Ch3 Discrete Time Fourier Transform

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1 Ch3 Discrete Time Fourier Trasform 3. Show that the DTFT of [] is give by ( k). e k 3. Determie the DTFT of the two sided sigal y [ ],. 3.3 Determie the DTFT of the causal sequece x[ ] A cos( 0 ) [ ], where A,, ad are real ad We showed that the iverse DTFT hlp[ ] ad its DTFT H ( e ) are give respectively LP by h LP sic [ ], ;, 0 c HLP ( e ) 0, c Determie ad plot the DTFT of g [ ] [ ] h [ ],. LP (Optioal) Determie ad plot the DTFT of the cascade of the LTI discrete time systems with si si two sided impulse resposes give by h[ ] [ ] ad h[ ], respectively, where 0. (Optioal)Let X ( e ) deote the DTFT of a real sequece x [], a) Show that if x [] is eve, the it ca be computed from ( e ) [ ] x X ( e )cos( )d. 0 X usig b) Show that if x [] is odd, the it ca be computed from ( e ) [ ] x X ( e )si( )d Determie the DTFT of the followig sequeces: a) x [ ] [ ], b) [ ] x [ ], X usig

2 , M, c) x3[ ] 0, otherwise d) x4 [ ] ( [ ] [ 4]), 3.6 Determie the DTFT of the followig fiite legth sequeces: a), 0 N, x[ ] 0, otherwise, b), N N, x[ ] N 0, otherwise, c) N, N N, x3[ ] 0, otherwise, d) cos( / N), N N, x4[ ] 0, otherwise, 3.7 Evaluate the iverse DTFT of each of the followig DTFTs: e ( e a) X ( e ) e N b) X ( e ) + cos( l) N l0 ) e ( ), ( e ) c) X3 e d) ( X ) ( ) 4 e k k 3.8 Evaluate the iverse DTFT of each of the followig DTFTs: a) ( X ) si(4 ) e b) X ( e ) cos(4 ) c) X3 ( e ) 43cos 4cos d) ( X e ) ( 4 3cos 4cos )si( / ) e 4 /

3 3.9 Determie the iverse DTFTs of the followig Fourier trasforms.4e.3e 0.4e a) X( e ) 3.5e 0.5e b) X ( e ) 3.0 Let X ( e ) deote the DTFT of a real sequece x [], a) Express the iverse DTFT y [] of Ye / / b) Defie Y ( e ) X ( e ) X ( e ) Y ( e ). i terms of x []; 3 ( ) Xe ( ), determie the iverse DTFT y [] of 3. The magitude fuctio X ( e ) of the discrete time sequece x [] is show i Fig. for a portio of the agular frequecy axis. Sketch the magitude fuctio for the frequecy rage. What type of sequece is x []? Fig. 3. Without computig the DTFT, determie which of the followig sequeces have real valued DTFTs ad which have imagiary valued DTFTs: a), N N, x[ ] 0, otherwise, 0, for eve, b) x[ ], for odd, 0, 0, c) x3[ ] cos, 0, 3.3 Without computig the DTFT, determie which of the DTFT of Fig. has a iverse that is a eve sequece ad which has a iverse that is a odd sequece:

4 Fig. (Optioal) Let x[] ad u [ ] be two real valued sequeces with DTFTs give by X ( e ) ad Ue ( ), respectively. Defie a complex valued y [ ] x [ ] u [ ]. Express the X ( e ) ad Ue ( ) i terms of the DTFT Ye ( ) of y [ ]. If we wish to force Ye ( ) to be equal to zero i the frequecy rage 0 x[] ad u [ ]., how would we geerate [ ] y from (Optioal) Let X ( e ) deote the DTFT of a complex sequece x [], determie the DTFT Y ( e ) of the sequece y[ ] x[ ] x*[ ] i terms of X ( e ), ad show that it is a real valued fuctio of ω. 3.4 Usig Parseval s relatio, evaluate the followig itegrals: 4 a) d 0 5 4cos b) 4 5 4cos 0 d 3.5 Let X ( e ) deote the DTFT of a legth 9 sequece give by x [ ] { 3,,, 0,,, 3}, 3 3 Evaluate the followig fuctios of X ( e ) without computig the trasform itself: 0 a) X ( e ) b) arg{ X( e )} c) X ( e ) d

5 d) X ( e ) d e) f) dx ( e ) d d dx ( e ) d Let G ( e ) deote the DTFT of the sequece g [ ] show i the Fig. 3. Express the DTFTs of the remaiig sequeces i Fig.3 i terms of G ( e ). Do ot evaluate G ( e ). g 4[] Fig Show that the fuctio u [ ] z, where z is a complex costat, is a eigefuctio of a LTI discrete time system. Is v [ ] z[ ] with z a complex costat also a eigefuctio of a LTI discrete time system? 3.8 Cosider the sequece g [ ] (0.4) [ ] with a DTFT Ge ( ). Make use of the symmetry relatios give i Table 3.&3. ad theorems i Table 3.3, determie without evaluatig Ge ( ), the iverse DTFTs of the followig fuctios of Ge ( ) : a) b) X e e G e ( ) 4 ( ) X e G e ( ) ( ( 0.5 ) ) dg( e ) c) X3( e ) d d) ( ) ( X ) 4 e G e im

6 3.9 a) Cosider a LTI discrete time systems with a real ad causal impulse respose h [] ad a frequecy respose H ( e ). Show that h [] ca be determied uiquely from the real part H ( e ) of H ( e ). re b) The real part of the frequecy respose of a real ad causal LTI discrete time system is give by H re ( e ) cos 3cos 4cos3. Determie the impulse respose h [] of the system. 3.0 Evaluate the liear covolutio of the followig sequeces usig the DTFT based method. a) x [ ] { } b) x [ 0 } 0 3. If the iput to each of the followig discrete time systems is x[ ] cos( 0), determie the frequecies preset i their outputs: a) y [ ] si( / 3) x[ ] 3 b) y [ ] x [ ] c) y [ ] x[3 ] 3 3. Determie a closed form expressio for the frequecy respose H ( e ) of the LTI discrete time system characterized by a impulse respose h [ ] [ ] [ R] where. What are the maximum ad miimum values of its magitude respose? How may peaks ad dips of the magitude respose occur i the rage? What are the locatios of the dips? Sketch the magitude ad the phase respose for R a) A ocausal LTI FIR discrete time system is characterized by a impulse respose h[ ] a[ ] [ ] a [ ] [ ]. For what value of the impulse 3 4 respose samples { a}, i4will its frequecy respose H ( e ) have a zero phase? i

7 b) A causal LTI FIR discrete time system is characterized by a impulse respose h [] a[] [ ] a[ ] [ 3] [ 4]. For what value of the impulse respose samples { a}, i 5will its frequecy respose H ( e ) have a liear phase? i (Optioal) Determie the iput output relatio of a factor L up sampler i the frequecy domai. 3.4 The frequecy respose H ( e ) of a legth 4 FIR filter with a real impulse respose has the 0 / followig specific values: He ( ), He ( ) 7 3ad He ( ) 0 Determie its impulse respose h []. (Optioal) a) Desig a legth 3 FIR otch filter with a symmetric impulse respose h [] that is h[ ] h[ ], 0, ad with a otch frequecy at 0.4π ad a 0 db dc gai; b) Determie the exact expressio for the expressio for the frequecy respose of the filter desiged, ad plot its magitude ad phase resposes. 3.5 a) Desig a legth 5 FIR badpass filter with a atisymmetric impulse respose h [] that is h [ ] h[4 ], 0 4 satisfyig the followig magitude respose values: He 0.4 ( ) , ad He ( ) 0.; b) Determie the exact expressio for the expressio for the frequecy respose of the filter desiged, ad plot its magitude ad phase resposes. (Optioal) Cosider the two LTI causal digital filters with impulse resposes give by h [ ] 0.3 [ ] [ ] 0.3 [ ], h [ ] 0.3 [ ] [ ] 0.3 [ ] A B a) Sketch the magitude resposes of the two filters ad compare their characteristics. b) Let ha[] be the impulse respose of a causal digital filter with a frequecy respose H ( e ). Defie aother digital filter whose impulse respose h [] is give by A h [ ] ( ) h [ ], for all. C A What is the relatio betwee the frequecy respose H ( e ) of this ew filter ad the frequecy respose He ( ) of the paret filter? A C C

8 3.6 Show that the group delay ( ) of a LTI discrete time system characterized by a frequecy respose H( e ) ca be expressed as dh ( e ) ( ) Re d He ( ) 3.7 The frequecy respose of a LTI FIR discrete time system is give by He ( ) e e e e. For what relatios betwee the coefficiets , 0i 4will He ( ) have a costat group delay? i 3.8 Determie the expressios for the group delay of each of the LTI systems whose frequecy resposes are give below: a) H ( e ) a e e b) Hb ( e ), < e (Optioal) Usig the equatio i questio above, determie the group delay of the LTI discrete time systems with frequecy resposes give below: a) H ( e ) 0.5e a b) e Hb( e ), < 0.5e (Optioal) Let He ( ) deote the frequecy respose of a LTI discrete time system with a impulse respose h [ ]. LetG( e ) deote the Fourier trasform of the sequece h[ ]. Show that the group delay of the LTI system ca be computed usig Hre ( e ) Gre ( e )+ Him ( e ) Gim ( e ) ( ) He ( ) where ( H ) re e ad ( H ) im e deote the real ad imagiary parts of He ( ), respectively, ad ( G ) re e ad ( G ) im e deote the real ad imagiary parts of Ge ( ), Which oe of the followig fuctio of ca be the DTFT of a discrete time sequece? Justify your aswers. a) ( X ) cos(0.4 ) e

9 b) X ( e ) 3cos(0.75 ) 4cos(0.5 ) c) X3 ( e ) cos(0. ) 3si( ) Let{ x [ ]}, 0 N, be a legth N sequece with a DTFT give by X ( e ) c) Let { xa[ ]}, be a legth M sequece obtaied by zero paddig {[]} x with M N zeros at the ed, i.e., x [ ], for 0 N { xa[ ]} 0, for N M with a DTFT give by X ( e ). What is the relatio betwee X ( e ) ad X ( e ). a d) Let { xb [ ]}, be a legth M sequece obtaied by zero paddig {[]} x with M N zeros at the begiig, i.e., 0, for 0 M N { xa[ ]} x [ ], for M N M with a DTFT give by X ( e ). What is the relatio betwee X ( e ) ad X ( e ). b b a

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