CS2403 DSP

EC 2311 – DIGITAL SIGNAL PROCESSING 1

JEPPIAAR ENGINEERING COLLEGE DEPARTMENT OF E.C.E

QUESTION BANK

EC 2311 – DIGITAL SIGNAL PROCESSING

UNIT I SIGNALS AND SYSTEMS 9 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.

UNIT II FREQUENCY TRANSFORMATIONS 9 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.

UNIT III IIR FILTER DESIGN 9 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 9 Structures of FIR – Linear phase FIR filter – Filter design using windowing techniques, Frequency sampling techniques – Finite word length effects in digital Filters

UNIT V APPLICATIONS 9 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:

3. Alan V.Oppenheim, Ronald W. Schafer & Hohn. R.Back, “Discrete Time Signal Processing”, Pearson Education.

4. Andreas Antoniou, “Digital Signal Processing”, Tata McGraw Hill.


UNIT I - SIGNALS ANS SYTEMS

1. What are energy and power signals?

The energy signal is one in which has finite energy and zero

average power. The power signal is one in which has finite average

power and infinite energy.

T

E = Lt ∫ │x(t)│2 dt joules .

T→∞ -T

P = Lt T

T→∞ 1 / 2T ∫ │x(t)│2 dt joules .

2. What are Random signals?

The signal that takes on random value at any given instant and it

cannot be predicted. Non deterministic signals are also called as

Random signals.

Example: ECG signals, Noise in electric circuits.

3. Distinguish between continuous and discrete time signals.

If the signal amplitude can be defined for all values of time t is

Called as continuous time signals and it is denoted by x(t).

If the signal amplitude can be defined for particular integer values of

time period n is called as discrete time signals and it is denoted by

x(n).

4. Find the period of x(n) = cos [8πn/7 +2].

ω = 8π/7 2πf = 8π/7

f= 4/7 ; here K= 4 & N =7

It is periodic and the fundamental period is N =7 samples.

5. Define System.

A system is a physical device that performs an operation on the

signal. The input signal is called as excitation and output signal is

called as response.

6. State the classification of system. (Nov/Dec 2004)

I. Linear and non linear system


II. Time variant and in variant system

III. Causal and non causal system.

IV. Static and Dynamic system.

V. Stable and unstable system.

7. Distinguish between linear and non linear system.

a1 y1(t) + a2 y2(t) = f[a1x1(t) + a2x2(t)]

If the above equation satisfies then the system is said to be linear

system. If the above equation does not satisfy then the system is said

to be non linear system.

8. State the properties of linear system.

A system is said to be linear it should satisfy the principle of

Superposition.

9. Define time invariant system. (May/June 2006)

A system is time invariant if it satisfies the following relation

F[ x(t – t1)] = y(t – t1).

10. What is meant by causal & non causal system? (May/June 2006)

A system is said be causal if it’s output at anytime depends

upon present and past input only. A system is said be non causal if it’s

output at anytime depends upon present and future input only.

11. State the condition for the BIBO stable? May/June 2005

The condition for the BIBO stable is given by

∞

∫ │h (t)│dt < ª

12. Define Z- Transform.

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

the power series α

Z[x(n)] = X(z) = Σ x(n) z-n

n= -α

13. Define Sampling Theorem. Nov/Dec 2008

Sampling is the process by which analog signal is converted

into corresponding sequence of samples that are spaced uniformly in

time.


14. What is aliasing?

The phenomenon of high frequency sinusoidal components

acquiring the identity of low frequency sinusoidal components after

sampling is called aliasing.

15. Check whether the system y(n) = e

x(n) is linear. May/June 2007

Consider two signals x1(n) and x2(n).Let

y1(n) & y2(n) be the

response of the system for inputs x1(n) and x2(n) respectively.

y1(n) = H[x1(n)] = e x1

(n)

y2(n) = H[x2(n)] = e x2

(n)

y3(n) = H[a1x1(n) +a2 x2(n)] = e (a

1x1

(n) + a2

x2

(n)) = e

a1

x1

(n) .

e

a2

x2

(n))

Therefore, a1y1(n) + a2y2(n) = a1ex1

(n) + a2 e

x2

(n)

y3(n) is not equal to a1y1(n) + a2y2(n). Hence the system is non linear.

PART-B

1) Determine whether the following signals are linear,time

variant, causal and stable. Nov/Dec 2008

Y(n) = Cos[x(n)]

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

Y(n) = x(2n)

Y(n) = x(n) + nx(n+1)

Ans: Ref Pg.No 31-45 , DSP by Nagoor Kani.A

2) Find the response of the system for the input signal

X(n) = {1,2,2,3} and h(n) = {1,0,3,2} May/June 2007

Ans: Ref Pg.No 164 , DSP by Nagoor Kani.A

3) Explain sampling theorem

Ans: Ref Pg.No 26 , DSP by Proakis

4) Find the inverse z- transform of May/June 2007

1/ (1-0.5 z-1

) (1-z-1

)

Ans: Ref Pg.No 461, DSP by Nagoor Kani.A

5) Perform circular convolution of the two sequences.

X1(n) = {2,1,2,1}

X2(n) = {1,2,3,4}

Ans: Ref Pg.No 175, DSP by Nagoor Kani.A


UNIT II - FREQUENCY TRANSFORMATION

PART-A

1. State and prove Parseval’s Theorem. Nov/Dec 2007

Parseval’s theorem states that

If x(n) ↔ X(K) and y(n) ↔ Y(K) , Then

N-1 N-1

∑ x(n) y*(n) = 1/N ∑ X(K) Y*(K)

n=0 K =0

When y(n) = x(n), the above equation becomes

2. . What do you mean by the term “bit reversal” as applied to FFT?

Nov/Dec 2007

Re-ordering of input sequence is required in decimation – in –time.

When represented in binary notation sequence index appears as

reversed bit order of row number.

3. Draw the basic butterfly diagram of radix -2 FFT.

April/May 2008.

1 1

a A = a+ WN

nk b

1

1

WN

nk

b B = a - WN

nk b

-1

4. Distinguish between DIT and DIF –FFT algorithm. Nov/Dec 2008

S.No DIT –FFT Algorithm DIF –FFT Algorithm

1. The input is in bit reversed

order; the output will be

normal order.

The input is in normal order;

the output will be bit reversed

order.

2. Each stage of computation the

phase factor are multiplied

before add subtract operation.

Each stage of computation the


after add subtract operation.

N-1 N-1

∑ │x(n)│2 = 1/N ∑ │X(K)│

2

n=0 k=0


5. If H(K) is the N-point DFT of a sequence h(n) , Prove that H(K)

and H(N-K) are comples conjugates. Nov/Dec 2008

This property states that, if h(n) is real , then

H(N-K) = H*(K) = H(-K)

Proof:

By the definition of DFT;

N-1

X(K) = ∑ x(n) e (–j2πnk)/N

n=0

Replace ‘K’ by ‘N-K’

N-1

X(N-K) = ∑ x(n) e (–j2πn(N-K))/N

n=

6. The first five DFT coefficients of a sequence x(n) are X(0) = 20,

X(1) = 5+j2,X(2) = 0,X(3) = 0.2+j0.4 , X(4) = 0 . Determine the

remaining DFT coefficients. May/June 2007

Solution:

X (K) = [20, 5+j2, 0, 0.2+j 0.4 , 0,X(5),X(6),X(7)]

X (5) = 0.2 – j0.4

X (6) = 0

X (7) = 5-j2

7. What is FFT? Nov/Dec 2006 The Fast Fourier Transform is a method or algorithm for computing

the DFT with reduced number of calculations. The computational

efficiency can be achieved if we adopt a divider and conquer

approach. This approach is based on decomposition of an N-point

DFT in to sucessively smaller DFT’s. This approach leads to a family

of an efficient computational algorithm is known as FFT algorithm.

8. What are the advantages of FFT algorithm over direct

computation of DFT? May/June 2007

1. Reduces the computation time required by DFT.

2. Complex multiplication required for direct computation is N2

and for FFT calculation is N/2 log 2 N.

Speed calculation

X(N-K) = X*(K)


9. Define the properties of convolution. April/May 2008.

3. Commutative property: x(n)*h(n) = h(n) *x(n)

4. Associative Property: [x(n)*h1(n)] *h2(n) = x(n)*[h1(n)*h2(n)]

5. Distributive Property: x(n)*[ h1(n)+ h2(n)] = [x(n)* h1(n)]+ [x(n)*

h1(n)]

10. Distinguish between linear & circular convolution.

s.no Linear convolution circular convolution

1 The length of the input sequence

can be different.

The length of the input

sequence should be same.

2 Zero Padding is not required. Zero padding is required if

the length of the sequence is

different.

11. Define Zero padding? Why it is needed?

Appending zeros to the sequence in order to increase the size or length

of the sequence is called zero padding.In circular convolution , when

the two input sequence are of different size , then they are converted

to equal size by zero padding.

12. State the shifting property of DFT.

Time shifting property states that

DFT {x(n-n0)} = X(K) e (–j2πn0k)/N

13. Why do we go for FFT?

The FFT is needed to compute DFT with reduced number of

calculations.The DFT is required for spectrum analysis on the sinals

using digital computers.

14. What do you mean by radix-2 FFT?

The radix -2 FFT is an efficient algorithm for coputing N- point DFT

of an N-point sequence .In radix-2 FFT the n-point is decimated into

2-point sequence and the 2-point DFT for each decimated sequence is

computed. From the results of 2-point DFT’s, the 4-point DFT’s are

computed. From the results of 4 –point DFT’s ,the 8-point DFT’s are

computed and so on until we get N - point DFT.


15. Give any two application of DFT?

1. The DFT is used for spectral analysis of signals using a digital

computer.

2. The DFT is used to perform filtering operations on signals

using digital computer.

16. How many multiplications & addition are involved in radix-2

FFT?

For performing radix-2 FFT, the value of Nshould be such that, N=

2m. The total numbers of complex additions are Nlog 2 N and the total

number of complex multiplication are (N/2) log 2 N.

17. What is Twiddle factor?

Twiddle factor is defined as WN = e –j2π/N.

It is also called as weight

factor.

18. What is main advantage of FFT?

FFT reduces the computation time required to compute Discrete

Fourier Transform.

PART-B

1. Derive the equation for Decimation – in time algorithm for

FFT. Nov/Dec 2006 & April /May 2008 & Nov/Dec 2008

Ans: Ref Pg.No 207-209 , DSP by Nagoor Kani.A

2. (i)From first principles obtain the signal flow graph for

Computing 8-point using radix -2 DIF –FFT algorithm.

ii) Using the above signal flow graph compute DFT of

x(n) = cos (nπ/4) ,0 ≤ n ≤ 7.

May/June 2007 & Nov/Dec 2007 & Nov/Dec 2008

Ans: Ref Pg.No 219 , DSP by Nagoor Kani.A

X(K) = {0, 3, 0, 2.7-j0.7, 0, 1, 0, 1.293-j0.7}

3. i)Discuss in detail the important properties of the DFT.

ii)Find the 4-point DFT of the sequence x(n) = cos (nπ/4)

iii)Compute an 8-point DFT using DIF FFT radix -2 algorithm.

x(n) = { 1,2,3,4,4,3,2,1} April /May 2008

Ans: i)Ref Pg.No 308-311, DSP by Salivahanan.

ii) X(K) = {1, 1-j1.414, 1, 1+j1.414}

X(K) = {20,-5.8-j2.4, 0, 0.17-j0.414, 0, -0.17+j0.414, 0,-5.82+j2.414}.


4. i)Prove the following properties of DFT when H(k) is the DFT

of an N-point sequence h(n). H(k) is real and even when h(n) is

real and even.H (k) is imaginary and odd when h(n) is real and

odd.Compute the DFT of x(n) = e-0.5n

, 0≤ n≤ 5.

May/June 2007

Ans: i) Ref Pg.No 309, DSP by Salivahanan.

X(K) = { 2.414, 0.87-j0.659, 0.627-0.394j, 1.202,0.62-j0.252, 0.627-

j0.252}.

5. Two finite duration sequence are given by

x(n) = sin (nπ/2) for n = 0,1,2,3

h(n) = 2 n for n = 0,1,2,3 Determine circular convolution using

DFT &IDFT method. Nov/Dec 2007

Ans: X(K) = {0, -2j, 0, 2j}

H(K) = {15, -3+6j, -5, -3-6j}

y(n) = {6, -3, -6, 3}

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

response of LTI system by radix -2 DIT FFT. Nov/Dec 2006

Ans:i) X(K) = { 0, -1-j,6,-1+j}

Ref Pg.No 320-328, DSP by Salivahanan

UNIT III - IIR FILTER DESIGN

1. What are the different types of filters based on impulse response?

Based on impulse response the filters are of two types

1. IIR filter

2. FIR filter

The IIR filters are of recursive type, whereby the present output

sample depends on the present input, past input samples and output samples.

The FIR filters are of non recursive type, whereby the present output

sample depends on the present input sample and previous input samples.

2. What are the different types of filters based on frequency

response?

Based on frequency response the filters can be classified as

1. Lowpass filter

2. Highpass filter

3. Bandpass filter

4. Bandreject filter


3. Distinguish between FIR filters and IIR filters.

FIR filter IIR filter

1. These filters can be easily designed tohave perfectly linear phase.These

filters do not have linear phase.

2. FIR filters can be realized recursive and non-recursively.

3. Greater flexibility to control the shapeof their magnitude response.

4. Errors due to round off noise are less severe in FIR filters, mainly because

feedback is not used.IIR filters are easily realized recursively.Less

flexibility, usually limited to specific

kind of filters.The round off noise in IIR filters is

more.

4. What are the design techniques of designing FIR filters?

There are three well known methods for designing FIR filters with

linear phase .They are (1.)Window method (2.) Frequency sampling method

(3.) Optimal or minimax design.

5. What is Gibb’s phenomenon?

One possible way of finding an FIR filter that approximates H(ejw)

would be to truncate the infinite Fourier series at n=±(N-1/2).Direct

truncation of the series will lead to fixed percentage overshoots and

undershoots before and after an approximated discontinuity in the frequency

response.

6. Find the digital transfer function H(Z) by using impulse

invariant method for the analog transfer function H(S) = 1/

(S+2).Assume T=0.5sec May /June 2007 &Nov/Dec 2007

Solution:

H(S) = 1/ (S+2).

H(Z) = 1/[1-e-1

Z-1

]

H(Z) = 1/ [1-0.368Z-1

]

7. What is the relationship between analog and digital

frequency in impulse invariant transformation? April/May 2008

Digital Frequency: ω = Ω T

Ω = analog frequency

T= Sampling interval


8. What is Prewarping? Why is it needed? Nov/Dec 2008

In IIR design using bilinear transformation the conversion of

specified digital frequencies to analog frequencies is called Pre-

warping. The Pre-Warping is necessary to eliminate the effect of

warping on amplitude response.

9. How one can design digital filters from analog filters?

1) Map the desired digital filter specifications into those for an

equivalent analog filter.

2) Derive the analog transfer function for the analog prototype.

3) Transform the transfer function of the analog prototype into an

equivalent digital filter transfer function.

10. Mention the procedures for digitizing the transfer function of an

analog filter.

The two important procedures for digitizing the transfer function of an

analog filter are

Impulse invariance method.

Bilinear transformation method.

11. What do you understand by backward difference?

One of the simplest method for converting an analog filter into a

digital filter is to approximate the differential equation by an equivalent

difference equation. d/dt y(t)=y(nT)-y(nT-T)/T

The above equation is called backward difference equation.

12. What is the mapping procedure between S-plane & Z-plane in the

method of mapping differentials? What are its characteristics?

The mapping procedure between S-plane & Z-plane in the method of

mapping of differentials is given by

H(Z) =H(S)|S=(1-Z-1

)/T

The above mapping has the following characteristics

The left half of S-plane maps inside a circle of radius ½ centered at Z= ½ in

the Z plane.

The right half of S-plane maps into the region outside the circle of radius ½

in the Z-plane.

The j-axis maps onto the perimeter of the circle of radius ½ in the Z-plane.


13. What is meant by impulse invariant method of designing IIR

filter?

In this method of digitizing an analog filter, the impulse response of

resulting digital filter is a sampled version of the impulse response of the

analog filter.

14. Give the bilinear transform equation between S-plane & Z-plane.

S=2/T(1-Z-1

/1+Z-1

)

15. What is bilinear transformation?

The bilinear transformation is a mapping that transforms the left half

of S-plane into the unit circle in the Z-plane only once, thus avoiding

aliasing of frequency components.The mapping from the S-plane to the Z-

plane is in bilinear transformation is

S=2/T(1-Z-1/1+Z-1)

16. What are the properties of bilinear transformation?

1) 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.

2) 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.

17. What are the advantages & disadvantages of bilinear

transformation?

Advantages:

The bilinear transformation provides one-to-one mapping.

Stable continuous systems can be mapped into realizable, stable digital

systems.

There is no aliasing.

Disadvantage:

The mapping is highly non-linear producing frequency, compression

at high frequencies.

Neither the impulse response nor the phase response of the analog

filter is preserved in a digital filter obtained by bilinear transformation.

18. Define signal flow graph.

A signal flow graph is a graphical representation of the relationships

between the variables of a set of linear difference equations.


PART-B

1) Derive the equation for designing IIR filter using bilinear

transformation.

ANS: Ref: Page No.336….Digital Signal Processing by A.Nagoorkani

2) Derive the equation for designing IIR filter using impulse

invariant method.

ANS: Ref: Page No .330….Digital Signal Processing by A.Nagoorkani

3) Draw all the possible realization of FIR system.

Ref : Page No 502 Digital Signal Processing by John G.Proakis

4) Design a digital Butterworth filter satisfying the constraints

0.707 ≤ | H(ω)| ≤ 1.0 ; 0 ≤ ω ≤ π/2

| H(ω)| ≤ 0.2 ; 3π/4 ≤ ω ≤ π.

Nov/Dec 2006

Ans: Ref Pg.No 435-437, DSP by Salivahanan.

5) Design a digital Butterworth filter satisfying the constraints

0.8 ≤ | H(ω)| ≤ 1.0 ; 0 ≤ ω ≤ π/4

| H(ω)| ≤ 0.2 ; π/2 ≤ ω ≤ π.

May/June2007 & Nov/Dec 2008

Ans: Ref: Pg.No: 359-362, DSP by Nagoorkani.

6) Design a digital BUTTERWORTH filter that satisfies the

following constraint using BILINEAR Transformation.

Assume T = 1 sec.

0.9 ≤ | H(ω)| ≤ 1 ; 0 ≤ ω ≤ π /2

| H(ω)| ≤ 0.2 ; (3 π /4) ≤ ω ≤ π

ANS: Ref: Page No .370….Digital Signal Processing by A.Nagoorkani

7) For the analog transfer function H(S) = 2/ (S+1)(S+2) .

Determine H(Z) using impulse invariant technique.

April /May 2008 ANS: Ref: Page No .341.Digital Signal Processing by A.Nagoorkani


UNIT IV - FIR FILTER DESIGN

PART-A

1) Give any two properties of Butterworth and Chebyshev filter.

Nov/Dec 2006

Properties of Butterworth:

The butterworth filters are all pole design.

The filter order N completely specifies the filter

The magnitude is maximally flat at the origin.

The magnitude is monotomically decreasing function of ohm.

Properties of Chebyshev:

The magnitude reponse of the filter exhibits ripples in the

pass band or stop band

The pole of the filter lies on an ellipse.

2) Show that the filter with h(n) = [-1,0,1] is a linear phase filter.

May /June 2007 & Nov/Dec 2008

Solution:

h(n) = [ -1,0,1]

h(0) = -1 = -h(N-1-n) = -h(3-1-0) = -h(2)

h(1) = 0 = -h(N-1-n) = -h(3-1-1) = -h(1)

h(2) = 1 = -h(N-1-n) = -h(3-1-2) = -h(0)

It is a linear phase filter.

3) In the design of FIR digital filter, how is Kaiser Window

different from other windows? Nov/Dec 2007

In all other windows a trade off exists between ripple ratio and main

lobe width. In Kaiser Window both ripple ratio and main lobe width

can be varied independently.

4) What are the merits and demerits of FIR filter? April/May 2008

Merits :

Linear phase filter.

Always Stable

Demerits:

The duration of the impulse response should be large

Non integral delay.


5) What are the advantages and disadvantages of FIR filters?

Advantages:

1. FIR filters have exact linear phase.

2. FIR filters are always stable.

3. FIR filters can be realized in both recursive and non recursive

structure.

4. Filters with any arbitrary magnitude response can be tackled using

FIR sequence.

Disadvantages:

1. For the same filter specifications the order of FIR filter design can

be as high as 5 to 10 times that in an IIR design.

2. Large storage requirement is requirement

3. Powerful computational facilities required for the implementation.

6) What are the desirable characteristics of the window function?

The desirable characteristics of the window are

1. The central lobe of the frequency response of the window should

contain most of the energy and should be narrow.

2. The highest side lobe level of the frequency response should be

small.

3. The side lobes of the frequency response should decrease in energy

7) Mention the windows of FIR filters.

a. Rectangular window

b. Hamming window

c. Hanning window

d. Bartlett window

e. Kaiser window

8) What is the necessary and sufficient condition for linear phase

characteristic in FIR filter?

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.,H(n) = h(N-1-n)where N is the duration of the

sequence.

9) What are the advantages of Kaiser Window?


It provides flexibility for the designer to select the side lobe level

It has the attractive property that the side lobe level can be varied

continuously from the low value in the Blackman window to the

high value in the rectangular window.

10) What is the principle of designing FIR filter using

frequency sampling method?

In frequency sampling method the desired magnitude response is

sampled and a linear phase response is specified .The samples of

desiredfrequency response are identified as DFT coefficients. The filter

coefficients are then determined as the IDFT of this set of samples.

11) For what type of filters frequency sampling method is

suitable?

Frequency sampling method is attractive for narrow band frequency

selective filters where only a few of the samples of the frequency response

are non zero.

12) When cascade form realization is preferred in FIR filters?

The cascade form realization is preferred when complex zeros with

absolute magnitude is less than one.

13) Compare Rectangular & Hamming window.

S.No Rectangular Window Hamming window.

1. The width of the main

lobe in window spectrum

is 4π/N

The width of the main lobe

in window spectrum is 8π/N

2. The maximun side lobe

magnitude in window

spectrum is -13 dB

The maximun side lobe

magnitude in window

spectrum is -41 dB

14) Compare Hamming window & Kaiser Window.

S.No Kaiser Window Hamming window.

1. The width of the main

lobe in window spectrum

depends on the value of α

and N.

The width of the main lobe

in window spectrum is 8π/N

2. The maximun side lobe The maximun side lobe


magnitude with respect to

peak of main lobe is

variable using the

parameter α.

magnitude in window

spectrum is -41 dB

15) List the steps involved in the design of FIR filters using

windows.

1. For the desired frequency response Hd(w), find the impulse response

hd(n) using Equation

hd(n)=1/2 Hd(w)ejwndw

2 .Multiply the infinite impulse response with a chosen window sequence

w(n) of length N to obtain filter coefficients h(n),i.e.,

h(n)= hd(n)w(n) for |n|._1-1)/2

= 0 otherwise

Find the transfer function of the realizable filter

(N-1)/2

ANS: Reference : Page No .294….Digital Signal Processing by

A.Nagoorkani [First Edition]

16) What is meant by limit cycle oscillation in digital filter?

May /June 2007 & Nov/Dec 2007 &April/May 2008

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 theoutput. Thus the oscillation is

called as Limit cycle.

17) Express the fraction 7/8 and – 7/8 in sign magnitude, 2’s

complement and 1’s complement. Nov/Dec 2006

Solution:

7/8 = 0.875 = (0.111)2 is sign magnitude

1’s Complement = (0.111)2


7/8 = -0.875

Sign magnitude: (1.111)2



18) Identify the various factors which degrade the performance

of the digital filter implementation when finite word length is

used. May /June 2007 & April/May 2008 & Nov/Dec 2008


Input quantization error

Coefficient quantization error

Product quantization error

19) a) What are the quantization errors due to finite word

length register in digital filter.

b) What are the different quantization methods? Nov/Dec 2006

Quantization Error :




Quantization methods

Truncation

Rounding

20) Express the fraction (-7/32) in signed magnitude and 2’s

complement notations using 6 bits. Nov/Dec 2007 &Nov/Dec

2008

In Signed Magnitude: 1.001110

In 2’s complement: 1.110010

PART-B

1. a) Design a high pass filter hamming window by taking 9

samples of w(n) and with a cutoff frequency of 1.2 radians/sec

Nov/Dec 2006


2. a) Describe the design of FIR filter using frequency sampling

technique.

b) The desired frequency response of a low pass filter is given by

Hd(ω) ={ e –j2ω

; -π/4 ≤ ω ≤ π/4

0 ; other wise.

3. Obtain the filter coefficient, h(n) using RECTANGUAR window

define by W(n) = { 1; 0 ≤ n ≤ 4

0; otherwise.


Nov/Dec 2007

Ans: a) Ref Pg.No 389-391, DSP by Salivahanan.

b) Ref Pg.No 399, DSP by Salivahanan.

4. ii) Determine the magnitude response of the FIR filter (M=11)

and show that phase and group delay are conatant.

iii) The desired frequency response of a low pass filter is given

by

Hd(ω) ={ e –j3ω

; -3π/4 ≤ ω ≤ 3π/4

0 ; other wise.

Determine H(ejω

) for M= 7using HAMMING window.

Ans: a) i) Ref Pg.No 437-439, DSP by Salivahanan.

ii) Ref Pg.No 383-384, DSP by Salivahanan.

iii) Ref Pg.No 400-401, DSP by Salivahanan.

iv) Ref Pg.No 426, DSP by Salivahanan.

5. Explain the design of lowpass digital butterworth filter.

ANS: Reference : Page No.347….Digital Signal Processing by


6. Explain the design of lowpass digital Chebyshev filter.



7. Derive the frequency response of linear phase FIR filter when

impulse response is Symmetric when N is ODD.




impulse response is Symmetric when N is EVEN.




impulse response is Anti symmetric when N is ODD.





impulse response is Antisymmetric when N is EVEN.



11. Analyze the Windows specifying the FIR filters

a. Rectangular window

b. Hamming window

c. Hanning window

d. Bartlett window

e. Kaiser window

ANS: Reference : Page No .281….Digital Signal Processing by


12. Draw the structures of FIR filters

Reference : Page No 502Digital Signal Processing by John G.Proakis[Third

Edition]

13. .Explain the design of FIR filters by frequency sampling

technique.



14. a)i) Explain the characteristics of a Limit cycle oscillation w.r.t

the

system described by the difference equation

y(n) = 0.95y(n-1)+x(n).Determine the dead band of the filter.

ii) Draw the product quantisation noise model of second order

IIR fihter. Nov/Dec 2006 & Nov/Dec 2007

Ans: a) i) Dead band = [+0.625.-0.625 ]


15. For the given transfer function H(Z) = H1(Z) .H2(Z) ,where 1 1

H1(Z) = and H2(Z) =

1 – 0.5 Z-1

1 – 0.6 Z-1


Find the output round off noise power. Nov/Dec 2006

Ans: 2-2b

/12(5.4315)

16. a)i) Consider the truncation of negative fraction number

represented in(β+1) bit fixed point binary form including sign bit

. Let (β-b) bits be truncated .Obtain the range of truncation errors

for signed magnitude ,2’s complement and 1’s complement

representation of negative numbers. Nov/Dec 2007

ii) The coefficients of a system defined by

1

H(Z) =

(1-0.4Z-1

)(1-0.55Z-1

)

are represented in anumber with a sign bit and 3 data bits.

iii) Consider the (b+1) bit bipolar A/D converter.Obtain an

expression for signal to quantization noise ratio . May /June 2007& Nov/Dec 2007&April/May2008 & Nov/Dec 2008


ii) Direct form: 1/ [1-0.875z-1

+0.125Z-2

]

Cascade form:1/[1-0.375Z-1

][1-0.5Z-1

]

17. With the neat diagram explain the operation of limit cycle

oscillations.

Ref Pg.No 513-514, DSP by Salivahanan.

UNIT V - APPLICATIONS

PART-A

1. Define multirate digital signal processing.

The process of converting a signal from a given rate to a

different rate is called sampling rate conversion. The system that employs

multiple sampling rates in the processing of digital signals are called

digital signal processing systems.

2. Give the advantages of multirate digital signal processing.

Computational requirements are less

Storage for filter coefficients is less

Finite arithmetic effects are less

Sensitivity to filter coefficients lengths are less


3. Give the applications of multirate digital signal processing.

Communication systems

Speech and audio processing systems

Antenna systems

Radar systems

4. Define Decimation.

The process of reducing the sampling rate of the signal is called

decimation (sampling rate compression).

5. Define Interpolation

The process of increasing the sampling rate of the signal is

called interpolation (sampling rate Expansion).

PART-B 1. Explain briefly: Multi rate signal processing May/June 2007

Ref Pg.No 751, DSP by Proakis

2. Explain briefly: Vocoder May/June 2007 Ref Pg.No 754, DSP by Proakis

3. Explain decimation of sampling rate by an integer factor D and

derive spectra for decimated signal May/June 2006


4. Explain interpolation of sampling rate by an integer factor I and

derive spectra for decimated signal May/June 2006


5. Explain about adaptive filters



B.E/B.TECH DEGREE EXAMINATION ,NOV/DEC 2006 FIFTH SEMESTER (REGULATION – 2004)

ELECTRONICS AND COMMUNICATION ENGINEERING

EC 1302 – DIGITAL SIGNAL PROCESSING TIME: 3 hrs Max. Mark: 100

PART A (10*2 =20 MARKS) Answer all Question

1. Determine the DTFT of a sequence x(n) = an u(n).

Solution:

x(n) = an u(n)

∞

X(ejω

) = ∑ x(n) e -jωn

n=-∞

∞

X(ejω

) = ∑ an u(n) e

-jωn

n=-∞

∞

X(ejω

) = ∑ an e

-jωn

n=0

∞

X(ejω

) = ∑ (a e –jω

)n

n=0

2. What is FFT?

The Fast Fourier Transform is a method or algorithm for computing

the DFT with reduced number of calculations. The computational

efficiency can be achieved if we adopt a divider and conquer

approach. This approach is based on decomposition of an N-point

DFT in to successively smaller DFT’s. This approach leads to a

family of an efficient computational algorithm is known as FFT

algorithm.

X(ejω

) = 1 / (1-a-ejω

)


3. Obtain the block diagram representation of a FIR system? X(Z)

h(N-1)

h(1)

h(0) h(2) h(N-2)

Y(Z)

4. Give any two properties of Butterworth and Chebyshev filter.

Properties of Butterworth:

The butterworth filters are all pole design.

The filter order N completely specifies the filter

The magnitude is maximally flat at the origin.

The magnitude is monotomically decreasing function of ohm.

Properties of Chebyshev:

The magnitude response of the filter exhibits ripples in the pass

band or stop band

The pole of the filter lies on an ellipse.

5. Express the fraction 7/8 and – 7/8 in sign magnitude, 2’s

complement and 1’s complement.

Solution:

7/8 = 0.875 = (0.111)2 is sign magnitude



7/8 = -0.875

Sign magnitude: (1.111)2



6. a) What are the quantization errors due to finite word length

register in digital filter.

b) What are the different quantization methods?

Z-1

Z-1

Z-1

Z-1

+ + + +


Quantization Error :




Quantization methods

Truncation

Rounding

7. What is zero padding? Does Zero padding improve the frequency

resolution in the spectral estimate?

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) we have to add (N-L) zeros to the

sequence x(n).This is known as Zero padding. Yes , it will improve

the frequency resolution in the spectral estimate.

8. Explain Deterministic and random signal with examples.

Deterministic signals are signals that are completely specified in time.

Deterministic signals can be predicted. Ex: Sin ωn

Non deterministic signals are defined as the signal that takes on

random value at any given instant and it cannot be predicted. It is also

called as Random signal. Ex: ECG signal.

9. Give the digital signal processing application with the TMS 320

family.

Communication system like voice coder, Speech recognization, Audio

signal processing, Control and data acquisition, Biometric information

processing and image /video processing.

10. What is the advantage of Harvard architecture of TMS 320

series?

Harvard architecture has two or three memory buses allowing

access to filter coefficients and input signals in the same cycle. Since it

possesses two independent bus system. The Harvard architecture is

capable of Simultaneous reading an instruction code and reading or

writing a memory or peripheral as part of the execution of the

previous instruction.


PART B (5*16 =80MARKS)

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

ii) Determine the response of LTI system by radix -2 DIT FFT. Ans:i) X(K) = { 0, -1-j,6,-1+j}

ii) Ref Pg.No 320-328 , DSP by Salivahanan .

(OR)

b) i) Derive the equation for Decimation – in time algorithm for

FFT.

ii) How do you linear filtering by FFT using Save –add method?

Nov/Dec 2006 & April /May 2008 & Nov/Dec 2008

Ans:i) Ref Pg.No 320-328 , DSP by Salivahanan .

ii) Ref Pg.No 369, DSP by Salivahanan.

a) Design a high pass filter hamming window by taking 9 samples

of w(n) and with a cutoff frequency of 1.2 radians/sec.


(OR)

b) Design a digital Butterworth filter satisfying the constraints

0.707 ≤ | H(ω)| ≤ 1.0 ; 0 ≤ ω ≤ π/2

| H(ω)| ≤ 0.2 ; 3π/4 ≤ ω ≤ π.

Ans: Ref Pg.No 435-437, DSP by Salivahanan.

a) For the given transfer function H(Z) = H1(Z) .H2(Z) ,where 1 1

H1(Z) = and H2(Z) =

1 – 0.5 Z-1

1 – 0.6 Z-1

Find the output round off noise power. Ans: 2

-2b/12(5.4315)

(OR)

b) i) Explain the characteristics of a limit cycle oscillation w.r.t

the

system described by the difference equation

y(n) = 0.95y(n-1)+x(n).Determine the dead band of the filter.


ii) Draw the product quantisation noise model of second order

IIR filter.

Ans: a) i) Dead band = [-10,10]

ii) Ref Pg.No 513-514, DSP by Salivahanan

a) Determine the performance characteristics of Non –

parametric

Power spectrum estimator Welch, Bartlett and Blackman

and

Tukey.

Ans: a) Ref Pg.No 594-603, DSP by Salivahanan.

(OR)

b) i) Give the key features of the digital signal processor

ii) Write Short notes on:

a) 32- bit accumulator

b) 16*16 bit parallel multiplier

c) Shifter

Ans: a) i) Ref Pg.No 8-9, Digital Signal Processor by

B.Venkataramani &M.Bhaskar

ii) Ref Pg.No 40, 59,266, Digital Signal Processor by


15. a) i) Explain the function of auxillary register in the indirect

Addressing mode to point the data memory location.

ii) Write a program to use the auxillary register in memory

pointing and looping.

Ans: Ref Pg.No 48-49, Digital Signal Processor by


(OR)

b) i) Write a program to compute the following equation .

Y= A*X1 + B* X2 +C*X3

ii) Write a program to perform addition of two 64 bit numbers

Ans: Ref Pg.No 265, Digital Signal Processor by



B.E/B.TECH DEGREE EXAMINATION, MAY/JUNE 2007 FIFTH SEMESTER (REGULATION – 2004)




1. The first five DFT coefficients of a sequence x(n) are X(0) = 20,

X(1) = 5+j2,X(2) = 0,X(3) = 0.2+j0.4 , X(4) = 0 . Determine the

remaining DFT coefficients

Solution:

X (K) = [20, 5+j2, 0, 0.2+j 0.4 , 0,X(5),X(6),X(7)]

X (5) = 0.2 – j0.4

X (6) = 0

X (7) = 5-j2

2. What are the advantages of FFT algorithm over direct

computation of DFT?

6. Reduces the computation time required by DFT.

7. Complex multiplication required for direct computation is N2

and for FFT calculation is N/2 log 2 N.

8. Speed calculation.

3. Show that the filter with h(n) = [-1,0,1] is a linear phase filter.

Solution:

h(n) = [ -1,0,1]

h(0) = -1 = -h(N-1-n) = -h(3-1-0) = -h(2)

h(1) = 0 = -h(N-1-n) = -h(3-1-1) = -h(1)

h(2) = 1 = -h(N-1-n) = -h(3-1-2) = -h(0)


4. Find the digital transfer function H(Z) by using impulse invariant

method for the analog transfer function H(S) = 1/ (S+2).Assume

T=0.5sec

Solution:


H(S) = 1/ (S+2).

H(Z) = 1/[1-e-1

Z-1

]

H(Z) = 1/ [1-0.368Z-1

]

5. Identify the various factors which degrade the performance of the

digital filter implementation when finite word length is used.




6. What is meant by limit cycle oscillation in digital filter?

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.

7. Define Power spectral density and cross power spectral density?

The spectral characteristics of a random process is obtained according

to wiener khinchine theorem, by computing the Fourier transform of

the autocorrelation function. The power spectral density is given by

∞

γxx (F) = ∫ γxx (τ) e –j2πft

dτ

-∞

The power density spectrum for two jointly stationary random process

X(t) and Y(t) gives the cross correlation function γxy (τ) .The Fourier

transform of γxy (F) is

∞

γxy (F) = ∫ γxx (τ) e –j2πft

dτ.

-∞

8. What are the disadvantages of non-parametric methods of power

spectral estimation?

1. It requires long data sequence to obtain the necessary frequency

resolution.

2. Spectral leakage effects because of windowing.

9. Differentiate between Von Neumann and Harvard architecture.


Von Neumann architecture possesses one bus system. The same bus

carries all the information exchanged between the CPU and the

peripherals including the instruction codes as well as data processed

by the CPU. Harvard architecture possesses two independent bus

systems. It is capable of simultaneous reading an instruction code and

reading or writing a memory or peripheral as part of the execution of

precious instruction.

10. State the merits and demerits of multi ported memories.

A Mutiported memory operates as on chip memory and off chip

memory.

1. Higher performance because no wait states are required.

2. Lower cost than external memory.

3. Lower Power than external memory.

4. Higher performance because better flow with in the pipeline at

lential arithmetic logic unit.

5. Ability to access large memory space.

PART B (5*16 =80MARKS)

11. a) i) Prove the following properties of DFT when H(k) is the DFT

of

an N-point sequence h(n).

1. H(k) is real and even when h(n) is real and even.

2. H(k) is imaginary and odd when h(n) is real and odd.

ii) Compute the DFT of x(n) = e-0.5n

, 0≤ n≤ 5.

Ans: i) Ref Pg.No 309, DSP by Salivahanan.

ii) X(K) = { 2.414, 0.87-j0.659, 0.627-0.394j, 1.202,

0.62-j0.252, 0.627-j0.252}.

(OR)

b) i) From first principles obtain the signal flow graph for



x(n) = cos (nπ/4) ,0 ≤ n ≤ 7.

Ans: i) Ref Pg.No 334-340, DSP by Salivahanan.

ii) X(K) = {0, 3, 0, 2.7-j0.7, 0, 1, 0, 1.293-j0.7}


12. a) Design a digital Butterworth filter satisfying the constraints

0.8 ≤ | H(ω)| ≤ 1.0 ; 0 ≤ ω ≤ π/4

| H(ω)| ≤ 0.2 ; π/2 ≤ ω ≤ π.


(OR)

b) A band pass FIR filter of length 7 is required. It is to have

lower

and upper cutt frequencies of 3 KHZ and 6 Khz respectively

and indended to be used with a sampling frequency of 24 KHZ

Determine the filter coefficient using HANNING window

Consider the filter to be causal. (16)

13. a) i) Consider the truncation of negative fraction number

represented in(β+1) bit fixed point binary form including sign

bit . Let (β-b) bits be truncated .Obtain the range of

truncation

errors for signed magnitude ,2’s complement and 1’s

complement representation of negative numbers.

ii) The coefficients of a system defined by

1

H(Z) =

(1-0.4Z-1

)(1-0.55Z-1

)


Determine the new pole location for 1) Direct realization and

2)

Cascade realization of first order systems. Compare the

movements of the new pole away from the original ones in

both the cases.

(OR)

b) Consider the (b+1) bit bipolar A/D converter.Obtain an

expression for signal to quantization noise ratio .



ii) Direct form: 1/ [1-0.875z-1

+0.125Z-2

]


][1-0.5Z-1

]

b) Ref Pg.No 499-503, DSP by Salivahanan.

14. a)i) Determine the performance characteristics of Non –

parametric

Power spectrum estimator Welch, Bartlett and Blackman

and

Tukey.

ii) Determine the frequency resolution of the Bartlett, Welch &

Blackman Tukey methods of power spectrum estimates for a

Quality factor Q = 20. Assume that overlap in Welch’s

method

is 50% and the length of the sample sequences is 2000.


ii)Ref Pg.No 606, DSP by Salivahanan.

(OR)

b) i) Compute the auto correlation & power spectral density for

the signal x(t) = K cos (2πft + θ ) .

ii) Explain briefly the periodogram method of power spectral

Estimation.

Ans: b) i) Ref Pg.No 590-591, DSP by Salivahanan.


15. a) i) Explain what is meant by instruction pipelining. Explain with

an example ,how pipelining increase the through put

efficiency.

ii) Explain the operation of TDM serial port in P-DSP’s. Ans: a) i) Ref Pg.No 45-46, Digital Signal Processor by


ii) Ref Pg.No 50-51, Digital Signal Processor by


(OR)

b) i) With a suitable diagram describe the function of

Multiplier/adder units of TMS 320 C54X.

ii) Explain the operation of CSSU of TMS 320 C54X and

explain


its use considering the viteri operator.

Ans: b) i) Ref Pg.No 267-268, Digital Signal Processor by




B.E/B.TECH DEGREE EXAMINATION, NOV/DEC 2007 FIFTH SEMESTER (REGULATION – 2004)




1. State and prove Parseval’s Theorem.

Parseval’s theorem states that

If

x(n) ↔ X(K) and y(n) ↔ Y(K) ,

Then

N-1 N-1

∑ x(n) y*(n) = 1/N ∑ X(K) Y*(K)

n=0 K =0

When y(n) = x(n), the above equation becomes

2. What do you mean by the term “bit reversal” as applied to FFT?

Re-ordering of input sequence is required in decimation – in –time.

When represented in binary notation sequence index appears as

reversed bit order of row number.

3. Find the digital transfer function H(Z) by using impulse invariant

method for the analog transfer function H(S) = 1/ (S+2).Assume

T=0.5sec

N-1 N-1

∑ │x(n)│2 = 1/N ∑ │X(K)│

2

n=0 k=0


Solution:

H(S) = 1/ (S+2).

H(Z) = 1/[1-e-1

Z-1

]

H(Z) = 1/ [1-0.368Z-1

]

4. In the design of FIR digital filter, how is Kaiser Window different

from other windows?

In all other windows a trade off exists between ripple ratio and main

lobe width. In Kaiser Window both ripple ratio and main lobe width

can be varied independently.




may cause periodic oscillation in the output. Thus the oscillation is


6. Express the fraction (-7/32) in signed magnitude and 2’s

complement notations using 6 bits. In Signed Magnitude: 1.001110


7. Define the terms: autocorrelation sequence and power spectral

density.

Autocorrelation sequence:

Rxx (n,n+m) = E[X(n) X(n+m)]

Power spectral density: ∞

Sxx (ejω

) = ∑ Rxx (m) e-jmω

m=-∞

8. Define unbiased estimate and consistent estimate.

Expected value of the estimate is equal to actual value is called as

unbiased estimate.

Consistent estimate: Lim Varience (estimate) = 0.

N→∞

9. What are the factors that may be considered when selecting a DSP

processor for an application?


1. Architectural features

2. Exection speed

3. Type of arithmetic

4. Word length

10. What is pipelining?

A task is broken down in to a number of distinct subtasks which are

then overlapped during execution. It is used in digital signal

processors to increase speed.

PART –B (5*16 =80 MARKS)

11.a) Two finite duration sequence are given by

x(n) = sin (nπ/2) for n = 0,1,2,3

h(n) = 2 n for n = 0,1,2,3 Determine circular convolution using

DFT &IDFT method.

Ans: X(K) = {0, -2j, 0, 2j}

H(K) = {15, -3+6j, -5, -3-6j}

y(n) = {6, -3, -6, 3}

(OR)




x(n) = cos (nπ/4) ,0 ≤ n ≤ 7.


ii) X(K) = {0, 3, 0, 2.7-j0.7, 0, 1, 0, 1.293-j0.7}

12.a) i) Describe the design of FIR filter using frequency sampling

technique.

ii) The desired frequency response of a low pass filter is given

by

Hd(ω) ={ e –j2ω

; -π/4 ≤ ω ≤ π/4

; other wise.


Obtain the filter coefficient, h(n) using RECTANGUAR

Window define by W(n) = { 1; 0 ≤ n ≤ 4

0; otherwise.



(OR)


0.7 ≤ | H(ω)| ≤ 1.0 ; 0 ≤ ω ≤ 0.2π

| H(ω)| ≤ 0.004 ; 0.6π ≤ ω ≤ π.

Apply impulse invariant transformation method.


13. a) i) Consider the truncation of negative fraction number

represented

in(β+1) bit fixed point binary form including sign bit . Let (β-b)

bits be truncated .Obtain the range of truncation errors for

signed magnitude ,2’s complement and 1’s complement

representation of negative numbers.

ii) A 8-bit ADC feeds a DSP system characterised by the

following

Transfer function H(Z) = 1/(Z+0.5) .Estimate the steady

state

Quantisation noise power at the output of the system. Ans: a) i) Ref Pg.No 496-499, DSP by Salivahanan.


(OR)

b) i) The coefficients of a system defined by

1

H(Z) =

(1-0.4Z-1

)(1-0.55Z-1

)



2)

Cascade realization of first order systems.Compare the



both

the cases.

ii) An IIR causal filter has the system function

H(Z)=Z/Z-0.97,Determine the dead band of the filter.

Ans: b) i) Direct form: 1/ [1-0.875z-1

+0.125Z-2

]


][1-0.5Z-1

]


14. a)i) With Suitable relation ,explain briefly the periodogram method

of

power spectral estimation.Examine the consistency and bias of

the

periodogram.

ii) Explain power spectrum estimation using the bartlett method.



(OR)

b) i) Explain how the Black man and Tukey is used in smoothing the

periodogram? Derive the mean and variance of the power

spectral

estimate of the blackman and Tukey method.

ii) Determine the frequency resolution of the Bartlett, Welch and

Blackman and Tukey methods of the power spectral estimation

for a quality factor Q =15 and sample is 1500.



15.a) i) Explain how Harvard architecture as used by the TMS 320

family differs from the Strict Harvard architecture

.Compare

this with the architecture of a satndard Von Neumann


Processor.

ii) Explain the operation of MAC unit.

Ans: Ref Pg.No 40-44, Digital Signal Processor by


(OR)

b) i) In relation to DSP processor, explain the following technique:

SMID, VLIW.

ii) Explain the operation of CSSU of TMS 320 C54X and

explain

its use considering the viterbi operator.

Ans: Ref Pg.No 40-44 &269-270 , Digital Signal Processor by


B.E/B.TECH DEGREE EXAMINATION, APRIL /MAY 2008 FIFTH SEMESTER (REGULATION – 2004)




1. Define the properties of convolution.

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)*

h1(n)]

2. Draw the basic butterfly diagram of radix -2 FFT.

1 1

a A = a+ WN

nk b

1


1

WN

nk

b B = a - WN

nk b

-1

3. What are the merits and demerits of FIR filter?

Merits :

Linear phase filter.

Always Stable

Demerits:

The duration of the impulse response should be large

Non integral delay.

4. What is the relationship between analog and digital frequency in

impulse invariant transformation?

Digital Frequency: ω = Ω T

Ω = analog frequency

T= Sampling interval



8 Input quantization error

8 Coefficient quantization error

8 Product quantization error




may cause periodic oscillation in the output. Thus the oscillation is


7. Define Periodogram.

F {rxx (m)} = 1/N │X (f)│2

The above equation gives the estimation of power spectral density

(PSD). This equation is also called as Periodogram.


8. Determine the frequency resolution of the Bartlett method of

power spectrum estimates for a quality factor Q =15 .Assume that

the length of the sample sequence is 1500.

Solution:

Q Bart = 1.11N Δf

Frequency resolution: Δf = 15 /(1.11 *1500)

Δf = 0.0009

9. What is pipelining?

A task is broken down in to a number of distinct subtasks which are

then overlapped during execution. It is used in digital signal

processors to increase speed.

10. What is the principal feature of the Harvard architecture?

The program and data memory lies in two separate spaces, permitting

full overlap of instruction, Fetch and execute.

PART –B (5*16 = 80 MARKS)

11.a) i) Disuss in detail the imporatnt properties of the DFT.

ii) Find the 4 –point DFT of the sequence x(n) = cos nπ/4.

Ans: i)Ref Pg.No 308-311, DSP by Salivahanan.

ii) X(K) = {1, 1-j1.414, 1, 1+j1.414}

iii) X(K) = {20,-5.8-j2.4, 0, 0.17-j0.414, 0, -0.17+j0.414, 0,

-5.82+j2.414}.

(OR)

b)i) Using DIT draw the butterfly line diagram for 8-point FFT

calculation and explain.

ii)Compute an 8-point DFT using DIF FFT algorithm.

X(n) = {1,2,3,4,4,3,2,1}

Ans: i) X(K) = {20,-5.8-j2.4, 0, 0.17-j0.414, 0, -0.17+j0.414, 0,

-5.82+j2.414}.


12. a) i) Determine the magnitude response of an FIR filter (M=11) and


show that the phase and group delays are constant.

ii) The desired frequency response of a low pass filter is given by

Hd(ω) ={ e –j3ω

; -3π/4 ≤ ω ≤ 3π/4

0 ; other wise.

Determine H(ejω

) for M= 7using HAMMING window.

(OR)

b)i) For the analog transfer function H(S) = 1/ (S+1)(S+2) .

Determine H(Z) using impulse invariant technique.

ii) Design a digital Butterworth filter satisfying the constraints

0.9 ≤ | H(ω)| ≤ 1.0 ; 0 ≤ ω ≤ π/2

| H(ω)| ≤ 0.2 ; 3π/2 ≤ ω ≤ π.

Apply Bilinear transformation method. (16)



b)i) Ref Pg.No 400-401, DSP by Salivahanan.


13.a) i) Discuss in detail the truncation error and round off error for

sign

Magnitude and 2’s complement.

ii) Explain the quantization effects in converting analog signal in to

digital signal



(OR)

b) i) A digital system, is characterized by the difference equation

y(n) = 0.9y(n-1) +x(n) .Determine the dead band of the filter.

ii) What is meant by coefficient quantization? Explain

Ans: b) Ref Pg.No 510, DSP by Salivahanan.

14.a) i) Explain the Bartlett method of averaging Periodogram.

ii) What is the relation ship between autocorrelation and power

Spectrum? Prove it.




(OR)

b)i) Derive the mean and variance of the power spectral estimate of

the Blackman and Tukey method.

ii) Obtain the expression for mean and variance of the

autocorrelation function of random signals.

Ans: b) Ref Pg.No 596-599, DSP by Salivahanan.

15.a) i) Describe the MAC unit in DSP Processor.

ii) Explain the architecture of TMS 320 C5X DSP Processor.

Ans: Ref Pg.No 40-44, &256 Digital Signal Processor by


(OR)

b) i) Discuss in detail the four phases of the pipeline techniques.

ii) Write short notes on:

1. Parallel Logic

2. Circular register.

Ans: Ref Pg.No 40-41,56-65,60,48, Digital Signal Processor by


B.E/B.TECH DEGREE EXAMINATION, NOV /DEC 2008 FIFTH SEMESTER (REGULATION – 2004)



PART A (10*2 =20 MARKS) Answer all Questions

1. Distinguish between DIT and DIF –FFT algorithm.

S.No DIT –FFT Algorithm DIF –FFT Algorithm

1. The input is in bit reversed

order; the output will be

normal order.

The input is in normal order;

the output will be bit reversed

order.

2. Each stage of computation the


before add subtract operation.

Each stage of computation the


after add subtract operation.


2. If H(K) is the N-point DFT of a sequence h(n) , Prove that H(K)

and H(N-K) are complex conjugates.

This property states that, if h(n) is real , then H(N-K) = H*(K) = H(-

K)

Proof:

By the definition of DFT;

N-1

X(K) = ∑ x(n) e (–j2πnk)/N

n=0

Replace ‘K’ by ‘N-K’

N-1

X(N-K) = ∑ x(n) e (–j2πn(N-K))/N

n=

3. Show that the filter with h(n) = [-1,0,1] is a linear phase filter.

Solution:

h(n) = [ -1,0,1]

h(0) = -1 = -h(N-1-n) = -h(3-1-0) = -h(2)

h(1) = 0 = -h(N-1-n) = -h(3-1-1) = -h(1)

h(2) = 1 = -h(N-1-n) = -h(3-1-2) = -h(0)


4. What is Prewarping? Why is it needed?

In IIR design using bilinear transformation the conversion of

specified digital frequencies to analog frequencies is called Pre-

warping. The Pre-Warping is necessary to eliminate the effect of

warping on amplitude response.



X(N-K) = X*(K)


1. Input quantization error

2. Coefficient quantization error

3. Product quantization error

6. Express the fraction (-7/32) in signed magnitude and 2’s

complement notations using 6 bits. In Signed Magnitude: 1.001110


7. What are the disadvantages of non-parametric methods of power

spectral estimation?

1. It requires long data sequence to obtain the necessary frequency

resolution.

2. Spectral leakage effects because of windowing.

8. Define unbiased estimate and consistent estimate. Expected value of the estimate is equal to actual value is called as

unbiased estimate.

Consistent estimate: Lim Varience (estimate) = 0.

N→∞

9. State the merits and demerits of multi ported memories.

A Mutiported memory operates as on chip memory and off chip

memory.

1. Higher performance because no wait states are required.

2. Lower cost than external memory.

3. Lower Power than external memory.

4. Higher performance because better flow with in the pipeline at

lential arithmetic logic unit.

5. Ability to access large memory space.

10. What are the factors that may be considered when selecting a

DSP processor for an application?

1. Architectural features

2. Exection speed

3. Type of arithmetic

4. Word length


PART –B ( 5*16 = 80 MARKS)

11. a) Two finite duration sequence are given by

x(n) = Cos (nπ/2) for n = 0,1,2,3

h(n) = 0.5 n for n = 0,1,2,3 Determine circular convolution using

DFT &IDFT method.

Ans: X(K) = {0, -2j, 0, 2j}

H(K) = {15, -3+6j, -5, -3-6j}

y(n) = {6, -3, -6, 3}

(OR)


Computing 8-point using radix -2 DIT –FFT algorithm.


x(n) = cos (nπ/4) ,0 ≤ n ≤ 7.


ii) X(K) = {0, 3, 0, 2.7-j0.7, 0, 1, 0, 1.293-j0.7}

12. a) A band pass FIR filter of length 7 is required. It is to have

lower

and upper cutt frequencies of 3 KHZ and 5 Khz respectively

.

and indended to be used with a sampling frequency of 20

KHZ

Determine the filter coefficient using HANNING window

Consider the filter to be causal.

(OR)


0.8 ≤ | H(ω)| ≤ 1.0 ; 0 ≤ ω ≤ 0.2π

| H(ω)| ≤ 0.2 ; 0.6π ≤ ω ≤ π.

Apply Bilinear transformation method.



13. a) i) Consider the (b+1) bit bipolar A/D converter. Obtain an

expression for signal to quantization noise ratio .

ii) Consider the truncation of negative fraction number

represented

in(β+1) bit fixed point binary form including sign bit . Let (β-b)

bits be truncated .Obtain the range of truncation errors for

signed magnitude ,2’s complement and 1’s complement

representation of negative numbers.



(OR)

b) The coefficients of a system defined by

1

H(Z) =

(1-0.3Z-1

)(1-0.65Z-1

)

are represented in a number with a sign bit and 3 data bits.


2)

Cascade realization of first order systems. Compare the


both the cases.

Ans: b) Direct form: 1/ [1-0.875z-1

+0.125Z-2

]


][1-0.5Z-1

]

14. a)i) With Suitable relation ,explain briefly the periodogram

method of power spectral estimation. Examine the consistency

and bias of the periodogram.

ii) Explain power spectrum estimation using the bartlett method.



(OR)

b) i) Explain how the Black man and Tukey is used in smoothing the

periodogram? Derive the mean and variance of the power


spectral

estimate of the blackman and Tukey method.

ii) Determine the frequency resolution of the Bartlett, Welch and

Blackman and Tukey methods of the power spectral estimation

for a quality factor Q =15 and sample is 1500.



15. a) i) Explain what is meant by instruction pipelining. Explain

with

an example ,how pipelining increase the through put

efficiency.

ii) Describe the MAC unit in DSP Processor.

Ans: a) i) Ref Pg.No 45-46, Digital Signal Processor by


ii) Ref Pg.No 40-44, &256 Digital Signal Processor by


(OR)

b)i) With a suitable diagram describe the function of

Multiplier/adder units of TMS 320 C54X.

ii) In relation to DSP processor, explain the following

technique:

SMID, VLIW.

Ans: b) i) Ref Pg.No 267-268, Digital Signal Processor by




Documents

CS2403 DSP