Fakultät Iformatik Istitut für Systemarchitektur Professur Recheretze Itroductio to Digital Sigal Processig Walteegus Dargie Walteegus Dargie TU Dresde Chair of Computer Networks
I 45 Miutes
Refereces Discrete-Time Sigal Processig. Ala V. Oppeheim ad Roald W. Schafer. McGraw Hill. Pearso Educatio. 3rd editio (2009). Uderstadig Digital Sigal Processig. Richard G. Lyos. Pretice Hall. 2d Editio (2004). Digital Sigal Processig. Iteratioal Versio. Joh G. Proakis Joh G. Proakis ad Dimitris K Maolakis. Pearso Educatio. 4th editio (2009) 5
Euler s formula/theorem ω e j Useful Equatios cos( ω) + j si( ω) 2 ( ω ) + si ( ω ) 2 cos Eve ad odd siusoidal fuctios cos ( ω ) cos ( ω ) ( ω) si( ω) si Time domai covolutio Frequecy domai multiplicatio ad vice versa y[ ] x[ k] h[ k k] Y [ Ω] X [ Ω] H[ Ω] 6
Useful Equatios The geometric series: ) ( + α α α i A A : ) ( 0 lim + α α α α α i Partial fractio decompositio (expasio) α α B A a x a x + + e x d x e x d x c bx x + + + ) ( ) ( 2 7
Motivatio Samplig Discrete sigals Discrete-time systems The Z-trasform Digital filters Outlie Discrete Fourier Trasform 8
Motivatio Why do we eed sigal processig? Sigal acquisitio ad propagatio etails sigal distortios ad corruptios at various stages 9
Motivatio Correct Distortio: De-blur Sigal Decompositio: Separatig messages or messages from oise Feature Ehacemet: Boost sigal compoets, sharpe images, etc. Noise Reductio: Smoothig Sigal Aalysis: Trasitios, patters, peaks, frequecy distributio, etc. Sigal Compressio: Sigal Ecryptio 0
Motivatio Aalogue sigal processig: Log term drift (ageig) Short term drift (temperature) Sesitivity to voltage istability. Digital sigal processig: No short or log term drifts Relative immuity to mior power supply variatios. Virtually idetical compoets. Software recofigurable
Motivatio A large umber of aturally occurrig ad mamade sigal iflueces are: Time ivariat Liear (obey the superpositio theorem) These properties are called Liear Time Ivariat Systems (LTI) 2
Motivatio LTI systems ca be fully characterised by covolutio Covolutio is greatly simplified by Fourier (harmoic) decompositio The Fast Fourier Trasform, which was rediscovered by Cooley ad Tukey i the 60's, eable the efficiet aalysis of spectral aspects of sigals ad systems I DSP, system aalysis ad sythesis are made by simple additios ad multiplicatios 3
x (t) (t) X Samplig s( t) δ ( t T ) Coversio from impulse trai to discrete time sequece x ( ) s x[ ] x T T s t x s ( t) x( t) s( t) s( t) δ ( t T ) x( T ) δ ( t X s ( jω) X ( jω) * S( jω) 2π 2π S ( jω) δ j( ω kωs ) ; T k ( ) X ( jω) T ) ( ) j k X X ( ω kω ) s ω 4 s
Samplig Xω(j) Fourier Trasform of cotiuous fuctio ω S > 2 ω N -ω N ω N X S (jω) ω Fourier Trasform of sampled fuctio -2ω S ω N -ωω S - ω N ω N ω S 2ω S ω Ω S < 2 Ω N (aliasig) X S (jω) -2 ω S - ω S ω S 2ω S 5 ω
Samplig Give a bad limited sigal, x(t), such that X(jω) 0 for ω ω N. The x(t) ca be uiquely determied from its samples [] x( T < < x ), 2π ωs 2ω T ω N ω N is called the Nyquist frequecy ad 2ω N Nyquist rate 6
Basic Discrete Sigals Delayig (Shiftig) a sequece y[] x[ Uit sample (impulse) sequece δ[] o ] 0 0 0 Uit step sequece u[] 0 Expoetial sequeces: < 0 0 x[ ] Aα 7
Siusoidal sequece Basic Discrete Sigals x [ ] cos( ω + φ) o Suppose the expoetial sequece has the followig compoets: jω α α e o ; A Ae jφ jω o j( ωo+φ) [] Aα A e α e A α e x x [] A α cos( ω + φ) + ja α si( ω + φ) o j φ o If α x[] becomes x [ ] A cos ( ω + φ ) + j A si ( ω + +φ ) o o 8
Basic Discrete Sigals Cosider a frequecy ( ω0 + 2π ) j( ωo + 2π ) jw0 j2π jw0 x[ ] Ae Ae e Ae More geerally, for ay iteger k, ( ω 2π ) k 0 + j( ωo + 2πk ) jw0 j2πk jw0 x[ ] Ae Ae e Ae The same is true for siusoidal sequeces: x [ ] Acos[ ( ω + 2πk ) + ϕ] Acos( ω + ϕ) o o 9
Basic Discrete Sigals So it is sufficiet to cosider frequecies i a iterval of: -π ω 0 π or 0 ω 0 2π However, discrete-time siusoid sigals are ot ecessarily yperiodic i : jωo ( + N ) jw0 x[ ] x[ + N] Ae Ae iff ω0n 2πK N 2πk ω o should be a iteger 20
Discrete-Time Systems A discrete time system y [] T{x[]} x[] T{.} y[] Ideal Delay System y [] x[ o ] Movig (Ruig) Average y[] x[] + x[ ] + x[ 2] + x[ 3] 2
Discrete-Time Systems Memory-less System The output y[] at every value of depeds oly o the iput x[] at the same value of Square y [] ( x[] ) 2 Couter example Ideal Delay System y[] x[ o] 22
Discrete-Time Systems Liear discrete system: obeys scalig ad the superpositio theorem T { x[ ] + x2[ ]} T{ x[ ] } + T{ x2[ ] } T { ax[ ] } at{ x[ ] } If the system is time-ivariat ivariat (shiftivariat), a time shift at the iput causes correspodig time-shift at the output T{x[] + x2[]} x[ o] + x2[ T{x []} + T{ x [] } x [ ] + x [ 2 T { ax[] } ax[ o] { x[] } ax [ ] at o 2 o o o ] ] 23
Discrete-Time Systems A liear time ivariat discrete system (LTI) ca completely l be characterised by its impulse respose: y[ ] T{ x[ ]} T x[ k] δ[ k] k k { δ[ ]} x[ k] T k x[ k] h[ k] k x[ ] * [ ] 24
Z-Trasform Ofte, oe is cofroted with questios pertaiig to a sigal s s property Does it coverge? To which value does it coverge? How fast does it coverge? The Z-Trasform Is a mappig from a discrete sigal to a fuctio of z Where: X ( z) 0 x[ ] z jω Ae A [ cos ( ω ) + j si ( ω ) ] z + 25
Z-Trasform I most cases X(z) ca be expressed as follows ROC X ( z) a i z i i 0 m j b j z j 0 Defies the poles ad zeros for which the system is coverget ROC z: 0 x[ ] z < 26
Z-Trasform Uit impulse δ ( ) X[0] X[] 0 X[2] 0 X[3] 0 X[4] 0 z 0 +0 z - +0 z -2 +0 z -3 +0 z -4 0.5 0-0 2 3 4 5 6 7 8 9 Δ(z) 27
Z-Trasform Delayed Uit Impulse Sigal δ ( ) x[0] 0 x[] x[2] 0 x[3] 0 x[4] 0 Δ(z) z 0 z 0 + z - +0 z -2 +0 z -3 +0 z -4 0.5 0-0 2 3 4 5 6 7 8 9 28
Uit Step Sigal u ( ) δ ( ) 0 Z-Trasform 0 0 U(z) x[0] x[] x[2] x[3] x[4] z 0 + z - + z -2 + z -3 + z -4 2 3 + z + z + z +... i z i 0 z 0.5 0-0 2 3 4 5 6 7 8 9 29
Expoetial sequece ( ) a x Z-Trasform x[0] x[] a x[2] a 2 x[3] a 3 x[4] a 4 z 0 +a z - +a 2 z -2 +a 3 z -3 +a 4 z -4 X(z) + az + a 2 z 2 + a 3 z 3 +... 6 5 - - az 4 a.2 3 2 0-0 2 3 4 5 6 7 8 9 30
Digital Filters Filters alter the spectral aspect of a iput sigal Digital filters are software recofigurable, ad hece, will ot drift with temperature or humidity ad do ot require precisio compoets 3
Digital Filters There are four basic types Lowpass, highpass, badpass ad badstop f c Lowpass f f c Highpass f f 2 f f 2 Badpass Badstop 32
Digital Filters I geeral, a filter ca be characterised by its trasfer fuctio H ( z ) Y X ( ) z i 0 m ( z) m j 0 a z i i b j z j 33
Digital Filters I terms of realisatio, they are classified ito Fiite impulse respose (FIR): Operate o the iput value Perform a covolutio of the filter coefficiets with a sequece of iput values, producig a equally umbered sequece of output values. Ifiite impulse respose (IIR) Operate o curret ad previous values of the iput as well as curret ad previous values of the output Also called auto regressive movig average (ARMA) The impulse respose is ifiite 34
Digital Filters FIR: [ ] [ ] [ ] [ ] 2 + + b b b [ ] [ ] [ ] [ ] 2 3 0 + + x b b x x b y 2 2 0 ] [ ] [ k k k x b y 2 0 ) ( ) ( k k k z b z X z Y 0 ) ( ) ( k b k z z X z Y 35
Digital Filters IIR [ ] [ ] + [ 2 ] + b [ ] y + + a y a y b x 2 0 H ( z) b 0 ( 2 z) a z a2 z Y ( z) X 36
Digital Filters IIR b 0 X[] + y[] Z - a + y[-] Z - a 2 y[-2] 37
DFT A sequece of N complex umbers x 0,..., x N is trasformed ito a sequece of N complex umbers X 0,..., X N Ulike the discrete-time Fourier trasform (DTFT), it oly evaluates eough frequecy compoets to recostruct the fiite segmet that was aalyzed. The iput to the DFT is a fiite sequece of real or complex umbers 38
DFT 2 N π...,, 0,, ] [ ] [ 0 2 N k e x k X N k N j π 0..., 0,,, ] [ ] [ 2 N e k X N x N k N j π 0 N k 39
Summary Samplig is the begiig of everythig The Nyquist rate has to be respected A precoditio for frequecy samplig is N 2πk ω o This is also the basis for DFT ad FFT 40
Thaks for Listeig 4