Signals and Systems Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin

Transcription

1 EE 345S Real-Time Digial Signal Processing Lab Spring 26 Signals and Sysems Prof. Brian L. Evans Dep. of Elecrical and Compuer Engineering The Universiy of Texas a Ausin

2 Review Signals As Funcions of Time Coninuous-ime signals are funcions of a real argumen x() where ime,, can ake any real value x() may be for a given range of values of Discree-ime signals are funcions of an argumen ha akes values from a discree se x[k] where k {...-3,-2,-1,,1,2,3...} Ineger ime index, e.g. k, for discree-ime sysems Values for x may be real or complex 3-2

3 Analog: Review Analog vs. Digial Signals Coninuous in boh ime and ampliude Digial: Discree in boh ime and ampliude

4 The Many Faces of Signals A funcion, e.g. cos() or cos(π k), useful in analysis A sequence of numbers, e.g. {1,2,3,2,1} or a sampled riangle funcion, useful in simulaion A collecion of properies, e.g. even, causal, sable, useful in reasoning abou behavior A piecewise represenaion, e.g. A generalized funcion, e.g. δ() Wha everyday device uses wo sinusoids o ransmi a digial code? 1 1 u( ) = 2 1 u[ k] = for for > = for < for k oherwise 3-4

5 Telephone Touchone Signal Dual-one muliple frequency (DTMF) signaling Sum of wo sinusoids: one from low-frequency group and high-frequency group On for 4-6 ms and off for res of signaling inerval (symbol duraion): 1 ms for AT&T 8 ms for ITU Q.24 sandard Maximum dialing rae AT&T: 1 symbols/s (4 bis/s) Q.24: 12.5 symbols/s (5 bis/s) 129 Hz 1336 Hz 1477 Hz 1633 Hz 697 Hz A 77 Hz B 852 Hz C 941 Hz * # D Alphabe of 16 DTMF symbols, wih symbols A-D for miliary and radio signaling applicaions ITU is he Inernaional Telecommunicaion Union 3-5

6 Review Uni Impulse Mahemaical idealism for an insananeous even Dirac dela as generalized funcion (a.k.a. funcional) Seleced properies Uni area: Sifing provided g() is defined a = Scaling: 1 δ ( a) d = if a a δ () is undefined Noe ha δ ( ) d =1 g( ) δ ( ) d = g() 1 Pε ( ) = rec 2ε 2 ε δ 1 Pε ( ) = ε ε δ ε ( ) = lim P ( ) ε ε ε ε ( ) = lim P ( ) ε ε ε 1 2ε 1 ε 3-6

7 Uni Impulse By convenion, plo Dirac dela as arrow a origin Undefined ampliude a origin Denoe area a origin as (area) Heigh of arrow is irrelevan Review Direcion of arrow indicaes sign of area δ ( ) (1) Wih δ() = for, i is emping o hink φ() δ() = φ() δ() φ() δ(-t) = φ(t) δ(-t) Simplify uni impulse under inegraion only 3-7

8 Review Uni Impulse We can simplify δ() under inegraion φ( ) δ ( ) d = φ( ) Assuming φ() is defined a = Wha abou? 1 φ( ) δ ( ) d = Wha abou? φ( ) δ ( T ) By subsiuion of variables, φ? d =? ( + T ) δ ( ) d = φ( T ) Oher examples δ δ e ( ) ( 2) 2 e jϖ d = 1 π cos d = 4 ( x ) 2( x 2) ( ) δ 2 d Wha abou a origin? + δ δ δ ( ) ( ) ( ) d =? d = d = 1 = e 3-8

9 Uni Impulse Funcional Relaionship beween uni impulse and uni sep ( τ ) δ dτ =? = 1 = u( ) Wha happens a he origin for u()? < > du d u( - ) = and u( + ) = 1, bu u() can ake any value Common values for u() are, ½, and 1 = δ u() = ½ is used in impulse invariance filer design: L. B. Jackson, A correcion o impulse invariance, IEEE Signal Processing Leers, vol. 7, no. 1, Oc. 2, pp ( ) 3-9

10 Sysems Sysems operae on signals o produce new signals or new signal represenaions Coninuous-ime examples y() = ½ x() + ½ x(-1) y() = x 2 () Discree-ime sysem examples y[n] = ½ x[n] + ½ x[n-1] y[n] = x 2 [n] Review x() T{ } y() x[k] T{ } y[k] y ( ) = T{ x( ) } y [ k] = T{ x[ k] } Squaring funcion can be used in sinusoidal demodulaion Average of curren inpu and delayed inpu is a simple filer 3-1

11 Sysem Properies Le x(), x 1 (), and x 2 () be inpus o a coninuousime linear sysem and le y(), y 1 (), and y 2 () be heir corresponding oupus A linear sysem saisfies Review Addiiviy: x 1 () + x 2 () y 1 () + y 2 () Homogeneiy: a x() a y() for any real/complex consan a For a ime-invarian sysem, a shif of inpu signal by any real-valued τ causes same shif in oupu signal, i.e. x( - τ) y( - τ) for all τ 3-11

12 Review Sysem Properies Ideal delay by T seconds. Linear? x() T y() y ( ) = x( T ) Scale by a consan (a.k.a. gain block) Two differen ways o express i in a block diagram x() y() x() a y() ( ) a x( ) y = a Linear? 3-12

13 Sysem Properies Tapped delay line x( ) T x ( T ) a a 1 a M 2 a M 1 T Σ y( ) T Each T represens a delay of T ime unis There are M-1 delays y M 1 m= ( ) = a x( m T ) m Coefficiens (or aps) are a, a 1, a M-1 Linear? Time-invarian? 3-13

14 Sysem Properies Ampliude Modulaion (AM) y() = A x() cos(2π f c ) f c is he carrier frequency (frequency of radio saion) A is a consan x() A cos(2 π f c ) y() Linear? Time-invarian? AM modulaion is AM radio if x() = 1 + k a m() where m() is message (audio) o be broadcas 3-14

15 Sysem Properies Frequency Modulaion (FM) FM radio: y ( ) = A cos 2 π f + ( ) c k f x d f c is he carrier frequency (frequency of radio saion) A and k f are consans x() Linear Linear Nonlinear Nonlinear Linear ( ) dτ k f + cos ( ) A y() 2 π f c Linear? Time-invarian? 3-15

16 s [ k] = s( ) k T s Sampling Many signals originae as coninuous-ime signals, e.g. convenional music or voice. By sampling a coninuous-ime signal a isolaed, equally-spaced poins in ime, we obain a sequence of numbers k {, -2, -1,, 1, 2, } T s is he sampling period. s() T s T s Sampled analog waveform 3-16

17 Generaing Discree-Time Signals Uniformly sampling a coninuous-ime signal Obain x[k] = x(t s k) for - < k <. How o choose T s? Using a formula x[k] = k 2 5k + 3, for k would give he samples {3, -1, -3, -3, -1, 3,...} Wha does he sequence look like in coninuous ime? k sem plo 3-17

18 Sysem Properies Le x[k], x 1 [k], and x 2 [k] be inpus o a linear sysem and le y[k], y 1 [k], and y 2 [k] be heir corresponding oupus A linear sysem saisfies Addiiviy: x 1 [k] + x 2 [k] y 1 [k] + y 2 [k] Homogeneiy: a x[k] a y[k] for any real/complex consan a For a ime-invarian sysem, a shif of inpu signal by any ineger-valued m causes same shif in oupu signal, i.e. x[k - m] y[k - m], for all m 3-18

19 Sysem Properies Tapped delay line in discree ime See also slide 5-3 x[k] 1 z x[ k 1] 1 z a a 1 a M 2 a M 1 Linear? Time-invarian? Σ y[k] 1 z Each z -1 represens a delay of 1 sample There are M-1 delays y[ k] = M 1 m= a m x[ k m] Coefficiens (or aps) are a, a 1, a M

20 Sysem Properies Coninuous ime Discree ime f() d d ( ) y() f[k] dˆ d ( ) y[k] y d d ( ) = { f ( ) } = lim ( ) f ( ) Linear? Time-invarian? f y [ k] = y( kt ) = { f ( ) } = = lim T Linear? s f s ( kt ) f ( kt T ) [ k] f [ k 1] Time-invarian? s f d d s T s = kt s s See also slide

21 Conclusion Coninuous-ime versus discree-ime: discree means quanized in ime Analog versus digial: digial means quanized in ime and ampliude A digial signal processor (DSP) is a discree-ime and digial sysem A DSP processor is well-suied for implemening LTI digial filers, as you will see in laboraory #

22 Opional Signal Processing Sysems Speech synhesis and recogniion Audio CD players Audio compression: MPEG 1 layer 3 audio (MP3), AC3 Image compression: JPEG, JPEG 2 Opical characer recogniion Video CDs: MPEG 1 DVD, digial cable, HDTV: MPEG 2 Wireless video: MPEG 4 Baseline/H.263, MPEG 4 Adv. Video Coding/H.264 (emerging) Examples of communicaion sysems? Moving Picure Expers Group (MPEG) Join Picure Expers Group (JPEG) 3-22

23 Opional Communicaion Sysems Voiceband modems (56k) Digial subscriber line (DSL) modems ISDN: 144 kilobis per second (kbps) Business/symmeric: HDSL and HDSL2 Home/asymmeric: ADSL, ADSL2, VDSL, and VDSL2 Cable modems Cellular phones Firs generaion (1G): AMPS Second generaion (2G): GSM, IS-95 (CDMA) Third generaion (3G): cdma2, WCDMA 3-23