Code No: 45010

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1. State and prove the following properties of the Fourier Transform.

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(a) Linearity

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III B.Tech I Semester Regular Examinations,Nov/Dec 2009 LINEAR SYSTEMS ANALYSIS Electrical And Electronics Engineering Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

(b) Time Shifting (c) Scaling in the time domain (d) Frequency Shifting

[4+4+4+4]

(a) The average value of f(t) (b) The effective value of f(t) (c) Fundamental period of f(t)

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(d) The average power of f(t)

or

2. If f (t) = 2 + 3 cos(10πt + 300 ) + 4 cos(20πt + 600 ) + cos(30πt + 900 ) , then find

[4+4+4+4]

3. State and prove the following properties of the z- Transform. (a) Linearity

(b) Time Shifting

(c) Scaling in the z-domain (d) Time Reversal

[4+4+4+4]

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4. Write the state equations for the following network using as shown in figure 1 (a) Equivalent source method

(b) Network topological method

[8+8]

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5. Check whether the following impedance functions are LC, RL or RC? (a) Z(s) =

(s4 +4s2 +3) (s4 +6s2 +8)

(b) Z(s) =

3(s+7)(s+3) (s+1)(s+5)

[8+8]

6. (a) The signals g1 (t) = 10 cos(100πt) and g2 (t) = 10 cos(50πt) are both sampled at the rate of 75 samples per second. Show that the two sequences of samples thus obtained are identical. What is the reason for this phenomeon?

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or

Figure 1:

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Code No: 45010

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(b) The signal g(t) = 10 cos(60πt) cos2 (160πt) is sampled at the rate of 400 samples per second. Determine the range permissible cutoff frequencies for the ideal reconstruction filter that may be used to recover g(t) from its sampled version. [8+8] 7. (a) Use Laplace transform to find the output of the system described by the differential equation: dy(t) + 5y(t) = x(t) dt in response to the input x(t) = 3e−2t u(t) and initial conditions y(0+ ) = −2 .

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(b) Find the current i(t) in a series RLC as shown in figure 2 A voltage of 100 V is switched on across the terminals at t=0. [8+8]

Figure 2:

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Code No: 45010

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8. Given a linear time-invariant passive network and two of its nodes. Call Z(s) the impedance seen at these nodes. Justify the following properties of Z(s): (a) Z(s) cannot have poles in the open right-half plane. (b) If Z(s) has a pole on the jw axis, the poles must be simple.

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(c) For all ω where it is defined (i.e., except at jw-axis poles of Z), Re[Z(jw)] ≥ 0. [5+5+6]

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or

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Code No: 45010 III B.Tech I Semester Regular Examinations,Nov/Dec 2009 LINEAR SYSTEMS ANALYSIS Electrical And Electronics Engineering Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. Define convolution integral. With an example explain how convolution of a signal can be obtained graphically. [16]

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2. (a) Explain the concept of state, state variables and state model with the help of examples?

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(b) Explain about the Laplace transform method for solving the state equations. [8+8] 3. Clearly explain with examples the Sturm’s test to check positive real functions.[16]

4. A waveform consists of a single pulse extending from t=-1 to t=1 sec and has amplitude 5 V. Find autocorrelation function and energy spectral density. [8+8]

(a) Differentiation in z-domain

or

5. State and prove the following properties of the z-Transform.

(b) Multiplication of two sequences (c) Correlation of two sequences

2

[4+6+6]

2

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(s +3)(s +1) 6. An impedance function is given by Z(s) = s(s 2 +2)(s2 +5) . Determine the nature of the function. Synthesize the network if possible by

(a) Cauer Form I and (b) Cauer Form II.

[8+8]

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∞ P 7. An electric circuit is excited by a voltage v(t) as v(t) = v0 + vn cos(nω0 t + θn ) . n=1 P The corresponding steady state current is i(t) = I0 + In cos(nω0 t + φn ) . Define

the input power at the input terminals as P =

1 T

TR/2

v(t)i(t) dt; T = 2π/ω0 . Show

−T /2

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that the input power can also be written as P = V0 I0 +

P Vn In 2

cos(θn − φn )

[16]

8. Sketch the frequency spectrum of f(t), g(t) and y(t) for the following system. Where f(t)=2cos 20t+4 cos20t and g(t)=200 t. as shown in figure 3 [2+4+10] ?????

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Figure 3:

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III B.Tech I Semester Regular Examinations,Nov/Dec 2009 LINEAR SYSTEMS ANALYSIS Electrical And Electronics Engineering Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. One cycle of non sinusoidal periodic voltage wave has a value of 1V from 0 to π/2 , 0 V from π/2 to π and -0.5 V from π to 2π . Determine

(b) The coefficient of the third harmonic (c) rms value of third harmonic (d) average power of third hormonic

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(a) The dc term

[4+4+4+4]

2. (a) State and prove the shifting theorem.

or

(b) Obtain the impulse response of the following RLC network. as shown in figure .4 4 [8+8]

3. Determine the z- transform , including the region of convergence, for each of the following sequences :

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(a) (1/2)n u(n)

(b) −(1/2)n u(-n-1) (c) (1/2)n u(-n)

(d) (1/2)n [ u(n) - u(n-10)]

(e) (1/2)n n u(n) Where u(n) is the unit step sequence.

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4. Using the Foster form II, synthesize the following functions. 5

[4+10]

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(a) Z(s) =

s(s2 +4)(s2 +6) (s2 +3)(s2 +5)

(b) Z(s) =

(s2 +2)(s2 +7) (s2 +3)(s2 +5)

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Figure 4:

[8+8]

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5. An state equation. #  is characterized " LTI system   by the  homogeneous    dx1 (t) 0 −2 x1 (t) 2 1 dt = + r(t); x(0) = dx2 (t) 1 −3 x2 (t) 0 1 dt

(a) Compute the solution when r(t)=0 by the Laplace transform method.

or

(b) Also find the solution of the system for unit step input, i.e., r(t)=1.

[8+8]

6. Using Sturm’s test, check whether the following functions are positive real or not? If not, say why? s s−1 s s+1

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(a) Z(s) = (b) Z(s) =

[8+8]

7. State and prove the four properties of autocorrelation function of energy signals. [4+4+4+4] 8. (a) A signal x(t) = ASin w0 t, wo = 2π/T with T being the time period, is passed through a full- wave rectifier. Find the spectrum of the out put waveform.

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(b) If the Fourier Transform of h(t) is H(w), that ∆T  prove  1 ∆w1 = 1, where , R∞ R∞ ∆T1 = h (t)dt/h (0) and ∆w1 = 1/2π H (w) dw /H (0). [8+8] −∞

−∞

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Code No: 45010

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III B.Tech I Semester Regular Examinations,Nov/Dec 2009 LINEAR SYSTEMS ANALYSIS Electrical And Electronics Engineering Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Determine the Fourier transform of the unit step, ramp and sinusoidal signal. (b) State and prove Parsevals theorem.

[12+4]

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2. (a) An analog ECG signal contains useful frequencies up to 100Hz.

i. What is Nyquist rate for the signal? ii. Suppose that we sample this signal at a rate of 250 samples/sec. What is the highest frequency that can be represented uniquely at this sampling rate.

or

(b) An analog signal xa (t) = sin(480πt) + 3 sin(720πt) is sampled 600 times per second.

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i. Determine the Nyquist sampling rate. ii. Determine the folding frequency iii. What are the frequencies in radians, in the resulting discrete time signal x(n). iv. If x(n) is passed through an ideal D/A converter, find the reconstructed signal? [4+12] 3. (a) Distinguish between continuous and discrete time signals with appropriate examples.

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(b) Define discrete time sinusoidal and discrete time exponential signals with examples. (c) Distinguish between Laplace, Fourier and Z-transforms clearly making out the limitations of each. [4+4+8]

4. Obtain the response of the states of the following system using

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(a) Taylor series expansion

(b) Laplace transform method. #  "     dx1 (t) −7 1 x1 (t) 1 dt = + r(t) dx2 (t) 1 −2 x2 (t) 0 dt

[8+8]

Where r(t) is unit ramp function and xT0 = [1 0]. 5. (a) Using the Foster form I, synthesize the RC impedance function Z(s) = . 7

6(s+3)(s+9) s(s+6)

Code No: 45010

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(b) Using the Foster form I, synthesize the RL admittance function Z(s) = [8+8]

6(s+3)(s+9) s(s+6)

Determine the Trigonometric form of Fourier series of v(t).

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6. The output the equation  of a rectifier is givenπ by  V cos ωt, 0 ≤ ωt ≤   m 2 π 3π 0, ≤ ωt ≤ v(t) = 2 2   ≤ ωt ≤ 2π Vm cos ωt, 3π 2 [16]

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7. Find the convolution between f1 (t) = e−2t u(t) and f2 (t) = tu(t) using (a) Graphical convolution (b) Laplace transform.

[8+8]

8. (a) Give two examples to show that if Z1 (s) and Z2 (s) are positive real, then Z1 (s)/Z2 (s) need not be positive real.

or

(b) Show that, if a one-port is made of lumped passive linear time-invariant elements and if it has a driving-point impedance Z(s), then Z(s) is a positive real function. [8+8]

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