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EC2361DIGITAL SIGNAL PROCESSING

UNIT IINTRODUCTION 9

Classification of systems: Continuous, discrete, linear, causal, stable,dynamic, recursive, time variance; classification of signals: continuous and

discrete, energy and power; mathematical representation of signals; spectral

density; sampling techniques, quantization, quantization error, Nyquist rate,

aliasing effect. Digital signal representation.

UNIT II - DISCRETE TIME SYSTEM ANALYSIS 9

Z-transform and its properties, inverse z -transforms; difference equation

Solution by z-transform, application to discrete systems - Stability analysis,frequency response Convolution Fourier transform of discrete sequence

Discrete Fourier series.

UNIT III - DISCRETE FOURIER TRANSFORM & COMPUTATION 9

DFT properties, magnitude and phase representation - Computation of DFT using

FFT algorithmDIT & DIF - FFT using radix 2Butterfly structure.

UNIT IV - DESIGN OF DIGITAL FILTERS 9

FIR & IIR filter realization Parallel & cascade forms. FIR design:

Windowing Techniques Need and choice of windows Linear phase

characteristics. IIR design: Analog filter design - Butterworth and Chebyshev

approximations; digital design using impulse invariant and bilinear transformation

- Warping, prewarping -Frequency transformation.

UNIT V - DIGITAL SIGNAL PROCESSORS 9

Introduction Architecture Features Addressing Formats Functional

modes -Introduction to Commercial Processors

L = 45, T = 15, TOTAL: 60 PERIODS

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UNIT 1INTRODUCTION (SIGNALS AND SYSTEMS)

1. Define Signal.

A signal is defined as any physical quantity that varies with time, space or

any other independent variable or variables.

2. What is multidimensional signal? Give examples.

A signal which is a function of two or more independent variables is

called multidimensional signal. The intensity or brightness of black & white

photograph at each point is a function of two independent spacial coordinates x andy. Hence it is a two dimensional signal and can be denoted as I (x,y).The intensity

of black & white motion picture is a function of x & y coordinates and time.

Hence it is a three dimensional signal and can be denoted as I (x,y,t).

3. What is analog signal?

The analog signal is a continuous function of an independent variable such

as time, space, etc. The analog signal is defined for every instant of the

independent variable and so the magnitude (or the value) of analog signal iscontinuous in the specified range. Here both the magnitude of the signal and

the independent variable are continuous

4. Distinguish between energy and power signals.Energy signals:For discrete time signal, energy (E) of x(n) is given by,

E= ()2

Power signal:

Average power of x(n) is

P=(( ) ()

2

5.Define random and deterministic signal.

Random signal: A signal is called a random signal if it cannot bedescribed with certainty before it actually occurs. Probabilistic models describe

random signals.

E.g.: Thermal noise in resistors, transistors etc.

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Deterministic signal: A signal is called deterministic signal if it can

be described without any uncertainty.

6. Give the expressions for even and odd signals.

Any signal x(t) can be broken into a sum of two signals, Even and Oddsignals.

Even signal is equal to, E{x(n)}=1/2 [x(n)+x(-n)]

Odd signal is equal to, O{x(n)}=1/2 [x(n)x(-n)].

7. What are the properties of linear time invariant systems?

Linearity: this principle requires that the response of the system to a

weighted sum of i/p signals is equal to the corresponding weighted sum of the

responses of the system to each of the individual i/p signals (i.e.)

H[ 1x1(n)+ 2x2(n)] = 1H[x1(n)]+ 2 H[x2(n)]= 1y1(n)+ 2y2(n)

Time invariant system (or) shift invariant or fixed if its i/p-o/prelationship does not change with time (i.e.) if y(t) is the response when x(t) is the

i/p and if x(t) is delayed by time ,then the o/p of the system is also delayed by thesame amount of time . H[x(n-N)] = Y(n-N)

8. State whether the following systems are linear or not.

1.y(n)=x(n)+x(n-100)

2.g(t)= dx(t)/dt

Soln:

1.y(n)=x(n)+x(n-100)

H[ 1x1(n)+ 2x2(n)] = 1H[x1(n)]+ 2H[x2(n)]= 1y1(n)+ 2y2(n)

RHSy1(n)=x1(n)+x1(n-100)

y2(n)=x2(n)+x2(n-100)

LHS H[1x1(n) +2x2(n)] = 1x1(n) + 2x2(n) + 1x1(n-100) + 2x2(n-100)

RHS1H[x1(n) + x1(n-100)]+ 2H[x2(n) + x2(n-100)]=

1x1(n) + 1x1(n-100) + 2x2(n) + 2x2(n-100)

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LHS = RHS

It is linear.

2.g(t) = dx(t)/dt

LHS [1x1(t) + 2x2(t)]= 1dx1(t)/dt + 2dx2(t)/dt

RHS = 1y1(t) + 2y2(t) = 1dx1(t)/dt + 2dy2(t)/dt

LHS = RHS

It is linear.

9. Define the impulse response of a DISCRETE system.

In discrete time systems the unit impulse signal is represented by

(n) = 1, n=0

= 0, otherwise

when the unit impulse (also called delta function ) (n) is applied to a

linear time invariant at n=0,the impulse response of the system is denoted by

h(n) = h[(n)]

10.What is meant by aliasing effect?

The superimposition of high frequency behavior on to the low frequencybehaviour is referred as aliasing. This effect is also referred as folding.

11.Is the system described by the equationy(n) = x(2n)time invariant or not? why?

T[x(n-N)] =y(n-N)

LHS =T[x(n-N)] =x[2(n-N)]

RHS =y(n-N)=x[2n-N]

LHS = RHS.

The system is time variant.

12. Is the systemy(n) = nx(n)is shift invariant or not.

shift invariant condition:T[x(n-N)] =y(n-N)

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For non- recursive realization the present output y(n) is a function of only

past and present inputs. This form corresponds to a finite impulse response (FIR)digital filter.

18. What is a causal system? Give an example

A system is said to be causal if the output of the system at any time ndepends only on present and past input, but does not depend on future inputs. This

can be represented mathematically as, y (n) = F[x (n), x (n-1), x (n-2)] e.g.y(n) = x(n) + x(n-1).

19. What is quantization error?

The difference between the quantized signal and the original

unquantized signal is called quantization error

20. Define Nyquist rate

The sampling frequency must be greater than or equal to twice the

maximum frequency

2Fm fs

21. What is an anti-aliasing filter?

A filter that is used to reject high frequency signals before it is sampled to

remove the aliasing of unwanted high frequency signals is called an ant aliasing

filter.

22. What is meant by quantizer?

It is a process of converting discrete time continuous amplitude into discretetime discrete amplitude.

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UNIT II - DISCRETE TIME SYSTEM ANALYSIS

1. Define Z-Transform

Z-transform can be defined as

Z[x(n)]=X(Z)= () -n

This equation is referred to as Bilateral or two sided Z-transform of x(n).

The unilateral z-Transform can be defined as

Z[x(n)]=X(Z)= ()-n

2. Define Region of convergence

The range of values of Z for which X(Z) expressed by the equation

Z[x(n)]=X(Z)= ()-n

approaches to a finite value is called Region of convergence(ROC).

3. State Initial value theorem

If x(n) is causal then x(0) = () = ()

It can be used to find the initial value of x(n)

4. State Final value theorem

If Z[x(n)]=X(Z) then X()=Ltx(n)=Lt (1-Z-1) X(Z)

5.What are the properties of ROC?

1. ROC doesnot contain any poles.2. If x(n) is a finite duration then the ROC is the entire Z plane

except possibly Z=0 and/Or Z=

3. If x(n) is a r ight sided sequence and if the circle with |Z|=ro is inROC then all the finite values of Z for which Z> ro will also be inROC.

4. If x(n) is a left sided sequence and if the circle with |Z|=rois in ROCthen all the finite values of Z for which 0

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6. List out any 4 properties of Z transform.

1. Linearity: X(Z)=a x1(z)+bx2(z)

2. Time shifting : Z(x(nno))=ZnoX(Z)

3. Frequency Shift: Z(ejwnx(n))=X(e-jwZ)

4. Time reversal: Z(x(-n))=x(1/Z)=x(z-1)

7. What are the three methods to obtain the inverse Z transform ?

1.Long division method 2.Partial fraction expansion method

3.Residue method

8. Find the Z.T ofx(n)= (n)

From the definition of Z.T

X(Z)= ()-n = ()

-n= 1.z0 = 1

Where (n)=1 for n=0

=0 otherwise

9. Find the Z-transform ofx(n)=(1/2)n u(n)

We know that Z transform of anu(n)= z/(z-a).

Similarly here a = . Therefore X(Z) = z/(z-1/2) = 2z/(2z-1).

10. Define discrete linear convolution.

The discrete convolution of the two discrete variable function x(n)andh(n) is the discrete variable function y(n) given by the summation

y(n) = ()()

11. Define discrete circular convolution.

Given two real N periodic sequences ,x(n)&h(n) the circular orperiodic convolution sequence y(n) is also n periodic sequence given by ,

y(n) = ()(( ) )N

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12. What are the advantages of sectioned convolution?

If one of the sequences is very much larger than the other, then it is

very difficult to compute convolution using other types of convolution.

Long delay in getting output. Therefore entire sequence is required before

convolution operation. Memory required also very large to store thesequence. These problems can be eliminated in sectioned convolution.

13. What is sectioned convolution?

In sectioned convolution the larger sequence is sectioned in to smallersequences and then linear or circular convolution is done. Output sequence

obtained from convolution of all sections are combined to get overall output

sequence.

14. What are the different methods of sectioned convolution?

1. overlap add method 2. overlap save method

15. What is the draw back in Fourier transform and how it is overcome?

The drawback in Fourier transform is that it is a continuous function o

f and so it cannot be processed by digital system. This drawback isovercome by using Discrete Fourier Transform. The DFT converts the

continuous function of to a discrete function of ,

16. Calculate the DFT of the sequence x (n) = {1,1,-2,-2}.

The N-point DFT of x(n) is given by

DFT{x(n)} = X(k) = ()-j2kn/N ; for k = 0,1,2,(N-1)

Since x(n) a 4-point sequence, we can take 4-point DFT.

X(k) = ()-j2kn/4

= ()-jkn/2

=x(0) e0+ x(1) e

-jk/2+ x(2) e

-j2k+ x(3) e

-j3k/2

= 1+ e-jk/2- 2 e-j2k2 e-j3k/2 ; for k = 0,1,2,3

17. Give two applications of DFT?

i) The DFT is used for spectral analysis of signals using a digitalcomputer.

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ii) The DFT is used to perform filtering operations on signals using

digital computer.

18. What is the relation between Z-transform & DFT? (MAY 2011)

Let N-point DFT of x(n) be X(k) and the Z-transform of x(n) be

X(z).The N-point sequence X(k) can be obtained form X(z) byevaluating X(z) at N equally spaced points around the unit circle.

X(k) = X(z)|z=ej2nk/N

; for k = 0,1,2,(N-1)

19.State the conditions for the existence of fourier series.

(i). The function x(t) should be single valued in any finite time

interval T

(ii). The function x(t) should have atmost finite number of discontinuities inany finite time interval T.

(iii). The function x(t) should have finite number of maxima and minima in

any time interval T.

(iv) The function x(t) should be absolutely integrable.

20. Define fourier transform & inverse fourier transform of a discrete time

signal.

The fourier transform of a discrete time signal x(n) is defined as

F{x(n)}=X() = ()-jn

The fourier transform exists only if () <

The inverse fourier transform of X() is defined as

F-1

{X()}=

()

ejn

d

21.State Rayleighs energy theorem.

Rayleighs energy theorem states that the energy of the sign al may be

written in frequency domain as superposition of energies due to individual

spectral frequencies of the signal.

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7.Compare DIT & DIF FFT algorithms

8. What is twiddle factor?

It is a complex valued phase factor or weight vector represented as

WNKn= e-(j2kn/N)

9.What are the steps involved in computing IDFT through FFT?

1. take conjugate of x(k)

2. compute N point DFT of complex conjugate x*(k) using FFT.

3. take again the conjugate of the output sequence.

4. then the resultant sequence is divided by N.

10. Compute the DFT of x(n) =(n).

X(k) = () WNKn

= = () WNKn

DIT FFT DIF FFT

1. input is bit reversed order 1. input is normal order

2. output is normal order 2. output is bit reversed order

3.time domain sequence is decimated 3. freq.domain sequence

is decimated

4. total no. of multipliers required is

N/2(log2N)

4. multipliers required is

N/2(log2N)

adders required is N(log2N)

5. In butterfly diagram, each stage ofcomputation, the phase factor is

multiplied before add &subtractoperation.

5. In this case, in each stage ofcomputation, the phase factors

are multiplied after add &subtract operation.

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11.Define Duality property for DFT.

DFT: F (k) g (n)

F (n) 1/N g (-k)

12.What is the relation between Z Transform and DFT.

Let N point DFT of x(n) be X(K) and the Z transform of x(n) be X(Z).The N

point sequence of X(K) can be obtained from X(Z) at N equally spaced pointsaround the unit circle.

X(K)=X(Z)/Z=e j2k/Nfor K=0,1,2,..(N-1)

13.What is a decimation in time algorithm?

DIT algorithm is used to calculate the DFT of a N point sequence. Initially

the N point sequence is divided into two N/2 point sequences Xeven(n) and Xodd (n).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 combinationof N/4 point DFTs. This process is continued until left with 2 point DFT. This

algorithm is called decimation in time because the sequence X (n) is often splitted

into smaller sequences.

14. What is meant by radix-2 FFT?

The FFT algorithm is most efficient in calculating N point DFT. If the

number of point N can be expressed as a power of 2 i.e. N= 2M where M is aninteger, then this algorithm is known as radix-2 FFT algorithm.

15. What is decimation in frequency algorithm?

It is one of the FFT algorithms. In this the output sequence X(k) is divided

into smaller subsequence, that is why the name decimation in frequency. Initiallythe input sequence is divided into two consisting of the first N/2 samples of X(n)

and the last N/2 samples of X(n).The above procedure can now be iterated to

express each N/2 point DFT as a combination of two N/4 point DFTs.This process

is continued until we are left with 2 point and 1 DFT.

16. What are the differences and similarities between DIF and DIT algorithms?

Differences : For DIT the input is bit reversed while the output is in naturalorder , whereas for DIF the input is in natural order while the output is bit reversed.

The DIF butterfly is slightly different from the DIT butterfly, the difference beingthat the complex multiplication takes place after the add-subtract operation in DIF.

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Similarities: Both algorithms require same number of operations to

compute the DFT. Both algorithms can be done in place and both need toperform bit reversal at some place during the computation.

17. Explain In-place computation.

To compute the elements p and q of the mth array , it is required to haveelements in the p and q of the (m-1) array. If Xm(p) and Xm(q) are stored in the

same register as Xm-1(p) and Xm-1(q) respectively ,it is possible to implement theabove computation with only N array of complex storage registers. This kind of

computation is commonly referred to as In-place computation.

18. Calculate the number of multiplications needed in the calculation of DFT and

FFT with 64 point sequence.

Number of complex multiplications required using direct computation is

N2= 642= 4096

Number of complex multiplications required using FFT is

(N/2) log N = ((64/2) log 64 = 192

speed improvement factor (4096/192) = 21.33.

19. Define discrete linear convolution.

The discrete convolution of the two discrete variable function x(n) and h(n)is the discrete variable function y(n) given by the summation

y(n) = ()()

20. What are the properties of DIT FFT?

1.Computation are done in place.

Once a butterfly structure operation is performed on a pair of complex

numbers(a,b) to produce (A,B) there is no need to save the input pair (a,b).

Hence we can store the results(A,B) in the same location as(a,b).

2. Data x(n) after decimation is stored in reverse order.

21. What are the advantages of FFT algorithm?

Fast fourier transform reduces the computation time. In DFT

computation, number of multiplication is N2and the number of addition is

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N(N-1). In FFT algorithm, number of multiplication is only N/2(log2N) .

Hence FFT reduces the number of elements (adder,multiplierZ&delayelements). This is achieved by effectively utilizing the symmetric and

periodicity properties of Fourier transform.

22. What are the computational savings in evaluation of DFT using radix-2 FFT?

Multiplications: N/2 logN2

Additions: N/ logN2

23.State the meaning of bit reversal in FFT algorithm.

DIT FFT: I/Pbit reversal order & O/Pnormal order

DIF FFT: O/Pbit reversal order & I/Pnormal order.

UNIT IV - DESIGN OF DIGITAL FILTERS

1. What are the disadvantages of Impulse invariant method?

Although this method is useful for implementing LPF and HPF the method

is unsuccessful for implementing digital filters for which |H(j)| does not approach

zero for large value of such as the high pass filter .

2. What are the advantages of Bilinear transformation method?

The Bilinear transform method provides non linear one to one mapping of

the frequency points on the jaxis in the S plane to those on the unit circle in the Z

plane .i.e. Entire jaxis for -< < maps uniquely on to a unit circle -/T