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“A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in difficulty.”

DEPARTMENT OF EI

SIGNALS & SYSTEMS ASSIGNMENT 1

1. Consider the DT signal = − ∑ ( − − ). Determine the value of the signals M and k so that x(n) may be expressed as x(n)=u(Mn-k).

2. Consider the CT signal = − − ( − ). Calculate the value E¥¥¥¥ for the signal = � � .

3. Find the even and odd component of the signal x(t)=ejt .

4. Which of the signals is/are periodic?

a) = � � � + � � � + � � �

b) = � � � � � � � � ( )

5. Let δδδδ(t) denotes the delta function. Find the value of the integral ( ) � � � ( )

6. If the signal f(t) has energy E, the energy of the signal f(2t) is equal to.

7. Give the period of the function � � � ( − ).

8. The time period of the signal = � � � � � + � � � � � .

9. The R.M.S value of the signal

= � � � � � � � � + + � � � � � ( � � � � ).

10. The signal x(t) is shown, draw x(3t).

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ASSIGNMENT 2

1. If | | = ( ) , then find the value of f(0) and f(¥¥¥¥).

2. Find the inverse laplace of

a) =( )

b) =

3. The laplace transform of

= � � � � � ( � � ).

4. Find the final value of [� �

] .

5.The output of the linear system to a unit step input u(t) is . Then find the value of function H(s).

6. What is the relationship between Z-transform and DTFT ?

7. Give the Z- transform of:

a) x(n)= an n³³³³0

= 0 for n<<<<0

b) f(nT)= anT

8. Find the final value

= ( )

( )

9. The z-transform of the system is =.

. If the ROC is|z|<0.2, then find the impulse response of the system.

10. Find the ROC for

= ( ) − ( ) (− − ).

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ASSIGNMENT 3

1. The Fourier transform [ ] is equal to � �

. Then find the value of

[� �

] .

2. If x(t) is a odd signal with fundamental period T and Fourier series coefficients Cn, then show that a0 =0 and Cn are imaginary and odd.

3. Find the Fourier Transform of the following:

a) � � � � � ( )

b) | | � � �

4. Find the fourier transform of a gate pulse of unit height, unit width and centered at t=0.

5. Using Fourier transform properties find Fourier transform of the signal

= � � �� �

6. If DFS[x(n)] = ck, then find DFS [x(n)+x(n+N/2)] in terms of ck.

7. Find the spectral coefficient and plot amplitude and phase spectrum

= . + � � � � � + + � � � ( � �� �

+ )

8. Find Fourier Transform of the following

a) − − ( − )

b) (− − )

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ASSIGNMENT 4 1. Check whether the following systems are causal or non causal

a) = ( − ) , b) = � � � [ ]

2. Check whether the system is linear or not

a) � �� �

+ = ( ) , b) y(n)= x(n) cosωωωωn

3. Check whether the given systems are time invariant or not

a) = � �

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

c) y(n) = sin[x(n)]

4. Check whether the given systems are:

®®®® static/dynamic

®®®® linear/non-linear

®®®® causal/ non-causal

®®®® time variant/ time-invariant

a) y(n) = x(n+1) – x(n-1) d) y(t)=10 x(t) +5

b) ( )� �

+ ( )� �

+ � � ( )� �

+ = ( + ) e) y(n) = |x(n)|

c) y(t) =Ev[x(t)] f) y(n) = anx(n)

5. State Superposition principle.

6. How do you find whether the given system is linear or non-linear.

7. Explain the following properties of the system:

a) Stability b) Causality c) Time-invariance

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ASSIGNMENT 5

1. Determine the impulse response h(n) for the system described by the second-order difference equation

Y(n) = 0.6y(n-1) – 0.08y(n-1) + x(n)

2. Determine the impulse response h(n) for the system described by difference equation

Y(n) + y(n-1) -2y(n-2) = x(n-1) + 2x(n-2)

3. Determine the natural response of the given system

−� �

− = − ; − = ; − = −

4. Find the forced response of the following difference equation

Y(n) – y(n-1) = x(n) where x(n) = cos 2n

5. Find the solution of the equation: − − + − = ( ) for

n³³³³0 ; with initial conditions y(-1) =4 and y(-2) = 10.

6. Define FIR and IIR systems?

7. Show that the impulse of two systems connected in series is equal to the convolution of individual responses.

8. Show that the impulse response of two system connected in parallel is equal to the sum of individual responses.

9. Prove that for a causal system the impulse h(t)=0 for t<0.

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