CS2403 DIGITAL SIGNAL PROCESSING

UNIT I SIGNALS AND SYSTEMS

Basic elements of DSP – concepts of frequency in Analog and Digital Signals – sampling

theorem – Discrete – time signals, systems – Analysis of discrete time LTI systems – Z

transform – Convolution (linear and circular) – Correlation.

theorem – Discrete – time signals, systems – Analysis of discrete time LTI systems – Z

transform – Convolution (linear and circular) – Correlation.

UNIT II FREQUENCY TRANSFORMATIONS

Introduction to DFT – Properties of DFT – Filtering methods based on DFT – FFT

Algorithms Decimation – in – time Algorithms, Decimation – in – frequency Algorithms –

Use of FFT in Linear Filtering – DCT.

Algorithms Decimation – in – time Algorithms, Decimation – in – frequency Algorithms –

Use of FFT in Linear Filtering – DCT.

UNIT III IIR FILTER DESIGN

Structures of IIR – Analog filter design – Discrete time IIR filter from analog filter – IIR

filter design by Impulse Invariance, Bilinear transformation, Approximation of derivatives

– (HPF, BPF, BRF) filter design using frequency translation

UNIT IV FIR FILTER DESIGN

Structures of FIR – Linear phase FIR filter – Filter design using windowing techniques,

Frequency sampling techniques – Finite word length effects in digital Filters

25

UNIT V APPLICATIONS

Multirate signal processing – Speech compression – Adaptive filter – Musical sound

processing – Image enhancement.

TEXT BOOKS:

1. John G. Proakis & Dimitris G.Manolakis, “Digital Signal Processing – Principles,

Algorithms & Applications”, Fourth edition, Pearson education / Prentice Hall, 2007.

2. Emmanuel C..Ifeachor, & Barrie.W.Jervis, “Digital Signal Processing”, Second

edition, Pearson Education / Prentice Hall, 2002.

REFERENCES:

1. Alan V.Oppenheim, Ronald W. Schafer & Hohn. R.Back, “Discrete Time Signal

Processing”, Pearson Education, 2nd edition, 2005.

2. Andreas Antoniou, “Digital Signal Processing”, Tata McGraw Hill, 2001UNIT 1

1. Determine the energy of the discrete time sequence

x(n) = (½)n, n≥0

=3 n, n<0

2. Define multi channel and multi dimensional signals

3. Define symmetric and anti symmetric signals.

4. Differentiate recursive and non recursive difference equations.

5. What is meant by impulse response?

6. What is meant by LTI system?

7. What are the basic steps involved in convolution?

8. Define the Auto correlation and Cross correlation?

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

10. What is zero padding? What are it uses?

11. State the Sampling Theorem.

12 What is an anti imaging and anti aliasing filter?

13. Determine the signals are periodic and find the fundamental period

in√ 2 πt

i) sin 20πt+ sin5πt

14. Give the mathematical and graphical representations of a unit sample, unit step

sequence.

15. Sketch the discrete time signal x(n) =4 δ (n+4) + δ(n)+ 2 δ (n-1) + δ (n-2) -5 δ (n-3

**)**16

**.**Find the periodicity of x(n) =cost(2πn / 7)17. What is inverse system?

18. Write the relationship between system function and the frequency response.

19. Define commutative and associative law of convolutions.

20. What is meant by Nyquist rate and Nyquist interval?

21. What is an aliasing? How to overcome this effect?

22. What are the disadvantages of DSP?

23. State initial value theorem of Z transform.

24 What are the different methods of evaluating inverse z transform?

25 What is meant by ROC?

26 What are the properties of ROC?

27 What is zero padding? What are it uses?

28 State convolution property of Z transform.

29 State Cauchy residue theorem.

30 Define fourier transform.

31 Define discrete fourier series.

32 Compare linear and circular convolution.

33 Distinguish between Fourier series and Fourier transform.

34 What is the relation between fourier transform and z transform.

35 What is the use of Fourier transform?

36. Define system function.

37. State Parseval relation in z transform

**CLASSIFICATION OF SYSTEMS:**

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

i)y(n) = Ax(n) +B

ii)y(n) =x(2n)

iii)y(n) =n x2 (n)

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

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

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

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

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

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

v) y(n) =x(n) +n x (n+1)

3.a) For each impulse response determine the system is i) stable ii) causal (8)

i) h(n)= sin

**(**π n / 2)ii) h(n) = δ(n) + sin π n

iii) h(n) = 2 n u(-n)

. b)Find the periodicity of the signal x(n) =sin

**(2**πn / 3)+**cos (**π n / 2) (8)4. Explain in detail about A to D conversion with suitable block diagram and to

reconstruct the signal.

5 a) State and proof of sampling theorem. (8)

b)What are the advantages of DSP over analog signal processing? (8)

6 a)Explain successive approximation technique. (8)

b)Explain the sample and hold circuit. (8)

**Z TRANSFORM:**

1. a)State and proof the properties of Z transform. (8)

b)Find the Z transform of (8)

i) x(n) =[ (1/2)n – (1/4)n ] u(n)

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

iii) x(n) (-1)n cos (πn/3) u(n)

iv) x(n) = (½) n-5 u(n-2) +8(n-5)

2 a) Find the Z transform of the following sequence and ROC and sketch the pole zero

diagram (8)

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

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

b)Find the convolution of using z transform (8)

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

(1/2) - n n<0 }

x2(n) = (1/2) n

**INVERSE Z TRANSFORM**:

5. Find the inverse z transform

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.

6.. 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 )

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 n u(-n-1)

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

**ANALYSIS OF LTI SYSTEM:**

7. a)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) (8)

b). 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 (8)

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

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

9. a)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) (8)

b). 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) (8)

10. 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.

11. 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).

**CONVOLUTION:**

12. 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

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

= 2 n= -1

= 0 elsewhere

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

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

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

Convolution.

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

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

0, elsewhere } 0, elsewhere }

__UNIT 2__1. How many multiplication and additions are required to compute N point DFT using

radix 2 FFT?

2. Define DTFT pair.

3. What are Twiddle factors of the DFT?

4. State Periodicity Property of DFT.

5. What is the difference between DFT and DTFT?

6. Why need of FFT?

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

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

9. Compare DIT and DIF?

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

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

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

13. What is meant by radix 2 FFT?

14. State the properties of W N

k ?

15. What is bit reversal in FFT?

16. Determine the no of bits required in computing the DFT of a 1024 point sequence

with SNR of 30dB.

17. What is the use of Fourier transform?

18. What are the advantages FFT over DFT?

**FOURIER TRANSFORM**:

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

(8)

b) Determine the Inverse Fourier transform H (w) = (1-ae-jw) -1 (8)

2. State and proof the properties of Fourier transform

**FFT:**

3. Determine the Discrete Fourier transform x (n) = (1, 1, 1, 1) and

Proof x(n)*h(n) =X(z) H(z

4. Derive and draw the 8 point FFT-DIT butterfly structure.

5. Derive and draw the 8 point FFT-DIF butterfly structure.

6.Compute the DFT for the sequence.(0.5,0.5,0.5,0.5,0,0,0,0)

7.Compute the DFT for the sequence.(1,1,1,1,1,1,0,0)

8.Find the DFT of a sequence x(n)=(1,1,0,0) and find IDFT of Y(k) =(1,0,1,0)

9. If x (n) = sin (nΠ/2), n=0, 1, 2, 3

10. h (n) = 2 n , n=0,1,2,3.Find IDFT and sketch it.

11.Find 4 point DFT using DIF of x(n) =(0,1,2,3)

12.a)Discuss the properties of DFT. (10)

b).Discuss the use of FFT algorithm in linear filtering. (6)

__UNIT 3& 4__1. Define canonic and non canonic form realizations

**.**2. Draw the direct form realizations of FIR systems?

3. Mention advantages of direct form II and cascade structure?

4. Define Bilinear Transformation.

5. What is prewar ping? Why is it needed?

6. Write the expression for location of poles of normalized Butterworth filter.

7. Distinguish between FIR and IIR Filters.

8. What is linear phase filter?

9. What are the design techniques available for IIR filter?

10. What is the main drawback of impulse invariant mapping?

11. Compare impulse invariant and bilinear transformation.

12. Why IIR filters do not have linear phase?

13. Mention the properties of Butterworth filter?

14. Mention the properties of Chebyshev filter?

15. Why impulse invariant method is not preferred in the design of high pass IIR filter?

16. Give the transform relation for converting LPF to BPF in digital domain.

17. What are Gibbs oscillations?

18. Explain briefly Hamming window .

19. If the impulse response of the symmetric linear phase FIR filter of length 5 is h(n) =

{2, 3, 0, x, y), then find the values of x and y.

20. What are the desirable properties of windowing technique?

21. Write the equation of Bartlett window.

22.Why IIR filters do not have linear phase?

23.Why FIR filters are always stable?

24.Why rectangular window are not used in FIR filter design using window method?

25.What are the advantages of FIR filter?

26.What are the advantages and disadvantages of window?

27.What is the necessary condition and sufficient condition for the linear phase characteristic of a

FIR filter?

28.Compare Hamming and Hanning window?

29.Why triangular window is not a good choice for designing FIR Filter?

30.Why Kaiser window is most used for designing FIR Filter?

31.What is the advantages in linear phase realization of FIR systems?

**Structures of IIR systems:**

1. Obtain the cascade and parallel form realizations for the following systems

Y (n) = -0.1(n-1) + 0.2 y (n-2) + 3x (n) +3.6 x (n-1) +0.6 x (n-2)

2.a) Obtain the Direct form II

y (n) = -0.1(n-1) + 0.72 y(n-2) + 0.7x(n) -0.252 x(n-2) (8)

b) .Find the direct form II

H (z) =8z-2+5z-1+1 / 7z-3+8z-2+1 (8)

3. Obtain the i) Direct forms ii) cascade iii) parallel form realizations for the following systems

y (n) = 3/4(n-1) – 1/8 y(n-2) + x(n) +1/3 x(n-1)

4.Find the direct form –I, cascade and parallel form for

H(Z) = z -1 -1 / 1 – 0.5 z-1+0.06 z-2

**IIR FILTER DESIGN:**

6. Explain the method of design of IIR filters using bilinear transform method.

7. a)Derive bilinear transformation for an analog filter with system function H(s) = b/ s + a (8)

b) For the analog transfer function H(s) = 2 / (s+1) (s+3) .

Determine H (z) using bilinear transformation. With T=0.1 sec (8)

8. a)Convert the analog filter H(s) = 0.5 (s+4) / (s+1)(s+2) using impulse invariant transformation

T=0.31416s (8)

b)The normalized transfer function of an analog filter is given by H a (sn) = 1/ sn

2 +1.414 s n +1.

Convert analog filter to digital filter with cut off frequency of 0.4 π using bilinear transformation.

(8)

9. Design a single pole low pass digital IIR filter with -3db bandwidth of 0.2Π by using bilinear

transformation.

10. For the constraints

0.8 ≤ |H (e jw)| ≤1, 0 ≤ ω ≤ 0.2π

|H (e jw)| ≤0.2, 0.6π ≤ ω ≤π with T= 1 sec .Determine system function H(z) for a Butterworth

filter using Bilinear transformation.

11.Design a digital Butterworth filter satisfying the following specifications

0.7 ≤ |H (e jw)| ≤1, 0 ≤ ω ≤ 0.2π

|H (e jw)| ≤0.2, 0.6π ≤ ω ≤π with T= 1 sec .Determine system function H(z) for a Butterworth

filter using impulse invariant transformation.

12. Design a digital Chebyshev low pass filter satisfying the following specifications 0.707 ≤ |H (e

jw)| ≤1, 0 ≤ ω ≤ 0.2π

|H (e jw)| ≤0.1 0.5 ≤ ω ≤π with T= 1 sec using for bilinear transformation.

13.Design a digital Butterworth High pass filter satisfying the following specifications

0.9 ≤ |H (e jw)| ≤1, 0 ≤ ω ≤ π/2

|H (e jw)| ≤0.2, 3π/4 ≤ ω ≤π with T= 1 sec. using impulse invariant transformation

14. Design a realize a digital filter using bilinear transformation for the following specifications

i) Monotonic pass band and stop band

ii) -3.01 db cut off at 0.5 π rad

iii) Magnitude down at least 15 db at ω = 0.75 π rad.

**FIR FILTER**

15.a) Prove that an FIR filter has linear phase if the unit sample response satisfies the condition h(n)

= ± h(M-1-n), n =0,1,….. M-1.Also discuss symmetric and anti symmetric cases of FIR filter.

(8)

b) Explain the need for the use of window sequence in the design of FIR filter. Describe the

window sequence generally used and compare the properties. (8)

16. Design a HPF of length 7 with cut off frequency of 2 rad/sec using Hamming window. Plot the

magnitude and phase response.

17. Explain the principle and procedure for designing FIR filter using rectangular window

18. Design a filter with

H d (e

jώ) = e - 3 jώ , π/4 ≤ ω ≤ π/4

0. π/4 ≤ ω ≤ π using a Hamming window with N=7.

19. H (w) =1 for | ω | ≤ π/3 and | ω | ≥2 π/3

0 otherwise for N=11. and find the response.

20.Design a FIR filter whose frequency response

H (e jώ) = 1 π/4 ≤ ω ≤ 3π/4

0. | ω | ≤3 π/4.

Calculate the value of h(n) for N=11 and hence find H(z).

21.Design an ideal differentiator with frequency response H (e jώ) = jw -π ≤ ω ≤ π

using hamming window for N=8 and find the frequency response.

22.Design an ideal Hilbert transformer having frequency response

H (e jώ) = j -π ≤ ω ≤ 0

-j 0 ≤ ω ≤ π for N=11 using rectangular window.

**FIR structures**:

23.a) Determine the direct form of following system (8)

H (z) =1+2z-1 - 3z-2 + 4z-3 - 5z-4

b). Obtain the cascade form realizations of FIR systems (8)

H (z) = 1+5/2 z-1+ 2z-2 +2 z-3

**UNIT 5**

1) Give some example of DSP

2) Explain Interpolation

3) Explain decimation

4) What is know to be subband coding

5) Define sampling rate conversion

6) How the image enchancement is achieved using DSP

7) Define compression

8) What are various compression technique

9) Explain subband coding

10) Define vocoders

11) Explain adaptive filtering

1) Explain the concept of deciation by a factor D and interpolation by factor I

2) With help of equation explain sampling rate conversion by a rational factor I/D

3) Explain the following application

i) speech compression

ii) sound processing

4) With help diagram explain adaptive filtering process

5) Explain speech vocoders and subband coding

6) Explain how image enchancement was achieved

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