Hacia un marco regulativo de la participación

Participação como um Gerador e Garante da Democracia e da Cidadania

4. Hacia un marco regulativo de la participación

2013 ONE MARK

5.1 Two systems with impulse responses h t1^ h and h t2^ h are connected in cascade. Then the overall impulse response of the cascaded system is given by

5.4 A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency which is not valid is

(A) 5 kHz (B) 12 kHz

5.5 Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

(A) All the poles of the system must lie on the left side of the jw axis

(B) Zeros of the system can lie anywhere in the s-plane (C) All the poles must lie within s =1

(D) All the roots of the characteristic equation must be located on the left side of the jw axis.

5.6 Assuming zero initial condition, the response y t^ h of the system given below to a unit step input u t^ h is

(A) u t^ h (B) tu t^ h

2 ^ h (D) e u t^-^t ^ h

5.7 Let g t^ h=e^-^p^t², and h t^ h is a filter matched to g t^ h. If g t^ h is applied as input to h t^ h, then the Fourier transform of the output is

(A) e^-^p^f² (B) e^-^p^{f 2}²^/

2013 TWO MARKS

5.8 The impulse response of a continuous time system is given by h t^ h=d^t-1h+d^t-3h. The value of the step response at t=2 is

(A) 0 (B) 1

5.9 A system described by the differential equation dt

5.10 A system described by a linear, constant coefficient, ordinary, first

order differential equation has an exact solution given by y t^ h for t>0, when the forcing function is x t^ h and the initial condition is y 0^ h. If one wishes to modify the system so that the solution becomes -2y t^ h for t>0, we need to

(A) change the initial condition to -y 0^ h and the forcing function to 2x t^ h

(B) change the initial condition to 2 0y^ h and the forcing function to -x t^ h of its z-transform in the z-plane will be

(A) 31 < z <3 (B) 31 z 12

< <

2012 TWO MARKS

5.14 The input x t( ) and output y t( ) of a system are related as

( ) ( )cos(3 ) y t = ^tx t t td

-3

#

. The system is

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(A) time-invariant and stable (B) stable and not time-invari-ant

5.15 The Fourier transform of a signal h t( ) is H j( )w =(2cosw)(sin2w w)/

. The value of h 0( ) is

(A) 1 4 (B) / 1 2/

5.16 Let y n[ ] denote the convolution of h n[ ] and g n[ ], where

[ ] ( / ) [ ]

h n = 1 2 ⁿu n and g n[ ] is a causal sequence. If y[0]=1 and [1] 1/2,

y = then g[1] equals

(A) 0 (B) 1 2/

2011 ONE MARK

5.17 The differential equation 100^{d y}_dt2 20^dy_dt y x t( )

2 - + = describes a system with an input x t( ) and an output y t( ). The system, which is initially relaxed, is excited by a unit step input. The output y t^ h can be represented by the waveform

5.18 The trigonometric Fourier series of an even function does not have

the

(A) dc term (B) cosine terms

5.19 A system is defined by its impulse response h n( )=2ⁿu n( -2). The system is

(A) stable and causal (B) causal but not stable (C) stable but not causal (D) unstable and non-causal

5.20 If the unit step response of a network is (1-e^-^a^t), then its unit

impulse response is

(A) ae^-^a^t (B) a^-¹e^-^a^t

2011 TWO MARKS

5.21 An input x t( )=exp( 2 ) ( )- t u t +d(t-6) is applied to an LTI system with impulse response h t( )=u t( ) . The output is

(A) [1-exp(-2t u t)] ( )+u t( +6) (B) [1-exp(-2t u t)] ( )+u t( -6) (C) 0 5 1. [ -exp(-2t u t)] ( )+u t( +6)

(D) 0 5 1. [ -exp(-2t u t)] ( )+u t( -6)

5.22 Two systems H Z1( )andH Z2( ) are connected in cascade as shown

below. The overall output y n( ) is the same as the input x n( ) with a one unit delay. The transfer function of the second system H Z2( ) is

(A) ( . . )

z zz

1 0 4 1 0 6

1 1

-- (B)

( . )

z z

1 0 4 1 0 6

1 1

-(C) ( . )

( . )

z z

1 0 6 1 0 4

1 1

(D)

( . )

z z

z 1 0 6 1 0 4

1 1

-- -

-5.23 The first six points of the 8-point DFT of a real valued sequence are

5, 1-j3, 0, 3-j4, 0 and 3+j4. The last two points of the DFT are respectively

(A) 0, 1-j3 (B) 0, 1+j3

2010 ONE MARK

5.24 The trigonometric Fourier series for the waveform f t( ) shown below

contains

(A) only cosine terms and zero values for the dc components (B) only cosine terms and a positive value for the dc components (C) only cosine terms and a negative value for the dc components (D) only sine terms and a negative value for the dc components

5.25 Consider the z-transform x z( ) =5z²+4z^-¹+3; 0< z <3. The inverse z- transform x n[ ] is

(A) 5 [d n+2]+3 [ ]d n +4d[n-1]

(B) 5 [d n-2]+3 [ ]d n +4 [d n+1]

5.26 Two discrete time system with impulse response h n1[ ]=d[n-1]

and h n2[ ]=d[n-2] are connected in cascade. The overall impulse response of the cascaded system is

(A) d[n-1]+d[n-2] (B) d -[n 4]

5.27 For a N-point FET algorithm N=2^m which one of the following

statements is TRUE ?

(A) It is not possible to construct a signal flow graph with both input and output in normal order

(B) The number of butterflies in the m^th stage in N/m (C) In-place computation requires storage of only 2N data

(D) Computation of a butterfly requires only one complex multipli-cation.

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2010 TWO MARKS

5.28 Given ( )

5.29 A continuous time LTI system is described by

( ) 4 ( ) 3 ( )

Assuming zero initial conditions, the response y t( ) of the above system for the input x t( )=e u t^-²^t ( ) is given by

(A) (e^t-e u t³^t) ( ) (B) (e^-^t-e^-³^t) ( )u t (C) (e^-^t+e^-³^t) ( )u t (D) (e^t+e u t³^t) ( )

5.30 The transfer function of a discrete time LTI system is given by

( )

-Consider the following statements:

S1: The system is stable and causal for ROC: z >1 2/ S2: The system is stable but not causal for ROC: z <1/4 S3: The system is neither stable nor causal for ROC:

/ z /

1 4< <1 2

Which one of the following statements is valid ?

(A) Both S1 and S2 are true (B) Both S2 and S3 are true (C) Both S1 and S3 are true (D) S1, S2 and S3 are all true

2009 ONE MARK

5.31 The Fourier series of a real periodic function has only

(P) cosine terms if it is even (Q) sine terms if it is even (R) cosine terms if it is odd (S) sine terms if it is odd

Which of the above statements are correct ?

(A) P and S (B) P and R

5.32 A function is given by ( )f t =sin²t+cos2t. Which of the following is true ?

(A) f has frequency components at 0 and 21

p Hz (B) f has frequency components at 0 and 1

p Hz (C) f has frequency components at

p and 1pHz (D) f has frequency components at .

0 12

p and 1p Hz

5.33 The ROC of z -transform of the discrete time sequence

x n ( ) 31 u n( ) u( n 1)

2009 TWO MARKS

5.34 Given that ( )F s is the one-side Laplace transform of ( )f t , the Laplace Which of the following is correct?

(A) Only S2 is true

(B) Both S1 and S2 are false

5.36 Consider a system whose input x and output y are related by the

equation ( )y t = x t( -t) ( )g 2t td

3 3

#

- where ( )h t is shown in the graph.

Which of the following four properties are possessed by the system

BIBO : Bounded input gives a bounded output.

Causal : The system is causal, LP : The system is low pass.

LTI : The system is linear and time-invariant.

(A) Causal, LP (B) BIBO, LTI (C) BIBO, Causal, LTI (D) LP, LTI

5.37 The 4-point Discrete Fourier Transform (DFT) of a discrete time

sequence {1,0,2,3} is

5.39 The input and output of a continuous time system are respectively

denoted by ( )x t and ( )y t . Which of the following descriptions corresponds to a causal system ?

(A) ( )y t =x t( -2)+x t( + (B) 4) y t( )=(t-4) (x t+1) (C) ( )y t =(t+4) (x t- (D) 1) y t( )=(t+5) (x t+5)

5.40 The impulse response ( )h t of a linear time invariant continuous

time system is described by ( )h t =exp( ) ( )at u t +exp( ) (bt u - t) where (u - denotes the unit step function, and a and b are real t) constants. This system is stable if

(A) a is positive and b is positive (B) a is negative and b is negative (C) a is negative and b is negative (D) a is negative and b is positive

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2008 TWO MARKS

5.41 A linear, time - invariant, causal continuous time system has a

rational transfer function with simple poles at s=- and s2 =- 4 and one simple zero at s=- . A unit step ( )1 u t is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is

(A) [exp(-2t)+exp(-4t u t)] ( )

Two of the angular frequencies at which its Fourier transform be-comes zero are

(A) , 2p p (B) 0.5 , 1.5p p

(C) 0, p (D) 2 , 2.5p p

5.43 A discrete time linear shift - invariant system has an impulse

response [ ]h n with [ ]h 0 =1, [ ]h 1 =-1, [ ]h 2 = and zero otherwise 2, The system is given an input sequence [ ]x n with [ ]x 0 =x[ ]2 = , 1 and zero otherwise. The number of nonzero samples in the output sequence [ ]y n , and the value of [ ]y 2 are respectively

(A) 5, 2 (B) 6, 2

5.44 Let ( )x t be the input and ( )y t be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, R3, R4

Properties Relations and ( )X k form N-point Discrete Fourier Transform (DFT) pairs.

The DFT ( )Y k of the sequence ( )y n N1 x r x n( ) ( r)

Statement for Linked Answer Question 6.31 and 6.32:

In the following network, the switch is closed at t =0^- and the sampling starts from t= . The sampling frequency is 10 Hz.0

5.46 The samples ( ),x n n =(0, 1, 2,...) are given by

(A) 5(1-e^-^{0 05}^{. n}) (B) 5e^-^{0 05}^{. n}

5.47 The expression and the region of convergence of the z -transform of

the sampled signal are (A) , Statement for Linked Answer Question 6.33 & 6.34:

The impulse response ( )h t of linear time - invariant continuous time system is given by ( )h t =exp( 2 ) ( )- t u t , where ( )u t denotes the unit step function.

5.48 The frequency response ( )H w of this system in terms of angular

2007 TWO MARKS

5.51 The 3-dB bandwidth of the low-pass signal e u t^-^t ( ), where ( )u t is the unit step function, is given by

(A) 21

5.54 A Hilbert transformer is a

(A) non-linear system (B) non-causal system (C) time-varying system (D) low-pass system

5.55 The frequency response of a linear, time-invariant system is given by

( )

H f = ₁₊_j⁵₁₀_p_f. The step response of the system is (A) 5(1-e^-⁵^t) ( )u t (B) 561-e^-^t⁵@u t( ) (C) (112 -e^-5t) ( )u t (D) 51 1^ -e^-^t⁵hu t( )

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2006 TWO MARKS

5.60 Consider the function ( )f t having Laplace transform

( )

(A) linear, stable and invertible

(B) non-linear, stable and non-invertible (C) linear, stable and non-invertible (D) linear, unstable and invertible

5.62 The unit step response of a system starting from rest is given by

( ) 1 system the steady-state value of the output for unit step input is equal to

unit step function ( )u t are respectively,

(A) , x t( )

-5.66 The region of convergence of z - transform of the sequence

( ) ( 1)

5.67 Which of the following can be impulse response of a causal system ?

5.68 Let x n( )=( )₂¹ ⁿu n y n( ), ( )=x n²( ) and (Y e^jw) be the Fourier

2005 TWO MARKS

5.70 The output ( )y t of a linear time invariant system is related to its

input ( )x t by the following equations

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( )y t =0 5. (x t- +td T)+x t( -td)+0 5. (x t- +td T) The filter transfer function ( )H w of such a system is given by (A) (1+cos T ew ) ^-^{j t}^w^d

(B) (1+0.5cos T ew ) ^-^{j t}^w^d (C) (1-cos T ew ) ^-^{j t}^w^d (D) (1-0.5cos T ew ) ^-^{j t}^w^d

5.71 Match the following and choose the correct combination.

Group 1

E. Continuous and aperiodic signal F. Continuous and periodic signal G. Discrete and aperiodic signal H. Discrete and periodic signal

Group 2

1. Fourier representation is continuous and aperiodic 2. Fourier representation is discrete and aperiodic 3. Fourier representation is continuous and periodic 4. Fourier representation is discrete and periodic (A) E-3, F-2, G-4, H- 1

(B) E-1, F-3, G-2, H-4 (C) E-1, F-2, G-3, H-4 (D) E-2, F-1, G-4, H-3

5.72 A signal ( )x n =sin(w0n+f) is the input to a linear time- invariant system having a frequency response (H e^jw). If the output of the system Ax n( -n0) then the most general form of +H e( ^j^w) will be (A) n- 0 0w + for any arbitrary realb

(B) n- 0 0w +2pk for any arbitrary integer k (C) n0 0w +2pk for any arbitrary integer k (D) n- 0 0w f

Statement of linked answer question 6.59 and 6.60 : A sequence ( )x n has non-zero values as shown in the figure.

5.73 The sequence ( )y n

( ), ,

x n

n 1

For even For odd

*

- ^{will be}

5.74 The Fourier transform of ( )y n2 will be

(A) e^-^j2^w[cos4w+2cos2w+2] (B) cos2w+2cosw+2 (C) e^-^j^w[cos2w+2cosw+2] (D) e^-^j2^w[cos2w +2cos+2]

5.75 For a signal ( )x t the Fourier transform is ( )X f . Then the inverse Fourier transform of (X f3 + is given by2)

(A) x t e 21

2 ^{j t}^3p

` j (B) x t e

31 3

j t 3 - 4p

` j

2004 ONE MARK

5.76 The impulse response [ ]h n of a linear time-invariant system is given by [ ]h n =u n[ +3]+u n[ -2)-2n n[ - where [ ]7] u n is the unit step sequence. The above system is

(A) stable but not causal (B) stable and causal

5.77 The z -transform of a system is ( )H z = _z_-^z_{0 2}_. . If the ROC is z <0 2. , then the impulse response of the system is

(A) ( . )0 2 ⁿu n[ ] (B) ( . )0 2 ⁿu[- -n 1] (C) ( . )- 0 2 ⁿu n[ ] (D) -( . )0 2 ⁿu[- -n 1]

5.78 The Fourier transform of a conjugate symmetric function is always

(A) imaginary (B) conjugate anti-symmetric

2004 TWO MARKS

5.79 Consider the sequence [ ]x n = - -[ 4 j51+j25]

-. The conjugate anti-symmetric part of the sequence is

(A) [ 4- -j2.5, 2, 4j -j2.5] (B) [-j2.5, 1, 2.5]j (C) [-j2.5, 2, 0]j (D) [ 4, 1, 4]

-5.80 A causal LTI system is described by the difference equation

[ ] y n

2 = ay n[ -2]-2 [ ]x n +bx n[ -1]

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The system is stable only if

(A) a = , 2 b <2 (B) a >2, b >2

5.84 The Fourier series expansion of a real periodic signal with

fundamental frequency f0 is given by ( )g tp c e is applied as an input to a linear, time-invariant system with the impulse response ( )h n =2d(n- where3)

2003 TWO MARKS

5.87 Let P be linearity, Q be time-invariance, R be causality and S be

stability. A discrete time system has the input-output relationship,

( )

The system under consideration is an RC low-pass filter (RC-LPF) with R= kW and 1 C=1 0. mF.

5.88 Let ( )H f denote the frequency response of the RC-LPF. Let f1 be

the highest frequency such that

( )

5.91 Which of the following cannot be the Fourier series expansion of a

periodic signal?

2002 TWO MARKS

5.94 The Laplace transform of continuous - time signal ( )x t is ( )X s = _s2⁵_{- -}^-_s^s₂

. If the Fourier transform of this signal exists, the ( )x t is (A) e u t²^t ( )-2e u t^-^t ( ) (B) -e u²^t (- +t) 2e u t^-^t ( ) (C) -e u²^t (- -t) 2e u t^-^t ( ) (D) e u²^t (- -t) 2e u t^-^t ( )

5.95 If the impulse response of discrete - time system is

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- and the system is unstable (D) z z impulse response of the system is

(A) ( *t² e^-²^t) ( )u t (B) ( *t e u t²^t) ( ) (C) (te t u t^-² ) ( ) (D) (te^-²^t) ( )u t

5.97 The region of convergence of the z - transform of a unit step function

2001 TWO MARKS

5.100 The impulse response functions of four linear systems S1, S2, S3, S4

are given respectively by ( ) time invariant, causal, and stable?

(A) S1 (B) S2

(A) linear and time - invariant (B) linear and time varying

5.104 A linear time invariant system has an impulse response ,e t²^t >0. If the initial conditions are zero and the input is e³^t, the output for t> is0

(A) e³^t-e²^t (B) e⁵^t (C) e³^t+e²^t (D) None of these

2000 TWO MARKS

5.105 One period ( , )0 T each of two periodic waveforms W1 and W2 are shown in the figure. The magnitudes of the n^th Fourier series coefficients of W1 and W2, for n$1,n odd, are respectively proportional to

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(B) a imaginary and odd function of w (C) an imaginary and even function of w (D) a real and odd function of w

1999 TWO MARKS

5.111 The Fourier series representation of an impulse train denoted by

( )

5.112 The z-transform of a signal is given by

( )

-Its final value is

(A) 1/4 (B) zero

(A) cannot be determined (B) is zero (C) is unity (D) is infinite

5.114 The trigonometric Fourier series of a even time function can have

only

(A) cosine terms (B) sine terms

5.115 A periodic signal x t( ) of period T0 is given by

5.116 The unit impulse response of a linear time invariant system is the

unit step function u t( ). For t>0, the response of the system to an

fundamental, second harmonic, third harmonic,... respectively. The total harmonic distortion is

(A) ....

5.122 The trigonometric Fourier series of an even function of time does

not have the

(A) dc term (B) cosine terms

5.123 The Fourier transform of a real valued time signal has

(A) odd symmetry (B) even symmetry (C) conjugate symmetry (D) no symmetry

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SOLUTIONS

5.1 Option (C) is correct.

If the two systems with impulse response h t1^ h and h t2^ h are connected in cascaded configuration as shown in figure, then the overall response of the system is the convolution of the individual impulse responses.

5.2 Option (C) is correct.

Given, the input

x t^ h =u t^ -1h It’s Laplace transform is

X s^ h =es^-^s

The impulse response of system is given h t^ h =t u t^ h

Its Laplace transform is H s^ h

= 2

Hence, the overall response at the output is Y s^ h =X s H s^ h ^ h

s e ^s

= ^-3

its inverse Laplace transform is y t^ h t

2 u t

1 1

-^ h ^ - h

5.3 Option (A) is correct.

Given, the signal

v t^ h =30sin100t+10cos300t+6sin^500t+₄^ph So we have

w1 =100rad s/ w2 =300rad s/ w3 =500rad s/

Therefore, the respective time periods are

T1 2 sec

1002 wp1 p

= =

T2 2 sec

3002 wp2 p

= =

T3 = 5002p sec

So, the fundamental time period of the signal is L.C.M. ^T T T1, 2 3h

, , 2 ,2 ,2 HCF

LCM

100 300 500 p p p

= ^^

h h

or, T0 =1002p

Hence, the fundamental frequency in rad/sec is w0 = 102p =100rad s/

5.4 Option (A) is correct.

Given, the maximum frequency of the band-limited signal fm =5 kHz

According to the Nyquist sampling theorem, the sampling frequen-cy must be greater than the Nyquist frequenfrequen-cy which is given as

fN =2fm=2#5=10 kHz So, the sampling frequency fs must satisfy

fs $fN

fs $10 kHz

only the option (A) doesn’t satisfy the condition therefore, 5 kHz is not a valid sampling frequency.

5.5 Option (C) is correct.

For a system to be casual, the R.O.C of system transfer function H s^ h which is rational should be in the right half plane and to the right of the right most pole.

For the stability of LTI system. All poles of the system should lie in the left half of S-plane and no repeated pole should be on imaginary axis. Hence, options (A), (B), (D) satisfies an LTI system stability and causality both.

But, Option (C) is not true for the stable system as, S =1 have one pole in right hand plane also.

5.6 Option (B) is correct.

The Laplace transform of unit step funⁿ is U s^ h =s1

So, the O/P of the system is given as Y s^ h =b bs s1 1l l

= 2

For zero initial condition, we check u t^ h dt

=dy t^ h

& U s^ h =SY s^ h-y 0^ h

& U s^ h s

s1₂ y 0

= c m- ^ h or, U s^ h =s1

y 0^ =0

^ h h

Hence, the O/P is correct which is Y s^ h s

its inverse Laplace transform is given by y t^ h =tu t^ h

5.7 No Option is correct.

The matched filter is characterized by a frequency response that is given as

H f^ h =G*^fhexp^-j fT2p h where g t^ h ^f G f^ h

Now, consider a filter matched to a known signal g t^ h. The fourier transform of the resulting matched filter output g t0^ h will be

G f0^ h =H f G f^ h ^ h

* exp

G f G f j fT2p

= ^ h ^ h ^- h exp

G f ² j fT2p

= ^ h ^- h

T is duration of g t^ h Assume exp^-j fT2p h=1

So, G f0^ h = G f_ i²

Since, the given Gaussian function is g t^ h =e^-^p^t² Fourier transform of this signal will be

g t^ h =e^-^p^t² ^f e^-^p^f²=G f^ h

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Therefore, output of the matched filter is G f0^ h = e^-^p^{f 2}²

5.8 Option (B) is correct.

Given, the impulse response of continuous time system h t^ h =d^t-1h+d^t-3h

From the convolution property, we know x t^ h*d -^t t0h =x t^ -t0h So, for the input

x t^ h =u t^ h (Unit step funⁿ) The output of the system is obtained as

y t^ h =u t^ h*h t^ h

5.9 Option (B) is correct.

Given, the differential equation dt 5

d y²₂ + dydt +6y t^ h =x t^ h

Taking its Laplace transform with zero initial conditions, we have s Y s² ^ h+5sY s^ h+6Y s^ h =X s^ h

....(1)

Now, the input signal is

x t^ h otherwise Taking its Laplace transform, we obtain

X s^ h = -s1 e^-s²^s se 1 ²^s

= - -Substituting it in equation (1), we get

Y s^ h

5.10 Option (D) is correct.

The solution of a system described by a linear, constant coefficient, ordinary, first order differential equation with forcing function x t^ h is y t^ h so, we can define a function relating x t^ h and y t^ h as below

P dtdy +Qy+K =x t^ h

where P, Q, K are constant. Taking the Laplace transform both the sides, we get

P sY s^ h-Py^0h+Q Y s^ h =X s^ h ....(1) Now, the solutions becomes

y t1^ h =-2y t^ h or, Y s1^ h =-2Y s^ h So, Eq. (1) changes to

P sY s1^ h-P y1^0h+Q Y s1^ h =X s1^ h

or, -2PSY s^ h-P y1^0h-2QY s1^ h =X s1^ h ....(2) Comparing Eq. (1) and (2), we conclude that

X s1^ h =-2X s^ h y 01^ h =-2 0y^ h

Which makes the two equations to be same. Hence, we require to

change the initial condition to -2 0y^ h and the forcing equation to x t

-2 ^ h

5.11 Option (A) is correct.

Given, the DFT of vector 8a b c dB as

For matrix circular convolution, we know x n6 @*h n6 @ ^hh trans-formation to Eq(1), we get

p q r s Now, we know that

x n16 @*x n26 @ =X1DFT6k X@ 2,DFT6k@

5.12 Option (D) is correct.

Using s-domain differentiation property of Laplace transform.

If f t( ) ^L F s( )

5.13 Option (C) is correct.

[ ]

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Series I converges if 31z <1 or z >31 Series II converges if 31z <1 or z <3 Series III converges if 21z <1 or z > 21

Region of convergence of X z( ) will be intersection of above three So, ROC : 21 < z <3

5.14 Option (D) is correct.

( )

5.15 Option (C) is correct.

( )

We know that inverse Fourier transform of sin c function is a rectangular function.

So, inverse Fourier transform of H jw( ) ( )

5.16 Option (A) is correct.

Convolution sum is defined as [ ]

5.17 Option (A) is correct.

We have

d y dydt y

100 ²₂ -20 + =x t( ) Applying Laplace transform we get

100s Y s² ( )-20sY s( )+Y s( ) =X s( )

5.18 Option (C) is correct.

For an even function Fourier series contains dc term and cosine term (even and odd harmonics).

5.19 Option (B) is correct.

Function h n( ) =a u nⁿ ( ) stable if a <1 and Unstable if a H1 We We have h n( ) =2ⁿu n( -2); Here a =2 therefore h n( ) is unstable and since h n( )=0 for n<0 Therefore h n( ) will be causal. So h n( ) is causal and not stable.

5.20 Option (A) is correct.

Impulse response =dtd (step response)

5.21 Option (D) is correct.

We have x t( ) =exp( 2 ) ( )- t m t +s t( -6) and h t( )=u t( ) Taking Laplace Transform we get

( )

5.22 Option (B) is correct.

( )

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5.23 Option (B) is correct.

For 8 point DFT, x 1^*[ ] =x[ ]; [ ]7 x^* 2 =x[ ]; [ ]6 x^* 3 =x[ ]5 and it is conjugate symmetric about x 4[ ], x 6[ ] =0; x[ ]7 = +1 j3

5.24 Option (C) is correct.

For a function x t( ) trigonometric fourier series is ( )

#

T0"fundamental period

and An T2 x t( )cosn t dt

Since given function is even function so coefficient Bn=0, only cosine and constant terms are present in its fourier series representation Constant term A0 T1 x t dt( ) Constant term is negative.

5.25 Option (A) is correct.

We know that aZ^!^a Inverse Z transform [n a] ad !

-Given that X z( ) =5z²+4z^-¹+3

Inverse z-transform x n[ ] =5d[n+2]+4d[n-1]+3d[ ]n

5.26 Option (C) is correct.

We have h n1[ ] =d[n-1]or H Z1[ ]=Z^-1

and h n2[ ] =d[n-2]or H Z2( )=Z^-2

Response of cascaded system

H z( ) =H z1( ):H z2( ) z ¹:z ² z ³

= ^- ^- =

-or, h n[ ] =d[n-3]

5.27 Option (D) is correct.

For an N-point FET algorithm butterfly operates on one pair of samples and involves two complex addition and one complex multiplication.

5.28 Option (D) is correct.

We have f t( ) By final value theorem

( )

5.29 Option (B) is correct.

System is described as

( ) ( ) ( )

Taking Laplace transform on both side of given equation

( ) ( ) ( )

s Y s² +4sY s +3Y s =2sX s( )+4X s( ) (s²+4s+3) ( )Y s =2(s+2) ( )X s s Transfer function of the system

( ) Taking inverse Laplace transform

( )

y t =(e^-^t-e^-³^t) ( )u t

5.30 Option (C) is correct.

We have

5.31 Option (A) is correct.

The Fourier series of a real periodic function has only cosine terms if it is even and sine terms if it is odd.

5.32 Option (B) is correct.

Given function is

The function has a DC term and a cosine function. The frequency of cosine terms is

w = =2 2pf"f= p1 Hz

The given function has frequency component at 0 and 1p Hz.

5.33 Option (A) is correct.

[ ]

x n 31 u n( ) u( n 1) 21

n n

=_b _l -_b _l -Taking z transform we have

( )

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Thus its ROC is the common ROC of both terms. that is 3 z

1 < < 21

5.34 Option (B) is correct.

By property of unilateral Laplace transform

( ) ( )

5.35 Option (A) is correct.

We have (2)h =1, (3)h =- otherwise ( )1 h k = . The diagram of 0 response is as follows :

It has the finite magnitude values. So it is a finite impulse response filter. Thus S2 is true but it is not a low pass filter. So S1 is false.

5.36 Option (B) is correct.

Here ( )h t ! for t0 < . Thus system is non causal. Again any 0 bounded input ( )x t gives bounded output ( )y t . Thus it is BIBO stable.

Here we can conclude that option (B) is correct.

5.37 Option (D) is correct.

We have x n { , , , )[ ]= 1 0 2 3 and N=4

5.38 Option (A) is correct.

5.39 Option (C) is correct.

The output of causal system depends only on present and past states only.

In option (B) ( )y 0 depends on ( )x 1 . In option (C) ( )y 0 depends on (x - .1) In option (D) ( )y 0 depends on ( )x 5 .

Thus only in option (C) the value of ( )y t at t = depends on (0 x - 1) past value. In all other option present value depends on future value.

5.40 Option (D) is correct.

We have h t ( ) =e u t^a^t ( )+e u^b^t (-t)

This system is stable only when bounded input has bounded out-put For stability ta <0 for t > that implies 0 a<0 and tb >0

In document Revista Internacional de Educación para la Justicia Social (RIEJS) (página 99-103)

Hacia un marco regulativo de la participación

Participação como um Gerador e Garante da Democracia e da Cidadania

4. Hacia un marco regulativo de la participación

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