CONTENTS Contents Linea timeshiftinva riant systems L TI ztransfo rm
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CONTENTS Contents Linea timeshiftinva riant systems L TI ztransfo rm

1 Region of convergence ROC 22 Examples 11 23 Prop erties of the ROC 19 24 Rational functions 22 25 Finite impulse resp onse FIR systems 25 26 In57356nite impulse resp onse I IR systems 31 27 The inverse ztransfo rm 32 Filters 34 CONTENTS 31 Ideal fr

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CONTENTS Contents Linea timeshiftinva riant systems L TI ztransfo rm




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CONTENTS Contents Linea time-/shift-inva riant systems (L TI) z-transfo rm 2.1 Region of convergence (ROC) 2.2 Examples 11 2.3 Prop erties of the ROC 19 2.4 Rational functions 22 2.5 Finite impulse resp onse (FIR) systems 25 2.6 Innite impulse resp onse (I IR) systems 31 2.7 The inverse z-transfo rm 32 Filters 34 CONTENTS 3.1 Ideal frequency-selective lters 35 3.2 Ideal dela lters 36 3.3 Group dela 37 3.4 Inversion of TI systems 38 3.5 Eects of single oles and zeros 40 3.6 Minimal-phase systems 53 LINEAR TIME-/SHIFT-INV ARIANT SYSTEMS (L TI)

Linea time-/shift-inva riant systems (L TI) linea r: )) Lf Lf time-/shift-inva riant (system do es not change over time): Lf Lf LINEAR TIME-/SHIFT-INV ARIANT SYSTEMS (L TI) Consider signal as sequence of eighted pulses 1 dx Lf 1 {z dx )( with impulse resp onse L shift inva riance from L By the convolution theo rem, and the system function transfer function ecomes
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LINEAR TIME-/SHIFT-INV ARIANT SYSTEMS (L TI) The resp onse of linea time-inva riant system to excitation with signal is given the convolution with impulse resp onse Complex exp onentials re eigenvecto

rs of all convolution op erato rs Le 1 dx 1 dx {z eigenvalue {z eigenvecto LINEAR TIME-/SHIFT-INV ARIANT SYSTEMS (L TI) Systems can classied as causal: acausal: rbitra ry anticausal: stable (b ounded input, ounded output (BIBO)): Lf is ounded if and 1 dx (1) ecause Lf 1 dx sup 1 dx Z-TRANSF ORM z-transfo rm The discrete-time ourier transfo rm and its inverse re given (cho osing notation dierent from last time): 1 d! The rgument of the exp onential is purely imagina ry and can generalized to contain real pa rt: 1 1

z-transfo rm can interp reted as DTFT of multiplied exp onential. Coincides with DTFT fo Z-TRANSF ORM −2 −1.5 −1 −0.5 0.5 1.5 −2 −1.5 −1 −0.5 0.5 1.5 PSfrag replacements −2 −1.5 −1 −0.5 0.5 1.5 −2 −1.5 −1 −0.5 0.5 1.5 PSfrag replacements colo black 0.1 blue 0.1 green 0.2
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Z-TRANSF ORM 2.1 Region of convergence (ROC) The z-transfo rm ma converge fo signals whose ourier series do es not converge. Example: unit step function is not absolutely summable and has no DTFT; multiplication

with exp onential deca mak es it summable and z-transfo rm exists within ROC most signals, convergence is in an (annula r) region of the -plane only ROC 1 The DTFT converges if unit circle is within ROC. Z-TRANSF ORM 10 ecause with 1 1 1 1 In general, er series will converge in annula region of the -plane. Z-TRANSF ORM 11 2.2 Examples Causal signals can transfo rmed using the one-sided z-transfo rm: =0 impulse ( ROC entire plane Z-TRANSF ORM 12 function ( elsewhere =0 ROC unit step ROC
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Z-TRANSF ORM 13 real and imagina ry

pa rt of z-transfo rm -2 -1 -2 -1 -4 -2 -2 -1 -2 -1 -2 -1 -4 -2 -2 -1 -2 -1 -2 -1 -2 -1 -2 -1 -2 -1 0.2 0.4 0.6 0.8 -2 -1 -2 -1 absolute of z-transfo rm Z-TRANSF ORM 14 one-sided exp onential deca 1 =0 =0 =z fo ROC: and FT exists if Z-TRANSF ORM 15 0.5 1.5 −2 −1 −2 −1 Re Im −1 −0.8 −0.6 −0.4 −0.2 −2 −1 −2 −1 Re Im Z-TRANSF ORM 16 one-sided exp onential gro wth 1) is unit step 1 1) 1 =1 =0 = fo ROC: and FT exists if ull sp ecication of z-transfo rm requires algeb raic exp ression fo and

ROC. Dierent choices fo the ROC lead to dierent impulse resp onses, but all co rresp ond to the same constant-co ecient dierence equation|alb eit with dierent initial conditions.
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Z-TRANSF ORM 17 another example (2 (3 12) 2)( 3) −1 −1 −0.5 0.5 Re Im −1 −1 −0.5 0.5 Re Im −1 −1 −0.5 0.5 Re Im −1 −1 −0.5 0.5 Re Im ROC of outer ole is blue, ROC of inner ole is ello w, common ROC is green 1) 1) 1) 1) Z-TRANSF ORM 18 The oles and zeros alone do not sp ecify the lter

uniquely: kno wledge of the ROC is required. constant-co ecient dierence equation do es not sp ecify the signal uniquely: the initial conditions re required. Z-TRANSF ORM 19 2.3 Prop erties of the ROC If the z-transfo rm can exp ressed as rational function, and if is ounded fo nite then the ROC dep ends on the distance from the rigin only ROC 1 ecause with 1 1 1 1 DTFT exists if unit circle ROC Z-TRANSF ORM 20 denition, ROC cannot contain oles (if ecomes innite, the series cannot converge). rom the ab ove rop

erties, the ROC is disc round ring et een oles. if fo only then ROC ecause the sum then has nite numb er of nite terms. if is right-sided, i.e. (shifted) causal series, then ROC outermost ole ecause each term in the fractional expansion of has to converge if is left-sided, i.e. (shifted) anticausal series, then ROC innermost ole
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Z-TRANSF ORM 21 ROC must connected region. Intuitively fo o-sided series, need to \balance" damping and amplication of left-and right-sided pa rt. oles can elong to right-sided to left-sided pa rt. ROC is ey ond the outermost

\right-sided" ole and within the innermost \left-sided" ole disc et een these is free of oles. Assumption of causalit ecomes right-sided ROC is ey ond outermost ole. Assumption of stabilit impulse resp onse is absolutely summable ROC must comp rise unit circle If ant causalit and stabilit all oles must reside within unit circle. Z-TRANSF ORM 22 2.4 Rational functions In this imp rtant class of z-transfo rms, olynomial olynomial ro ots oles 1 The ROC cannot contain oles. Consider system that is linea r, time-inva riant, causal, ractically realizable (ie requiring nite memo ry

and computation fo each output sample). Its input/output relation is describ ed constant-co ecient, linea dierence equation: =1 =0 Z-TRANSF ORM 23 =0 =0 =1 If re function of time time-va rying system. If re function of input output non-linea system. =0 =0 =0 )) =0 )) =0 =0 Z-TRANSF ORM 24 But )( so that =0 =0 Numerato and denominato olynomials can each sp ecied their ro ots zeros and oles. If all co e ís re nonzero, =1 (1 =1 (1 By eq. 1, the system is stable if the unit circle is within the ROC.
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Z-TRANSF ORM 25 2.5 Finite impulse resp onse (FIR) systems

Let =0 elsewhere trivial oles at zeros \all-zero system" The follo wing slides sho examples fo FIR lters Z-TRANSF ORM 26 2-tap moving average numerator 0.5 0.5 denominator 1 0 zeros −1 poles 0 gain 0.5 0.5 0.1 0.2 0.3 0.4 0.5 −1 −1.5 −1 −0.5 0.5 1.5 Real Part Imaginary Part −50 −40 −30 −20 −10 Normalized Frequency ( rad/sample) Magnitude (dB) −100 −50 50 100 Normalized Frequency ( rad/sample) Phase (degrees) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Frequency ( rad/sample) Group delay (samples) Z-TRANSF ORM 27 3-tap

moving average numerator 0.33333 0.33333 0.33333 denominator 1 0 0 zeros −0.5+0.86603i −0.5−0.86603i poles 0 0 gain 0.33333 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −1 −1.5 −1 −0.5 0.5 1.5 Real Part Imaginary Part −60 −50 −40 −30 −20 −10 Normalized Frequency ( rad/sample) Magnitude (dB) −150 −100 −50 50 100 150 Normalized Frequency ( rad/sample) Phase (degrees) Normalized Frequency ( rad/sample) Group delay (samples) Z-TRANSF ORM 28 3-tap binomial numerator 0.25 0.5 0.25 denominator 1 0 0 zeros −1

−1 poles 0 0 gain 0.25 0.1 0.2 0.3 0.4 0.5 −1 −1.5 −1 −0.5 0.5 1.5 Real Part Imaginary Part −100 −80 −60 −40 −20 Normalized Frequency ( rad/sample) Magnitude (dB) −200 −100 100 200 Normalized Frequency ( rad/sample) Phase (degrees) Normalized Frequency ( rad/sample) Group delay (samples)
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Z-TRANSF ORM 29 3-tap derivative numerator −1 0 1 denominator 1 0 0 zeros −1 1 poles 0 0 gain −1 −1 −0.5 0.5 −1 −1.5 −1 −0.5 0.5 1.5 Real Part Imaginary Part −40

−30 −20 −10 10 Normalized Frequency ( rad/sample) Magnitude (dB) −700 −600 −500 −400 −300 −200 −100 Normalized Frequency ( rad/sample) Phase (degrees) 0.2 0.4 0.6 0.8 1.2 1.4 Normalized Frequency ( rad/sample) Group delay (samples) Z-TRANSF ORM 30 3-tap second derivative numerator −1 2 −1 denominator 1 0 0 zeros 1 1 poles 0 0 gain −1 −1 −0.5 0.5 1.5 −1 −1.5 −1 −0.5 0.5 1.5 Real Part Imaginary Part −80 −60 −40 −20 20 Normalized Frequency ( rad/sample)

Magnitude (dB) −400 −300 −200 −100 Normalized Frequency ( rad/sample) Phase (degrees) Normalized Frequency ( rad/sample) Group delay (samples) Z-TRANSF ORM 31 2.6 Innite impulse resp onse (I IR) systems Example: 1) By pa rtial fraction expansion (P rtialb ruchzerlegung) and if oles re distinct and transfer function can exp ressed as =1 giving an impulse resp onse =1 Each nontrivial ole contributes to the innite impulse resp onse. Z-TRANSF ORM 32 2.7 The inverse z-transfo rm 1 1 dz 1 dz 1 dz dz
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Z-TRANSF ORM

33 the Cauchy integral theo rem: dz Integration must ca rried out counter-clo ckwise along contour contained in the ROC. In ractice, numb er of sho rt cuts re kno wn: if the z-transfo rm is rational: use residue theo rem pa rtial fractional expansion if the z-transfo rm can exp ressed as er series: match app rop riate co ecients (Ko ezientenvergleich) FIL TERS 34 Filters Impact of lters on signal: )( with transfer function system function In ola co rdinates magnitude resp onse phase resp onse FIL TERS 35 3.1 Ideal frequency-selective lters induce no phase shift,

but re not ractically realizable: sin sin sin sin FIL TERS 36 3.2 Ideal dela lters induce only phase shift: id dela the shift theo rem: id dela magnitude resp onse id phase resp onse id dela Dierent frequency comp onents re shifted dierent multiples of their erio to retain the sup erp osition pattern.
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FIL TERS 37 3.3 Group dela is measure fo the linea rit of the phase resp onse (\Grupp enlaufzeit"). grd( )) := ar )) where ar gives the continuous, unfolded, phase. linea phase resp onse, the group dela is constant. The va riation of the group dela is

measure of the non-linea rit FIL TERS 38 3.4 Inversion of TI systems inv )( inv Not all systems have an inverse, e.g. ideal high-pass lter. What do inverse systems lo ok lik e? If the transfer function exists, inv log inv log log inv log log inv log FIL TERS 39 systems with rational system function =1 (1 =1 (1 inv =1 (1 =1 (1 oles ecome zeros and vice versa. evaluate inv the ROC of oth and inv should overlap. stable causal TI system has stable causal inverse i all oles and zeros reside within the unit circle. Such systems re called \minimal phase". FIL TERS 40 3.5 Eects of

single oles and zeros Consider stable TI system with rational transfer function =1 (1 =1 (1 In ola co rdinates, =1 (1 =1 (1 Squa ring oth sides, taking log 10 and multiplying 10 gives the magnitude sp ectrum in dB 10 log 20 log 20 log =1 20 log (1 =1 20 log (1
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FIL TERS 41 Simila rly the phase resp onse and group dela re sum of contributions from individual oles and zeros that dier in their sign only =1 (1 =1 (1 Group dela grd( )) =1 arg (1 =1 arg (1 study single oles and zeros! FIL TERS 42 Single facto in the numerato denominato olynomial can written as amplitude resp

onse (1 )(1 cos( magnitude sp ectrum dB 10 log 10 log cos( phase resp onse (1 (1 cos( sin( )) arctan sin cos( group dela (1 (1 cos( (1 FIL TERS 43 In the follo wing gures, single oles at the follo wing lo cations re studied: −2 −1.5 −1 −0.5 0.5 1.5 −2 −1.5 −1 −0.5 0.5 1.5 In essence, oles build up the magnitude resp onse, zeros supp ress the magnitude resp onse. oles and zeros on the unit circle have linea phase resp onse, but only the zeros lead to stable systems. FIL TERS 44 2000 4000 6000 8000 0.8 0.9 1.1 1.2 1.3 Magnitude Frequency

2000 4000 6000 8000 −20 −10 10 20 Phase (degrees) Frequency 2000 4000 6000 8000 −1 −0.5 0.5 Group Delay Frequency −1 −0.5 0.5 1.5 −1 −0.5 0.5 Zeros & Poles Real Imaginary x 10 −4 0.2 0.4 0.6 0.8 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 2.5 Magnitude Frequency 2000 4000 6000 8000 −40 −20 20 40 Phase (degrees) Frequency 2000 4000 6000 8000 −1 Group Delay Frequency −1 −0.5 0.5 1.5 −1 −0.5 0.5 Zeros & Poles Real Imaginary 0.5 1.5 x 10 −3 0.2 0.4 0.6 0.8 Impulse Response Time 2000 4000

6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase (degrees) Frequency 2000 4000 6000 8000 10 15 20 Group Delay Frequency −0.5 0.5 1.5 2.5 −1 −0.5 0.5 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 0.2 0.4 0.6 0.8 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 500 600 Phase (degrees) Frequency 2000 4000 6000 8000 −20 −15 −10 −5 Group Delay Frequency −0.5 0.5 1.5 2.5 −1 −0.5 0.5 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02

0.025 0.03 10 x 10 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 500 600 Phase (degrees) Frequency 2000 4000 6000 8000 −3 −2 −1 Group Delay Frequency 0.5 1.5 2.5 −1 −0.5 0.5 Zeros & Poles Real Imaginary 0.5 1.5 2.5 x 10 −3 x 10 Impulse Response Time
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FIL TERS 45 2000 4000 6000 8000 0.5 1.5 2.5 Magnitude Frequency 2000 4000 6000 8000 −40 −20 20 40 Phase (degrees) Frequency 2000 4000 6000 8000 −1 Group Delay Frequency −0.5 0.5 −0.2 0.2 0.4 0.6 0.8 Zeros &

Poles Real Imaginary 0.5 1.5 2.5 x 10 −3 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase (degrees) Frequency 2000 4000 6000 8000 10 15 Group Delay Frequency 0.5 1.5 −0.2 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 −1 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 500 Phase (degrees) Frequency 2000 4000 6000 8000 −20 −15 −10 −5 Group Delay Frequency 0.5 1.5 0.2 0.4

0.6 0.8 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 0.025 0.03 −1 −0.5 0.5 x 10 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 500 Phase (degrees) Frequency 2000 4000 6000 8000 −3 −2 −1 Group Delay Frequency 0.5 1.5 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.5 1.5 2.5 x 10 −3 −2 x 10 Impulse Response Time FIL TERS 46 2000 4000 6000 8000 0.8 0.9 1.1 1.2 1.3 Magnitude Frequency 2000 4000 6000 8000 −20 −10 10 20 Phase (degrees) Frequency 2000 4000 6000 8000 −1

−0.5 0.5 Group Delay Frequency −1 −0.5 0.5 −0.4 −0.2 0.2 0.4 0.6 Zeros & Poles Real Imaginary x 10 −4 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 2.5 Magnitude Frequency 2000 4000 6000 8000 −40 −20 20 40 Phase (degrees) Frequency 2000 4000 6000 8000 −1 Group Delay Frequency −1 −0.5 0.5 −0.2 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.5 1.5 x 10 −3 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase

(degrees) Frequency 2000 4000 6000 8000 10 15 20 Group Delay Frequency −0.5 0.5 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 −1 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 500 Phase (degrees) Frequency 2000 4000 6000 8000 −20 −15 −10 −5 Group Delay Frequency −0.5 0.5 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 0.025 0.03 −1 −0.5 0.5 x 10 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 Magnitude Frequency 2000 4000

6000 8000 100 200 300 400 500 Phase (degrees) Frequency 2000 4000 6000 8000 −3 −2 −1 Group Delay Frequency −1.5 −1 −0.5 0.5 1.5 0.5 1.5 Zeros & Poles Real Imaginary 0.5 1.5 2.5 x 10 −3 −6 −4 −2 x 10 Impulse Response Time FIL TERS 47 2000 4000 6000 8000 0.8 0.9 1.1 1.2 1.3 Magnitude Frequency 2000 4000 6000 8000 −20 −10 10 20 Phase (degrees) Frequency 2000 4000 6000 8000 −1 −0.5 0.5 Group Delay Frequency −1 −0.5 0.5 −0.4 −0.2 0.2 0.4 0.6 Zeros & Poles Real Imaginary x 10 −4

−0.5 0.5 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 2.5 Magnitude Frequency 2000 4000 6000 8000 −40 −20 20 40 Phase (degrees) Frequency 2000 4000 6000 8000 −1 Group Delay Frequency −1 −0.5 0.5 −0.2 0.2 0.4 0.6 Zeros & Poles Real Imaginary 0.5 1.5 x 10 −3 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 25 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase (degrees) Frequency 2000 4000 6000 8000 10 15 Group Delay Frequency −1.5 −1 −0.5 −0.2 0.2 0.4 0.6 0.8 Zeros & Poles Real

Imaginary 0.005 0.01 0.015 0.02 −1 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 Phase (degrees) Frequency 2000 4000 6000 8000 −20 −15 −10 −5 Group Delay Frequency −1.5 −1 −0.5 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 0.025 0.03 −1 −0.5 0.5 x 10 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 Phase (degrees) Frequency 2000 4000 6000 8000 −3 −2 −1 Group Delay

Frequency −2 −1.5 −1 −0.5 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.5 1.5 2.5 x 10 −3 −6 −4 −2 x 10 Impulse Response Time FIL TERS 48 2000 4000 6000 8000 0.8 0.9 1.1 1.2 1.3 Magnitude Frequency 2000 4000 6000 8000 −20 −10 10 20 Phase (degrees) Frequency 2000 4000 6000 8000 −1 −0.5 0.5 Group Delay Frequency −1 −0.5 0.5 −0.5 0.5 Zeros & Poles Real Imaginary x 10 −4 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 2.5 Magnitude Frequency 2000 4000 6000 8000 −40 −20 20

40 Phase (degrees) Frequency 2000 4000 6000 8000 −1 Group Delay Frequency −1.5 −1 −0.5 −0.5 0.5 Zeros & Poles Real Imaginary 0.5 1.5 x 10 −3 −1 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase (degrees) Frequency 2000 4000 6000 8000 10 15 20 Group Delay Frequency −1.5 −1 −0.5 −0.5 0.5 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 −1 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000

6000 8000 100 200 300 400 Phase (degrees) Frequency 2000 4000 6000 8000 −20 −15 −10 −5 Group Delay Frequency −2 −1.5 −1 −0.5 −0.5 0.5 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 0.025 0.03 −1 −0.5 0.5 x 10 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 Magnitude Frequency 2000 4000 6000 8000 100 200 300 400 Phase (degrees) Frequency 2000 4000 6000 8000 −3 −2 −1 Group Delay Frequency −2.5 −2 −1.5 −1 −0.5 0.5 Zeros & Poles Real Imaginary 0.5 1.5 2.5 x 10 −3 −4

−2 x 10 Impulse Response Time
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FIL TERS 49 2000 4000 6000 8000 0.8 0.9 1.1 1.2 1.3 Magnitude Frequency 2000 4000 6000 8000 −20 −10 10 20 Phase (degrees) Frequency 2000 4000 6000 8000 −1 −0.5 0.5 Group Delay Frequency −0.5 0.5 −0.6 −0.4 −0.2 0.2 0.4 Zeros & Poles Real Imaginary x 10 −4 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 2.5 Magnitude Frequency 2000 4000 6000 8000 −40 −20 20 40 Phase (degrees) Frequency 2000 4000 6000 8000 −1 Group Delay Frequency −0.5 0.5 −0.6

−0.4 −0.2 0.2 Zeros & Poles Real Imaginary 0.5 1.5 x 10 −3 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 25 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase (degrees) Frequency 2000 4000 6000 8000 10 15 20 Group Delay Frequency 0.5 1.5 −0.8 −0.6 −0.4 −0.2 0.2 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 −1 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 −100 100 200 300 Phase (degrees) Frequency 2000 4000 6000 8000 −20

−15 −10 −5 Group Delay Frequency 0.5 1.5 −0.8 −0.6 −0.4 −0.2 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 0.025 0.03 −1 −0.5 0.5 x 10 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 Magnitude Frequency 2000 4000 6000 8000 −100 100 200 300 Phase (degrees) Frequency 2000 4000 6000 8000 −3 −2 −1 Group Delay Frequency 0.5 1.5 −1 −0.8 −0.6 −0.4 −0.2 Zeros & Poles Real Imaginary 0.5 1.5 2.5 x 10 −3 −6 −4 −2 x 10 Impulse Response Time FIL TERS 50 2000 4000 6000

8000 10 15 20 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase (degrees) Frequency 2000 4000 6000 8000 10 15 Group Delay Frequency 0.5 1.5 −0.2 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 −1 −0.5 0.5 Impulse Response Time 2000 4000 6000 8000 0.5 1.5 Magnitude Frequency 2000 4000 6000 8000 −100 −50 50 100 Phase (degrees) Frequency 2000 4000 6000 8000 −15 −10 −5 Group Delay Frequency 0.5 1.5 −0.2 0.2 0.4 0.6 0.8 Zeros & Poles Real Imaginary −1 x 10 −4 −1 −0.5 0.5 Impulse

Response Time 2000 4000 6000 8000 10 15 Magnitude Frequency 2000 4000 6000 8000 −200 −100 100 200 Phase (degrees) Frequency 2000 4000 6000 8000 10 15 Group Delay Frequency −0.5 0.5 1.5 −0.5 0.5 Zeros & Poles Real Imaginary 0.005 0.01 0.015 0.02 −2 −1 Impulse Response Time 2000 4000 6000 8000 Magnitude Frequency 2000 4000 6000 8000 −200 −100 100 200 Phase (degrees) Frequency 2000 4000 6000 8000 −15 −10 −5 Group Delay Frequency −0.5 0.5 1.5 −0.5 0.5 Zeros & Poles Real Imaginary −1 x 10 −4 −1.5

−1 −0.5 0.5 Impulse Response Time FIL TERS 51 Single oles and zeros can combined to build up desired magnitude phase resp onse lter design aking lo w-pass lters as an example, Butterw rth IR lters have magnitude resp onse that deca ys monotonically Cheb yshev yp IR lters have magnitude ripples in the passband only Cheb yshev yp IR lters have magnitude ripples in the stopband only Elliptic IR lters have ripples in oth the pass- and the stopband. Windo ed FIR lters re obtained multiplying the impulse resp onse of an ideal lo

w-pass with windo of nite supp rt. FIL TERS 52 Considering stable systems only FIR lters can have linea phase, but ypically require mo re co ecients to meet sp ecication. IR lters, on the other hand, intro duce nonlinea phase that needs to co rrected using an all-pass lter. All-pass lters have constant magnitude resp onse, ap const: Consider ap cz which has recip ro cal ole and zero that is, ole and zero have the same phase but recip ro cal magnitude. The system is indeed all-pass ecause ap ce the no rm of the multiplier is one and the

numerato and denominato re each otherís conjugate. Mo re complex all-pass lters can obtained multiplying systems of the ab ove fo rm.
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FIL TERS 53 3.6 Minimal-phase systems system with can inverted =H if it exists. oles ecome zeros and vice versa. Assuming the riginal system is causal and stable (p oles inside the unit circle) then its zeros have to lie in the unit circle also to gua rantee causal and stable inverse. Systems with all oles and zeros inside the unit circle re denoted minimal-phase Any system can decomp osed into minimal-phase and an all-pass system: min

ap FIL TERS 54 Pro of: ak causal, stable system with just one zero, =c outside the unit circle with min )(1 =z min )(1 =z c=z c=z min )(1 c=z =z c=z min ap Argument can generalized to mo re than one zero outside the unit circle. Zero outside the unit circle has een mapp ed onto its complex conjugate recip ro cal inside the circle. FIL TERS 55 Comp ensating disto rtions If the disto rting system is causal and stable and the co rrecting system should causal and stable, then erfect reconstruction is ossible i the disto rting system is minimal-phase. If the disto rting system is kno wn can app ro

ximated with rational system, then it can made minimal-phase in combination with an all-pass. The combined disto rtion/comp ensation system then is all-pass and causes phase shift only