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

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1 Review of Discrete-time Sigals ELEC 635 Prof. Siripog Potisuk 1

2 Discrete-time Sigals Discrete-time, cotiuous-valued amplitude (sampled-data sigal) Discrete-time, discrete-valued amplitude (digital sigal) I practice, we work with digital sigals Mathematical Represetatio represets a DT sigal, i.e., a sequece of umbers defied oly at iteger values of (udefied for oiteger values of ) Each umber is called a sample may be a sample from a aalog sigal d = a (T s ), where T s = samplig period (s) F s = samplig frequecy (samples/s) 2

3 Sigal Trasformatios Operatios performed o the idepedet ad the depedet variables 1) Reflectio or Time Reversal or Foldig 2) Time Shiftig 3) Time Scalig 4) Amplitude Scalig 5) Amplitude Shiftig Note: The idepedet variable is assumed to be represetig sample umber. 3

4 Eample Cosider the followig DT sigal: Determie the mathematical epressio for the reflected, shifted, ad time-scaled versios. Also, plot the resultig sigals. Time reversal or Foldig or Reflectio Replace by, resultig i y y is the reflected versio of about = (i.e., the vertical ais). 4

5 Time Shiftig (Advace or Delay) Replace by where I I resultig i y is the amout of shift is shifted to the right by is shifted to the left by, (Delay) (Advace) Time Scalig (Samplig rate alteratio) Dow - samplig: y D, the samplig rate of y is (1/ D) th D I that of 5

6 Time Scalig (Samplig rate alteratio) / D, if /D is a iteger Up-samplig: y, otherwise the samplig rate of y is D times that of Amplitude Scalig Multiply by a scalig factor A, where A is a real costat. y A If A is egative, y is the amplitude-scaled ad reflected versio of about the horizotal ais 6

7 Amplitude Shiftig Add a shiftig factor A to, where A is a real costat. y A Sigal Characteristics Determiistic vs. Radom Fiite-legth vs. Ifiite-legth Right-sided/ Left-sided/ Two-sided Causal vs. Ati-causal Periodic vs. Aperiodic (No-periodic) Real vs. Comple Cojugate-symmetric vs. Cojugate-atisymmetric Eve vs. Odd 7

8 A right-sided digital sigal A left-sided digital sigal Eve & Odd DT Sigals A comple-valued sequece is said to be Cojugate symmetricif, Cojugate atisymmetric if A real-valued sigal ) is said to be eve if, odd if. * *. 8

9 Origial sequece Eve part of Odd part of DT Periodic Sigals A DT sigal is said to be periodic if N kn,, k I N is the smallest positive iteger called the fudametal period (dimesioless) w 2p/N = the fudametal agular frequecy (radias) 9

10 A periodic square wave digital sigal Sum of three siusoidal sequeces DT Siusoidal Sigals Acos w is dimesioless (sample umber) w ad have uits of radias may or may ot be periodic periodic if w is a ratioal multiple of 2p,i.e., m w 2p, m I ad N N the fudametal period 1

11 Discrete-time periodic siusoidal sequeces for several differet frequecies A eample of a discrete-time o-periodic siusoidal sequece 11

12 12 DT Uit Impulse DT Uit impulse (Kroecker delta fuctio) Properties: 1,, 4) 3) 2) 1 1)

13 Uit Step Fuctio, u 1, u ca be writte as a sum of shifted uit impulses as u k k 13

14 Epoetial Sigals Br where B ad r are real or comple umbers By lettig r e, Be Eamples of causal real epoetial sequeces: (a) =.25(1.1) µ ad (b) y = 25(.89) µ. 14

15 Sigal Metrics Eergy Power Magitude Area Average Value Zero Crossig Eergy E 2 A ifiite-legth sequece with fiite sample values may or may ot have fiite eergy 15

16 Power P 1 lim 2N 1 N N N 2 - A fiite eergy sigal with zero average power is called a ENERGY sigal - A ifiite eergy sigal with fiite average power is called a POWER sigal Average Power Average over oe period for periodic sigal, e.g., P 1 N N 1 2 for ay Root-mea-square value: rms P 16

17 Magitude & Area Magitude: Bouded if M M ma Area : A Average value & ZCR 1 N avg lim N 2N 1 N ZCR Computatio: Icremet ZCR cout by 1 if there is a sig chage betwee two adjacet samples 17

18 Eample Cosider the followig DT sigal: (a) Determie its mathematical epressio. (b) List its characteristics (c) Compute all of its metrics. 18

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

Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1 Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits