ldsa

Code No: R05310205 Set No. 1

III B.Tech I Semester Regular Examinations, November 2008LINEAR AND DISCRETE SYSTEMS ANALYSIS

(Electrical & Electronic Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

1. (a) Obtain the state variable model for a system described by the following dif-ferential equation.10d

3y

dt3+ 5d

2y

dt2+ 2dy

dt+ 3y = 20dx

dt+ 10x

(b) Determine the state transition matrix for the state matrix.

A =

1 1 00 1 10 0 2

[8+8]

2. The input to the circuit of figure 2(a) is a rectified sine wave as shown in figure2(b).Determine the current following through 1 ohm resistor. = 1 rad/sec. Draw themagnitude spectrum and find out the nth harmonic of i(). [8+4+4]

(a) & (b) Figure 2

3. (a) Show that a periodic signal can be expressed as continuous sum of everlastingexponentials.

(b) Find the F.T. of

i. (t t0)

ii. rect(t/

). [8+8]

4. (a) Explain the graphical interpretation of convolution with the following func-tions. As shown in figure 4a.

1 of 2


Figure 4a

(b) A voltagete u(t) is applied to a series RC network as shown in figure 4b.

Figure 4bFind the voltage V0(t) using frequency domain analysis.

5. (a) State and explain the properties of positive real function.

(b) Check whether given polynomial H(s) = 2s4 +5s3 +6s2 +2s +1 is Hurwitzor not. [8+8]

6. Find the networks for the following functions in one Foster and one Cauer form

(a) Y (s) = (s+1)(s+3)(s+4)(s+2)

(b) Z(s) = 2(s+0.5)(s+4)s(s+2)

. [2 8]

7. (a) The signals m1(t) = 10cos100t and m2(t) = 10cos50t are both sampledwith fs = 75 Hz. Show that the two sequences of samples so obtained areidentical.

(b) The signal g(t) = cos10 t+0.5cos 20 t is sampled with the interval betweensamples is Ts. Find the maximum allowable time for Ts.

(c) Determine the sampling rate for the band pass signal whose centre frequencyfc is 5 fm with a signal band width of 2fm. [5+5+6]

8. How many different sequences have a Z-transform given byH(z) = 12z

1+3z2

(1 18z1+ 1

4z2)(1+ 1

3z1)

. [16]

2 of 2





1. (a) Develop the state variable model equation for the circuit as shown in figure1a.

Figure 1a

(b) A system matrix is given by

A =

[1/2

5/21/2

7/5

]

obtain the state transition matrix. [8+8]

2. A rectangular waveform of magnitude 10V,duty ratio 75% and frequency 50Hzis applied across a resistance of 1ohm in series with an inductance of 100mH.Determine the steady state current in the circuit. Also find the power and P.F ofthe load current. [8+4+4]

3. (a) Use the duality property of F.T, find the transform ofg(t) = A

1+(at)2

(b) Using the superposition and time shifting properties, Find the F.T. of thesignals shown in figure 3b. Sketch the amplitude spectrum assuming


i. f(t) = Kt K is a real constant >1

ii. f(t) = t(t)

(b) The impulse response of a certain linear system is given by

h(t) =2te u(t) t 0

= 0 t < 0using the convolution integral, determining the response y(t) due to the rampinputx(t) = 0 t < 0= t t 0. [8+8]

5. (a) List the properties of positive real function.

(b) A function is given by N(s) = s3+5s2+9s+3s3+4s2+7s+9

.Determine the positive realness of the function. [8+8]

6. (a) Given the driving-point impedance function

Z(s) = s(s2+2)

(s2+1)(s2+4)

Synthesize a ladder network of the first Cauer form for this impedance func-tion.

(b) A network is made up of a series connection of an RL network and RC network.Assuming that neither of the networks is a short circuit find the location ofpoles and also the location of zeros. What is the behavior at the origin andat infinity? [8+8]

7. Find the mean square value of the output voltage vo(t) of an RC network shownin figure 7 if the input voltage has a power density spectrum Si() given by

(a) Si ()= k

(b) Si ()= G2() [gate function with cutoff at =1]

(c) Si ()= [( + 1)+( 1)]In each case, also calculate the power of the input signal. [16]

Figure 7

8. For a causal discrete-time LTI system, if the input x(n) is

x(n) =(1/2

)nu(n) 1/4

(1/2

)n1u(n 1)

Then the output is y(n) =(1/3

)nu(n)

(a) Determine the impulse response h(n) and the system function H(z)

2 of 3


(b) Find the difference equation that characterizes this system. [2 8]

3 of 3





1. (a) Show that the inductor current iL(t) in the circuit as shown in figure 1a isgiven by

iL (t) = 3

1 20t/3

20t/3e

u(t).

Figure 1a

(b) A dynamical system is described by the differential equation...y + 4y + 5y + 2y = ushow that the state variable formulation is

x1x3

x2

=

0 1 00 0 12 5 4

x1x2

x3

+

00

1

u [8+8]

2. (a) Derive an expression for the effective value of non-sinusoidal periodic wave-form.

(b) A periodic current source given by i(t) = 5+3 cos (100t+ 45)+2 cos (200t 10)is applied to a parallel RL circuit as shown in figure 2b. Calculate the responseV(t) and average power. [8+8]

Figure 2b

1 of 3


3. (a) Determine the F.T. of a trapzoidal function and triangular RF pulse f(t) shownin figure 3(a)i and figure 3(a)ii. Draw its spectrum.

i. Figure 3(a)i

Figure 3(a)i

ii. figure 3(a)ii

Figure 3(a)ii

(b) Show that a normalized Gaussian pulse is its own fourier transform. [8+8]

4. (a) Find the convolution for the signals.

i. x(t) =

{1 0 < t < T0 otherwise

ii. h(t) =

{t 0 < t < 2T0 otherwise

(b) Given LT, how do you obtain its F.T. [8+8]

5. Given z(s) = (s2+Xs)

s2+5s+4

(a) What are the conditions on X for Z(s) to be a positive real function?

(b) Find X for Re (Z (j)) to have a second order zero at = 0. [8+8]

6. (a) Given the driving point admittance function Y (s) = s(s2+1)(s2+4)s(s2+2)

. Synthesizeladder network of the first Cauer form.

(b) State and explain Fosters reactance theorem for LC networks. [8+8]

7. (a) How does flat top sampling differ from impulse sampling? Discuss the meritsand drawbacks of both types of sampling.

(b) Show that the continuous time signal xa(t) = A cos (ot+) can be uniquelyrecovered from its sampled version x[n]=xa(nT) if the sampling frequency iss = 2 /T > 2wo. [8+8]

2 of 3


8. The output y(n) of a discrete-time LTI system is found to be 2(1/3)nu(n) when the

input x(n) is u(n)

(a) Find the impulse response h(n) of the system.

(b) Find the output y(n) when the input x(n) is(12

)nu(n). [8+8]

3 of 3





1. (a) Write the state equation for the circuit as shown in figure 1a.

Figure 1a

(b) Find the state response of the system shown in figure 1b.[x1x2

]=

[0 11 0

] [x1x2

]+

[01

]

[x1(0)x2(0)

]=

[00

] [8+8]

Figure 1b

2. Find the trigonometric form of the following voltage waveform shown figure 2 andhence compute average power and power factor of the load if voltage is applied toseries RL circuit with R = 1, L = 1H. [8+4+4]

1 of 3


Figure 2

3. (a) State and prove the time scaling property of F.T.

(b) Find the fourier transform of the pulse functions shown in figure 3b.

Figure 3b

4. (a) Evaluate the following convolution integrals.

i. u(t) t

e u(t)

ii. u(t) tu(t)

(b) Find the Inverse LT of the following functionF (s) = S+4

2S2+5S+3. [8+8]

5. (a) Explain Sturms theorem.

(b) Test whether the following function is a positive real function and the poly-nomials are Hurwitz or not using Sturms test.

F (s) = 2s4+7s3+11s2+12s+4s4+5s3+9s2+11s+6

. [8+8]

6. Indicate which of the following functions are either RC, RL, or LC impedancefunctions. Give reasons.

(a) Z(s) = (s+1) (s+3)s(s+4)

(b) Z(s) = (s+3) (s+7)(s+2) (s+5)

(c) Z(s) = s2+4s+3s2+6s+8

(d) Z(s) = s2+5s+6s2+s

. [4 4]

2 of 3


7. For a low pass signal with a bandwidth of 6000Hz, what is the minimum samplingfrequency for perfect reconstruction of the signal? What is the minimum requiredsampling frequency if a guard band of 200Hz is required? What is the minimumrequired sampling frequency and the value of K for perfect reconstruction if thereconstruction filter has the following frequency response

H(f) =

K, |f | < 7000K K(|f | 7000)/3000, 7000 < |f | < 10000

0, otherwise. [16]

8. Using the relation anu(n) zza

, |z| > |a|. Find the Z-transform of the following:

(a) x1(n) = nan1u[n]

(b) x2(n) = n(n 1)an2u[n]

(c) x3(n) = n(n 1)......(n k + 1)anku[n]. [5+5+6]

3 of 3

ldsa

Documents