SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY



SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY

COIMBATORE-10

DEPARTMENT OF IT

UNIT – I

CLASSIFICATION OF SIGNALS AND SYSTEMS

PART – A

1. Define step function and delta function.

Ans : CT unit step function u(t) = 1 for (t ≥ 0)

0 for (t < 0)

DT unit step function u(n) = 1 for (n ≥ 0)

0 for (n < 0)

CT delta function ∫ ((t) = 1

((t) = 1 for t=0

0 for t≠0

DT delta function ((n) = 1 for n=0

0 for n≠0

2. What is the total energy of the discrete time sinal x(n) which takes the value of unity at n= -1,0,1?

Ans : Energy of the signal is given as,

∞ 1

E = ∑ │x(n)│2 = ∑ │x(n)│2

n = -∞ n = -1

= │x(-1)│2 + │x(0)│2 + │x(1)│2 = 3

2. Classify the signals as, periodic or nonperiodic signal ean , a>1

Ans : Since it is exponential the given signal is non periodic.

3. Draw the waveform x(-t) and x(2-t) of the signal x(t) = t 0≤t≤3

0 t>3

Ans:

[pic]

4. Define power signal.

Ans : A signal is said to be power signal if tis normalized power is nonzero and finite. i.e.. (0-a

8. What is the laplace transform of e-at sin((t) u(t)

Ans:

LT

e-at sin((t) < ------ > ( / [ (s+a)2 + (2 ]

9. A signal x(t) = cos(2 πf t) is passed through a device whose input – output is related by y(t)=x2(t). What are the frequency components in the output?

Ans:

Since an input is squared,

Y(t) = ( cos(2πft) )2

= (1+cos(4πft) ) / 2

= (1/2) + (1/2) cos[2π(2f)t]

The output present in the output is ‘ 2f ‘

10. Define the fourier transform pair for continuous time signal.

Ans:

Fourier Transform :

∞

X(() = ∫ x(t) e-j(t dt

-∞

Inverse Fourier Transform :

∞

X(f) = ( 1 / 2 π ) ∫ X(() ej(t d(

-∞

11. Find the laplace transform of x(t) = t e-at u(t) , where (a>0)

Ans:

LT

e-at u(t) < ---- > ( 1 / (s+a’) ) , ROC: Re(s) > (-a)

Differentiation in S-Domain property gives,

LT

-t x(t) < --- > (d X(s) / ds )

LT

-t e-at u(t) < --- > (d(1/(s+a)) / ds )

LT

t e-at u(t) < --- > 1/(s+a)2 ROC: Re(s) > (-a)

12. Find the Fourier transform of x(t) = t e-at u(t) , where (a>0)

Ans:

∞

X(f) = ∫ x(t) e-j2πf t dt

-∞

∞

= ∫ e-at e-j2πf t dt

0

∞

= ∫ e-(a+j2πf) t dt

0

= (1 / (a+j2πf) )

13. State the initial and final value theorem of laplace transform.

Ans:

initial value theorem

lt

f(0+) = s(∞ [ s F(s) ]

final value theorem

lt lt

t(∞ f(t) = s(0 [ s F(s) ]

14. Find the laplace transform of signal u(t).

Ans:

∞

LT [ u(t) ] = ∫ u(t) e-st dt

-∞

∞

= ∫ e-st dt [ since u(t) =0 for t 1 / (s+a) n’ , Re(s) > -a

n=2

LT

{ t2-1 / (n-1)! } e-at u(t) < ----- > 1 / (s+a) n’ , Re(s) > -a

a=2

LT

t e-at u(t) < ----- > 1 / (s+2) n’

LT

-t e-at u(t) < ----- > - 1 / (s+2) n’ , Re(s) > -2

18. let x(t)=t, 0≤t≤1 be a periodic signal with fundamental period T=1 and fourier series coefficients ak , Find the value of a0

Ans:

t+T0

C0 = (1/T0) ∫ x(t) dt

t

1 1

= (1/1) ∫ t dt = [ t2 / 2 ] = 1/2

1 0

19. What is the relationship between fourier transform and laplace transform ?

Ans:

X(s) = X(j() when s=j(

20. Define fourier series and fourier transform

Ans:

Fourier series

∞

X(t) = ∑ X(k) e jk(0 t

k = -∞

where , X(k) = (1/T) ∫ x(f) e jk(0 t dt

Fourier Transform :

∞

X(() = ∫ x(t) e-j(t dt

-∞

Inverse Fourier Transform :

∞

X(f) = ( 1 / 2 π ) ∫ X(() ej(t d(

-∞

21. What is difference between fourier transform and laplace transform?

Ans:

- Laplcae transform is evaluated over complte s-plane , but fourier transform is evaluated over j( axis in s-plane.

- Laplcae transform is broader compared to fourier transform. In other words, fourier transform is the special case of laplace transform.

22. obtain the fourier transform of x(t)= e j2( fc t .

Ans:

∞

X(f) = ∫ x(t) e - j2( fc t dt

-∞

∞

= ∫ e j2( f t e - j2( fc t dt

-∞

∞

= ∫ e j2( (f- fc) t dt

-∞

= ((f - fc)

23. obtain the fourier transform of inmpulse function.

Ans:

∞

X(f) = ∫ x(t) e - j2( f t dt

-∞

∞

= ∫ ( (t) e - j2( f t dt

-∞

The shifting property of impulse function is given as,

∞

∫f(t) f(t-t0) = f(t0)

-∞

f(t)= e - j2( f t and t0=0

∞

X(f) = ∫ e - j2( f t ( (t-0) dt = e - 2( f . 0 = 1

-∞

24. State the Time scaling property of fourier transform.

Ans:

Let x(t0 and X(f) be a fourier transform pair and ‘a’ is some constant.

Then by time scaling property

X(at) < -- > (1/│a│) X(f/a)

25. obtain the fourier transform of unit step function.

Ans:

CT unit step function u(t) = 1 for (t ≥ 0)

0 for (t < 0)

∞

X(f) = ∫x(t) e - j2( f t dt

-∞

∞

= ∫1. e - j2( f t dt

-∞

∞

= (1/- j2( f ) [e - j2( f t]

0

=(1/- j2( f )

PART-B

1. (i)Find the fourier series for the periodic signal x(t)=t , 0≤t≤1 and repeats every 1sec. (10 Marks)

(ii) Determine the fourier series representation for x(t)=2sin(2(t-3)+sin(6(t).

(6 Marks)

2. (i) Determine the trigonometric fourier series representation of the half wave rectifier output. (10 Marks)

(ii) State and prove parsaval’s theorem for complex exponential fourier series. (6 Marks)

3. Find the fourier transform of the signal x(t) and plot the amplitude spectrum. (16 Marks)

X(t) = 1 , (-ح/2) ≤t≤(ح/2)

0 , otherwise

2(

4. Find the fourier series of the signal x(t) = ∫ sin(2(f0 m)t . cos(2(f0 n)t dt, where 0

f0 is the fundamental frequency and m and n are any positive integer.

(16 Marks)

5. (i) Determine the trigonometric fourier series representation of the full wave

rectified output. (10 Marks)

(ii) Find the Inverse laplace transform of

(2s2 + 9s – 47 ) / { (s+1)(s2 +6s +25) } . (6 Marks)

6 (i) Find the laplace transform of x(t) = e-b│t│ for (b0)

(10 Marks)

(ii) Determine the fourier transform of x(t) = 1 for (-1≤t≤1) and zero for other

value of t. (6 Marks)

.

7. (i) State and prove convolution theorem of laplace transform. (8 Marks)

(ii) Prove that the convolution of two signals is equivalent to multiplication of

their respective spectrum in frequency domain. (8 Marks)

8. (i) Find the Laplace transform of tx(t) and x(t-t0) where t0 is a constant term and

x(t)< --- > X(s). (10 Marks)

(ii) Determine the laplace transform of x(t) = 2t for (0≤t≤1)

0. Otherwise

(6 Marks)

9. State and explain following properties of fourier transform. (16 Marks)

10. (i) Find the laplace transform of x(t) = e-at u(t) (8 Marks)

(ii) Determine the fourier series coefficients of (a) x(t) = sin (0t

(b) x(t) = sin (0t (8 Marks)

Unit-III

Linear Time Invariant – Continuous Time System

PART –A

1.Give four steps to compute convolution integral.

Ans:

a) Folding: One of the signal is first folded at t=0

b) Shifting: The folded signal is shifted right or left depending upon time at which output is to be calculated.

c) Multiplication: The shifted signal is multiplied other signal.

d) Integration: The multiplied signals are integrated to get the convolution output.

2. What is the overall impulse response h(t) when two systems with impulse response h1(t) and h2(t) are in parallel and in series?

Ans:

For parallel connection h(t)= h1(t) + h2(t)

For series connection h(t)= h1(t)*h2(t)

3. Write down the input-output relation of LTI system in time and frequency domain.

Ans:

y(t) = h(t) * x(t) :time domain

Y(f) = H(f) X(f) :frequency domain

Or Y(s) = H(s) . X(s) : frequency domain

4. Define transfer function in CT systems.

Ans:

H(f) = Y(f)/X(f)’ using fourier transform

Or

H(s) = Y(s) / X(s)’ using laplace transform.

5. Define linear time invariant system.

Ans:

The output response of linear time invariant system is linear and time invariant.

6. Define impulse response of a linear time invariant system.

Ans:

The impulse response of LTI system is denoted by h(t). It is the response of the system to unit impulse input.

7. State the properties of convolution.

Ans:

1. Commutative : x(t) * h(t) = h(t) *x(t)

2. Associative : [x(t) *h1(t)] *h2(t) =x(t) * [h1(t)* h2(t)]

3. Distributive : x(t) *h1(t) +x(t) * h2(t) =x(t) *{h1(t) +h2(t) }

8. What is the relationship between input and output of an LTI system?

Ans:

Input and output of an LTI systems are related by,

Y(t) = ∫ x(ι) h(t-ι) dι (i.e., ) convolution.

9. What is the transfer function of a system whose poles are at -0.3 ± j0.4 and a zero at -0.2 ?

Ans:

P1 = -0.3 + j 0.4, p2 = -0.3 – j 0.4

Z = -0.2

H(s) = ( s-z) /(s-p1)(s-p2)

= (s+0.2) / (s+0.3-j0.4)(s+0.3+j0.4)

= (s+0.2) / ( s2 +0.6s + 0.25)

10. Find the impulse response of the system given by H(s) = 1/(s+9)

Ans:

e-at u(t) -z d/dz X(z)

11. Define Parseval’s theorem.

Ans:

E = ∑| x(n)| 2 = 1/2∏ ∫|X(Ω)|2dΩ.

The above equation gives the energy of the signal.

12. Write the properties of ROC of z-transform.

Ans:

1. ROC of causal sequence is exterior of some circle of radius r=a.

2. ROC of non causal sequence is interior of some circle of radius r=b.

3. If the sequence is both sided then its ROC is a disc lying between

a e-jΩn0 X(Ω)

FT { δ (n-n0) } = e-jΩn 0

15. Write the differentiation and integration property of Fourier transform.

Ans:

-jnx(n)(> d/dΩ X(Ω) for differentiation.

16. Find the z-transform of δ (n-2).

Ans:

δ (n-2) (> z-2 .1

Thus δ (n-2) (> z-2, ROC: Entire z-plane expect z=0.

17. Define discrete time fourier transform.

Ans:

DTFT , X(Ω) = ∑ x(n) e-jΩn ,

IDTFT x(n) = 1/2∏ ∫X(Ω) e j Ωn dΩ.

18. Define z-transform.

Ans:

X(z) = ∑ x(n) z-n

The z- transform pair pair is denoted by,

x(n) (> X(z)

19. What is the z-transform of u(n) and δ (n)?

Ans:

u(n) (> 1/(1-z-1) , ROC: |z|>1

δ(n) (> 1 , ROC: Entire z plane

PART --B

1.a) What is ROC? State some properties of z transform. (6 Marks)

b) Find the inverse z- transform of x(z) = (z+4)/(z2-4z+3) 10 Marks)

2. a) How will you evaluate fourier transform from pole zero plot of z-transform. (6 Marks)

b) Find the inverse z- transform of X(z) = 1/(1-1.5z-1+0.5z-2) for ROC :

0.5=0

3 n ,n |

| |  |1 |2 |3 |4 |5 |6 |

|Y(n) | | | | | | | |

| | | | | | | | |

|| | | | | | | | |

|| | | | | | | | |

| | | | | | | | |

| |2 |2 |4 |6 |8 |10 |12 |

| |-4 |-4 |-8 |-12 |-16 |-20 |-24 |

| |6 |6 |12 |18 |24 |30 |36 |

| |-8 |-8 |-16 |-24 |-32 |-40 |-48 |

Y(n)= {2 0 4 0 -4 -8 -25 -4 -48 }

4. Determine the system function of the discrete time system described by the difference equation.

Y(n) - { (1/2) y(n-1) } + { (1/4) y(n-2) } = x(n) – x(n-1)

Ans:

Y(z) - { (1/2) z-1 y(z) } + { (1/4) z-2 y(z) } = x(z) – z-1x(z)

H(z) = Y(z) / X(z) = (1-z-1) / (1 – [(1/2) z-1] + [(1/4) z-2]

5. What is the linear convolution of two signals. [2,3,4] and [1,-2,1]?

Ans:

Y[n] = {2 -1 0 -5 4}

6. What is the response of an LSI system with impulse response

h(n)= ((n)+2 ((n-1) for the input x(n)={1,2,3}?

Ans:

Y[n] = {1 4 7 6}

7. Write the general difference equation relating input and output of a system.

Ans:

The generalized difference equation is given as

N M

Y(n) = - ∑ ak y(n-k) = ∑ bk y(n-k)

K=1 k=0

8. Write the difference equation and the transfer function of the system in fig.

Ans:

H(z) = Y(z) / X(z) = ( b2 z-2 ) / { (1- a1z-1) - (a1z-1 ) }

9. Draw the direct form-II realization of the system decribed by the differential equation,

[d2y(t) / dt2] + 5 [dy(t) / dt] + 4y(t) = [dx(t) / dt]

Ans:

[pic]

10. Determine the transfer function of the system described by

y(n)= a y(n-1) + x(n)

Ans:

H(z) = Y(z) / X(z) = 1 / (1 - az-1)

11. State two advantages of FFT computations.

Ans:

1. FFT algorithms are extremely fast.Hence they are computationally efficient.

2. FFT algorithms require less memory.

12. Draw direct form-II representation of H(z) = (1+z-1 + 3z-2 ) / (1+z-2 + z-3 )

Ans:

[pic]

13. Find the convolution sum for x(n) = {1,1,1,1} and h(n)={1,2,2,1}

Ans:

Y(n) = { 1 3 5 6 5 3 1}

14. Draw the radix-2 basic butterfly diagram.

Ans:

[pic]

15. Draw the black diagram for H(z) = (1+2z-1+4 z-4) / (1-z-1+4 z-2)

Ans:

[pic]

11jhdfDraw the block diagram for the system specified by the difference equation y[n]+ay[-2]=b0x[n]+b1x[n-1]

Ans:

[pic]

17. For a state space representation of the system. Find the transfer function of the system.

A = 0 1 B = 0 C = 1 2

-3 -2 1

Ans:

H(z)= ( 2z-1 + z-2 ) / (1+ 2z-1 + 3z-2)

18. Draw the block diagram of state variable equation.

Ans:

[pic]

19. What are the properties of convolution?

Ans:

1. Commutative property of convolution

2. Associative property of convolution

3. Distributive property of convolution

20. What are Impulse response and properties of LTI systems?

Ans:

1. Causality

2. Stability

PART – B

1. Find the output of the system whose input-output is related by,

y(n) = 7 y(n-1) – 12 y(n-2) + 2 x(n) – x(n-2) for the input x(n) = u(n).

(16 Marks)

2. Find the impulse response of the stable system whose input-output relation is given by the equation y(n) - 4 y(n-1) + 3 y(n-2) = x(n) + 2 x(n-1) (16 Marks)

3. (i) Find the linear convolution of x(n) = {1,2,3,4} and h(n)={2,3,4,1}

(6 Marks)

(ii) Compute the convolution of the two sequences given and plot the output. (10 Marks)

4. Find the output sequence y(n) of the system described by the equation

Y(n) = 0.7 y(n-1) – 0.1 y(n-2) + 2 x(n) – x(n-2).

For the input sequence x(n) = u(n). (16 Marks)

5. (i) What is the impulse response x(n) of the system if the poles and zeros are

radially moved k times their original location? (3 Marks)

(ii) Find the overall impulse response of the causal system in fig.

h1(n) = (1/3)n u(n) , h2(n) = (1/2)n u(n) and h3(n) = (1/5)n u(n) (12 Marks)

6. Realize direct form-I , direct form-II , cascade and parallel realization of the

discrete time system having system function

H(z) = 2(z+2) / {z(z-0.1) (z+0.5)0 (z+0.4)} (16 Marks)

7. (i) The difference equation of the system is

y(n) – (3/4) y(n-1) + (18) y(n-2) = x(n) + (1/2) x(n-1).

Draw the direct form-I and II structures. (10 Marks)

(ii) Find the convolution of x(n) = {1,2,3,4,5} with h(n) = {1,2,3,3,2,1}

(6 Marks)

8. (i) Find the impulse of the discrete time system described by the difference

equation. Y(n-2) – 3y(n-1) + 2 y(n) = x(n-1) (6 Marks)

(ii) Describe radix-2 DIT FFT algorithm (10 Marks)

9. (i) Explain the state variable description of discrete time system. (8 Marks)

(ii) Compute the linear convolution of x(n) = { 1,1,0,1,1} and

h(n) ={1,-2,-3,4} (8 Marks)

10. Given H(z) = (0.3 +z-1 – 0.47 z--2) / (1-0.5 z-1 + z-2 + 6 z-3) . Draw the block

diagram representation using DFI and DF II realization. (16 Marks)

-----------------------

X(2-t)

-1 0 1 2 t

X(-t)

-3 -2 -1 0 t

X(t)

0 1 2 3 t

X(n)=u(n)-u(n-3)

1

0 1 2 n

u(n-3)

1

0 1 2 3 4 5 6 7 8 9 n

u(n)

1

0 1 2 3 4 5 6 7 n

Fig. (a)

X(t)

1

-2 -1 0 1 2 3 4 t

A2

A1

X(n)

Z-1

Z-1

Z-1

Z-1

Y(n)

B2

B0=0

B1=1

-4

-5

∫

∫

Y(t)

X(t)

B2=0

X(n) 1 y(n)

-1 1

-1 -3

Z-1

Z-1

a A=a+ WNrb

WNr

b B=a- WNrb

-1

1

1

2

Y(n)

X(n)

Z-1

Z-1

Z-1

Z-1

Z-1

Z-1

4

X(n)

-a

B1

B0

Y(n)

z-1

z-1

-a2

-a1

Q1(n+1)

Q1(n) b1

Q2(n+1)

Q2(n) b2

X(n)

Y(n)

Z-1

Z-1

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