7.17. Determine the z-transform and ROC for the following time signals: Sketch the ROC, poles, and zeros in the z-plane. X(z) = x[n]z n.

Size: px
Start display at page:

Download "7.17. Determine the z-transform and ROC for the following time signals: Sketch the ROC, poles, and zeros in the z-plane. X(z) = x[n]z n."

Transcription

1 Solutions to Additional Problems 7.7. Determine the -transform and ROC for the following time signals: Sketch the ROC, poles, and eros in the -plane. (a) x[n] δ[n k], k > 0 X() x[n] n n k, 0 Im k multiple Re Figure P7.7. (a) ROC (b) x[n] δ[n + k], k > 0 X() k, all Im k multiple Re Figure P7.7. (b) ROC

2 (c) x[n] u[n] X() n0 n, > Im Re Figure P7.7. (c) ROC (d) x[n] ( 4) n (u[n] u[n 5]) X() 4 n0 ( ) n 4 ( 4 ) 5 4 [ ) 5 ] 5 ( 4 4 ( 4 ), all 4 poles at 0, pole at 0 5 eros at 4 ejk π 5 k 0,,,, 4 Note ero for k 0 cancels pole at 4

3 Im 0.5 Re Figure P7.7. (d) ROC (e) x[n] ( 4) n u[ n] X() 0 n (4) n n0 ( ) n 4 4, < 4 Im 0.5 Re Figure P7.7. (e) ROC (f) x[n] n u[ n ] X() n n ( ) n ( ) n

4 Im, < Pole at Zero at 0 Re Figure P7.7. (f) ROC (g) x[n] ( ) n X() n ( ) n + n0 + ( ) n 5 6 ( )( ), < < Im Re Figure P7.7. (g) ROC (h) x[n] ( ) n u[n]+ ( 4) n u[ n ] 4

5 X() n0 ( ) n + n ( ) n 4 +, > and < 4 No region of convergence exists Given the following -transforms, determine whether the DTFT of the corresponding time signals exists without determining the time signal, and identify the DTFT in those cases where it exists: (a) X() 5 +, > ROC includes, DTFT exists. X(e jω ) 5 + e jω (b) X() 5 +, < ROC does not include,, DTFT does not exist. (c) X() ( )(+ ), < ROC does not include,, DTFT does not exist. (d) X() ( )(+ ), < < ROC includes, DTFT exists. X(e jω ) e jω ( e jω )(+e jω ) 7.9. The pole and ero locations of X() are depicted in the -plane on the following figures. In each case, identify all valid ROCs for X() and specify the characteristics of the time signal corresponding to each ROC. (a) Fig. P7.9 (a) X() C( ) ( + 4 )( ) There are 4 possible ROCs () > 4 x[n] is right-sided. 5

6 () < < 4 x[n] is two-sided. () < x[n] is left-sided. (b) Fig. P7.9 (b) X() C( 4 ) ( e j π 4 )( e j π 4 ) There are possible ROCs () > x[n] is right-sided. () < x[n] is two-sided. (c) Fig. P7.9 (c) X() ( )( + )( + 9 )C, < 6 x[n] is stable and left-sided Use the tables of -transforms and -transform properties given in Appendix E to determine the -transforms of the following signals: (a) x[n] ( ) n u[n] n u[ n ] a[n] ( ) n u[n] b[n] n u[ n ] x[n] a[n] b[n] A() B() X(), >, < X() A()B() ( )( ), < < (b) x[n] n (( n ( ) u[n] ) n ) 4 u[n ] a[n] ( ) n u[n] A(), > 6

7 b[n] ( ) n u[n] 4 c[n] b[n ] x[n] n[a[n] b[n]] B(), > 4 4 C() 4 X() d d A()B() X() ), > (c) x[n] u[ n] X() ) (, < (d) x[n] n sin( π n)u[ n] x[n] n sin( π n)u[ n] X() d ( ) d + [ ] + ( ) ( + ) + + ( + ) (e) x[n] n u[n] cos( π 6 n + π/)u[n] a[n] 9 n u[n] 9 A() b[n] cos( π 6 n + π )u[n] [ cos( π 6 n) cos(π) sin(π 6 n) sin(π) ] u[n] b[n] X() A()B() B() cos( π )( + cos( π 6 )) ( 9 cos( π sin( π )( sin( π 6 )) 6 )+ cos( π 6 ) )+ cos( π )( + cos( π 6 )) sin( π ) sin( π 6 ) cos( π 6 )+ 7.. Given the -transform pair x[n] to determine the -transform of the following signals: (a) y[n] x[n ] 6 with ROC < 4, use the -transform properties 7

8 y[n] x[n ] Y () X() 6 (b) y[n] (/) n x[n] y[n] ( )n x[n] Y () X() 4 (c) y[n] x[ n] x[n] y[n] x[ n] x[n] Y () X( )X() (d) y[n] nx[n] y[n] nx[n] Y () d d X() ( 6) (e) y[n] x[n +]+x[n ] y[n] x[n +]+x[n ] Y () ( + )X() + 6 (f) y[n] x[n] x[n ] y[n] x[n] x[n ] Y () X() X() ( 6) 7.. Given the -transform pair n n u[n] X(), use the -transform properties to determine the time-domain signals corresponding to the following transforms: (a) Y () X() Y () X() y[n] ( )n x[n] ( )n n n u[n] (b) Y () X( ) Y () X( ) y[n] x[ n] n n u[ n] 8

9 (c) Y () d d X() Y () d d X() ( d d X() ) y[n] (n )x[n ] (n ) n u[n ] (d) Y () X() (e) Y () [X()] Y () X() y[n] (x[n +] x[n ]) [ (n +) n+ u[n +] (n ) n u[n ] ] y[n] Y () X()X() y[n] x[n] x[n] n y[n] u[n] k k (n k) n k k0 n u[n] n [ k n nk + k 4] k0 7.. Prove the following -transform properties: (a) Time reversal x[n] X( ) y[n] x[ n] Y () x[ n] n let l n n l X( ) x[l]( ) l (b) Time shift x[n n o ] no X() y[n] x[n n o ] Y () x[n n o ] n let l n n o 9 n

10 l ( l no X() x[l] (l+no) x[l] l ) no (c) Multiplication by exponential sequence α n x[n] X( α ) y[n] α n x[n] Y () α n x[n] n n n X( α ) x[n]( α ) n (d) Convolution Let c[n] x[n] y[n] x[n] y[n] C() X()Y () (x[n] y[n]) n n n p ( p ( x[p] p ( x[p]y[n p] n ) n y[n p] (n p) ) } {{ } Y () ) x[p] p Y () } {{ } X() X()Y () p (e) Differentiation in the -domain. nx[n] X() d d X() x[n] n n 0

11 d d X() Therefore nx[n] Differentiate with respect to and multiply by. nx[n] n n d d X() 7.4. Use the method of partial fractions to obtain the time-domain signals corresponding to the following -transforms: + (a) X() 7 6, > ( )(+ ) x[n] is right-sided X() A + B + A + B 7 6 A B X() + + x[n] [( )n ( ] )n u[n] (b) X() ( )(+ ), < same as (a), but x[n] is left-sided x[n] [ ( )n +( )n ] u[ n ] (c) X() ( )(+ ), < < same as (a), but x[n] is two-sided x[n] ( )n u[ n ] ( )n u[n] (d) X() x[n] is two-sided +, < < X() + A + B +

12 A + B A B X() + + x[n] ( )n u[n] () n u[ n ] (e) X() 4 6, > 4 x[n] is right-sided X() A +4 + B 4 A + B 4 4A +4B X() [ 49 x[n] ( 4)n + 47 ] 4n u[n] (f) X() , < x[n] is left-sided X() x[n] δ[n ] + [( ] )n ( ) n u[ n ] (g) X() 4, > x[n] is right-sided X() ( ) x[n] δ[n +]+[( ) n ] u[n +] 7.5. Determine the time-domain signals corresponding to the following -transforms: (a) X() , > 0

13 x[n] δ[n]+δ[n 6]+4δ[n 8] (b) X() 0 k5 k k, > 0 x[n] 0 k5 δ[n k] k (c) X() ( + ), > 0 X() (δ[n]+δ[n ]) (δ[n]+δ[n ]) (δ[n]+δ[n ]) x[n] δ[n]+δ[n ]+δ[n ] + δ[n ] (d) X() , > 0 x[n] δ[n +6]+δ[n +]+δ[n]+δ[n ] + δ[n 4] 7.6. Use the following clues to determine the signal x[n] and rational -transform X(). (a) X() has poles at / and, x[],x[ ], and the ROC includes the point /4. Since the ROC includes the point /4, the ROC is < <. A X() + B + ( ) n x[n] A u[n] B( ) n u[ n ] ( ) x[] A A x[ ] B( ) B ( ) n x[n] u[n] ( ) n u[ n ] (b) x[n] is right-sided, X() has a single pole, and x[0],x[]/. x[n] c(p) n u[n] where c and p are unknown constants. x[0] c (p) 0

14 x[] c (p) p x[n] ( ) n u[n] (c) x[n] is two-sided, X() has one pole at /4, x[ ],x[ ] /4, and X() /. A X() + B 4 c ( ) n x[n] A u[n] B(c) n u[ n ] 4 x[ ] Bc x[ ] 4 Bc c B X() A + 4 A 5 4 x[n] 5 4 ( ) n u[n] + () n u[ n ] Determine the impulse response corresponding to the following transfer functions if (i) the system is stable, or (ii) the system is causal: (a) H() ( )(+ ) H() + + (i) h[n] is stable, ROC < <, ROC includes. h[n] () n u[ n ]+( )n u[n] (ii) h[n] is causal, ROC > h[n] [ n +( )n ] u[n] 4

15 (b) H() 5 6 H() + + (i) h[n] is stable, ROC <, ROC includes. h[n] [ () n ( ) n ] u[ n ] (ii) h[n] is causal, ROC > h[n] [ n +( ) n ] u[n] (c) H() H() 4 ( 4 ) (i) h[n] is stable, ROC > 4, ROC includes. h[n] 6n( 4 )n u[n] (ii) h[n] is causal, ROC > 4 h[n] 6n( 4 )n u[n] 7.8. Use a power series expansion to determine the time-domain signal corresponding to the following -transforms: (a) X() 4, > 4 X() ( 4 ) k k0 5

16 x[n] ( 4 )k δ[n k] k0 { ( ) n 4 n even and n 0 0 n odd (b) X() 4, < 4 X() 4 () k x[n] k0 (k+) (k+) k0 (k+) δ[n +(k + )] k0 (c) X() cos( ), > 0 Note: cos(α) k0 ( ) k (k)! (α)k X() x[n] k0 k0 k0 ( ) k (k)! ( ) k ( ) k (k)! 6k ( ) k δ[n 6k] (k)! (d) X() ln( + ), > 0 Note: ln( + α) ( ) k (α) k k k0 X() x[n] ( ) k ( ) k k ( ) k δ[n k] k k k 6

17 7.9. A causal system has input x[n] and output y[n]. Use the transfer function to determine the impulse response of this system. (a) x[n] δ[n]+ 4 δ[n ] 8 δ[n ], y[n] δ[n] 4δ[n ] X() Y () 4 H() Y () X() h[n] [5( )n ( 4 )n ] u[n] (b) x[n] ( ) n u[n], y[n] 4() n u[n] ( ) n u[n] X() + Y () ( )( ) H() Y () X() h[n] [0() n 7( ] )n u[n] 7.0. A system has impulse response h[n] ( ) n u[n]. Determine the input to the system if the output is given by H() (a) y[n] δ[n 4] Y () 4 X() Y () H() 4 5 x[n] δ[n 4] δ[n 5] 7

18 (b) y[n] u[n]+ ( ) n u[n] Y () + + X() Y () H() x[n] δ[n]+ 6 u[n]+4 ( )n u[n] 7.. Determine (i) transfer function and (ii) impulse response representations for the systems described by the following difference equations: (a) y[n] y[n ] x[n ] Y () ( ) X() H() Y () X() h[n] ( )n u[n ] (b) y[n] x[n] x[n ] + x[n 4] x[n 6] Y () ( + 4 6) X() H() Y () X() h[n] δ[n] δ[n ] + δ[n 4] δ[n 6] (c) y[n] y[n ] 5y[n ] x[n]+x[n ] ( Y () ) 5 (+ )X() H() Y () X() h[n] ( 4 5 ) [( 45 )n + 4 ] n(45 )n u[n] 8

19 7.. Determine (i) transfer function and (ii) difference-equation representations for the systems with the following impulse responses: (a) h[n] ( 4) n u[n ] h[n] H() ( )( ) n u[n ] Y () X() Taking the inverse -transform yields: y[n] 4 y[n ] x[n ] 4 (b) h[n] ( ) n u[n]+ ( ) n u[n ] ( ) n ( ) n h[n] u[n]+ u[n ] H() Y () X() Taking the inverse -transform yields: y[n] 5 6 y[n ] + 6 y[n ] x[n]+ x[n ] x[n ] (c) h[n] ( n ( ) u[n ] + ) n 4 [cos( π 6 n) sin( π 6 n)]u[n] Taking the -transform yields: y[n] ( H() + 4 cos( π 6 ) sin( π 6 ) cos( π 6 )+ 6 +( 5 ) 8 + ( 6 ) 4 + ( + ) 4 + ( 6 + ) 6 4 ) y[n ] + ( x[n]+ ( 5 8 ) x[n ] + ( 6 4 ) y[n ] 4y[n ] ) x[n ] + x[n ] 9

20 (d) h[n] δ[n] δ[n 5] H() 5 Taking the -transform yields: y[n] x[n] x[n 5] 7.. (a) Take the -transform of the state-update equation Eq. (.6) using the time-shift property Eq. (7.) to obtain q() (I A) bx() where q() Q () Q (). Q N () is the -transform of q[n]. Use this result to show that the transfer function of a LTI system is expressed in terms of the state-variable description as H() c(i A) b + D q[n +] Aq[n]+bx[n] q() A q()+bx() q()(i A) bx() q() (I A) bx() y[n] cq[n]+dx[n] Y () c q()+dx() Y () c(i A) bx()+dx() H() c(i A) b + D (b) Determine transfer function and difference-equation representations for the systems described by the following state-variable [ ] descriptions. [ ] Plot the pole and ero locations in the -plane. (i) A 0 0 [ ], b, c, D [] 0 H() c(i A) b + D

21 Imaginary Part Real Part Figure P7.. [ (b)-(i) Pole-Zero ] [ Plot] (ii) A [, b, c 4 0 ], D [0] H() Imaginary Part Real Part Figure P7.. (b)-(ii) Pole-Zero Plot

22 (iii) A [ ] [, b ] [, c 0 ], D [0] H() Imaginary Part Real Part Figure P7.. (b)-(iii) Pole-Zero Plot 7.4. Determine whether each of the systems described below are (i) causal and stable and (ii) minimum phase. (a) H() ero at: poles at: 5 4, 4 (i) Not all poles are inside, the system is not causal and stable. (ii) Not all poles and eros are inside, the system is not minimum phase. (b) y[n] y[n ] 4y[n ] x[n] x[n ] eros at: 0, poles at: ±

23 (i) Not all poles are inside, the system is not causal and stable. (ii) Not all poles and eros are inside, the system is not minimum phase. (c) y[n] y[n ] x[n] x[n ] H() ( ) eros at: 0, poles at: ± (i) Not all poles are inside, the system is not causal and stable. (ii) Not all poles and eros are inside, the system is not minimum phase For each system described below, identify the transfer function of the inverse system, and determine whether it can be both causal and stable. (a) H() H() ( 4) ( ) H inv () ( ) ( 4) poles at: 4 (double) For the inverse system, not all poles are inside, so the system is not causal and stable. (b) H() 8 00 H inv () poles at: (double) For the inverse system, all poles are inside, so the system can be causal and stable. (c) h[n] 0 ( ) n ( u[n] 9 ) n 4 u[n] ( ) H() ( + )( + 4 ) H inv () ( + )( + 4 ) ( )

24 poles at: 0, For the inverse system, not all poles are inside, so the system cannot be both causal and stable. (d) h[n] 4 ( ) n u[n ] 0 ( ) n u[n ] ( + H() ) ( )( ) H inv () ( )( ) ( + ) pole at: For the inverse system, all poles are inside, so the system can be both causal and stable. (e) y[n] 4y[n ] 6x[n] 7x[n ]+x[n ] H inv () X() Y () ( )( + ) poles at: 7 ± j For the inverse system, all poles are inside, so the system can be both causal and stable. (f) y[n] y[n ] x[n] H inv () pole at: 0 For the inverse system, all poles are inside, so the system can be both causal and stable A system described by a rational transfer function H() has the following properties: ) the system is causal; ) h[n] is real; ) H() has a pole at j/ and exactly one ero; 4) the inverse system has two eros; 5) n0 h[n] n 0;6)h[0]. By ) and ) 4

25 poles at ± j By 4) H inv H H has two poles. e±j π H() By 5) h[n] n n0 A( C ) cos( π )+ 4 A( C ) + 4 h[n] n H() n0 Since H() A( C ) +, 4 H() 0 implies C h[n] [ A h[0] A H() h[n] ( ) n cos( π n) A + 4 [( ) n cos( π n) ( ) n sin( π ] n) u[n] ( ) n sin( π ] n) u[n] (a) Is this system stable? The poles are inside, so the system is stable. (b) Is the inverse system both stable and causal? No, the inverse system has a pole at, which is not inside. (c) Find h[n]. h[n] [( ) n cos( π ( ) n n) sin( π ] n) u[n] (d) Find the transfer function of the inverse system. H inv () H() + 4 5

26 7.7. Use the graphical method to sketch the magnitude response of the systems having the following transfer functions: (a) H() H() H(e jω ) ( + j 7 8 )( j 7 8 ) (e jω + j 7 8 )(ejω j 7 8 ) Im 7 8 Re Figure P7.7. (a) Graphical method. 4.5 P7.7 (a) Magnitude Response 4.5 H(e jω ) Ω Figure P7.7. (a) Magnitude Response (b) H() + + 6

27 H() + + H(e jω ) ejω + e jω + e jω poles at: 0, (double) eros at: e ±j π Im Re Figure P7.7. (b) Graphical method. P7.7 (b) Magnitude Response H(e jω ) Ω Figure P7.7. (b) Magnitude Response (c) H() + +(8/0) cos( π 4 ) +(8/00) 7

28 H() + ( 9 0 ej 4 π )( 9 0 e j 4 π ) H(e jω ) e jω + e jω e jω + (8/0) cos( π 4 )ejω + (8/00) eros: poles: 9 0 e±j π 4 e jπ Im 4 Re Figure P7.7. (c) Graphical method. P7.7 (c) Magnitude Response 5 4 H(e jω ) Ω Figure P7.7. (c) Magnitude Response 7.8. Draw block-diagram implementations of the following systems as a cascade of second-order sec- 8

29 tions with real-valued coefficients: (a) H() ( 4 ej π 4 )( 4 e j π 4 )(+ 4 ej π 8 )(+ 4 e j π 8 ) ( ej π )( e j π )( 4 ej 7π 8 )( 4 e j 7π 8 ) H() H ()H () H () cos( π 4 ) + 6 cos( π ) + 4 H () + cos( π 8 ) + 6 cos( 7π 8 ) X() Y() cos( ) cos( ) 4 cos( 7 8 ) cos( ) Figure P7.8. (a) Block diagram. (b) H() (+ ) ( π ej )( π e j ) ( 8 )( π 8 ej )( π 8 ej )(+ 4 ) H() H ()H () H () H () cos( π ) X() Y() cos( ) Figure P7.8. (b) Block diagram. 4 9

30 7.9. Draw block diagram implementations of the following systems as a parallel combination of secondorder sections with real-valued coefficients: (a) h[n] ( n ( ) u[n]+ j ) n ( u[n]+ j ) n ( u[n]+ ) n u[n] H() + j + + j H() H ()+H () X() Y() 0.5 Figure P7.9. (a) Block diagram. (b) h[n] ( ) ej π n ( 4 u[n]+ ) 4 ej π n ( u[n]+ ) 4 e j π n ( u[n]+ ) e j π n 4 u[n] H() ej π ej π + 4 e j π + e j π H() H ()+H ()

31 X() Y() /6 Figure P7.9. (b) Block diagram Determine the transfer function of the system depicted in Fig. P7.40. X() H () H () Y() H () Figure P7.40. System diagram H () 4 H () H () H() H ()H ()+H ()H ()

32 7.4. Let x[n] u[n + 4]. (a) Determine the unilateral -transform of x[n]. x[n] u[n +4] X() u X() n0 n x[n] n n0 (b) Use the unilateral -transform time-shift property and the result of (a) to determine the unilateral -transform of w[n] x[n ]. w[n] x[n ] u W () x[ ] + x[ ] + X() W () Use the unilateral -transform to determine the forced response, the natural response, and the complete response of the systems described by the following difference equations with the given inputs and initial conditions. (a) y[n] y[n ] x[n], y[ ], x[n] ( )n u[n] X() + Y () ( Y ()+ ) X() Y () ( ) +X() Y () + }{{ X() }}{{} Y (n) () Y (f) () Natural Response Y (n) () y (n) [n] ( ) n u[n] Forced Response 6 4 Y (f) 5 () [ ( 6 y (f) [n] ) n + 4 ( ) n ] u[n] 5 5

33 Complete Response y[n] [ ( 6 ) n ( ) n ] u[n] (b) y[n] 9y[n ] x[n ], y[ ],y[ ] 0, x[n] u[n] X() Y () ( Y ()+ ) X() 9 Y () ( 9 ) 9 + X() Y () 9 + X() 9 }{{ 9 }}{{ } Y (n) () Y (f) () Natural Response y (n) [n] 6 Forced Response Y (n) () [( Y (f) () y (f) [n] Complete Response y[n] ) n ( [ ( + ) n ] u[n] + ) n ( ) n ] u[n] [ 9 4 ( ) n 4 ( ) n ] u[n]+ 6 [( ) n ( ) n ] u[n] (c) y[n] 4 y[n ] 8y[n ] x[n]+x[n ], y[ ],y[ ], x[n] n u[n] X() Y () ( Y ()+ ) ( Y ()+ ) X()+ X() 4 8 Y () ( 4 8 ) (+ )X() Y () + ( + )X() 8 ( )( ) ( }{{} )( + 4 ) }{{} Y (n) () Y (f) () Natural Response Y (n) ()

34 y (n) [n] 8 Forced Response Y (f) () y (f) [n] Complete Response y[n] [ ( ) n ( ) n ] u[n] [ ()n ( 5 [ ()n 0 + ) n + 4 ( 4 ) n ] u[n] ( ) n ( ) n ] u[n] 04 4 Solutions to Advanced Problems 7.4. Use the -transform of u[n] and the differentiation in the -domain property to derive the formula for evaluating the sum n a n assuming a <. n0 X() x[n] n n0 let x[n] a n u[n] d d X() nx[n] (n ) n d d X() d ( ) d a Evaluate at d ( ) d a n0 n n n0 n(n )x[n] (n ) n x[n] (n ) n n a n (n ) na n (n ) n a n na n n0 ( n a n d d n0 }{{} d d X() n0 a ) d d a ( a) + a ( a) 4 ( nx[n] (n ) a )

35 a a ( a) A continuous-time signal y(t) satisfies the first-order differential equation d y(t)+y(t) x(t) dt Use the approximation d dt y(t) [y(nt s) y((n )T s )] /T s to show that the sampled signal y[n] y(nt s ) satisfies the first-order difference equation y[n]+αy[n ] v[n] Express α and v[n] in terms of T s and x[n] x(nt s ). y[n] y[n ] +y[n] x[n] T ( ) s y[n] + y[n ] x[n] T s T s y[n] +T s y[n ] T s +T s x[n] α +T s T s v[n] x[n] +T s The autocorrelation signal for a real-valued causal signal x[n] is defined as r x [n] x[l]x[n + l] Assume the -transform of r x [n] converges for some values of. Find x[n] if R x () ( α )( α) where α <. l0 r x [n] x[n] x[ n] Let y[n] x[ n] r x [n] y[k]x[n k] let p k k k 5 x[ k]x[n k]

36 r x [n] x[p]x[n + p] p l x[l]x[n + l] x[l]x[n + l] l0 since x[l] 0, for l<0 R x () X( )X() ( ) α α Implies X() α x[n] α n u[n] The cross-correlation of two real-valued signals x[n] and y[n] is expressed as r xy (n) x[l]y[n + l] l (a) Express r xy [n] as a convolution of two sequences. r xy [n] x[n] y[ n], see previous problem (b) Find the -transform of r xy [n] as a function of the -transforms of x[n] and y[n]. R xy () X()Y ( ) A signal with rational -transform has even symmetry, that is, x[n] x[ n]. (a) What constraints must the poles of such a signal satisfy? x[n] x[ n] Implies X() X( ) X() X( ) 6

37 This means if there is a pole at o, there must also be a pole at o. Hence poles occur in reciprocal pairs. (b) Show that the -transform corresponds to a stable system if and only if n x[n] <. The reciprocal poles are ( ) o, o assume they take the following form: o r o e jθo o e jθo r o If o is inside, its -transform is right-sided and stable. For the pole at o, its corresponding -transform is either right-sided unstable, or left-sided stable. For convergence, the ROC must include the unit circle,, which means the -transforms are exponentially decaying as they approach, respectively. (c) Suppose Determine the ROC and find x[n]. X() (7/4) ( (/4) )( 4 ) For the system to be stable, the pole at 4 must be right-sided, and the pole at 4 must be left sided so their -transforms are exponentially decaying as they approach, respectively. This implies the ROC is 4 < < Consider a LTI system with transfer function H() a a, a < Here the pole and ero are a conjugate reciprocal pair. (a) Sketch a pole-ero plot for this system in the -plane. Im Let a a e j then a * a j e a a Re Figure P7.48. (a) Pole-Zero plot. 7

38 (b) Use the graphical method to show that the magnitude response of this system is unity for all frequency. A system with this characteristic is termed an all-pass system. H(e jω ) a e jω e jω a a e jω e jω a As shown below, H(e jω ) for all Ω. Im H (e j ) +a a a a Re + Im a H (e j ) +a a a Re + Figure P7.48. (b) Magnitude Response. 8

39 H (e j ) H (e j ) + Figure P7.48. (b) Magnitude Response. (c) Use the graphical method to sketch the phase response of this system for a. H(e jω ) ejω e jω arg { H(e jω ) } arg { } { ejω arg e jω } π + arg { e jω } { arg e jω } Im Im pole ero 0.5 Re Re Figure P7.48. (c) Pole/Zero graphical method. 9

40 ero Figure P7.48. (c) Zero phase response. pole Figure P7.48. (c) Pole phase response. H(e j ) Figure P7.48. (c) Phase response. (d) Use the result from (b) to prove that any system with a transfer function of the form H() P k a k a k a k < 40

41 corresponds to a stable and causal all-pass system. Since a k <, if the system is causal, then the system is stable. H(e jω ) H(e jω ) P k a k ejω e jω a k a e jω e jω a a e jω e jω a a pe jω e jω a p P H(e jω a k ) ejω e jω a k k H(e jω ) a e jω a e jω e jω a e jω a a pe jω e jω a p The system is all-pass. (e) Can a stable and causal all-pass system also be minimum phase? Explain. For a stable and causal all-pass system, a k < for all k. Using P: H() a a The ero is a Which implies a > This system cannot also be minimum phase Let H() F ()( a) and G() F ()( a) where 0 <a< is real. (a) Show that G(e jω ) H(e jω ). G() H() G(e jω ) H(e jω ) H(e jω ) 4 a a }{{} all-pass term a a

42 Since a a, see prob 7.48 (b) Show that g[n] h[n] v[n] where V () a a V () is thus the transfer function of an all-pass system (see Problem P7.48). G() H()V () V () a a a a g[n] h[n] v[n] (c) One definition of the average delay introduced by a causal system is the normalied first moment k0 d kv [k] k0 v [k] Calculate the average delay introduced by the all-pass system V (). V () a a a v[k] a k u[k ] aa k u[k] for k 0 v[0] a v [0] a for k v[k] a k a k+ v [k] a k + a k+ a (k )+(k+) a k (a + a ) kv [k] (a + a ) k(a ) k k0 k0 k0 +a4 a ( a ) v [k] a +(a + a ) (a ) k a + +a4 a a k a a, which also follows from Parseval s Theorem. k0 d kv [k] k0 v [k] 4

43 +a4 a ( a ) The transfer function of a LTI system is expressed as H() b M 0 k ( c k ) N k ( d k ) where d k <,k,,...,n, c k <,k,,...,m, and c M >. (a) Show that H() can be factored in the form H() H min ()H ap () where H min () is minimum phase and H ap () is all-pass (see Problem P7.48). H() ( c M ) b M 0 k ( c k ) N k ( d k ) b M 0 k ( c k )( c M ) N k ( d k ) }{{} H min() H min ()H ap () c M c M }{{ } H ap() (b) Find a minimum phase equalier with transfer function H eq () chosen so that H(e jω )H eq (e jω ) and determine the transfer function of the cascade H()H eq (). H eq () H min () so H()H eq () H ap () H()H eq () H ap () H ap () c M c M 7.5. A very useful structure for implementing nonrecursive systems is the so-called lattice stucture. The lattice is constructed as a cascade of two input, two output sections of the form depicted in Fig. P7.5 (a). An M th order lattice structure is depicted in Fig. P7.5 (b). (a) Find the transfer function of a second-order (M ) lattice having c and c 4. X() 0.5 A() Figure P7.5. (a) Lattice Diagram. B() 0.5 Y() 4

44 Y () 4 A() + B() A() X()( + ) B() X()( +) ( Y () 4 + ) 8 + X() H() Y () X() (b) We may determine the relationship between the transfer function and lattice structure by examining the effect of adding a section on the transfer function, as depicted in Fig. P7.5(c). Here we have defined H i () as the transfer function between the input and the output of the lower branch in the i th section and H i () as the transfer function between the input and the output of the upper branch in the i th section. Write the relationship between the transfer functions to the (i ) st and i th stages as [ Hi () H i () ] T() [ Hi () H i () where T() is a two-by-two matrix. Express T() in terms of c i and. ] H i () H i () + c i H i () H i () H i () c i + H i () [ ] [ ][ Hi () c i Hi () H i () c i H i () [ ] c i T() c i ] (c) Use induction to prove that H i () i H i ( ). i Hi () H i () H () + c H () c + H ( )+c H () H ( ) i k assume H k () k H k ( ) i k + Hk+ () Hk ()+c k+ H k () 44

45 H k+ () c k+ Hk ()+H k () substitute H k () k H k ( ) H k+ () (k+) H k ( )+c k+ H k () H k+ () (k+) c k+ H k ( )+H k () H k+ ( ) (k+) c k+ H k ()+H k ( ) H ( ) k+ () (k+) (k+) c k+ H k ()+H k ( ) H k+ () (k+) H k+ ( ) Therefore H i () i H i ( ) (d) Show that the coefficient of i in H i () is given by c i. [ Hi H i ] [ c i c i ][ Hi H i ] H i c i Hi + H i H i () (i ) c i The highest order of ( )inh i () is(i), and the coefficients of i is (c i ), since H i () doesnot contribute to i, therefore the coefficient of i is H i () is given by (c i ). (e) By combining the results of (b) - (d) we may derive an algorithm for finding the c i required by the lattice structure to implement an arbitrary order M nonrecursive transfer function H(). Start with i M so that H M () H(). The result of (d) implies that c M is the coefficient of M in H(). By decreasing i, continue this algorithm to find the remaining c i. Hint: Use the result of (b) to find a two-by-two matrix A() such that [ Hi () H i () ] A() [ Hi () H i () ] [ ] [ ] Hi () Hi () T H i () H i () [ ] [ ] Hi () Hi () A H i () H i () where A T [ c i A ( c i ) c i let H() M b k k k0 () ] 45

46 H M () Where H() is given. From this we have c M b M. The following is an algorithm to obtain all c i s. () c M b M () for i M to, descending compute H i () and H i () from () get c i from H i () end Thus we will have c,c,...c M Causal filters always have a nonero phase response. One technique for attaining ero phase response from a causal filter involves filtering the signal twice, once in the forward direction and the second time in the reverse direction. We may describe this operation in terms of the input x[n] and filter impulse response h[n] as follows. Let y [n] x[n] h[n] represent filtering the signal in the forward direction. Now filter y [n] backwards to obtain y [n] y [ n] h[n]. The output is then given by reversing y [n] to obtain y[n] y [ n]. (a) Show that this set of operations is equivalently represented by a filter with impulse response h o [n] as y[n] x[n] h o [n] and express h o [n] in terms of h[n]. y[n] y [n] h[n] (x[n] h[n]) h[ n] x[n] (h[n] h[ n]) x[n] h o [n] h o [n] h[n] h[ n] (b) Show that h o [n] is an even signal and that the phase response of any system with an even impulse response is ero. h o [ n] h[ n] h[n] h[n] h[ n] h o [ n] h o [n] Which shows that h o [n] is an even signal. Since h o [n] is even, the phase response can be found from the following: H o (e jω ) H o (e jω ) h o [n]e jωn n n h o [n]e jωn 46

47 h o [ n]e jωn n n Ho (e jω ) H o (e jω ) arg { H o (e jω ) } 0 h o [n]e jωn (c) For every pole or ero at β in h[n], show that h o [n] has a pair of poles or eros at β and β. h o [n] h[n] h[ n] so H o () H()H( ) let H() c β H o () c c β β ( c)( c) ( β)( β )β Therefore, H o () has a pair of poles at β, β let H() β p H o () β β p p β ( β)( β ) p ( p)( p )β Therefore, H o () has a pair of eros at β, β 7.5. The present value of a loan with interest compounded monthly may be described in terms of the first-order difference equation ( where ρ + r/ 00 y[n] ρy[n ] x[n] ), r is the annual interest rate expressed as a percent, x[n] is the payment credited at the end of the n th month, and y[n] is the loan balance at the beginning of the n + st month. The beginning loan balance is the initial condition y[ ]. If uniform payments of $c are made for L consecutive months, then x[n] c{u[n] u[n L]}. 47

48 (a) Use the unilateral -transform to show that Y () y[ ]ρ c L n0 n ρ Hint: Use long division to show that L L n0 n Y () which implies y[ ]ρ X() ρ X() c L c ( L+) L c n0 n Y () y[ ]ρ c L n0 n ρ (b) Show that ρ must be a ero of Y () if the loan is to have ero balance after L payments. Y () y[ ]p + c L n0 n ρ The pole at p results in an infinite length y[n] in general. If the loan reaches ero after the L th payment, we have: y[n] 0,n L So, Y () L y[n] n n0 Thus we must have: L L y[ ]ρ c n ( ρ ) y[n] n to cancel the pole at p n0 n0 From polynomial theory, the first term is ero if f( ρ) 0, or ρ + r is a ero of Y () (c) Find the monthly payment $c as a function of the intitial loan value y[ ] and the interest rate r assuming the loan has ero balance after L payments. L y[ ]ρ c ρ n 0 n0 48

49 c ρ L ρ y[ ]ρ c ρ y[ ] ρ L Solutions to Computer Experiments Use the MATLAB command plane to obtain a pole-ero plot for the following systems: (a) H() P7.54(a) Imaginary Part Real Part Figure P7.54. (a)pole-zero plot of H() (b) H()

50 P7.54(b) Imaginary Part Real Part Figure P7.54. (b) Pole-Zero plot of H() Use the MATLAB command residue to obtain the partial fraction expansions required to solve Problem 7.4 (d) - (g). P7.55 : Part (d) : r - p k 0 Part (e) : 50

51 r p 4-4 k 0 Part (f) : r - p k 0 Part (g) : r - p - 5

52 k Use the MATLAB command tfss to find state-variable descriptions for the systems in Problem 7.7. P7.56 : Part (a) : A B 0 C D Part (b) : A 6 0 B 0 5

53 C 5 0 D 5 Part (c) : A B 0 C 4 0 D Use the MATLAB command sstf to find the transfer functions in Problem 7.. P7.57 : Part (b)-(i) : Num Den

54 Part (b)-(ii) : Num 0 0 Den Part (b)-(iii) : Num Den Use the MATLAB command plane to solve Problem 7.5 (a) and (b). 54

55 P7.58(a) Poles Zeros of the inverse system.5 Imaginary Part Real Part Figure P7.58. (a) Pole-Zero plots of the inverse system. P7.58(b) Poles Zeros of the inverse system Imaginary Part Real Part Figure P7.58. (b) Pole-Zero plots of the inverse system. 55

56 7.59. Use the MATLAB command freq to evaluate and plot the magnitude and phase response of the system given in Example 7.. P Magnitude Omega Phase (rad) 0 0 Omega Figure P7.59. Magnitude Response for Example Use the MATLAB command freq to evaluate and plot the magnitude and phase response of the systems given in Problem

57 P7.60(a) Magnitude Omega Phase(rad) 0 0 Omega Figure P7.60. (a) Magnitude and phase response P7.60(b) 0.8 Magnitude Omega Phase(rad) 0 0 Omega Figure P7.60. (b) Magnitude and phase response 57

58 P7.60(c) Magnitude Omega.5 Phase(rad) Omega Figure P7.60. (c) Magnitude and phase response 7.6. Use the MATLAB commands filter and filtic to plot the loan balance at the start of each month n 0,,...L+ for Problem 7.5. Assume that y[ ] $0, 000, L 60, r 0. and the monthly payment is chosen to bring the loan balance to ero after 60 payments. From Problem 7.5: [ c ρ y[ ] ρ L y[ ] 0, 000 L 60 ρ + r.008 c 0.05 b [y[ ]ρ c, c(ones(, 59))] a, ρ] 58

59 0000 P Balance Month Figure P7.6. Monthly loan balance Use the MATLAB command psos to determine a cascade connection of second-order sections for implementing the systems in Problem 7.8. Part (a) : sos Part (b) : sos

60 7.6. A causal discrete-time LTI system has the transfer function H() ( ) ( +) ( j0.5889)( j0.5889)( j0.8)( j0.8) (a) Use the pole and ero locations to sketch the magnitude response. Im a tan a tan double a a double Re Figure P7.6. (a) Pole Zero plot symmetric a a Figure P7.6. (a) Sketch of the Magnitude Response. (b) Use the MATLAB commands ptf and freq to evaluate and plot the magnitude and phase response. 60

61 P7.6(b) 0.8 H(Omega) Omega < H(Omega) : rad 0 0 Omega Figure P7.6. (b) Plot of the Magnitude Response. (c) Use the MATLAB command psos to obtain a representation for this filter as a cascade of two second-order sections with real-valued coefficients. H() ( 0.4( + + ) )( ( + ) ) (d) Use the MATLAB command freq to evaluate and plot the magnitude response of each section in (c). 6

62 P7.6(d) H (Omega) Omega.5 H (Omega) Omega Figure P7.6. (d) Plot of the Magnitude Response for each section. (e) Use the MATLAB command filter to determine the impulse response of this system by obtaining the output for an input x[n] δ[n]. 0. P7.6(e) Impulse Resp Figure P7.6. (e) Plot of the Impulse Response. 6

63 (f) Use the MATLAB command filter to determine the system output for the input x[n] ( + cos( π4 ) n) + cos(π n) + cos(π4 n) + cos(πn) u[n] Plot the first 50 points of the input and output. 6 P7.6(f) 5 4 Input Output Figure P7.6. (f) System output for given input. The system eliminates and cos(πn) terms. It also greatly attenuates the cos( π 4 n) term. The cos( π n) term is also attenuated, so the output is dominated by cos( π 4 n). 6

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Chapter 2 Review of Fundamentals of Digital Signal Processing 2.1 (a) This system is not linear (the constant term makes it non linear) but is shift-invariant (b) This system is linear but not shift-invariant

More information

Chapter 7: The z-transform

Chapter 7: The z-transform Chapter 7: The -Transform ECE352 1 The -Transform - definition Continuous-time systems: e st H(s) y(t) = e st H(s) e st is an eigenfunction of the LTI system h(t), and H(s) is the corresponding eigenvalue.

More information

Generalizing the DTFT!

Generalizing the DTFT! The Transform Generaliing the DTFT! The forward DTFT is defined by X e jω ( ) = x n e jωn in which n= Ω is discrete-time radian frequency, a real variable. The quantity e jωn is then a complex sinusoid

More information

Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )

Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 ) Review: Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q()=0 Zeros:

More information

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Solution Manual for Theory and Applications of Digital Speech Processing by Lawrence Rabiner and Ronald Schafer Click here to Purchase full Solution Manual at http://solutionmanuals.info Link download

More information

Z-TRANSFORMS. Solution: Using the definition (5.1.2), we find: for case (b). y(n)= h(n) x(n) Y(z)= H(z)X(z) (convolution) (5.1.

Z-TRANSFORMS. Solution: Using the definition (5.1.2), we find: for case (b). y(n)= h(n) x(n) Y(z)= H(z)X(z) (convolution) (5.1. 84 5. Z-TRANSFORMS 5 z-transforms Solution: Using the definition (5..2), we find: for case (a), and H(z) h 0 + h z + h 2 z 2 + h 3 z 3 2 + 3z + 5z 2 + 2z 3 H(z) h 0 + h z + h 2 z 2 + h 3 z 3 + h 4 z 4

More information

Digital Signal Processing. Lecture Notes and Exam Questions DRAFT

Digital Signal Processing. Lecture Notes and Exam Questions DRAFT Digital Signal Processing Lecture Notes and Exam Questions Convolution Sum January 31, 2006 Convolution Sum of Two Finite Sequences Consider convolution of h(n) and g(n) (M>N); y(n) = h(n), n =0... M 1

More information

Digital Signal Processing

Digital Signal Processing Digital Signal Processing The -Transform and Its Application to the Analysis of LTI Systems Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Cech

More information

Digital Signal Processing. Midterm 1 Solution

Digital Signal Processing. Midterm 1 Solution EE 123 University of California, Berkeley Anant Sahai February 15, 27 Digital Signal Processing Instructions Midterm 1 Solution Total time allowed for the exam is 8 minutes Some useful formulas: Discrete

More information

Module 4 : Laplace and Z Transform Problem Set 4

Module 4 : Laplace and Z Transform Problem Set 4 Module 4 : Laplace and Z Transform Problem Set 4 Problem 1 The input x(t) and output y(t) of a causal LTI system are related to the block diagram representation shown in the figure. (a) Determine a differential

More information

If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable

If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable 1. External (BIBO) Stability of LTI Systems If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable g(n) < BIBO Stability Don t care about what unbounded

More information

VI. Z Transform and DT System Analysis

VI. Z Transform and DT System Analysis Summer 2008 Signals & Systems S.F. Hsieh VI. Z Transform and DT System Analysis Introduction Why Z transform? a DT counterpart of the Laplace transform in CT. Generalization of DT Fourier transform: z

More information

ECE-314 Fall 2012 Review Questions for Midterm Examination II

ECE-314 Fall 2012 Review Questions for Midterm Examination II ECE-314 Fall 2012 Review Questions for Midterm Examination II First, make sure you study all the problems and their solutions from homework sets 4-7. Then work on the following additional problems. Problem

More information

Discrete Time Systems

Discrete Time Systems Discrete Time Systems Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz 1 LTI systems In this course, we work only with linear and time-invariant systems. We talked about

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 30 Signals & Systems Prof. Mark Fowler Note Set #26 D-T Systems: Transfer Function and Frequency Response / Finding the Transfer Function from Difference Eq. Recall: we found a DT system s freq. resp.

More information

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE)

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE) QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE) 1. For the signal shown in Fig. 1, find x(2t + 3). i. Fig. 1 2. What is the classification of the systems? 3. What are the Dirichlet s conditions of Fourier

More information

Stability Condition in Terms of the Pole Locations

Stability Condition in Terms of the Pole Locations Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., 1 = S h [ n] < n= We now develop a stability

More information

Signals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 2 Solutions

Signals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 2 Solutions 8-90 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 08 Midterm Solutions Name: Andrew ID: Problem Score Max 8 5 3 6 4 7 5 8 6 7 6 8 6 9 0 0 Total 00 Midterm Solutions. (8 points) Indicate whether

More information

SIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals

SIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals SIGNALS AND SYSTEMS Unit IV Analysis of DT signals Contents: 4.1 Discrete Time Fourier Transform 4.2 Discrete Fourier Transform 4.3 Z Transform 4.4 Properties of Z Transform 4.5 Relationship between Z

More information

UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z).

UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z). Page no: 1 UNIT-II Z-TRANSFORM The Z-Transform The direct -transform, properties of the -transform, rational -transforms, inversion of the transform, analysis of linear time-invariant systems in the -

More information

Ch. 7: Z-transform Reading

Ch. 7: Z-transform Reading c J. Fessler, June 9, 3, 6:3 (student version) 7. Ch. 7: Z-transform Definition Properties linearity / superposition time shift convolution: y[n] =h[n] x[n] Y (z) =H(z) X(z) Inverse z-transform by coefficient

More information

The Discrete-Time Fourier

The Discrete-Time Fourier Chapter 3 The Discrete-Time Fourier Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 3-1-1 Continuous-Time Fourier Transform Definition The CTFT of

More information

Digital Signal Processing Lecture 3 - Discrete-Time Systems

Digital Signal Processing Lecture 3 - Discrete-Time Systems Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015 Overview 1 2 3 4 5 6 7 8 Introduction Three components of

More information

Signals and Systems. Spring Room 324, Geology Palace, ,

Signals and Systems. Spring Room 324, Geology Palace, , Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn Chapter 10 The Z-Transform 1) Z-Transform 2) Properties of the ROC of the z-transform 3) Inverse z-transform

More information

Discrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections

Discrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections 3. - 3.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:

More information

Topic 4: The Z Transform

Topic 4: The Z Transform ELEN E480: Digital Signal Processing Topic 4: The Z Transform. The Z Transform 2. Inverse Z Transform . The Z Transform Powerful tool for analyzing & designing DT systems Generalization of the DTFT: G(z)

More information

Z-Transform. 清大電機系林嘉文 Original PowerPoint slides prepared by S. K. Mitra 4-1-1

Z-Transform. 清大電機系林嘉文 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 Chapter 6 Z-Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 z-transform The DTFT provides a frequency-domain representation of discrete-time

More information

University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing

University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing University of Illinois at Urbana-Champaign ECE 0: Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem. Hz z 7 z +/9, causal ROC z > contains the unit circle BIBO

More information

Very useful for designing and analyzing signal processing systems

Very useful for designing and analyzing signal processing systems z-transform z-transform The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems Very useful for designing and analyzing signal processing

More information

EE 3054: Signals, Systems, and Transforms Spring A causal discrete-time LTI system is described by the equation. y(n) = 1 4.

EE 3054: Signals, Systems, and Transforms Spring A causal discrete-time LTI system is described by the equation. y(n) = 1 4. EE : Signals, Systems, and Transforms Spring 7. A causal discrete-time LTI system is described by the equation Test y(n) = X x(n k) k= No notes, closed book. Show your work. Simplify your answers.. A discrete-time

More information

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz Discrete Time Signals and Systems Time-frequency Analysis Gloria Menegaz Time-frequency Analysis Fourier transform (1D and 2D) Reference textbook: Discrete time signal processing, A.W. Oppenheim and R.W.

More information

Solutions: Homework Set # 5

Solutions: Homework Set # 5 Signal Processing for Communications EPFL Winter Semester 2007/2008 Prof. Suhas Diggavi Handout # 22, Tuesday, November, 2007 Solutions: Homework Set # 5 Problem (a) Since h [n] = 0, we have (b) We can

More information

z-transform Chapter 6

z-transform Chapter 6 z-transform Chapter 6 Dr. Iyad djafar Outline 2 Definition Relation Between z-transform and DTFT Region of Convergence Common z-transform Pairs The Rational z-transform The Inverse z-transform z-transform

More information

Properties of LTI Systems

Properties of LTI Systems Properties of LTI Systems Properties of Continuous Time LTI Systems Systems with or without memory: A system is memory less if its output at any time depends only on the value of the input at that same

More information

VALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF INFORMATION TECHNOLOGY. Academic Year

VALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF INFORMATION TECHNOLOGY. Academic Year VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur- 603 203 DEPARTMENT OF INFORMATION TECHNOLOGY Academic Year 2016-2017 QUESTION BANK-ODD SEMESTER NAME OF THE SUBJECT SUBJECT CODE SEMESTER YEAR

More information

ESE 531: Digital Signal Processing

ESE 531: Digital Signal Processing ESE 531: Digital Signal Processing Lec 6: January 30, 2018 Inverse z-transform Lecture Outline! z-transform " Tie up loose ends " Regions of convergence properties! Inverse z-transform " Inspection " Partial

More information

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,

More information

Signal Analysis, Systems, Transforms

Signal Analysis, Systems, Transforms Michael J. Corinthios Signal Analysis, Systems, Transforms Engineering Book (English) August 29, 2007 Springer Contents Discrete-Time Signals and Systems......................... Introduction.............................................2

More information

Linear Convolution Using FFT

Linear Convolution Using FFT Linear Convolution Using FFT Another useful property is that we can perform circular convolution and see how many points remain the same as those of linear convolution. When P < L and an L-point circular

More information

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous

More information

Discrete-Time Fourier Transform (DTFT)

Discrete-Time Fourier Transform (DTFT) Discrete-Time Fourier Transform (DTFT) 1 Preliminaries Definition: The Discrete-Time Fourier Transform (DTFT) of a signal x[n] is defined to be X(e jω ) x[n]e jωn. (1) In other words, the DTFT of x[n]

More information

Digital Signal Processing

Digital Signal Processing COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #21 Friday, October 24, 2003 Types of causal FIR (generalized) linear-phase filters: Type I: Symmetric impulse response: with order M an even

More information

Chapter 13 Z Transform

Chapter 13 Z Transform Chapter 13 Z Transform 1. -transform 2. Inverse -transform 3. Properties of -transform 4. Solution to Difference Equation 5. Calculating output using -transform 6. DTFT and -transform 7. Stability Analysis

More information

Question Paper Code : AEC11T02

Question Paper Code : AEC11T02 Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)

More information

ESE 531: Digital Signal Processing

ESE 531: Digital Signal Processing ESE 531: Digital Signal Processing Lec 6: January 31, 2017 Inverse z-transform Lecture Outline! z-transform " Tie up loose ends " Regions of convergence properties! Inverse z-transform " Inspection " Partial

More information

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science. Fall Solutions for Problem Set 2

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science. Fall Solutions for Problem Set 2 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Issued: Tuesday, September 5. 6.: Discrete-Time Signal Processing Fall 5 Solutions for Problem Set Problem.

More information

Lecture 4: FT Pairs, Random Signals and z-transform

Lecture 4: FT Pairs, Random Signals and z-transform EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 4: T Pairs, Rom Signals z-transform Wed., Oct. 10, 2001 Prof: J. Bilmes

More information

Chap 2. Discrete-Time Signals and Systems

Chap 2. Discrete-Time Signals and Systems Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,

More information

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform.

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Inversion of the z-transform Focus on rational z-transform of z 1. Apply partial fraction expansion. Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Let X(z)

More information

EC Signals and Systems

EC Signals and Systems UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J

More information

Discrete-time Signals and Systems in

Discrete-time Signals and Systems in Discrete-time Signals and Systems in the Frequency Domain Chapter 3, Sections 3.1-39 3.9 Chapter 4, Sections 4.8-4.9 Dr. Iyad Jafar Outline Introduction The Continuous-Time FourierTransform (CTFT) The

More information

Complex symmetry Signals and Systems Fall 2015

Complex symmetry Signals and Systems Fall 2015 18-90 Signals and Systems Fall 015 Complex symmetry 1. Complex symmetry This section deals with the complex symmetry property. As an example I will use the DTFT for a aperiodic discrete-time signal. The

More information

Need for transformation?

Need for transformation? Z-TRANSFORM In today s class Z-transform Unilateral Z-transform Bilateral Z-transform Region of Convergence Inverse Z-transform Power Series method Partial Fraction method Solution of difference equations

More information

Z Transform (Part - II)

Z Transform (Part - II) Z Transform (Part - II). The Z Transform of the following real exponential sequence x(nt) = a n, nt 0 = 0, nt < 0, a > 0 (a) ; z > (c) for all z z (b) ; z (d) ; z < a > a az az Soln. The given sequence

More information

A system that is both linear and time-invariant is called linear time-invariant (LTI).

A system that is both linear and time-invariant is called linear time-invariant (LTI). The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped

More information

Discrete-Time Signals and Systems. Frequency Domain Analysis of LTI Systems. The Frequency Response Function. The Frequency Response Function

Discrete-Time Signals and Systems. Frequency Domain Analysis of LTI Systems. The Frequency Response Function. The Frequency Response Function Discrete-Time Signals and s Frequency Domain Analysis of LTI s Dr. Deepa Kundur University of Toronto Reference: Sections 5., 5.2-5.5 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:

More information

Final Exam January 31, Solutions

Final Exam January 31, Solutions Final Exam January 31, 014 Signals & Systems (151-0575-01) Prof. R. D Andrea & P. Reist Solutions Exam Duration: Number of Problems: Total Points: Permitted aids: Important: 150 minutes 7 problems 50 points

More information

EE123 Digital Signal Processing. M. Lustig, EECS UC Berkeley

EE123 Digital Signal Processing. M. Lustig, EECS UC Berkeley EE123 Digital Signal Processing Today Last time: DTFT - Ch 2 Today: Continue DTFT Z-Transform Ch. 3 Properties of the DTFT cont. Time-Freq Shifting/modulation: M. Lustig, EE123 UCB M. Lustig, EE123 UCB

More information

EEL3135: Homework #4

EEL3135: Homework #4 EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]

More information

LTI Systems (Continuous & Discrete) - Basics

LTI Systems (Continuous & Discrete) - Basics LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying

More information

Module 4. Related web links and videos. 1. FT and ZT

Module 4. Related web links and videos. 1.  FT and ZT Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link

More information

EECS20n, Solution to Mock Midterm 2, 11/17/00

EECS20n, Solution to Mock Midterm 2, 11/17/00 EECS20n, Solution to Mock Midterm 2, /7/00. 5 points Write the following in Cartesian coordinates (i.e. in the form x + jy) (a) point j 3 j 2 + j =0 (b) 2 points k=0 e jkπ/6 = ej2π/6 =0 e jπ/6 (c) 2 points(

More information

Signals and Systems Lecture 8: Z Transform

Signals and Systems Lecture 8: Z Transform Signals and Systems Lecture 8: Z Transform Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2012 Farzaneh Abdollahi Signal and Systems Lecture 8 1/29 Introduction

More information

DIGITAL SIGNAL PROCESSING UNIT 1 SIGNALS AND SYSTEMS 1. What is a continuous and discrete time signal? Continuous time signal: A signal x(t) is said to be continuous if it is defined for all time t. Continuous

More information

Digital Filters Ying Sun

Digital Filters Ying Sun Digital Filters Ying Sun Digital filters Finite impulse response (FIR filter: h[n] has a finite numbers of terms. Infinite impulse response (IIR filter: h[n] has infinite numbers of terms. Causal filter:

More information

Final Exam 14 May LAST Name FIRST Name Lab Time

Final Exam 14 May LAST Name FIRST Name Lab Time EECS 20n: Structure and Interpretation of Signals and Systems Department of Electrical Engineering and Computer Sciences UNIVERSITY OF CALIFORNIA BERKELEY Final Exam 14 May 2005 LAST Name FIRST Name Lab

More information

Z-Transform. x (n) Sampler

Z-Transform. x (n) Sampler Chapter Two A- Discrete Time Signals: The discrete time signal x(n) is obtained by taking samples of the analog signal xa (t) every Ts seconds as shown in Figure below. Analog signal Discrete time signal

More information

Digital Signal Processing, Homework 1, Spring 2013, Prof. C.D. Chung

Digital Signal Processing, Homework 1, Spring 2013, Prof. C.D. Chung Digital Signal Processing, Homework, Spring 203, Prof. C.D. Chung. (0.5%) Page 99, Problem 2.2 (a) The impulse response h [n] of an LTI system is known to be zero, except in the interval N 0 n N. The input

More information

ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing ELEG 305: Digital Signal Processing Lecture 18: Applications of FFT Algorithms & Linear Filtering DFT Computation; Implementation of Discrete Time Systems Kenneth E. Barner Department of Electrical and

More information

2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.

2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5. . Typical Discrete-Time Systems.1. All-Pass Systems (5.5).. Minimum-Phase Systems (5.6).3. Generalized Linear-Phase Systems (5.7) .1. All-Pass Systems An all-pass system is defined as a system which has

More information

Problem Value

Problem Value GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 2-May-05 COURSE: ECE-2025 NAME: GT #: LAST, FIRST (ex: gtz123a) Recitation Section: Circle the date & time when

More information

Discrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section The (DT) Fourier transform (or spectrum) of x[n] is

Discrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section The (DT) Fourier transform (or spectrum) of x[n] is Discrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section 5. 3 The (DT) Fourier transform (or spectrum) of x[n] is X ( e jω) = n= x[n]e jωn x[n] can be reconstructed from its

More information

! z-transform. " Tie up loose ends. " Regions of convergence properties. ! Inverse z-transform. " Inspection. " Partial fraction

! z-transform.  Tie up loose ends.  Regions of convergence properties. ! Inverse z-transform.  Inspection.  Partial fraction Lecture Outline ESE 53: Digital Signal Processing Lec 6: January 3, 207 Inverse z-transform! z-transform " Tie up loose ends " gions of convergence properties! Inverse z-transform " Inspection " Partial

More information

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals. Z - Transform The z-transform is a very important tool in describing and analyzing digital systems. It offers the techniques for digital filter design and frequency analysis of digital signals. Definition

More information

y[n] = = h[k]x[n k] h[k]z n k k= 0 h[k]z k ) = H(z)z n h[k]z h (7.1)

y[n] = = h[k]x[n k] h[k]z n k k= 0 h[k]z k ) = H(z)z n h[k]z h (7.1) 7. The Z-transform 7. Definition of the Z-transform We saw earlier that complex exponential of the from {e jwn } is an eigen function of for a LTI System. We can generalize this for signals of the form

More information

EE Homework 5 - Solutions

EE Homework 5 - Solutions EE054 - Homework 5 - Solutions 1. We know the general result that the -transform of α n 1 u[n] is with 1 α 1 ROC α < < and the -transform of α n 1 u[ n 1] is 1 α 1 with ROC 0 < α. Using this result, the

More information

Z-Transform. The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = g(t)e st dt. Z : G(z) =

Z-Transform. The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = g(t)e st dt. Z : G(z) = Z-Transform The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = Z : G(z) = It is Used in Digital Signal Processing n= g(t)e st dt g[n]z n Used to Define Frequency

More information

Chapter 2 Time-Domain Representations of LTI Systems

Chapter 2 Time-Domain Representations of LTI Systems Chapter 2 Time-Domain Representations of LTI Systems 1 Introduction Impulse responses of LTI systems Linear constant-coefficients differential or difference equations of LTI systems Block diagram representations

More information

New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Final Exam

New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Final Exam New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Name: Solve problems 1 3 and two from problems 4 7. Circle below which two of problems 4 7 you

More information

EE102B Signal Processing and Linear Systems II. Solutions to Problem Set Nine Spring Quarter

EE102B Signal Processing and Linear Systems II. Solutions to Problem Set Nine Spring Quarter EE02B Signal Processing and Linear Systems II Solutions to Problem Set Nine 202-203 Spring Quarter Problem 9. (25 points) (a) 0.5( + 4z + 6z 2 + 4z 3 + z 4 ) + 0.2z 0.4z 2 + 0.8z 3 x[n] 0.5 y[n] -0.2 Z

More information

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform Signals and Systems Problem Set: The z-transform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem - Transfer functions in MATLAB A discrete-time, causal LTI system is described by the

More information

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Spring 2018 Exam #1

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Spring 2018 Exam #1 New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Spring 2018 Exam #1 Name: Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total / 30 points / 20 points / 25 points /

More information

Frequency-Domain C/S of LTI Systems

Frequency-Domain C/S of LTI Systems Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the

More information

Discrete-Time Signals & Systems

Discrete-Time Signals & Systems Chapter 2 Discrete-Time Signals & Systems 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 2-1-1 Discrete-Time Signals: Time-Domain Representation (1/10) Signals

More information

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals z Transform Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals (ii) Understanding the characteristics and properties

More information

DIGITAL SIGNAL PROCESSING. Chapter 3 z-transform

DIGITAL SIGNAL PROCESSING. Chapter 3 z-transform DIGITAL SIGNAL PROCESSING Chapter 3 z-transform by Dr. Norizam Sulaiman Faculty of Electrical & Electronics Engineering norizam@ump.edu.my OER Digital Signal Processing by Dr. Norizam Sulaiman work is

More information

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002.

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002. The Johns Hopkins University Department of Electrical and Computer Engineering 505.460 Introduction to Linear Systems Fall 2002 Final exam Name: You are allowed to use: 1. Table 3.1 (page 206) & Table

More information

Discrete-time signals and systems

Discrete-time signals and systems Discrete-time signals and systems 1 DISCRETE-TIME DYNAMICAL SYSTEMS x(t) G y(t) Linear system: Output y(n) is a linear function of the inputs sequence: y(n) = k= h(k)x(n k) h(k): impulse response of the

More information

Digital Signal Processing Lecture 4

Digital Signal Processing Lecture 4 Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 4 Begüm Demir E-mail:

More information

Digital Signal Processing I Final Exam Fall 2008 ECE Dec Cover Sheet

Digital Signal Processing I Final Exam Fall 2008 ECE Dec Cover Sheet Digital Signal Processing I Final Exam Fall 8 ECE538 7 Dec.. 8 Cover Sheet Test Duration: minutes. Open Book but Closed Notes. Calculators NOT allowed. This test contains FIVE problems. All work should

More information

University Question Paper Solution

University Question Paper Solution Unit 1: Introduction University Question Paper Solution 1. Determine whether the following systems are: i) Memoryless, ii) Stable iii) Causal iv) Linear and v) Time-invariant. i) y(n)= nx(n) ii) y(t)=

More information

E : Lecture 1 Introduction

E : Lecture 1 Introduction E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation

More information

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform z Transform Chapter Intended Learning Outcomes: (i) Represent discrete-time signals using transform (ii) Understand the relationship between transform and discrete-time Fourier transform (iii) Understand

More information

Lecture 2 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE

Lecture 2 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE EEE 43 DIGITAL SIGNAL PROCESSING (DSP) 2 DIFFERENCE EQUATIONS AND THE Z- TRANSFORM FALL 22 Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr

More information

Review of Discrete-Time System

Review of Discrete-Time System Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.

More information

Lecture 19 IIR Filters

Lecture 19 IIR Filters Lecture 19 IIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/10 1 General IIR Difference Equation IIR system: infinite-impulse response system The most general class

More information

Final Exam ECE301 Signals and Systems Friday, May 3, Cover Sheet

Final Exam ECE301 Signals and Systems Friday, May 3, Cover Sheet Name: Final Exam ECE3 Signals and Systems Friday, May 3, 3 Cover Sheet Write your name on this page and every page to be safe. Test Duration: minutes. Coverage: Comprehensive Open Book but Closed Notes.

More information

UNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.

UNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything. UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 9th, 011 Examination hours: 14.30 18.30 This problem set

More information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE EE Topic wise Questions SIGNALS & SYSTEMS www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)

More information

ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin

ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM Dr. Lim Chee Chin Outline Introduction Discrete Fourier Series Properties of Discrete Fourier Series Time domain aliasing due to frequency

More information