2D DSP Basics: Systems Stability, 2D Sampling

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1 - Digital Iage Processig ad Copressio D DSP Basics: Systes Stability D Saplig

2 Stability ty Syste is stable if a bouded iput always results i a bouded output BIBO For LSI syste a sufficiet coditio for stability: h S < D FIR filters always stable Stability test is difficult for IIR filters Z-trasfor is used to study stability

3 D Z rasfor x akes sese whe < { valuesof ad for which < } ROC fiite su coverges N D d fiite su

4 D Z rasfor Note: oly if uit circle ROS a s o ω ω ω ω e e j j y stable syste rasfer fuctio: { } b a Y H h Z Liear costat-coefficiets differece equatio to ipleet IIR syste: x a y b

5 D Z rasfor Aalogous to D -trasfor pole-ero plot is the Root Map i D: H : hold cost. o the Uit e.g. ad fid the roots i e.g. H 0 repeat for all o the Uit Circle Repeat the sae thig with o Uit Circle ad fid roots i We get a pair of plots: Root Map cosists of two parts I{ } I{ } Re{ } Re{ } Stable if root aps stay i Uit Circle

6 Saplig Saplig i oe doai Periodicity i other Replicas ay overlap ad we loose sigal Aliasig If o overlap we ca recover sigal by keepig oe replica -D Saplig: A badliited sigal i oe doai ca be periodically sapled with o loss of iforatio provided d it is sapled ofte eough Miiu saplig rate Nyquist rate is proportioal to the sigal badwidth

7 Saplig D sigals sg as here are ay ways to saple D sigals Aalog Iage Saplig exaple Saplig exaple

8 Saplig D sigals sg as he siplest ad ost coo of this is rectagular saplig Saple t ad t x a t t x a x Cotiuous-doai D sigal discrete tie at regular itervals idices of saples : Horiotal saplig iterval F.. replicated with period / i horiotal directio : Vertical saplig iterval F.. replicated with period / i vertical directio

9 Saplig D sigals sg as he siplest ad ost coo of this is rectagular saplig t x a t t 0 t

10 Saplig D sigals sg as he Bed of Nails t p t t δ t t 0 t

11 Saplig D sigals sg as We ca saple x a t t as follows: x s a t t x t t δ t t a x x a t t δ t t δ t t

12 Saplig D sigals Sa p g sg as the Fourier trasfor is 4 ** 4 a s δ ** 4 a δ a

13 Saplig D sigals sg as Aliasig occurs if we saple below the Nyquist rate critical saplig rate Nyquist rate o aliasig oversaplig rate > Nyquist rate o aliasig usually oversaple saple sice oideal atialiasig filter does ot have to have sharp cutoff replicas ore spaced out udersaplig rate < Nyquist rate Aliasig caot recover sigal

14 Saplig D sigals Sa p g sg as I the frequecy doai DF saple r r t t x a a ω ω r r ω ω b a

15 Saplig D sigals sg as I the frequecy doai ω / - ω ω / - ω -

16 Saplig D sigals sg as Udersaplig gives aliasig: / b / -/ -/ -/ / b -/ / Aliasig

17 Saplig D sigals sg as I the frequecy doai f s f b s b b b

18 Saplig D sigals sg as Hexagoal saplig y 0 x

19 Note about polar frequecy doai Agular frequecy Radial frequecy Relatio to rectagular frequecies

20 Note about polar frequecy doai f cosω 0 f θ cosω +ω ω +ω cst θ he cut of fuctio alog horiotal will be of period he vertical cat will be of period

21 Note about polar frequecy doai cosθ cosθ ω ω0 cosθ cosθ si θ si θ ω ω0 si θ siθ ω + +ω ω 0 horiotal frequecy vertical frequecy radial frequecy Note: Fuctio rotated by agle θ F.. rotated by agle θ

22 Note about polar frequecy doai Relatio betwee cosω 0 old ad cosω ew + ω ew : rotatio by agle θ For a poit rotate back by agle θ to get value of fuctio at that poit old old old ew cosθ + ew siθ cos cos θ si θ siθ cosθ or to express old i ters of ew ew ew ω cos ω cosθ + siθ cos ω + ω 0 old 0 ew ew ew ew ew ew cosθ siθ siθ cosθ old old

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece 6.34 Discrete Time Sigal Processig Fall 24 BACKGROUND EXAM September 3, 24. Full Name: Note: This exam is closed