CS2403_QB _franxavier.pdf

Unit I Signals and system PART A

1. Define Signal. A Signal is defined as any physical quantity that varies with time, space or any other

independent variables.

2. Define a system. A System is a physical device (i.e., hardware) or algorithm (i.e., software) that performs an

operation on the signal.

3. What are the steps involved in digital signal processing? Converting the analog signal to digital signal, this is performed by A/D converter Processing Digital signal by digital system. Converting the digital signal to analog signal, this is performed by D/A converter.

4. Give some applications of DSP? Speech processing Speech compression & decompression for voice storage system Communication Elimination of noise by filtering and echo cancellation. Bio-Medical Spectrum analysis of ECG,EEG etc.

5. Write the classifications of DT Signals. Continuous & Discrete signals Energy & Power signals Periodic & Aperiodic signals Even & Odd signals Deterministic & Random signals

6. What is an Energy and Power signal? MAY 2012 Energy signal: A finite energy signal is periodic sequence, which has a finite energy but zero average power.

Power signal: An Infinite energy signal with finite average power is called a power signal.

7. What is Discrete Time Systems? The function of discrete time systems is to process a given input sequence to generate output

sequence. In practical discrete time systems, all signals are digital signals, and operations on such signals also lead to digital signals. Such discrete time systems are called digital filter.

8. Write the Various classifications of Discrete-Time systems. Linear & Non linear system Causal & Non Causal system Time variant & Time invariant system Stable & Un stable system Static & Dynamic system FIR & IIR system

9. Define Linear system A system is said to be linear system if it satisfies Super position principle. Let us consider

x1(n) & x2(n) be the two input sequences & y1(n) & y2(n) are the responses respectively + = +

10. Define Static & Dynamic systems When the output of the system depends only upon the present input sample, then it is called

static system, otherwise if the system depends past values of input then it is called dynamic system

11. Define causal system. When the output of the system depends only upon the present and past input sample, then it is

called causal system, otherwise if the system depends on future values of input then it is called non-causal system

12. What are FIR and IIR systems? The impulse response of a system consist of infinite number of samples are called IIR system

& the impulse response of a system consist of finite number of samples are called FIR system.

13. What are the basic elements used to construct the block diagram of discrete time system? The basic elements used to construct the block diagram of discrete time Systems are Adder,

Constant multiplier &Unit delay element.

14. What is ROC in Z-Transform? The values of z for which z transform converges is called region of convergence (ROC).

The z-transform has an infinite power series; hence it is necessary to mention the ROC along with z-transform.

15. List the properties of Z-Transform. Linearity property Time Shifting property Frequency shift or Frequency translation Time reversal property Scaling property Differentiation property Convolution property Parsevals theorem Initial value theorem Final value theorem

16. What are the different methods of evaluating inverse z-transform? Partial fraction expansion Long division method Contour integration (Residue method)

17. Define sampling theorem. MAY2011/NOV2013 A continuous time signal can be represented in its samples and recovered back if the

sampling frequency Fs > 2f. Here Fs is the sampling frequency and f is the maximum frequency present in the signal.

18. What are the properties of convolution? NOV2013 1. Commutative property x(n) * h(n) = h(n) * x(n) 2. Associative property [x(n) * h1(n)]*h2(n) = x(n)*[h1(n) * h2(n)] 3. Distributive property x(n) *[ h1(n)+h2(n)] = [x(n)*h1(n)]+[x(n) * h2(n)]

19. Why the result of circular and linear convolution is not same? Circular convolution contains same number of samples as that of x (n) and h (n), while in linear

convolution, number of samples in the result (N) are, N=L+M-1 Where L= Number of samples in x (n) M=Number of samples in h (n)

20. What is meant by aliasing? How it can be avoided? Nov2012 When the sampling frequency is less than twice of the highest frequency content of the signal,

then the aliasing is frequency domain takes place. In aliasing, the high frequencies of the signal mix with lower frequencies and create distortion in frequency spectrum.

Part B

1. State and Explain sampling theorem. NOV 2010/NOV 2013

It is a process of converting a continuous time signal to discrete time Signal. The continuous time signal is sampled at regular interval.

The sampling interval is defined as time interval between two successive samples. It is also called as sampling time.

= Where, Sampling Rate Sampling Time

Sampling Theorem states that a band limited signal with highest frequency component (fm) can be determined from its samples. If the sampling frequency is greater than or equal to twice the maximum frequency of the signal,

2 If < 2, Aliasing problem will occurs.

What is Aliasing effect?

Aliasing is a problem due to interference of information between two band of frequencies.

To avoid Aliasing effect, using sampling theorem, sampling continuous time signal at high rate.

Proof of Sampling Theorem:

Let x(t) be the continuous time signal(band limited) to fm. the sampling function samples the signal regularly at the rate of fs samples per seconds.

x(t) is sampled by function.

Impulse function is given as,

= After sampling, we get

= Fourier transform of equation 1 & 2, we get

= =

Equation 4 can be written as

= 0 + = +

Fourier Transform of x(t): If we take Fourier transform of continuous time x(t), we get

Xf = $ xt e()*dt,

For discrete put t=nTs

= e()-./ fs = 2f

= +

Rearranging,

= 1

= 1 If frequency lies between f to +f

Substitute equation 7 in 9

= 1 1 23

45677 8 , +

Reconstruction of x(t): x(t) can be represented from 10 by putting fs = 2f.

= 12 1 2 23

456778

Taking Inverse Fourier Transform(IFT),

xt = $ ; 12f x < n2f>

- e()-)? e()*df7

7

xt = x < n2f>

-12f $ e()df

77

Interchanging order of summation and integration,

xt = x < n2f>

- @sin 2ft n2ft n C

Since,

sin DE = FG HEHE sin 2ft n2ft n = sin C2ft n

xt = x < n2f>

- sin C2ft n , < n < This is known as interpolation formula, by expanding 14, we get

x(t)=x(0) sin C2ft + x sin C2ft 1 +

2. List various properties of z-transform. 1.) Linearity property 2.) Time shifting Property 3.) Time reversal Property 4.) Scaling Property 5.) Differentiation Property 6.) Convolution Property 7.) Parsevals Theorem 8.) Initial Value Theorem 9.) Final Value Theorem

1.) Linearity Property: Let , Discrete sequence M O M O

Then,

+ M O + O Proof:

By definition of Z transform,

Xz = xnz- z + = + z-

= z- + z-

= z- + z-

z + = X + X

2.) Time Shifting Property: Let , Discrete sequence M O Q M ORS

Proof:

zxn = xnz- zxn k = xn kz-

Put, n k = m n = m + k When = , V = = , V = V WXG4F XYV Y +

zxn k = xmz[\] = xmz[z] zxn k = z]Xz

3.) Time Reversal Property: Let , Discrete sequence M O M O

Proof:

zxn = xnz- Put, n = m m = n When = , V = = , V = V WXG4F XYV Y +

zxn = xmz[ = xmz[ zxn = Xz

4.) Scaling Property: Let , Discrete sequence M O M

Proof:

za-xn = a-xnz- = xnaz-

= xn -

za-xn = X

5.) Differentiation Property: Let , Discrete sequence M O

M R aab R Proof:

zxn = Xz = xnz- ccR Xz = ccR ; xnz-

? = ccR xnz-

= xnnz- = xnnz-z

ccR Xz = 1z nxnz-

z ccR Xz = nxnz-

6.) Convolution Property: Let , Discrete sequence M O M O

Then,

R = O O

Proof:

= Q Q]

R = 1 Q Q] 8 z-

Changing order of summation,

R = Q Q- z-z] z\]

S = Qz] Q- z-]

S

= Xz 1 Q- z-]8 Put, n k = m n = m + k When = , V = = , V = V WXG4F XYV Y +

R = Xz 1 V,[, z[8 R = Xz Xz

7.) Parsevals Theorem: Let , Discrete sequence M O M O Then,

,, = 12He f R 2 1R3 R cR

8.) Initial Value Theorem: Let , Discrete sequence M O 4,

0 = limb, O Proof:

zxn = Xz = xnz-,, Xz = x0zj + x1z + x2z +

Apply the limit R limb, Xz = limb, x0zj + limb, x1z + limb, x2z + limb, Xz = 0

9.) Final Value Theorem: If M O

Then,

lim, = lim|b|1 RO

3. Check whether the following systems are static or dynamic, linear or non-linear, time variant or invariant, causal or non-causal, stable or unstable. Nov/Dec 2013

i. y(n) = Cos[x(n)] ii. y(n) = x(-n+2)

iii. y(n) = x(2n) iv. y(n) = x(n). coslj(n)

Solution:

A) static: depends on present input state, dynamic:- depends on past and future input state B) causal:- depends on present and past inputs, non-causal:- depends on future input C) Linear:- satisfies the super position principle + = +

Otherwise non-linear D) Time invariant: system do not change with time

m, Q = m Q Time variant: system change with time. m, Q m Q

E) Stable: it produces a bounded output sequence for every bounded input sequence Unstable: it produces a unbounded(infinite) output sequence for every bounded input sequence

i. y(n) = Cos[x(n)] static, non linear, time invariant, causal, stable ii. y(n) = x(-n+2)

iii. y(n) = x(2n) iv. y(n) = x(n). cosop(n)

4. Describe the classification of system. i. Linear & Non linear system

ii. Causal & Non Causal system iii. Time variant & Time invariant system iv. Stable & Un stable system v. Static & Dynamic system

vi. FIR & IIR system

i. static: depends on present input state, dynamic:- depends on past and future input state ii. causal:- depends on present and past inputs, non-causal:- depends on future input

iii. Linear:- satisfies the super position principle + = + Otherwise non-linear

iv. Time invariant: system do not change with time m, Q = m Q Time variant: system change with time.

m, Q m Q v. Stable: it produces a bounded output sequence for every bounded input sequence

Unstable: it produces a unbounded(infinite) output sequence for every bounded input sequence

vi. FIR:- The system is of finite duration, IIR:- The system is of infinite duration.

5. Describe the different types of digital signal representation. MAY 2013 There are different types of representations for signal

a) Graphical representation b) Functional representation c) Tabular representation d) Sequence representation

6. Perform the convolution of the signals

: 7.7.7.7. Find the inverse Z-transform of st = uv\uuwux, ROC: |Z|>3, using a) Residue Method b) Convolution Method NOV 2011

Solution: a) Residue Method:

yz{|}~z = w w! u }w}uw u stuw Where m order of pole at Z=a

~ yz{|}~z = w + vx~ b) Convolution Method: = w v

= w. v ,, = w + vx~

8. A causal system is represented y the following difference equation MAY 2011/NOV 2012

+ w w = + wv w Find the system transfer function H(z), unit sample response, magnitude Response and phase

function of the system.

Solution:-

I) Transfer function:

R = 1 + 12 O1 + 14 O

45 = 1 + 12 451 + 14 45 II) Magnitude response

|| = l l || = 1.25 + YFl1.06 + 0.5YFl

III) Phase response:

l = tan GVGXm Y l4 Y l l = tan 2 0.25 sin l1 + 0.25 cos l3 tan 2 0.5 sin l1 + 0.5 cos l3

9. A) Determine the Z transform and ROC of the signal. = ~ w Solution: -

From the definition

R = O,, R = O,

UNIT II Frequency Transformations

PART A

1. What is DFT? NOV2013 It is a finite duration discrete frequency sequence, which is obtained by sampling one period of

Fourier transform. Sampling is done at N equally spaced points over the period extending from l =0Y 2H.

2. Define N point DFT. The DFT of discrete sequence x(n) is denoted by X(K). It is given by, Here k=0,1,2N-1 Since

this summation is taken for N points, it is called as N-point DFT.

3. What is DFT of unit impulse (n)? The DFT of unit impulse (n) is unity.

4. List the properties of DFT. Periodicity Linearity Symmetry property Circular convolution of two sequences Time reversal of sequence Circular Time shift of sequence Circular Frequency shift of sequence Circular correlation of two sequences Multiplication of two sequences Parsevals theorem.

5. State Linearity property of DFT. DFT of linear combination of two or more signals is equal to the sum of linear combination of

DFT of individual signal. 6. What is circular time shift of sequence?

Shifting the sequence in time domain by 1 samples is equivalent to multiplying the sequence in frequency domain by WNkl

7. What is the disadvantage of direct computation of DFT? For the computation of N-point DFT, N2 complex multiplications and N[N-1] Complex additions

are required. If the value of N is large than the number of computations will go into lakhs. This proves inefficiency of direct DFT computation.

8. What is the way to reduce number of arithmetic operations during DFT computation? Number of arithmetic operations involved in the computation of DFT is greatly reduced by using

different FFT algorithms as follows. 1. Radix-2 Decimation in Time (DIT) algorithm. 2. Radix-2 Decimation in Frequency (DIF) algorithm. 3. Radix-4 FFT algorithm.

9. What is the computational complexity using FFT algorithm? Complex multiplications = N/2 log2N Complex additions = N log2N

10. How linear filtering is done using FFT? Correlation is the basic process of doing linear filtering using FFT. The correlation is nothing but

the convolution with one of the sequence, folded. Thus, by folding the sequence h (n), we can compute the linear filtering using FFT.

11. What is zero padding? What are its uses? Let the sequence x (n) has a length L. If we want to find the N point DFT (N>L) of the sequence x

(n). This is known as zero padding. The uses of padding a sequence with zeros are (i) We can get better display of the frequency spectrum. (ii) With zero padding, the DFT can be used in linear filtering.

12. Why FFT is needed? The direct evaluation of the DFT using the formula requires N2 complex multiplications and N

(N-1) complex additions. Thus for reasonably large values of N (inorder of 1000) direct evaluation of the DFT requires an inordinate amount of computation. By using FFT algorithms the number of computations can be reduced. For example, for an N-point DFT, The number of complex multiplications required using FFT is N/2log2N. If N=16, the number of complex multiplications required for direct evaluation of DFT is 256, whereas using DFT only 32 multiplications are required.

13. What is the speed of improvement factor in calculating 64-point DFT of a sequence using direct computation and computation and FFT algorithms?

The number of complex multiplications required using direct computation is N2=642=4096. The number of complex multiplications required using FFT is N/2 log2N = 64/2log264=192. Speed improvement factor = 4096/192=21.33

14. What is FFT? MAY2012 The fast Fourier transforms (FFT) is an algorithm used to compute the DFT. It makes use of the

Symmetry and periodically properties of twiddles factor WKN to effectively reduce the DFT computation time. It is based on the fundamental principle of decomposing the computation of the DFT of a sequence of length N into successively smaller discrete Fourier transforms. The FFT algorithm provides speed-increase factors, when compared with direct computation of the DFT, of approximately 64 and 205 for 256-point and 1024-point transforms, respectively.

15. How many multiplications and additions are required to compute N-point DFT using redix-2 FFT?

The number of multiplications and additions required to compute N-point DFT using redix-2 FFT are N log2N and N/2 log2N respectively.

16. What is meant by radix-2 FFT? The FFT algorithm is most efficient in calculating N-point DFT. If the number of output points N

can be expressed as a power of 2, that is, N=2M, where M is an integer, Then this algorithm is known as radix-s FFT algorithm.

17. What is a decimation-in-time algorithm? Decimation-in-time algorithm is used to calculate the DFT of a N-point Sequence. The idea is to

break the N-point sequence into two sequences, the DFTs of which can be combined to give the DFT of the original N-point sequence. Initially the N-point sequence is divided into two N/2-point sequences xe(n) and x0(n), which have the even and odd members of x(n) respectively. The N/2 point DFTs of these

two sequences are evaluated and combined to give the N point DFT. Similarly the N/2 point DFTs can be expressed as a combination of N/4 point DFTs. This process is continued till we left with 2-point DFT. This algorithm is called Decimation-in-time because the sequence x(n) is often splitted into smaller sub sequences.

18. What are the differences and similarities between DIF and DIT algorithms? Differences:

1. For DIT, the input is bit reversal while the output is in natural order, whereas for DIF, the input is in natural order while the output is bit reversed.

2. The DIF butterfly is slightly different from the DIT butterfly, the difference being that the complex multiplication takes place after the add-subtract operation in DIF. Similarities: Both algorithms require same number of operations to compute the DFT. Both algorithms can be done in place and both need to perform bit reversal at some place during the computation.

19. What are the applications of FFT algorithms? NOV2012 1. Linear filtering 2. Correlation 3. Spectrum analysis

20. What is a decimation-in-frequency algorithm? In this the output sequence X (K) is divided into two N/2 point sequences and each N/2 point

sequences are in turn divided into two N/4 point sequences.

21. Distinguish between DFT and DTFT.

DFT DTFT Obtained by performing

sampling operation in both the time and frequency domains.

Discrete frequency spectrum

Sampling is performed only in time domain.

Continuous function of

22. Distinguish between Fourier series and Fourier transform.

Fourier Series

Fourier transform

Gives the harmonic content of a periodic time function.

Discrete frequency spectrum

Gives the frequency information for an aperiodic signal.

Continuous frequency spectrum

23. How linear filtering is done using FFT? Nov 2011 Correlation is the basic process of doing linear filtering using FFT. The correlation is nothing

but the convolution with one of the sequence, folded. Thus, by folding the sequence h (n), we can compute the linear filtering using FFT.

Part B

1. A)Find eight point DFT of the following sequence using direct method: MAY 2013 {1,1,1,1,1,1,0,0}

Solution:

Q = 456S/; Q = 0,1, 1j For the given sequence N=8, by substituting k and N values in above equation, we get

X(0)= 6 X(1)= -0.707- j1.707 X(2)= 1- j X(3)= 0.707+j0.293 X(4)= 0 X(5)= 0.707- j0.293 X(6)= 1+ j X(7)= - 0.707+j1.707

B) State any six properties of DFT (6 marks) MAY 2013 1. Periodicity 2. Linearity 3. Symmetry Property 4. Circular Convolution of two sequences. 5. Time reversal of sequence. 6. Parsevals Theorem.

2. A) Compute eight point DFT of the following sequence using radix 2 Decimation in time FFT algorithm. MAY 2011(16 mark)/MAY 2013(8 mark)

X(n)={1,-1,-1,-1,1,1,1,-1} Solution:- Butterfly diagram should be drawn (2 marks) The output of each stage are given below (each stage 2 marks)

Input of I stage Output of I stage Output of II stage Output of III stage

1 2 2 0

1 0 2j -1.414+j3.414 -1 0 2 2-2j 1 -2 -2j 1.414 - j0.586 -1 0 -2 4

1 -2 -2 1.414+j0.586 -1 -2 2 2+2j -1 0 -2 -1.414 j3.414

B) Discuss the use of FFT in linear filtering MAY 2013

The overlap-save and overlap-add methods are used for filtering a long data sequence with an FIR filter based on the use of DFT. FFT algorithm can be used for computing DFT and IDFT. (2 marks) Comparison of Overlap-save and overlap-add method in filtering. (4 marks)

3. Compute the FFT of the sequence = v + w p w, where N=8 using DIT algorithm. NOV 2012 Solution: - N=8 = v + w p x(0)=1 x(1)=2 x(2)=5 x(3)=10 x(4)=17 x(5)=26 x(6)=37 x(7)=50

Butterfly diagram should be drawn The output of each stage are given below

Input of I stage Output of I stage Output of II stage Output of III stage 1 18 60 148 17 -16 -16+32j -4.688+13.248j 5 42 -24 -24+32j 37 -32 -16-32j -27.312+13.248j 2 28 88 -28 26 -24 -24+40j -27.312-13.248j 10 60 -32 -24-32j 50 -40 -24-40j -4.688-77.248j

4. Find DFT for {1,1,2,0,1,2,0,1} using FFT DIT butterfly algorithm. NOV 2013

Butterfly diagram should be drawn The output of each stage are given below

Input of I stage Output of I stage Output of II stage Output of III stage 1 2 4 8 1 0 -2j -0.586j 2 2 0 -2j 0 2 2j 3.414j 1 3 4 0 2 -1 -1+j -3.414j 0 1 2 2j 1 -1 -1-j 0.586j

5. Discuss the properties of DFT. NOV 2013 1. Periodicity 2. Linearity 3. Symmetry Property 4. Circular Convolution of two sequences 5. Time Reversal of sequence 6. Circular time shift of sequence 7. Circular frequency shift of sequence 8. Circular Correlation of two sequences 9. Multiplication of two sequences 10. Parsevals Theorem

1. Periodicity: If discrete time signal is periodic, then its DFT is also periodic. If x(n) is a discrete signal of length N, then

i. x(n+N)=x(n) ii. X(k+N)=x(k)

Where x Input signal (time domain) X DFT[x(n)] (frequency domain)

2. Linearity: It satisfies superposition principle

YGQ Similarly,

YGQ

YGQ

By Linearity Property,

+ YGQ + Q Where a,b constants

3. Symmetry Property: If signal or sequence repeats its waveform in negative direction after N/2 number of

samples, then it is called symmetric signal or sequence. From periodicity property,

If x(n)=x(N-n) Then X(k)=X(N-k)

For Symmetric signal,

If YGQ Then YG Q

4. Circular Convolution of two sequences: Circular Convolution is defined as

V = . V j Where m=0,1,2,..N-1

It is denoted by = [ DGXXDYWYGY]

YGQ + Q 5. Time reversal of a sequence:

If YGQ Then = YG Q = Q

6. Circular Time Shift of a Sequence:

If YGQ Then V = 4 Q

i.e. V YG4 Q 7. Circular Frequency Shift:

If YGQ

Then 4 YGQ V

8. Circular Correlation of two sequences: Let

YGQ

YGQ RV = QQ

Then circular correlation is given by

V = j V

Circular correlation property is V = QQ (i.e)

j VYGQQ

9. Multiplication of two sequences: Let

YGQ

YGQ Then

YG 1 QQ [ DGXXDYWYGY]

10. Parsevals Theorem: For complex valued sequences c YGQ

YGQ Then

j =1 Q Q

Sj

is the general form of Parsevals theorem if = = then Q = Q = Q and . = || Q. Q = |Q|

||j =1 |Q|

Sj

6. Draw the flow chart for N=8 using radix 2 DIF algorithm for finding DFT coefficient. (Nov 2010)

Decimation-in-frequency (DIF) is another important radix 2 FFT algorithm. In case of Decimation-in-Time, the input data sequence x(n) are decimated but in case of Decimation-in-frequency, the data coefficients X(k) are decimated. In this algorithm, we first divide the DFT

formula into two summations. The first term contains first - points and the second term contains the last points.

Xk = xnW-]j , k = 0,1,2, , N 1

Xk = xnW-] j + xnW-]

Xk = xnW-] j + x 2n +

23 W]

j

Xk = xnW-] j + W

] x 2n + 23 W-]

j

Since W] = 1]

Xk = xnW-] j + 1] x 2n +

23 W-]

j

Xk = @xn + 1]x 2n + 23C

jW-] w

Now decimate X9K) into odd and even-indexed samples.

For even decimation,

X2k = @xn + x 2n + 23C

jW-], k = 0,1,2, , 22 13

X2k = @xn + x 2n + 23C

jW-], k = 0,1,2, , 22 13 v

For odd decimation,

X2k + 1 = @xn x 2n + 23C

jW-]\, k = 0,1,2, , 22 13

X2k + 1 = @xn x 2n + 23C W-

jW-], k = 0,1,2, , 22 13 x

The equation (2) and (3) can be redefined as

X2k = gn j W-], k = 0,1,2, , 2

2 13 Where

gn = + 2 + 23 X2k + 1 = gn

j W-], k = 0,1,2, , 2

2 13 Where

gn = @xn x 2n + 23C W-

Decimation in frequency stands for splitting the sequences in terms of frequency. That means we have split output sequences into smaller subsequences. This decimation is done as follows. First stage of decimation: first stage of decimation as shown in fig. below.

Second stage of decimation: In the first stage of decimation we have used 4-point DFT. We can further decimate the sequence by using 2 point DFT. The second stage of decimation is shown in fig below.

Third stage of decimation : In the second stage of decimation we have used 2- point DFT. So further decimation is not possible. Now we will use a butterfly structure to obtain 2-point DFT. Thus the total flow graph of 8 point DIF-FFT is shown below.

7. Develop a Radix-2, 8-point DIT FFT algorithm

The principle of decimation-in-time (DIT) can be obtained by considering the special case of N, an integer power of 2, i.e.

= X (1) If r=2, radix-2 = 2 (2)

In radix-2,N is an even integer. Let us consider computing X(k) by separating x(n) into two N/2 point subsequences consisting of the even-numbered points in x(n) and the odd-numbered points in x(n).

Xk = xnW-]j , k = 0,1,2, , N 1 x Separate x(n) into odd-numbered points and even numbered points.

Xk = xnW-]-- + xnW-]- , k = 0,1,2, , N 1 For even-numbered points, replace n=2r

For odd-numbered points, replace n=2X + 1.

Xk = x2rW] j + x2r + 1W\]

j

S = S = 45 = 45

6 = S = S

Xk = x2rS

j+ W] x2r + 1S

j

Q = Q + W]Hk, k = 0,1,2, , N 1 Where,

Q = x2rS

j

Q = x2r + 1S

j

Each term [G(k) and H(k)] in equation (5) is a - point DFT of 2X and 2X + 1 respectively. G(k) and H(k) are periodic, with period . Therefore,

2Q + 23 = Q 2Q + 23 = Q

= S

Hence, equation (5) becomes,

Q = Q + W]Hk, k = 0,1,2, , 22 13 2Q + 23 = 2Q + 23 + WS\ H 2Q + 23

2Q + 23 Q W]Hk, k = 0,1,2, , 22 13 Now k ranges between 0 to

Fig 1: Decimation in time of a length N-DFT into two lengths DFT followed by a combined stage

Fig 2: Radix-2 Decimation-in-Time FFT algorithm for a length-8 signal

8. Draw a 8 point radius 2 FFT DIT flow graphs and obtain DFT of the following sequence x(n)={0,1,-1,0,0,2,-2,0}

UNIT III - IIR FILTER DESIGN PART A

1. What is bilinear transformation? Nov2013 The bilinear transformation is conformal mapping that transforms the s-plane to z-plane. In

this mapping the imaginary axis of s-plane is mapped into the unit circle in z-plane, The left half of s-plane is mapped into interior of unit circle in z-plane and the right half of s-plane is mapped into exterior of unit circle in z-plane

2. What are the characteristics of Chebyshev filter? May 2013 The magnitude response of the chebyshev filter exhibits ripples either in pass band or in

stop band according to type The poles of the filter lies on an ellipse

3. What is impulse invariant transformation? May2012 The transformation of analog filter to digital filter without modifying the impulse response of

the filter is called impulse invariant transformation. 4. What is the importance of poles in filter design? Nov 2011

The stability of a filter is related to the location of the poles. For a stable analog filter the poles should lie on the left half of s-plane. For a stable digital filter the poles should lie inside the unit circle in the z-plane

5. Mention the properties of Butterworth filter? Nov 2013 All pole design. The poles lie on a circle in s-plane. The magnitude response is maximally flat at the origin and monotonically decreasing function

of . The normalized magnitude response has a value of 1 / 2 at the cutoff frequency c. Only few parameters have to be calculated to determine the transfer function.

6. What are the properties of bilinear transformation? May 2011 The mapping for the bilinear transformation is a one-to-one mapping that is for every point Z,

there is exactly one corresponding point S, and vice-versa. The j -axis maps on to the unit circle |z|=1,the left half of the s-plane maps to the interior of

the unit circle |z|=1 and the half of the s-plane maps on to the exterior of the unit circle |z|=1.

7. What are the various methods to design IIR filters? Approximation of derivatives Impulse invariance Bilinear transformation.

8. Write the transformation equation to convert low pass filter into band stop filter? MAY 2013

O O 21 + Q O + 1 Q1 + Q1 Q1 + Q O 21 + Q O + 1

Where,

= cos l + l2 cos l l2 Q = tan @l + l2 C tan l2

9. Distinguish between FIR and IIR filters. FIR filter IIR filter These filters can be easily designed to

have perfectly linear phase. FIR filters can be realized recursively

and non-recursively. Greater flexibility to control the shape of

their magnitude response. Errors due to roundoff noise are less

severe in FIR filters, mainly because feedback is not used.

These filters do not have linear phase.

IIR filters can be realized recursively.

Less flexibility,usually limited to kind of filters.

The roundoff noise in IIR filters are more.

10. What is prewarping? Prewarping is the method of introducing nonlinearly in frequency relationship to compensate

warping effect.

11. What are the various methods to design IIR filters? Approximation of derivatives Impulse invariance Bilinear transformation.

12. What is meant by warping effect? MAY 2011 The relation between the analog and digital frequencies in bilinear transformation is given by = 2 tan l2 For smaller values of l there exist linear relationship between l and . But for large values

of l the relationship is non-linear. The non-linearity introduces distortion in the frequency axis. This is known as wrapping effect.

PART B

1. For given analog filter system function H(s)=

\j.\j.\ into digital IIR filter by means of bilinear z transformation. Digital filter is to have resonant frequency o = v Nov2012/13 Ans:

From above eqn f=4 Value of o = v

Substitute s= MM\

2 A chebyshev low pass filter has the following specifications: Nov2013/2012 (a) Order of the filter = 3 (b) Ripple in pass-band = 1 db (c) Cut off frequency = 100 Hz (d) Sampling frequency = 1 kHz. Determine H(z) of the corresponding IIR digital filter using bilinear transformation technique

Ans Oder of the filter = N= 3

Step 1:Calculation of required design specification of digital filter

Given =1 db For normalize filter F sample = 1 Khz

Calculate sk=acosQ+jbsinQ, where Q= +S Calculate =-1+(1+-2)

Calculate a = p [ ] = 0.297 and b = p [ \ ] = 1.043 .Calculate Poles

ES = 6 + S\6 , Q = 0,1,2 k=0, Ej = 6 + 6 = 6 k=1, E = 6 + 6 = H k=2, E = 6 + 6 = 6

S k=a cosk+jbsink S 0=a cos0+jbsin0,

S 1=a cos1+jbsin1,

S 2=a cos2+jbsin2,

Calculate system function

3. Convert the analog filter transfer function to digital filter using Impulse Invariance methodH(S) = {\v{\w{\x Ans

Using Partial Fraction

Apply Impulse invariance transformation

R = F/ F = b

4. Design a digital chebyshev filter for the following specifications using bilinear transformation

MAY 2012

Given:

5. Design a digital chebeshev filter for the following specifications using Impulse in variance transformation

Ans:

6.Design a digital lowpass butterworth filter where transfer function is given by

p. z w p p. v z p. x p.

Ans:

ww\v = p. ww\v = p. x wp = p. v ws = p. =3.179 =1.02 Using Impulse invariant transform w= T ,T=1 Sec

p = p. v s= p. v N= log(( /)/log1(s/p) =1.03 =2

Ha(s) = \ \ It is a LPF s = s/ c

c = p / 1/N =0.622 rad/sec

s = s/ 0.622

= p. xv + p. + p. x Using partial fraction A = 0.439 j B = -0.439 j

= p. xv + p. + p. x = p. x p. x p. x p. x p. x + p. x

Using impulse invariance method w{| = wwz| uw H(Z)= 0.2401 Z-1 /1- 1.165Z-1+0.414 Z -2

7. Design a digital butterworth loepass filter using bilinear transformation with passband and stop band frequencies 800 rad/sec and 1800 rad/sec.The passband and stop band attenuation are -3dB and -10 dB respectively.

Ans :

p =-3dB s =-3dB wp =800 rad/sec ws =1800 rad/sec

Using Bilinear transformation = , p = 3.2397 rad/sec s =30.12 rad/sec N= log(( /)/log1(s/p) =1.0.491 =1

= 10j. 1 = 3 = 10j. 1 = 0.997 H(s) = 1/(S+1) S.S S= S/c c = p / 1/N =3.24

H(s) = 1/ (s/3.24)+1 Using Bilinear transformation Substitute s by MM\ H(Z) =3.24 (1+Z-1)/2-2Z-1 +3.24(1+Z-1)

UNIT IV - FIR FILTER DESIGN PART A

1. What is frequency warping? Nov 2011 Because of the non-linear mapping: the amplitude response of digital IIR filter is expand at

lower frequencies and compressed at higher frequencies in comparison to the analog filter.

2. What is the frequency response of Butterworth filter? Nov 2011 Butterworth filter has monotonically reducing frequency response

3. What is Gibbs phenomenon (or Gibbs Oscillation)? Nov 2013/2012 In FIR filter design by Fourier series method the infinite duration impulse response is

truncated to finite duration impulse response. The abrupt truncation of impulse response introduces oscillations in the pass band and stop band. This effect is known as Gibbs phenomenon.

4. What is the condition for linear phase of a digital filter? May 2012 The necessary and sufficient condition for linear phase characteristic in FIR filter is, the

impulse response h(n) of the system should have the symmetry property i.e., = 1 , where N is the duration of the sequence. 5. What is meant by limit cycle oscillation? May 2008, May 2011, Nov 2012

In recursive system when the input is zero or same non-zero constant value the non linearities due to finite precision arithmetic operation may cause periodic oscillation in the output. Thus the oscillation is called as Limit cycle.

6. What are the different types of arithmetic in digital systems? Nov2011 There are three types of arithmetic used in digital systems. They are fixed point arithmetic,

floating point, block floating point arithmetic.

7. What is meant by fixed point number? May2013 In fixed point number the position of a binary point is fixed. The bit to the right represent the

fractional part and those to the left is integer part.

8. What is zero input limit cycle oscillation? May2013 When a stable IIR filter is excited by a finite input sequence, the output will ideally decay to

zero. But due to non linearities in the finite precision arithmetic operation cause periodic oscillation to occur in the output

9. Write the equation for blackman window. May2013 = 0.42 + 0.5 cos 2 2H 13 + 0.08 cos 2 4H 13 , 2 12 3 2 12 3

10. State the condition for a digital filter to be causal and stable. A digital filter is causal if its impulse response h(n) = 0 for n

12. How phase distortion and delay distortions are introduced? The phase distortion is introduced when the phase characteristics of a filter is not linear within the

desired frequency band. The delay distortion is introduced when the delay is not constant within the desired frequency range.

13. Write the steps involved in FIR filter design. Choose the desired (ideal) frequency response Hd(w). Take inverse fourier transform of Hd(w) to get hd(n). Convert the infinite duration hd(n) to finite duration h(n). Take Z-transform of h(n) to get the transfer function H(z) of the FIR filter.

14. What are the advantages of FIR filters? Linear phase FIR filter can be easily designed. Efficient realization of FIR filter exist as both recursive and nonrecursive structures. FIR filters realized nonrecursively are always stable. The roundoff noise can be made small in nonrecursive realization of FIR filters.

15. What are the disadvantages of FIR filters? The duration of impulse response should be large to realize sharp cutoff filters. The non-integral delay can lead to problems in some signal processing applications.

PART B 1. Prove that an FIR filter has Linear phase if the unit sample response satisfies the condition

h(n)=h(N-1-n).Also discuss the symmetric and anti symmetric cases of FIR filter when N is even. NOV 2013

Case 1:- Symmetric Impulse Response with Even Length:

The frequency response of h(n) is 45 = 45j

If filter length is even,

45 = 45j + 45

45 = 45j + N 1 n45-

j

We know that h(n)=h(N-1-n)

45 = 45j + n45-

j

45 = 45 45

j + 45j

45 = 45 2 cos @2 12 3 C

j

45 = 45 2 22 3 cos @2 123 lC

j

45 = 45 cos @2 123 lC

j

Where = 2 45 = 4545 = 4545

45 = cos 2 123 l

El = 2 + 12 3 l Case 2:- Anti Symmetric Impulse Response with Even Length:

The frequency response of h(n) is 45 = 45j

If filter length is even,

45 = 45j + 45

45 = 45j + N 1 n45-

j

We know that h(n)= h(N-1-n) 45 = 45

j n45-

j

45 = 45 45

j 45j

45 = 45456 2 22 3 sin @2 123 lC

45 = 45456 c cos @2 123 lC

Where c= 2 45 = 4545645 = 4545

45 = cos 2 123 l

El = H2 2 12 3 l 2. Prove that an FIR filter has Linear phase if the unit sample response satisfies the condition

h(n)=h(N-1-n).Also discuss the symmetric and anti symmetric cases of FIR filter when N is odd. Case 1:- Symmetric Impulse Response with Odd Length

The frequency response of h(n) is 45 = 45j

If filter length is odd,

45 = 45j + 2

12 3 45 + 45

We know that h(n)=h(N-1-n)

45 = 45j + 2

12 3 45 + 45-j

45 = 45 45

j + 2

12 3 + 45j

45 = 45 2 45

j + 2

12 3

Let = , 45 = 45

2. 2 12 3 cos l + 2

12 3

45 = 45 2. 2 12 3 cos l

+ 2

12 3

45 = 45 cos lj

Where 0 = c = 2. 2 3 45 = 4545 = 4545

45 = cos lj

El = 2 12 3 l Case 2:- Anti Symmetric Impulse Response with Odd Length:

For this type of sequence,

2 12 3 = 0 The frequency response of h(n) is

45 = 45j If filter length is odd,

45 = 45j + 2

12 3 45 + 45

We know that h(n)= h(N-1-n) 45 = 45

j 45-

j

45 = 45 45

j 45j

45 = 45e 2 sin @2 12 3 C

j

45 = 45456 2. 2 12 3 sin l

45 = 45456 cos lj

Where c = 2. 2 3

45 = 45 456 45 = 4545 45 = sin l

El = H2 2 12 3 l

3. Design an ideal band pass filter with a frequency response. NOV 2011/MAY 2012

Hd4el = 1 for 4 |l| 3H40 Y4XGF4

Find the values of h(n) for N=7. Find the realizable filter transfer function and magnitude function of H4el

Soln. Step 1. Draw the ideal desired frequency response of bandpass filter.

Form the desired frequency response, we can find that the given response is symmetric N odd

Step 2. To find

Step 3. To find h(n). For symmetry response

Step 4. To find filter transfer function,

Step 5. To find the realizable filter transfer function

Therefore, the filter co-efficients of the causal filters are,

Step 6. To find the magnitude response of H45

4. Design an ideal high pass filter with a frequency response MAY 2013/MAY 2011

H45 = 1 for 4 |l| H0 YX |l| < H4 Find the value of h(n) for N=11 using

a. Hamming window b. Hanning window

Sol. (a) Hamming Window

Step 1. Draw the desired frequency response of ideal highpass filter.

Step 2. To find hd(n)

We know that,

Step3. To find the Hamming window sequence.

Step 4. To find the filter co-efficient hd(n)

Step 5. To find the filter co-efficients using Hamming window sequence.

Step 6. To find the transfer function of the filter.

Step 7. To find transfer function of the realizable filter

The filter co-efficients of causal filters are,

(b) Hanning Window

Step 1. The filter co-efficient can be obtained from part (a), step (2) and step (5)

Step 2. To find the Hanning window sequence

The Hanning window sequence is given by

Step 3. To find the filter co-efficients using Hanning window.

The filter co-efficients using Hanning window are

Step 4. To find the transfer function of the filter.

The transfer function of the filter is given by,

Step 5. To find the transfer function of realizable filter.

5. Design an ideal low pass filter with a frequency response. NOV 2011/NOV 2013

Hd4el = 1 for 2 l H20 YX H2 l H

Find the values of h(n) for N=1. Find the realizable filter transfer function and magnitude function of H4el

Sol. Step 1. Draw the desired frequency response:

From the frequency response, we can find that the given response is a symmetrical N odd response. Step 2. To find

In general,

Step 3. To find h(n): For symmetric response,

Step 4. To find the filter transfer function.

6. Design an ideal low pass filter with a frequency response. NOV 2013/MAY 2012

Hd4el = 1 for 0 |l| H60 YX H6 |l| H

Use 10 tap filter and obtain the impulse response of the desired filter

Ans. The filter co-efficients are given by :

7. Explain Finite word length effects in FIR filters

Parameter quantization in digital filters

The common method of quantization is Truncation and Rounding.

Truncation: - Truncation is the process of discarding all bits less significant than least significant bits that is retained.

Rounding: - Rounding of a number of b bits is choosing the rounded results as the b bit closet to the original number unrounded.

In the realization of FIR and IIR filters hardware or in software on a general purpose computer, the accuracy with which filter coefficients can be specified is limited by word length of the computer. Since the coefficients used in implementing a given filter are not exact, the poles and zeros of system function will be different from desired poles and zeros. Consequently, we obtain a filter having a frequency response that is different from the frequency response of the filter with unquantized coefficients. Also it sometimes affects stability of filter.

Round off noise in multiplication As already explained when a signal is sampled or a calculation in the computer is performed, the results must be placed in a register or memory location of fixed bit length. Rounding the value to the required size introduces an error in the sampling or calculation equal to the value of the lost bits, creating a nonlinear effect. Round-off error is a characteristic of computer hardware.

Sampling/Digitization Error There is another, different, kind of error that is a characteristic of the program or algorithm used, independent of the hardware on which the program is executed. Many numerical algorithms compute discrete approximations to some desired continuous quantity. For example, an integral is evaluated numerically by computing a function at a discrete set of points, rather than at every point. Or, a function may be evaluated by summing a finite number of leading terms in its infinite series, rather than all infinity terms. In cases like this, there is an adjustable parameter, e.g., the number of points or of terms, such that the true answer is obtained only when that parameter goes to infinity. Any practical calculation is done with a finite, but sufficiently large, choice of that parameter. The difference between the true answer and the answer obtained in a practical calculation is called the truncation error. Truncation error would persist even on a hypothetical, perfect computer that had an infinitely accurate representation and no round off error.

Overflow in addition Overflow in addition of two or more binary numbers occurs when the sum exceeds the word size available in the digital implementation of the system.

Limit cycles Since quantization inherent in the finite precision arithmetic operations render the system nonlinear, in recursive system these nonlinearities often cause periodic oscillation to occur in the output, even when input sequence is zero or some nonzero value. Such an oscillation in recursive systems are called limit cycles.

As explained in above paragraphs finite word length affects LTI system in many ways. We have concentrated on effects due to coefficient quantization on filter response and in that also on IIR filters. Later we have given brief overview of effects of coefficient quantization in FIR system for the sack of completeness.

8. Explain about limit Cycle oscillations

When a stable IIR digital filter is excited by finite input sequence that is constant the output will ideally zero. However non linearity's due to finite precision arithmetic operation often cause periodic oscillation to occur in the output. Such oscillations in recursive systems are called zero input limit cycles Consider a first order IIR filter the difference equation) y(n) =x(n) +a y(n-1)

Let us assume a=1/2 and the data register length is 3 bits plus a sign bit. If the input is

x(n) =0.875 YX = 0 0 Y4XGF4

n x(n) y(n-1) ay(n-1) Q[ay(n-1] Y(n) 0 0.875 0.0 0.0 0.000 7/8 1 0 7/8 7/16 0.110 1/2 2 0 1/2 1/4 0.010 1/4 3 0 1/4 1/8 0.001 1/8 4 0 1/8 1/16 0.001 1/8 5 0 1/8 1/16 0.001 1/8

Note that beyond n=4 the value of ay(n-1) is 1/16 and in binary 0.000100 an 0.000100 which whenrounded gives 1.001 and 0.001 exhibiting oscillatory.

Dead ban d will b calculated by using y(n-1) wvvw||

UNIT V APPLICATIONS PART A

1. Define multi rate digital signal processing? NOV 2012 Digital signal processing system handles processing at multiple sampling rates and then it is

called multi-rate signal processing.

2. What are the two techniques of sampling rate conversion? NOV2012 i. D/A conversion and resampling at required rate. ii. Sampling rate conversion in digital domain (multi-rate processing)

3. What are the applications of sampling rate conversion? MAY 2013 i. Narrow band filters.

ii. Quadrature mirror filters. iii. Digital filter banks.

4. What is meant by decimation? MAY 2013 Decimation by a factor D, means to reduce the sampling rate by a factor D. It is also called

down sampling.

5. What is meant by interpolation? Write the interpolation equation? MAY 2012 Interpolation by a factor I, means to increase the sampling rate by a factor I. It is also called as

up sampling by I.

6. Define speech compression and decompression. MAY2012 Speech analysis by a vocoder becomes the compression and synthesis by a vocoder becomes

decompression. Vocoder extracts the spectral envelope of speech and information regarding voicing and pitch. This data is coded and transmitted. The synthesizer generates speech from the received data.

7. Write the principle of adaptive filters. NOV 2013 The coefficients of the filter are changed automatically according to the changes in input

signal. This means the filtering characteristics of the adaptive filter are changed or adapted according to the changes in input signal.

8. How the image enhancement is achieved using DSP? NOV 2013 i. Local neighbourhood operations as in convolution.

ii. Transform operations as in DFT. iii. Mapping operations as in pseudo coloring and gray level mapping.

9. List the different methods of image enhancement. MAY 2011/NOV 2011 i. Contrast and edge enhancement.

ii. Pseudo coloring. iii. Noise filtering. iv. Sharpening. v. Magnifying.

10. What are the applications of image enhancement? NOV 2011

i. Feature extraction in an image. ii. Image analysis.

iii. Visual information display.

11. What is adaptive equalization? NOV 2012 Adaptive equalization is the technique used to reliably transmit data through a communication

channel. Ideally, if the channel is ideal (without and channel distortion and additive noise), we can demodulate the signal perfectly at the output without causing any error.

12. State a few applications of adaptive filter. NOV 2013/NOV 2010

Noise cancellation Signal prediction Adaptive feedback cancellation Echo cancellation

13. List various special audio effects that can be implemented digitally. MAY 2013 Echo effect Reverberation Chorus effect Phasing effect Flanging

PART B 1. Discuss sub band coding process in detail. MAY 2013

Digitized speech signals may be transmitted over a limited bandwidth channel or it can be stored. Reducing the size of the signal before transmission or storage is known as speech compression. The signal compression can be achieved by sub band coding. This method is making use of the non uniform distribution of signal energy in the frequency component. Transmitter:

The signal is spit into many narrow band signals which occupy continuous frequency bands using analysis filter bank.

Down sampling these signals gives sub band signals. Then compress it using encoders and the compressed signal is multiplexed and transmtted.

Receiver: The received signal is demultiplexed, decoded, up sampled and then passed through a

synthesis filter. The output of synthesis filter bank are combined to get the original uncompressed signal.

Block diagram of analysis and synthesis section of sub band coding 2. With lock diagram explain adaptive filtering based adaptive channel equalization. MAY 2013

Filters with adjustable coefficients are called adaptive filters In digital communication system the adaptive equalizer is used to compensate for the distortion

caused by the transmission medium. Block diagram with explanation of each block.

3. A.) Explain how the speech compression is achieved. NOV 2013 The human speech in its pristine form is an acoustic signal. For the purpose of communication

and storage, it is necessary to convert it into an electrical signal. This is accomplished with the help of certain instruments called transducer.

This electrical representation of speech has certain properties: 1. It is one- dimensional signal, with time as its independent variable. 2. It is random in nature 3. It is non stationary, that is, all the characters of the signal changes with time.

With the advent of digital computing mechanism, it was propounded to exploit the powers of the same for processing of speech signals. This required a digital representation of speech. To achieve this, the analog signal is sampled at some frequency and then quantized at discrete levels. Thus parameters of digital speech are

1. Sampling rate 2. Bits per second 3. Number of channels

Compression is the process of converting an input speech data stream into another data stream that has a smaller size. Compression is possible only because data is normally represented in the computer in a format that is longer than necessary that is, the input data has some amount of redundancy associated with it. The main objective of compression system is to eliminate the redundancy.

Some application of speech compression:

B.) Discuss about multirate signal processing. NOV 2013 1. Decimation: Decimation is a process of reducing the sampling rate by a factor M. Prove the Decimator (Down- sampler)

Let x(n) be a sequence which has been sampled at rate 1(unity), i.e. x(n) is obtained by sampling a continuous time sequence x(t) at Nyquist rate

X(n)=x(t)|t=n The decimation (or down sampling) operator M converts the input sequence

x(n) into a new sequence y(n), having the rate 1/M. y(n)=x(Mn)

1. Interpolation: Interpolation is a process of increasing the sampling rate by a factor L. Prove the Interpolation (up- sampler)

The Interpolation pads L-1 new samples between successive values of the signal. The interpolation process increases the sampling rate from I to IFs. since interpolation process increases the sampling rate, it is symbolically represented by a up arrow (L) m =

4. How the image enhancement is achieved using DSP? NOV2013/2012 i) Local neighborhood operations as in convolution. ii) Transform operations as in DFT. iii) Mapping operations as in pseudo coloring and gray level mapping.

Different methods of image enhancement. i) Contrast and edge enhancement. ii) Pseudo coloring.

iii) Noise filtering. iv) Sharpening. v) Magnifying.

Applications of image enhancement i) Feature extraction in an image. ii) Image analysis. iii) Visual information display.

5. Explain Adaptive noise cancellation with a neat diagram. NOV 2012 Linear Filtering will be optimal only if it is designed with some knowledge about the input

data. If this information is not known, then adaptive filters are used. The adjustable parameters in the filter are assigned with values based on the estimated

statistical nature of the signals. Filters are adaptable to the changing environment. Adaptive filtering finds its application in adaptive noise cancelling, line enhancing, frequency

tracking, channel equalizations, etc.

The noise cancellers are used to eliminate intense background noise. This configuration is applied in mobile phones and radio communications, because in some situations these devices are used in high-noise environments. Figure 6 shows an adaptive noise cancellation system.

The canceller employs a directional microphone to measure and estimate the instantaneous amplitude of ambient noise r(n), and another microphone is used to take the speech signal which is contaminated with noise d(n) + r(n). The ambient noise is processed by the adaptive filter to make it equal to the noise contaminating the speech signal, and then is subtracted to cancel out the noise in the desired signal. In order to be effectively the ambient noise must be highly correlated with the noise components in the speech signal, if there is no access to the instantaneous value of the contaminating signal, the noise cannot be cancelled out, but it can be reduced using the statistics of the signal and the noise process.

6. Explain any one application using multirate processing of signals. NOV 2010 Some application of Multirate signal processing are 1. Sampling rate conversion 2. Design of phase shifters 3. Interfacing of digital systems with different sampling rate.

4. Improved digital-to-analog conversion (DAC) and analog-to-digital conversion(ADC) 5. Frequency division multiplexing (FDM) channel modulation and processing 6. Sub band coding of speech and images.

Sub band coding of speech and images.

Digitized speech signals may be transmitted over a limited bandwidth channel or it can be stored. Reducing the size of the signal before transmission or storage is known as speech compression. The signal compression can be achieved by sub band coding. This method is making use of the non uniform distribution of signal energy in the frequency component. Transmitter:

The signal is spit into many narrow band signals which occupy continuous frequency bands using analysis filter bank.

Down sampling these signals gives sub band signals. Then compress it using encoders and the compressed signal is multiplexed and transmtted.

Receiver: The received signal is demultiplexed, decoded, up sampled and then passed through a

synthesis filter. The output of synthesis filter bank are combined to get the original uncompressed signal.

Block diagram of analysis and synthesis section of sub band coding

7. Derive and explain the frequency domain characteristics of the decimator by the factor M and interpolator by the factor L. MAY 2011/MAY 2013

1. Decimation: Decimation is a process of reducing the sampling rate by a factor M. Prove the Decimator (Down- sampler): Let x(n) be a sequence which has been sampled at rate 1(unity), i.e. x(n) is obtained by sampling a continuous time sequence x(t) at Nyquist rate X(n)=x(t)|t=n The decimation (or down sampling) operator M converts the input sequence x(n) into a new sequence y(n), having the rate 1/M. y(n)=x(Mn)

Spectral analysis of Decimator: BLOCK DIAGRAM OF DECIMATOR AND DERIVATION 2. Interpolation: Interpolation is a process of increasing the sampling rate by a factor I.

Prove the Interpolation (up- sampler) The Interpolation pads L-1 new samples between successive values of the signal. The interpolation process increases the sampling rate from I to IFs. since interpolation process increases the sampling rate, it is symbolically represented by a up arrow (L) m =

Spectral analysis of Interpolator: BLOCK DIAGRAM OF

8. Explain the methods of speech analysis and synthesis in details. Vocoders (Voice Coder) were originally designed to reduce the bandwidth requirements of

transmission of normal voice signal. The vocoder implements analysis and sis nothing but application of multirate signal processing. In the analysis section, natural speech is analyzed, typically by a bank of filters as shown in figure. The output of each filter is coded by one of a variety of different methods, and this coded information is transmitted across the channel.

Block diagram of analysis and synthesis of subband encoded speech signal

Spectral analysis of Interpolator: BLOCK DIAGRAM OF INTERPOLATOR AND DERIVATION

Explain the methods of speech analysis and synthesis in details. Vocoders (Voice Coder) were originally designed to reduce the bandwidth requirements of

transmission of normal voice signal. The vocoder implements analysis and sis nothing but application of multirate signal processing. In the analysis section, natural speech is analyzed, typically by a bank of filters as shown in figure. The output of each filter is coded by one of

methods, and this coded information is transmitted across the channel.Block diagram of analysis and synthesis of subband encoded speech signal

AND DERIVATION

NOV 2011 Vocoders (Voice Coder) were originally designed to reduce the bandwidth requirements of

transmission of normal voice signal. The vocoder implements analysis and synthesis sections, which is nothing but application of multirate signal processing. In the analysis section, natural speech is analyzed, typically by a bank of filters as shown in figure. The output of each filter is coded by one of

methods, and this coded information is transmitted across the channel. Block diagram of analysis and synthesis of subband encoded speech signal