z TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability
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1 TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSP-G 6. Trnsform Bsics The definition of the trnsform for digitl signl is: -n X x[ n is complex vrile The trnsform of x[, clled X is sid to lie in the domin The trnsform my not e defined for ll vlue of Region of convergence ROC the vlues of for which the trnsform defines DSP-G 6.2
2 Trnsform Bsics cont. The -trnsform is considered n opertor tht trnsforms digitl signl into its domin: { x[ } Z x[ -n X n Z{ } indictes trnsform is tken The inverse trnsform - x[ Z { X } See Exmples 6. ~ 6.5 DSP-G 6.3 Trnsform Bsics cont. For ny signl with finite numer of smples, the ROC is, i.e., ll except See Tle 6. for the trnsform of sic signls DSP-G 6.4
3 Trnsform Bsics cont. DSP-G 6.5 Trnsform Bsics cont. Time shifting property if Z{x[} X, then Z{x[n-]} - X A fctor - in the domin corresponds to one -n smple dely in the time Z{x[ n ]} x[ n ] n domin - m Generl form: x[ m] x[ m] Z{x[n-k]} -k m m X - -m - x[ m] Z{ x[ } - m X DSP-G 6.6 -m -
4 Trnsform Bsics cont. x[ Dely x[n-] x[ Z - x[n-] See Exmples 6.6 & 6.7 DSP-G 6.7 Trnsfer Functions et X e the trnsform of the input x[, Y e the trnsform of the output y[, the trnsfer function H is defined s the rtio of Y to X: Y H X x[ impulse Response h[ y[ x[ h[ X Trnsfer Function H Y H X DSP-G 6.8
5 DSP-G 6.9 Trnsfer Functions cont. Given generl difference eqution: or Tke trnsform on oth sides: or k k k k k n x k n y ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ n x n x n x n y n y n y X X X Y Y Y X Y DSP-G 6. Trnsfer Functions cont. The trnsfer function is: See Exmples 6.8 ~ 6.2 k k k k k k X Y H
6 Trnsfer Functions cont. From the digitl convolution Tke trnsform on oth sides: H k y [ h[ x[ h[ k] x[ n k] Y k h[ k] -k X X Y -k h[ k] Z{ h[ } X k The trnsfer function is the trnsform of the impulse response k h[ k] -k DSP-G 6. Trnsfer Functions cont. The impulse response is the inverse trnsform of the trnsfer function: h[ Z { H } See Exmple 6.3 DSP-G 6.2
7 Trnsfer Functions cont. Finding filter outputs Time domin Computed from the difference eqution Computed y using digitl convolution x[ x[ X Time Domin Difference Eqution impulse Response h[ Domin Trnsfer Function H domin Computed y using the trnsfer function Y H Y H X X y[ y[ x[ h[ Y H X DSP-G 6.3 Trnsfer Functions cont. Summry The trnsform of the output, Y, is the product of the trnsfer function, H, in the domin nd the trnsform of the input, X The output of the filter cn e tken y the inverse trnsform of Y: y[ Z { Y } Convolution theorem convolution in the time domin corresponds to multipliction in the domin DSP-G 6.4
8 DSP-G 6.5 Trnsfer Functions cont. Cscde comintions of filters Prllel comintions of filters See Exmples 6.4 ~ 6.6 DSP-G 6.6 Bck to the Time Domin Stndrd form Stndrd form of the trnsfer function All exponents of in the trnsform e positive The coefficient of the highest power term in oth numertor nd the denomintor e one H
9 Bck to the Time Domin cont. A trnsfer function expressed in stndrd form is rtionl function consisting of numertor polynomil divided y denomintor polynomil Degree of polynomil the highest power in polynomil Proper rtionl function the degree of the numertor polynomil is less thn or equl to the degree of the denomintor polynomil Strictly proper rtionl function the degree of the numertor is less thn the degree of the denomintor Improper rtionl function the degree of the numertor is lrger thn the degree of the denomintor See Exmples 6.7 & 6.8 DSP-G 6.7 Bck to the Time Domin cont. Inverse trnsform Inspection method Power series expnsion long division Prtil frction expnsion Inspection method using sic trnsforms listed in Tle 6. See Exmples 6.9 ~ 6.23 DSP-G 6.8
10 Bck to the Time Domin cont. Power series expnsion using long division If the -trnsform is given s power series of the form: X x[ n -n x[-2] x[ ] x[] x[] x[2] The sequence vlue x[ x re the coefficients of -n See Exmples 6.24 & 6.25 DSP-G 6.9 Bck to the Time Domin cont. Prtil frction expnsion useful for strictly proper rtionl function in stndrd form An exmple: x[ u[n-], h[ -.25 n u[, y[? Step : X Z{ x[ } Z{ u[ n ]} Step 2: Step 3: Y H X n H Z{ h[ } Q Z{ β u[ } Q Z{ u[ } β DSP-G 6.2
11 Bck to the Time Domin cont. Step 4: represent Y y prtil frction expnsion Y Cover-up method for finding the coefficients A nd B A multiply oth side y.25 nd then set.25 B Y A.25 Set.25 : A.2.25 B multiply oth side y - nd then set.25 A A B, set : B DSP-G 6.2 B Bck to the Time Domin cont. Step 4: cont Y Step 5: tke the inverse trnsform of Y y[.2.25 n See Exmples 6.26 ~ 6.29 u[ n ].8u[ n ] DSP-G 6.22
12 Trnsfer Functions nd Stility Poles the vlues of tht mke the denomintor of trnsfer function ero Zeros the vlues of tht mke the numertor of trnsfer function ero Given generl form of trnsfer function H Y X eros:, 2,, poles: p, p 2,, p DSP-G 6.23 Trnsfer Functions nd Stility cont. The trnsfer function cn e written s H K 2 p p p i : eros of the filter p i : poles of the filter K: gin of the filter 2, K -plne the complex plne on which the poles nd eros of the trnsfer function re plotted See Exmples 6.3 ~ 6.32 DSP-G 6.24
13 Trnsfer Functions nd Stility cont. Stle system every ounded input finite in sie produces ounded output x[ < B y[ < B 2, n If filter is unstle, output grows without ound The output from n unstle filter cn chnge drmticlly even when the input chnges y only the smllest mount All useful filters re stle nd n importnt spect of filter design is to gurntee stility DSP-G 6.25 Trnsfer Functions nd Stility cont. If the input x[ is ounded, i.e., x[ < B, n then y [ h[ k] x[ n k] h[ k] x[ n k] B h[ k] k If the impulse response is solutely summle, i.e., if h [ k] < k The system is stle k k DSP-G 6.26
14 Trnsfer Functions nd Stility cont. Fourier trnsform: trnsform: H e n jω H h[ n h[ e -jωn If e jω with ω rel i.e.,, the trnsform ω of h[ corresponds to the Re discrete-time Fourier trnsform of h[ Unit circle circle with rdius one centered t the origin of the plne -n unit circle Im e jω -plne DSP-G 6.27 Trnsfer Functions nd Stility cont. If the Fourier trnsform of h[ converges The ROC of H must include the unit circle For cusl system with rtionl trnsfer function, the ROC is outside the outermost pole If the ROC includes the unit circle,, ll of the poles must e inside the unit circle A cusl digitl filter with rtionl trnsfer function H is stle if nd only if ll of the poles of H lie inside the unit circle i.e., they must ll hve mgnitude smller thn DSP-G 6.28
15 Trnsfer Functions nd Stility cont. Summry Stle ll the poles of the filter re inside the unit circle rginlly stle with some poles on the unit circle Unstle with some poles outside the unit circle The ROC for stle trnsfer function must include the unit circle See Exmples 6.33 & 6.34 DSP-G 6.29 DSP-G 6.3
16 Trnsfer Functions nd Stility cont. First order system A simple first order system is: α α H Hs just one pole t α Requirement for stility: α < n Impulse response: h[ α u[ When α >, h[ grows without ound s n increses When α <, h[ settles down to ero Difference eqution: y [ αy[ n ] x[ The step response settles to constnt vlue y ss in stedy stte: yss αyss yss α DSP-G 6.3 Trnsfer Functions nd Stility cont. DSP-G 6.32
17 Trnsfer Functions nd Stility cont. α > α < DSP-G 6.33 Trnsfer Functions nd Stility cont. α > α < See Exmples 6.35 & 6.36 DSP-G 6.34
18 Trnsfer Functions nd Stility cont. Second order system The trnsfer function of simple second order system is: 2 2 H 2 2 α β α β p p p nd p 2 re the two poles of the trnsfer function Hs two eros t Requirement for stility: p < nd p 2 < 2 DSP-G 6.35 Trnsfer Functions nd Stility cont. Second order system Difference eqution: y [ αy[ n ] βy[ n 2] x[ The stedy stte vlue y ss cn e predicted y: y ss αy βy yss See Exmples 6.37 ~ 6.4 ss ss α β DSP-G 6.36
19 Trnsfer Functions nd Stility cont. DSP-G 6.37 Trnsfer Functions nd Stility cont. DSP-G 6.38
20 Trnsfer Functions nd Stility cont. DSP-G 6.39 Trnsfer Functions nd Stility cont. DSP-G 6.4
21 Trnsfer Functions nd Stility cont. DSP-G 6.4 Trnsfer Functions nd Stility cont. DSP-G 6.42
22 Trnsfer Functions nd Stility cont. DSP-G 6.43 Trnsfer Functions nd Stility cont. DSP-G 6.44
23 Trnsfer Functions nd Stility cont. The mgnitudes of the poles hve lrge impct on the time it tkes the system to settle to its finl vlue The closer pole is to the edge of the unit circle, the longer it tkes for the output to settle The closer pole is to the center of the unit circle, the fster the output settles The mgnitudes of the eros cn modify the ehvior of the output drmticlly The closer the eros re to the poles, the greter their effect on system ehvior DSP-G 6.45 Trnsfer Functions nd Stility cont. DSP-G 6.46
24 Trnsfer Functions nd Stility cont. DSP-G 6.47 Trnsfer Functions nd Stility cont. DSP-G 6.48
25 Trnsfer Functions nd Stility cont. DSP-G 6.49
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