Em LAB

Em 2 mannual

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dsp notes

Part B

1. Determine the DFT of the sequence x(n) =1/4, for 0<=n <=2

0, otherwise

Ans: The N point DFT of the sequence x(n) is defined as

N-1

x(k)= ∑ x(n)e-j2πnk/N             K=0,1,2,3,…N-1 n=0

x(n) = (1/4,1/4,1/4)

X(k) = ¼ e-j2πk/3[1+2cos(2πk/3)]   where k= 0,1,……….,N-1

2. Derive the DFT of the sample data sequence x(n) = {1,1,2,2,3,3}and compute the corresponding amplitude and phase spectrum.

Ans: The N point DFT of the sequence x(n) is defined as

N-1

X(k)=  ∑ x(n)e-j2πnk/N             K=0,1,2,3,…N-1 n=0

X(0) = 12

X(1) = -1.5 + j2.598

X(2) = -1.5 + j0.866

X(3) = 0

X(4) = -1.5 – j0.866

X(5) =-1.5-j2.598

X(k) = {12, -1.5 + j2.598, -1.5 + j0.866,0, -1.5 – j0.866, -1.5-j2.598}

|X(k)|={12,2.999,1.732,0,1.732,2.999}

∟X(k)={0,- π/3,- π/6,0, π/6, π/3}

3.Given x(n) = {0,1,2,3,4,5,6,7} find X(k) using DIT FFT algorithm.

Ans: Given N = 8

W

N

k = e-j(2π/N)k

W

8

0  = 1

W

8

1 =0.707-j0.707

W

8

2 = -j

W

8

3 = -0.707-j0.707

Using butterfly diagram

X(k) = {28,-4+j9.656,-4+j4,-4+j1.656,-4,-4-j1.656,-4-j4,-4-j9.656}

4.Given X(k) = {28,-4+j9.656,-4+j4,-4+j1.656,-4,-4-j1.656,-4-j4,-4-j9.656} ,find x(n)

using inverse DIT FFT algorithm.

W

N

k = ej(2π/N)k

W

8

0  = 1

W

8

1 =0.707+j0.707

W

8

2 = j

W

8

3 = -0.707+j0.707

x(n) = {0,1,2,3,4,5,6,7}

5.Find the inverse DFT of X(k) = {1,2,3,4} Ans: The inverse DFT is defined as

N-1

x(n)=(1/N ) ∑ x(k)ej2πnk/N                           n=0,1,2,3,…N-1 k=0

x(0) = 5/2

x(1) = -1/2-j1/2 x(2) = -1/2

x(3) = -1/2+j1/2

x(n) = {5/2, -1/2-j1/2, -1/2, -1/2+j1/2}

6. Design an ideal low pass filter with a frequency response Hd(e jw) =1 for –

∏/2<=w<=∏/2

find the value of h(n) for N=11 find H(Z) plot magnitude response a.   Find h(n) by IDTFT

b.   Convert h(n) in to a fine length by truncation c.   H(0)=1/2,

0 otherwise

h(1)=h(-1)=0.3183 h(2)=h(-2)=0 h(3)=h(-3)= -0.106 h(4)=h(-4)=0 h(5)=h(-5)=0.06366

d.     Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)

e.   H’(Z) obtained is 0.06366-0.106z-2+.3183Z-4+.5Z-5+.3183Z-6-.106Z-8+0.06366Z-

10

f.   Find H (e jw) and plot amplitude response curve.

7. Design an ideal low pass filter with a frequency response Hd(e jw) =1 for –

∏/4<=|w|<=∏

find the value of h(n) for N=11 find H(Z) plot magnitude response

0 otherwise

g.   Find h(n) by IDTFT

h.   Convert h(n) in to a fine length by truncation i.       H(0)=0.75

h(1)=h(-1)=-.22 h(2)=h(-2)=-.159 h(3)=h(-3)= -0.075 h(4)=h(-4)=0 h(5)=h(-5)=0.045

j.      Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)

k.   H’(Z) obtained is 0.045-0.075z-2  -.159 Z-3-0.22Z-4+0.75Z-5-.22Z-6   -0.159Z-7   -.

075Z-8+0.045Z-10

l.    Find H (e jw) and plot amplitude response curve.

8. Design band pass filter with a frequency response Hd(e jw) =1 for –∏/3<=|w|<=2∏/3

0 otherwise

find the value of h(n) for N=11 find H(Z) plot magnitude response

m. Find h(n) by IDTFT

n.   Convert h(n) in to a fine length by truncation

o.     Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)

p.   H’(Z) obtained Find H (e jw) and plot amplitude response curve.

9. Design band reject filter with a frequency response Hd(e jw) =1 for ∏/4<=|w|<=3∏/4

0 otherwise

find the value of h(n) for N=11 find H(Z) plot magnitude response

q.   Find h(n) by IDTFT

r.    Convert h(n) in to a fine length by truncation

s.     Find the transfer function H(Z) which is not realizable conver in to realizable by multiplying by z-(N-1/2)

t.    H’(Z) obtained Find H (e jw) and plot amplitude response curve.

10. Derive the condition of FIR filter to be linear in phase. Conditions are

Group delay and Phase delay should be constant

And

And show the condition is satisfied

11Derive the expression for steady state I/P Noise Power and Steady state O/P Noise

Power.

Write the derivation.

12   Draw the product quantatization model for first order and second order filter

Write the difference equation and draw the noise model.

13  For the second order filter Draw the direct form II realization and find the scaling factor S0 to avoid over flow

Find the scaling factor from the formula

1+r2

I=    --------------------------------------- (1-r2)(1-2r2cos2ø =r4)

14  Explain Briefly about various number representation in digital computer.

1 Fixed point

2 Floating point

3 Block floating point

Signed magnitude represenation

1’s Complement

2’s Complement etc

15  Consider the transfer function H(Z)=H1(Z)H2(Z) where H1(Z) =1/1-a1Z-1

H2(z) =1/ 1-a2Z-1

Find the o/p Round of noise power Assume a1=0.5 and a2= 0.6 and find o.p round off noise power.

Draw the round of Noise Model. By using residue method find σ01

By using residue method find σ 02

= σ 01 2+ σ 02  2

2 –2b (5.43) Ans:                             

12

16.Explain the architecture of DSP processor . Diagram. & explanation.

17.Describe briefly the different methods of power spectral estimation?

1.         Bartlett method

2.         Welch method

3.         Blackman-Tukey method and its derivation.

18.what is meant by A/D conversion noise. Explain in detail?

A DSP contains a device, A/D converter that operates on the analog input x(t) to produce xq(t) which is binary sequence of 0s and 1s.

At first the signal x(t) is sampled at regular intervals to produce a sequence x(n) is of infinite precision. Each sample x(n) is expressed in terms of a finite number of bits given the sequence xq(n). The difference signal e(n)=xq(n)-x(n) is called A/D conversion noise.

+ derivation.

19  onsider the transfer function H(Z)=H1(Z)H2(Z) where H1(Z) =1/1-a1Z-1

H2(z) =1/ 1-a2Z-1

Find the o/p Round of noise power Assume a1=0.7 and a2= 0.8and find o.p round off noise power.

Draw the round of Noise Model. By using residue method find σ01

By using residue method find σ 02

= σ 01 2+ σ 02  2

20.Given X(k) = {1,1,1,1,1,1,1,1,} ,find x(n) using inverse DIT FFT algorithm.

W

N

k = ej(2π/N)k

Find x(n)

21.Find the inverse DFT of X(k) = {3,4,5,6} Ans: The inverse DFT is defined as

N-1

x(n)=(1/N ) ∑ x(k)ej2πnk/N                           n=0,1,2,3,…N-1 k=0

22. Explain various addressing modes of TMS processor.

Immediate. Register

Register indirect

Indexed

      & its detail explanation.

23  Derive the expression for steady state I/P Noise Variance and Steady state O/P Noise Variance

Write the derivation.

24.   Explain briefly the periodogram method of power spectral estimation?

Write the derivation with explanation.

      25. Explain various arithmetic instruction of TMS processor.

       All arithmetic instruction with explanation.