Application of Combined Fourier Series Transform (Sampling Theorem)
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1 Applicaion o Combined Fourier Serie ranorm Sampling heorem x X[] m m Sampling Frequency Deparmen o Elecrical and Compuer Engineering
2 Deparmen o Elecrical and Compuer Engineering X[] x We need Fourier Serie ]in [ ]co [ [0] X X X x c d d x X [0] in ] [ 0 0 d x X
3 X c 0 [ ] x co 0 d co 0 d co0 Le go bac o Fourier Serie repreenaion x X[0] X c [ ]co X [ ]in X[0] X c [ ]co x co Deparmen o Elecrical and Compuer Engineering
4 Wha abou m? need Fourier ranorm m Fourier ranorm i he andwidh o m Deparmen o Elecrical and Compuer Engineering
5 We now have x a requency repreenaion via Fourier Serie x co And m a Frequency repreenaion via Fourier ranorm m Le muliply m wih x m x m m co Wha i he requency repreenaion o mx? Deparmen o Elecrical and Compuer Engineering
6 Deparmen o Elecrical and Compuer Engineering m m x m co m m F x m F ] co [ ] [ We need o ae Fourier ranorm ] [ ] [
7 F[ m x ] [ ] Deparmen o Elecrical and Compuer Engineering
8 Deparmen o Elecrical and Compuer Engineering
9 Sampling heorem Deparmen o Elecrical and Compuer Engineering
10 Sampling heorem Sampling requency hould be a lea equal o or greaer han wice he bandwidh o he meage ignal or ucceul recovery o he ignal rom i ample Lowpa Filer bandwidh o lowpa iler = Hz Deparmen o Elecrical and Compuer Engineering
11 A pracical problem wih Sampling x X[] No pracical o generae! m m Sampling Frequency Deparmen o Elecrical and Compuer Engineering
12 Le ry Square Wave x X[] Problem Reolved m m Sampling Frequency Deparmen o Elecrical and Compuer Engineering
13 x X[] We need Fourier Serie and We now X[ 0] x X[0] X c X c 4 [ ] [ ]co in X X [ ] [ ]in 0 x 4 in co Deparmen o Elecrical and Compuer Engineering
14 We now have x a requency repreenaion via Fourier Serie x 4 in co And m a Frequency repreenaion via Fourier ranorm m Le muliply m wih x 4 m x m in m co Wha i he requency repreenaion o mx? Deparmen o Elecrical and Compuer Engineering
15 Deparmen o Elecrical and Compuer Engineering m m x m co in 4 We need o ae Fourier ranorm ] co in 4 [ ] [ m m F x m F ] [ in 4 ] [ in
16 F[ m x ] 4 in[ ] Deparmen o Elecrical and Compuer Engineering
17 F[ m x ] 4 in[ ] Lowpa Filer Sampling heorem ill wor wih pracical ample Deparmen o Elecrical and Compuer Engineering
18 Diorion and Linear yem h x y h - Impule Repone o a Linear ime Invarian Syem y x * h Y X H h H Deparmen o Elecrical and Compuer Engineering
19 Diorionle Syem y x d Y X e j d we now Y X H H e j d H H d Deparmen o Elecrical and Compuer Engineering
20 Filer Lowpa Filer H Cu-o Frequency cuo cuo Highpa Filer H Cu-o Frequency cuo cuo H andpa Filer Cener Frequency andwidh cener cener Deparmen o Elecrical and Compuer Engineering
21 Lowpa and Highpa combined o generae andpa Filer Lowpa Filer H Highpa Filer c, LP c, HP c, HP c, LP c, LP c, HP H cener c, LP c, HP cener cener Deparmen o Elecrical and Compuer Engineering
22 Lowpa and Highpa combined o generae andop Filer Lowpa Filer H Highpa Filer c, HP c, LP c, LP c, HP opband c, HP c, LP H cener, opband c, LP opband c, HP cener, opband cener, opband Deparmen o Elecrical and Compuer Engineering
23 How a ignal hould loo lie aer paing hrough a iler Deparmen o Elecrical and Compuer Engineering
24 Pracical Lowpa Filer aic Lowpa Filer x R i C y H jrc Deparmen o Elecrical and Compuer Engineering
25 Phae rad agniude R=0,000 Ohm RC 0mec C= mf R=, 000 Ohm RC mec R=, 000 Ohm C= mf R=0,000 Ohm Frequency in rad/ec Deparmen o Elecrical and Compuer Engineering
26 Single Sage Lowpa Filer R i C H jrc wo Sage Lowpa Filer x R i C i R C y H jrc jrc Deparmen o Elecrical and Compuer Engineering
27 Phae rad agniude Single Sage Dual Sage R=000 Ohm C= mf Dual Sage Single Sage R=000 Ohm C= mf Frequency in rad/ec Deparmen o Elecrical and Compuer Engineering
28 Pracical andpa Filer aic andpa Filer x R i C L y H j j / RC j / RC / LC 0 / LC Deparmen o Elecrical and Compuer Engineering
29 agniude and Phae Repone o PF Deparmen o Elecrical and Compuer Engineering
30 Deparmen o Elecrical and Compuer Engineering D= x[n] y[n] ] [ ] [ ] [ n x n y n y Dicree ime Filer ] [ ] [ n u n h n Impule Repone F e F H raner Funcion by aing DF o h[n]
31 C v. D Lowpa Filer Coninuou ime Dicree ime x R i C y x[n] D= y[n] RC dy d y x y[ n] y[ n ] x[ n] RC dh d h h[ n] h[ n ] [ n] h e / RC u h[ n] u[ n] n Deparmen o Elecrical and Compuer Engineering
32 C v. D Lowpa Filer con. ime Repone h e / RC u h[ n] u[ n] n n Frequency Repone H HF F Deparmen o Elecrical and Compuer Engineering
33 C v. D Lowpa Filer con. H agniude Repone HF F Phae Repone H HF F Deparmen o Elecrical and Compuer Engineering
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