**UNIT – I SIGNALS & SYSTEMS**

**PART A ( 2marks)**

1. Determine the energy of the discrete time sequence (2)

x(n) = (½)n, n_0 =3 n, n<0

2. Define multi channel and multi dimensional signals. (2)

3. Define symmetric and anti symmetric signals. (2)

4. Differentiate recursive and non recursive difference equations. (2)

5. What is meant by impulse response? (2)

6. What is meant by LTI system? (2)

7. What are the basic steps involved in convolution? (2)

8. Define the Auto correlation and Cross correlation? (2)

9. What is the causality condition for an LTI system? (2)

10. What are the different methods of evaluating inverse z transform? (2)

11. What is meant by ROC? (2)

12. What are the properties of ROC? (2)

13. What is zero padding? What are it uses? (2)

14. What is an anti imaging and anti aliasing filter? (2)

15. State the Sampling Theorem. (2)

16. Determine the signals are periodic and find the fundamental period (2)

i) sin_ 2 _t

ii) sin 20_t+ sin5_t

17. Give the mathematical and graphical representations of a unit sample, unit step sequence. (2)

18. Sketch the discrete time signal x(n) =4 _ (n+4) + _(n)+ 2 _ (n-1) + _ (n-2) -5 _ (n-3**) **(2)

19**. **Find the periodicity of x(n) =cost(2_n / 7) (2)

20. What is inverse system? (2)

21. Write the relationship between system function and the frequency response. (2)

22. Define commutative and associative law of convolutions. (2)

23. What is meant by Nyquist rate and Nyquist interval? (2)

24. What is an aliasing? How to overcome this effect? (2)

25. What are the disadvantages of DSP? (2)

26.Compare linear and circular convolution.(2)

27.What is meant by section convolution? (2)

28.Compare over lap add and save method. (2)

29. Define system function. (2)

30.State Parseval’s relation in z - transform. (2)

**PART B**

**CLASSIFICATION OF SYSTEMS:**

1. Determine whether the following system are linear, time-invariant (16)

(a)y(n) = Ax(n) +B. (4)

i(a)y(n) =x(2n). (4)

ii(a)y(n) =n x2 (n). (4)

iv)y(n) = a x(n) (4)

2. Check for following systems are linear, causal, time in variant, stable, static (16)

i) y(n) =x(2n). (4)

ii) y(n) = cos (x(n)). (4)

iii) y(n) = x(n) cos (x(n) (4)

iv) y(n) =x(-n+2) (4)

3. (a)For each impulse response determine the system is (a) stable i(a) causal

i) h(n)= sin **(**_ n / 2) . (4)

ii) h(n) = _(n) + sin _ n (4)

(b)Find the periodicity of the signal x(n) =sin **(2**_n / 3)+ **cos (**_ n / 2) (8)

4. (a)Find the periodicity of the signal

i) x(n) = **cos **(_ /4) **cos(**_n /4). (4)

ii) x(n) = cos (_ n 2 / 8) (4)

(b) State and proof of sampling theorem. (8)

5. Explain in detail about A to D conversion with suitable block diagram and to

reconstruct the signal. (16)

6. What are the advantages of DSP over analog signal processing? (16)

**CONVOLUTION:**

**8. **Find the output of an LTI system if the input is x(n) =(n+2) for 0_ n_ 3

and h(n) =an u(n) for all n (16)

9. Find the convolution sum of x(n) =1 n = -2,0,1

= 2 n= -1

= 0 elsewhere

and h(n) = _ (n) – _ (n-1) + _( n-2) - _ (n-3). (16)

10. (a)Find the convolution of the following sequence x(n) = u(n) ; h(n) =u(n-3). (8)

(b)Find the convolution of the following sequence x(n) =(1,2,-1,1) , h(n) =(1, 0 ,1,1). (8 )

12.Find the output sequence y(n) if h(n) =(1,1,1) and x(n) =(1,2,3,1) using a circular

Convolution. (16)

13. Find the convolution y(n) of the signals (16)

x(n) ={ _ n, -3 _ n _ 5 and h(n) ={ 1, 0 _ n _ 4

0, elsewhere } 0, elsewhere }

**Z TRANSFORM:**

14. State and proof the properties of Z transform.(16)

15.(a) Find the Z transform of

i) x(n) =[ (1/2)n – (1/4)n ] u(n) (2)

ii) x(n) = n(-1)n u(n) (2)

iii) x(n) (-1)n cos (_n/3) u(n) (2)

iv) x(n) = (½) n-5 u(n-2) +8(n-5) (2)

(b) Find the Z transform of the following sequence and ROC and sketch the pole zero

diagram.

i) x(n) = an u(n) +b n u(n) + c n u(-n-1) , |a| <|b| <| c| (4)

ii) x(n) =n2 an u(n) (4)

17. Find the convolution of using z transform (16)

x1(n) ={ (1/3) n, n>=0

(1/2) -n n<0 }

x2(n) = (1/2) n

**INVERSE Z TRANSFORM**:

18. Find the inverse z transform (8x2=16)

X(z) = log (1-2z) z < |1/2 |

X(z) = log (1+az-1) |z| > |a|

X(z) =1/1+az-1 where a is a constant

X(z)=z2/(z-1)(z-2)

X(z) =1/ (1-z-1) (1-z-1)2

X(z)= Z+0.2/(Z+0.5)(Z-1) Z>1 using long division method.

X(z) =1- 11/4 z-1 / 1-1/9 z-2 using residue method.

X(z) =1- 11/4 z-1 / 1-1/9 z-2 using convolution method.

19. A causal LTI system has impulse response h(n) for which Z transform is given by H(z)

1+ z -1 / (1-1/2 z -1 ) (1+1/4 z -1 ) (16)

i) What is the ROC of H (z)? Is the system stable?

ii) Find THE Z transform X(z) of an input x(n) that will produce the output y(n) = -

1/3 (-1/4)n u(n)- 4/3 (2) n u(-n-1)

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

**ANALYSIS OF LTI SYSTEM:**

20. The impulse response of LTI system is h(n)=(1,2,1,-1).Find the response of the system to

the input x(n)=(2,1,0,2) (16)

21. Determine the magnitude and phase response of the given equation

y(n) =x(n)+x(n-2) (16)

22. Determine the response of the causal system y(n) – y(n-1) =x(n) + x(n-1) to inputs

x(n)=u(n) and x(n) =2 –n u(n).Test its stability (16)

23. Determine the frequency response for the system given by

y(n)-y3/4y(n-1)+1/8 y(n-2) = x(n)- x(n-1) (16)

24. Determine the pole and zero plot for the system described difference equations

y(n)=x(n)+2x(n-1)-4x(n-2)+x(n-3) (16)

25.A system has unit sample response h(n) =-1/4 _(n+1)+1/2 _(n)-1/4 _(n-1).Is the system

BIBO stable? Is the filter is Causal? Find the frequency response? (16)

26. Find the output of the system whose input- output is related by the difference equation

y(n) -5/6 y(n-1) +1/6 y(n-2) = x(n) -1/2 x(n-1) for the step input. (16)

27. Find the output of the system whose input- output is related by the difference equation

y(n) -5/6 y(n-1) +1/6 y(n-2) = x(n) -1/2 x(n-1) for the x(n) =4 n u(n). (16)

**UNIT – II**

**FAST FOURIER TRANSFORM**

**PART A ( 2marks)**

1. How many multiplication and additions are required to compute N point DFT using

radix 2 FFT? (2)

2. Define DTFT pair. (2)

3. What are Twiddle factors of the DFT? (2)

4. State Periodicity Property of DFT. (2)

5. What is the difference between DFT and DTFT? (2)

6. Why need of FFT? (2)

7. Find the IDFT of Y (k) = (1, 0, 1, 0) (2)

8. Compute the Fourier transform of the signal x(n) = u(n) – u(n-1). (2)

9. Compare DIT and DIF? (2)

10. What is meant by in place in DIT and DIF algorithm? (2)

11. Is the DFT of a finite length sequence is periodic? If so, state the reason. (2)

12. Draw the butterfly operation in DIT and DIF algorithm? (2)

13. What is meant by radix 2 FFT? (2)

14. State the properties of W N

k ? (2)

15. What is bit reversal in FFT? (2)

16. Determine the no of bits required in computing the DFT of a 1024 point sequence with SNR of 30dB. (2)

17. What is the use of Fourier transform? (2)

18. What are the advantages FFT over DFT? (2)

19. What is meant by section convolution? (2)

20. Differentiate overlap adds and save method? (2)

21.Distinguish between Fourier series and Fourier transform. (2)

22.What is the relation between fourier transform and z transform. (2)

23. Distinguish between DFT and DTFT. (2)

**PART B**

1.(a) Determine the Fourier transform of x (n) =a |n|; -1<1 (8)