z < 0 Re(g[]) Complex z z < 1 Im(g[]) z > 1 z < -1 Re(g[]) z > 1 Im(g[]) 9/10/16 M. J. Roberts - All Rights Reserved 10

11 The Uit Impulse Fuctio δ[] 1 [ ] = 1, = 0 δ 0, 0 The discrete-time uit impulse (also kow as the Kroecker delta fuctio ) is a fuctio i the ordiary sese (i cotrast with the cotiuous-time uit impulse). It has a samplig property, [ ] Aδ [ 0 ]x = Ax 0 = but o scalig property. That is, δ [ ] [ ] = δ [ a] for ay o-zero, fiite iteger a. 9/10/16 M. J. Roberts - All Rights Reserved 11

19 Scalig ad Shiftig Fuctios Time expasio / K, K > 1 For all such that / K is a iteger, g[ / K ] is defied. For all such that / K is ot a iteger, g[ / K ] is ot defied. 9/10/16 M. J. Roberts - All Rights Reserved 19

20 Scalig ad Shiftig Fuctios There is a form of time expasio that is useful. Let [ ] = x [ / m ], / m a iteger y 0, otherwise All values of y are defied. This type of time expasio is actually used i some digital sigal processig operatios.

27 Symmetric Fiite Summatio... Eve Fuctio g[] Sum #1 Sum #2 -N N Sum #1 = Sum # Odd Fuctio Sum #1 -N g[] N Sum #2 Sum #1 = - Sum #2... N [ ] g = g 0 = N N [ ] + 2 g[ ] g = 0 =1 = N [ ] 9/10/16 M. J. Roberts - All Rights Reserved 27 N

28 Periodic Fuctios A periodic fuctio is oe that is ivariat to the chage of variable + mn where N is a period of the fuctio ad m is ay iteger. The miimum positive iteger value of N for which [ ] = g[ + N ] is called the fudametal period N 0. g 9/10/16 M. J. Roberts - All Rights Reserved 28

29 Periodic Fuctios Fid the fudametal period of [ ] = cos( π /18) + si( 10π / 24) [ ] = cos( 2π / 36) + si 2π ( 5 / 24 ) x x N 0 =36 N 0 = LCM( 36,24) = 72 Fid the fudametal period of x ( ) N 0 =24 [ ] = cos( 5π /13) + si( 8π / 39) [ ] = cos( 2π( 5 / 26) ) + si 2π ( 4 / 39 ) x N 0 =26 ( ) = 78 N 0 = LCM 26, 39 ( ) N 0 =39 9/10/16 M. J. Roberts - All Rights Reserved 29

32 Sigal Eergy ad Power Fid the sigal eergy of Usig E x = N 1 r =0 x[ ] = [ ] 2 ( 5 / 3) 2, 0 < 8 0, otherwise x = 5 / 3 = 5 / 3 = = 7 =0 N, r = 1 1 r N, r 1 1 r ( ) 2 ( )4 E x = 1 5 / / =0 ( ) 4 ( ) /10/16 M. J. Roberts - All Rights Reserved 32

33 Sigal Eergy ad Power Some sigals have ifiite sigal eergy. I that case It is usually more coveiet to deal with average sigal power. The average sigal power of a sigal x 1 P x = lim N 2N For a periodic sigal x P x = 1 N N 1 = N x[ ] 2 [ ] is [ ] the average sigal power is = N x[ ] 2 The otatio meas the sum over ay set of = N cosecutive 's exactly N i legth. 9/10/16 M. J. Roberts - All Rights Reserved 33

34 Sigal Eergy ad Power Fid the average sigal power of 1 P x = lim N 2N N 1 = N [ ] = 2sg[ ] 4 x x[ ] 2 1 = lim x 2N N 1 = N 2sg[ ] 4 2 N 1 N 1 N 1 1 P x = lim N 2N 4 [ sg2 ] sg[ ] = N = N =1 δ[ ] = N =2 N =N 1 N = 1 =N 1+N =2 N 1 1 P x = lim { 4 2N 1 N ( ) + 32N 16 ( 1 ) } = 20 2N 9/10/16 M. J. Roberts - All Rights Reserved 34

35 Sigal Eergy ad Power A sigal with fiite sigal eergy is called a eergy sigal. A sigal with ifiite sigal eergy ad fiite average sigal power is called a power sigal. 9/10/16 M. J. Roberts - All Rights Reserved 35

36 Fudametal Period of a Sum of Two Periodic Sigals δ 14 [!] 6δ! [ ] 8 N 0 =14 N 0 =8 " $$ # $$ % N 0 =LCM( 14,8)=56 ( ( )) 2cos( 3π /12) +11cos( 14π /10) = 2cos 2π 1/ 8 Impulses ad Periodic Impulses cos 2π 7 /10 " $$ # $$ % " $$ # $$ % N 0 =10 $$$$$$$ $$$$$$$ % " N 0 =8 # N 0 =LCM( 8,10)= δ [ + 6] = 1368, 12( 0.4) u[ ]δ 3 [ ] = = 18 Equivalece Property Scalig Property 7 = 4 ( ( )) ( ) 0 + ( 0.4) 3 + ( 0.4) 6 = ( ) δ [ 3] = 27( 0.3) 3 δ [ 3] = 0.729δ [ 3] [ ] = 13δ [ ], ( No scalig property for discrete-time impulses) 13δ 3 22, 4 = 3k 22δ 3 [ 4] = 22 δ [ 4 3k] = k= 0, otherwise = 22, 4 / 3 = k 0, otherwise Sice k is a iteger, impulses occur oly where 4 / 3 is a iteger & 22δ 3 [ 4] 22( k = 0) 0( 4 / 3 k) 0( 8 / 3 k) 22( k = 1) 0( 16 / 3 k) & 9/10/16 M. J. Roberts - All Rights Reserved 36

37 Sigal Eergy ad Sigal Power x[ ] = ( 1.3) u[ ] u[ 4] 3 ( ) 2 = ( ) E x = 1.3 E x = = = x =0 [ ] is periodic ad oe period of x[ ] is described by [ ] = ( 1 ), 3 < 6 x P x = 1 N [ ] 2 x, = N x[ ] P x = 1 [ ] = = ( ) u[ ] u[ 4] ( ) 2 9/10/16 M. J. Roberts - All Rights Reserved 37

Review of Discrete-time Signals. ELEC 635 Prof. Siripong Potisuk

Review of Discrete-time Signals. ELEC 635 Prof. Siripong Potisuk Review of Discrete-time Sigals ELEC 635 Prof. Siripog Potisuk 1 Discrete-time Sigals Discrete-time, cotiuous-valued amplitude (sampled-data sigal) Discrete-time, discrete-valued amplitude (digital sigal)