I. Review of 1D continuous and discrete convolution. A. Continuous form: B. Discrete form: C. Example interface and review:

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Lecture : Samplig Theorem ad Iterpolatio Learig Objectives: Review of cotiuous ad discrete covolutio Review of samplig with focus o sigal restoratio Applicatio of

1 Lecture : Samplig Theorem ad Iterpolatio Learig Objectives: Review of cotiuous ad discrete covolutio Review of samplig with focus o sigal restoratio Applicatio of sigal iterpolatio I. Review of D cotiuous ad discrete covolutio A. Cotiuous form: g f h f τ h τ dτ B. Discrete form: m g f h f m h m C. Eample iterface ad review:

2 Lecture : Samplig Theorem ad Iterpolatio II. Retur to the samplig theorem to focus o sigal iterpolatio A. Samplig theorem ad the DFT. For badlimited F with cutoff frequecy ma, the iverse Fourier trasform is fully specified for samples spaced. ma. For f with compact support over the iterval [ L ] specified for samples spaced L, L, the forward Fourier trasform is fully, where L ca be defied as the image field of view FOV.

3 Lecture : Samplig Theorem ad Iterpolatio III f III f where the comb fuctio uder multiplicatio The "siftig" property of δ δ Results i a sampled f. the comb fuctio uder covolutio Replicatio of I III F III f Imposig a bad-limited filter: Nyquist criterio The iff k - F ma ma ma ma - l ma ma f s F rect III F rect Figure : Graphic represetatio of space itervals for sampled fuctio f ad it trasform: B. Restoratio - sic - sic - - ma ma ma f f III F rect f I δ The true cotiuous f is restored by a sort of hybrid discrete to cotiuous covolutio with a cotiuous sic fuctio of width. The sic fuctio is referred to as the ideal restoratio kerel.

4 Lecture : Samplig Theorem ad Iterpolatio Figure : Graphic depictio of ideal restoratio: Small aside: the equal sig i ma does ot hold for restorig odd harmoics: cosπ φ cosπ cos φ + siπ si φ cosider period of cm. samples / cm s.5 cm cos π cos φ + si π si φ ad the odd compoet is lost. cos φ

5 Lecture : Samplig Theorem ad Iterpolatio C. Iterpolatio. Usig the restoratio kerel Imposig compact support o f: N / f N / - sic f For discrete fuctio f, the the restoratio kerel ca be used to iterpolate ay poit betwee [, +] i order to upsample f to fm. For eample, if fm is upsampled by a factor of the, f m' M / m M / δ - m' f m'. This ca be performed by accessig a stadardized look- where M N. up table of sic values at poits. Figure 3: Graphic represetatio of compact support i spatial domai implyig samplig i the trasform domai. Other iterpolatio kerels are possible. Cosiderig that fm is periodic: N / '- ' f circshift N φ f N / where the ' ca be, for eample, at itervals ' / such that f' fm'. I geeral the restoratio kerel ca be a rage of possible fuctios. For eample, the ideal sic kerel, a rect kerel i.e. piel replicatio, or the tri kerel i.e. liear iterpolatio.. Zero-paddig iterpolatio: Cosider a cojugate space represetatio:

6 Lecture : Samplig Theorem ad Iterpolatio Pad or apped zero values such that F is twice as wide F III k F - l - Bad - limitig filter is twice as wide : F III ma ' Note to avoid lots of algebra, I have ot treated F as a discrete fuctio. However, the effect of paddig Fk with zero values to double its legth is aalogous. The samples are ow sythesized at positios f f ' f δ - ' - where '. Figure 3: Graphic depictio of zero-padded iterpolatio i trasform domai leadig to sic iterpolatio i the trasform domai.

7 Lecture : Samplig Theorem ad Iterpolatio Appedi: Restoratio script % Restoratio script L ; d L/.5; % Samplig iterval -:d:-d; f_ si*pi.*/l; d_ew d/; ew -:d_ew:-d_ew; f_ew si*pi.*ew/l; kerel tripulsew,*d; % liear iterpolatio kerel %kerel sicew./d.*kerel_mod; figure;plot,f_,'o',ew,f_ew,'-'; title['sampled ad ',' Sample Rate ', umstr/d,' f_o ', umstr/l]; leged'samples','' figure;plotew,kerel,'-';title'restoratio Kerel' f_restore zeros,legthew; for i :legthf_-, f_cot f_i.*kerel; f_restore f_restore+circshiftf_cot,[ i-*]; ed figure;plotew,f_restore,'-',ew,f_ew,'-'; title['restored Fuctio',' Sample Rate ', umstr/d,' f_o ', umstr/l]; leged'','' fuctio b circshifta,p %CIRCSHIFT Shift array circularly. % B CIRCSHIFTA,SHIFTSIZE circularly shifts the values i the array A % by SHIFTSIZE elemets. SHIFTSIZE is a vector of iteger scalars where % the N-th elemet specifies the shift amout for the N-th dimesio of % array A. If a elemet i SHIFTSIZE is positive, the values of A are % shifted dow or to the right. If it is egative, the values of A % are shifted up or to the left. % % Eamples: % A [ 3;4 5 6; 7 8 9]; % B circshifta, % circularly shifts first dimesio values dow by. % B % 3 % % B circshifta,[ -] % circularly shifts first dimesio values % % dow by ad secod dimesio left by. % B % 3 % % % See also FFTSHIFT, SHIFTDIM. % Copyright The MathWorks, Ic. % $Revisio:..4. $ $Date: 3/5/ :4:8 $ Compariso of Sic Iterpolatio, si*pi.*/l, fs varied:

8 Lecture : Samplig Theorem ad Iterpolatio.8 Sampled ad Sample Rate f o.5 samples Restored Fuctio Sample Rate f o

9 Lecture : Samplig Theorem ad Iterpolatio.8 Sampled ad Sample Rate.5 f o.5 samples Restored Fuctio Sample Rate.5 f o

10 Lecture : Samplig Theorem ad Iterpolatio.8 Sampled ad Sample Rate.5 f o.5 samples Restored Fuctio Sample Rate.5 f o

11 Lecture : Samplig Theorem ad Iterpolatio Compariso of Sic Iterpolatio, si*pi.*/l, fs varied: Sampled ad Sample Rate f o.5 Sampled ad Sample Rate.5 f o.5 Sampled ad Sample Rate.5 f o.5 samples samples samples Restored Fuctio Sample Rate f o.5 Restored Fuctio Sample Rate.5 f o.5 Restored Fuctio Sample Rate.5 f o

12 Lecture : Samplig Theorem ad Iterpolatio Compariso of Trucated Sic at fied fs: Restoratio Kerel Restored Fuctio Sample Rate.5 f o

13 Lecture : Samplig Theorem ad Iterpolatio Restoratio Kerel Restored Fuctio Sample Rate.5 f o

14 Lecture : Samplig ad iterpolatio Compariso of Trucated Sic at fied fs: Restoratio Kerel Restoratio Kerel Restoratio Kerel Restored Fuctio Sample Rate.5 f o.5 Restored Fuctio Sample Rate.5 f o.5 Restored Fuctio Sample Rate.5 f o

15 Lecture : Samplig ad iterpolatio No-ideal Restoratio, NN ad Liear Iterpolatio:.8 Restored Fuctio Sample Rate.5 f o.5.9 Restoratio Kerel Restored Fuctio Sample Rate.5 f o Restoratio Kerel

Finite-length Discrete Transforms. Chapter 5, Sections

Finite-length Discrete Transforms. Chapter 5, Sections Fiite-legth Discrete Trasforms Chapter 5, Sectios 5.2-50 5.0 Dr. Iyad djafar Outlie The Discrete Fourier Trasform (DFT) Matrix Represetatio of DFT Fiite-legth Sequeces Circular Covolutio DFT Symmetry Properties