2025pr12b

2025pr12b - Echigesggm FWE‘RAQENG fimb bEfiFEngmNfi...

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Unformatted text preview: Echigesggm FWE‘RAQENG fimb bEfiFEngmNfi m;ng MATEWI'ES Wm mmwwwwmeWmWr A {43:553. fir WMX A1%OMM ($0 3‘3; 3 (£1; £3) m <50 +5; 39...;«53 $9....S’ gzflfgs M96915 www, ggelmam . W {NaoLm/wav, Wu? ) €253? Q @fififi‘fi? wgfiamiumfififlx :5} E“ Vim x z ‘ v E a .r Min. y? 32“ app. . : [AA LE m flu, t X [3; 2‘5 “New” 4 “him “~ W‘i 2; ~93 ‘* EN iv‘wmfiflg 1mg? ‘Efw , EA; “Jfi‘nfi? g; me. i“ Karm- i’éil, ‘vgg’fifwéwfl «W: \ \J r“: 5 "i 3- M L“? 3?; é“ ‘ E - z= - ' "h. viii: 9“! ¢M§fi§ W " ; 3&4; ' . * 2 " ““ _ “ ‘ " F M‘ ‘g «hfikvwgm'mj; E . 9: E 2 e ._ .vk .‘ s l f, "- 355? £3“ng a; avmmw waif?“ 5'; rm”; a Him? agfiwfiive gm: f W: E ‘ m3 §hgm fixifl @“§€%&€.€“ &€%' e2“? E‘meE-émmé: 5,63}, $5 v 9‘4 a“. r. . E i ‘3 “ W E r x W 51 . '9 a, i my: a Liz‘sflfi mag-Wig, Mfigfigfi 3 a???“ 2 $15: EA} if.) ? I g E r _ 1;; «—~ 131,} mmé‘ E e? _..._.__....u._..wmwi~}W‘m 2. The Discrete Fourier Transform “"“ F F g Exercises 2.3.]. Observe that (a+ib)(e+id) = (a ~b)d+(e way; + [(a w by: + (e + rifle] r. This means that to compute the product of two complex numbers, we need to compute oniy three res} multipiica~ tions, namely (.5: ~—" by}, (c — dja, and (c —£— dfli. 2.3.2. Let u x {1, 3}, v m (0, 4), andz 2 {1, 0, 3, 4). 2.3 i. Compute :1 and :3. ii. USe part i and equations {2.48) and {2.49) to compute 2. iii. Compute .55 directly and compare the answer with your answer to part ii. j iv. Let M» : (G,1,4,3). Use equations (2.48) and (2.49} to compute 1}}. 2.3.3. Let {69, 61,...18Nfl3} be the Eueiidean basis for 830239), and Jet {273,101, . . . , FN_1} be the Fourier basis. i. Show that émUC) : .ewz’m’m”V for all It. Notice that am is very nearly (up to a reflection and a normalization) an element of the Fourier basis. ii. Show that Fm 2 em. 2.3.4. Let w = (1,0,1,0,1,0,1,0}. Compute 12). Hint: consider .3 :(1, 1,1,1). 2.3.5. Let u 2 (1,2, —1, mi), :1 =(1,~—1,1,—1),and 22(1,},i,~v~1,——1,1,mi,—1). 2_3_ 1. Compute :2 and D. (Suggestion: Use Exercise 2.3.3.} ii. Compute 2. 2.3.6. Suppose u 2 (01,326,672), 12 m [05, t3, 3/, 6), and z = (6:, 05,29, ,8, c, 3/, (1,3). If fi:[2,i,—I,{)} and fi:(3,—2,O,41‘) J finale. EKeru‘Se g. m ottoman Wm M w: mm twwm (Q0; G913 muck?) $4.3m (Q7: agrujaagaa , [WW Movie; ergo 9WM1- EKeteesiflawtwls Dd; or“; was flu . _ .. . .. ofi {'5 N as‘flei: Trm$¥ofm s, Verges Fomfi‘eo 36 Math 2025, Quiz #6 Name: —> 1) Let f = (1, 3, ~11, 1:3,0,1{],3). Write down all the arrays: of length two that you get in the first step of the FFT . 2) Use the FFT to find the Fourier transfonn of the away = (2,4, #8, 6). ...
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2025pr12b - Echigesggm FWE‘RAQENG fimb bEfiFEngmNfi...

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