ECEN 644 HOMEWORK #5 SOLUTION SET

Transcription

1 ECE 644 HOMEWORK #5 SOUTIO SET

2 7. x is a real valued sequece. The first five poits of its 8-poit DFT are: {0.5, j 0.308, 0, j 0.058, 0} To compute the 3 remaiig poits, we ca use the followig property for real valued sequeces: * X X X (page 468 i the boo) I our case =8 ad therefore we have the equatios for X 5, X 6 ad 7 * X X X 0.5 j j * X : * 6 8 X X X 0 0 * * 7 8 X X X 0.5 j j * Hece the complete 8-poit DFT of x is: {0.5, j 0.308, 0, j 0.058, 0, j 0.058, 0, j 0.308}.

3 7.3 X,0, X as: is the -poit DFT of X x, 0. We defie the DFT X, 0 c, c, 0, c c From this defiitio, we ca represet X as the product of lowpass filter H where: H, 0 c, c, 0, c c X with the ideal Hece this leads to the coclusio that x, the iverse -poit DFT of lowpass versio of x. X, is a

4 7.7 X is the -poit DFT of the sequece DFTs of the two sequeces derived from x. We wat to determie the -poit x : xc x 0 cos, 0 xs x 0 si, 0 The DFT of x, X c c, is give by: 0 X c xcos exp j 0 Developig the cosie i the previous equality we get: 0 0 X c x exp j exp j exp j xexp j xexp j 0 0 From the properties of the DFT, this expressio simply becomes: X X X c 0 mod 0 mod Operatig the same way for the sequece x we get the correspodig DFT s X : s X X X j j s 0 mod 0 mod

5 7.3 a) xp of is a periodic sequece with fudametal period. We have the -poit DFT DFT DFT x : x X ad the 3-poit DFT of x : x X p p We wat to fid a expressio for as W exp j. We ca the write: X 3 as a fuctio of X 0 X x W X x W p p. 3. et s first defie 3 W If we develop the previous expressio for X 3 we get: X x W x W x W 3 ( ) ( ) xw xw3 xw xw xw3 W xw3 W x W W W W W xw 0 3 Fially the desired expressio is obtaied: X W W X 3

6 7.3 We have to compute the -poit DFT of 4 sigals: a) x b) x, c) x a, 0, eve x 0 0, eve h) a) x The -poit DFT of x is defied as: X exp j 0 (0) 0exp j Therefore: DFT x, 0 X, 0 b) x 0, 0 0 The -poit DFT of X 0 exp j 0 ( 0 ) 0 exp j ( 0 ) exp j Therefore: x, 0 is defied as: 0 0

7 DFT 0 x 0, 0 0 X exp j, 0 c) x a, 0 The -poit DFT of x a, 0 X a exp j 0 is defied as: a exp j 0 a exp j a exp j a a exp j Therefore: DFT a x a, 0 X, 0 a exp j, eve h) x 0 0, eve, eve The -poit DFT of x 0 is defied as: 0, eve X xexp j 0

8 If we assume odd, the - is eve ad we have: ( ) (4 ) ( ) X exp j exp j... exp j terms i.e., X ( ) exp j ( ) exp j exp j 4 exp j exp j exp j exp j exp j Therefore:, eve DFT x 0 X, 0 0, eve exp j

9 7.8 We are give a discrete-time sigal x a, 0, where a 0.95 ad 0. (a) Here we eed to compute ad plot we have: x The correspodig plot ca be foud below. x. Obviously from the give values of a ad, 0.95, 0 0, x() cos (b) We eed to show that X x x X xexp j which i our case becomes:. By defiitio, we have

10 exp X a j a exp j 0 a exp j a exp j (0) a exp j a exp j a exp j whe i st sum a cos a exp j exp j 0 cos x x The correspodig plot at, 0,,...,, ca be foud below X() (c) We eed to compute c for 30 with c defied as: c X, 0,,...,

11 For 30, c becomes: c X, 0,,..., Usig (b), we ca derive the desired expressio for c for 30 : c x0 x cos, 0,,..., cos, 0,,..., The correspodig plot ca be foud below. 0.6 = c (d) We eed to compute x with x defied as: x c exp j 0 Replacig c by its expaded expressio i the previous equality we get:

12 9 x X exp j X exp j X w 30 0 exp jw Therefore x is the iverse 30-poit DFT of the DFT of plot ca be foud below. x. The correspodig = x ~ () (e) We eed to compute x for 30 with x x x l, l For 30, x becomes: x x 30 l, l defied as:

13 From the correspodig plot below, we ca see that x of x. is a periodic/repeated versio = x ~ () (f) Here we just have to replace by 5 istead of 30 i the previously obtaied equatio. This is trivial so just the ew plots are beig show.. = c

14 .5 = x ~ () = x ~ ()

15 8. To show that exp j, 0 is a th root of uity we just have to show that X for X exp j, 0. This is fairly obvious sice: exp j exp j. Hece, exp j, 0 is a th root of uity. ow if we cosider the sum used i the orthogoality property, we ca rewrite that sum as: l l exp j exp j exp j 0 0 If l, the terms i the sum represet the equally spaced uity roots o the uit circle which add to zero. Mathematical proof if 0 l : l exp j l exp j l exp j l exp j 0

16 If l, the sum adds up to : l 0 exp j exp j A plot of the uitary roots for =4 is show below.

17 8.3 x is a real valued -poit sequece with. X x j 0 x is The -poit DFT of exp which, beig eve, ca be rewritte as: X xw x W, where W exp j 0 xw x W 0 0 ' ' X correspods to the odd harmoics of X, i.e., X X therefore: X x W x W ( ) ' ( ) 0 x W W x W W W 0 ad becausew W. We ca simplify further this expressio usig the fact that W : Fially we get the odd harmoics of ' X xw W x W W 0 X usig the followig formulae: ' X X x x W W 0

18 8.4 We wat to develop a method to compute a 4-poit DFT from three 8-poit DFTs. et Y deote the 4-poit DFT ad,, DFTs. We the have: 3 Y y W y W 0 0 Y Y Y deote the three 8-poit 3 We ca rewrite this sum as three sums that would tae values amog the sets {0, 3, 6,, }, {, 4, 7,, } ad {, 5, 8,, 3} respectively. 3 0,3,6,...,4,7,...,5,8,... Y y W y W y W y 3W y 3 W W y 3 W W y 3W y 3 W W y 3 W W pt DFT 8 pt DFT Y Y W Y W 3 8 pt DFT With three 8-poit DFTs, Y, Y, Y, we ca create a 4-poit DFT Y usig the followig formulae: 3 Y Y Y W Y W 3

19 8.7 We wat to derive the radix- decimatio i time usig the steps 8..6 to 8..8 i the boo. Page 59 i the boo already gives some guidelies o how to proceed such as selectig M ad. ) The first step to follow (8..6) maes us compute the M-poit DFTs F l, q defied as: M mq F l, q x l, m W, 0 l ; 0 q M m0 M Therefore we have two -poit DFTs to compute for l 0 ad l. M mq M m0 m0 mq 0, 0, 0, F q x m W x m W mq m0,, F q x m W ) The secod step (8..7) cosists i computig a ew rectagular array G l, q defied as: lq G l, q W F l, q, 0 l ; 0 q M Therefore we have two rectagular arrays to compute for l 0 ad l. (0) q 0, 0, 0, G q W F q F q M q q mq M M m0 () () G, q W F, q W x, mw

20 3) The third ad last step (8..8) cosists i computig the -poit DFT X p, q defied as: lp X p, q G l, q W, 0 p ; 0 q M l0 Therefore we have two -poit DFTs to compute for p 0 ad p. (0) l 0,,, X q G l q W G l q l0 l0 0,, () l,,, l0 l0 G q G q q 0,, F q W F q X q G l q W G l q W l 0,, G q W G q W 0 G 0, q G, q as W q 0,, F q W F q F 0, q ad F, q here are the same as F ad F boo ad therefore we get the desired radix- decimatio i time: i equatio 8..6 of the X 0, q X F W F X q X F W F,,0