SUDHARSAN ENGINEERING COLLEGE , SATHIYAMANGALAM

622501.
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
.
Sub Code:
CS2403
Subject Name:Digital Signal Processing (R

2008)
Class:III Year IT
Staff Incharge:RM.Senthilkumar,AP/ECE.
Question
Bank
UNIT I
Signals and systems
9
Part

A
1.A.1)
What is a linear time invariant system? (APRIL/MAY 2008)
A system is said to be time invariant system if a time delay or advance of the input signal
leads to an idenditical
shift in the output signal. This implies that a time invariant system
responds idenditically no matter when the input signal is applied. It also satisfies the
condition
R{x(n

k)}=y(n

k).
1.A.2)
What is known as aliasing?( APRIL/MAY 2008)
In
signal processing
and related disciplines,
aliasing
refers to an effect that causes different signals to become
indistinguishable (or
aliases
of one another) when
sampled
. It also refers to the
distortion
or
artifact
that results when the signal
reconstructed from samples is different from the original continuous signal.
Aliasing can occur in signals sampled in time, for instance
digital audio
, and is referred to as
temporal aliasing
. Aliasing can
also occur in spatially sampled signals, for instance
digital images
. Aliasing in spatially sampled signals is called
spatial aliasing
.
1.A.3)
Define ROC in Z

transform. ( APRIL/MAY 2008)
The region of convergence (ROC) of X(Z) the set of all values of Z for which X(Z)
attain final value.
1.A.4)
Determine the Z

transform
of the s
equence x(n)={2,1,

1,0,3} (APRI
L/MAY 2008)
X(Z)=2+1XZ

1

1XZ

2
+0XZ

4
+3XZ

4
X(z)=2+Z

1

Z

2
+3Z

4
1.A.5)
Check whether the system is
linear
y(n)=e
x(n)
(APRIL/MAY 2007)
Y1(n)=e
x
1
(n)
Y2(n)=
e
x
2
(n
)
y(n)=
a1
e
x
1
(n)
+a2
e
x
2
(n
)

1
y(n)=e
(a1x(n)+a2x2(n))

2
equn 1
≠ equn 2
So the system is non

linear systems.
1.A.6)
State sampling theorem.(APRIL/MAY 2007)
Sampling
is the process of converting a
signal
(for example, a function of continuous time or space) into a numeric sequ
ence (a
function of discrete time or space).
Shannon's
version of the theorem states:
[2]
If a function
x
(
t
) contains no frequencies higher than
B
hertz
, it is completely determined by giving its ordinates at a series of
points spaced 1/(2
B
) seconds apart.
A sufficient condition to reconstruct
x
(
t
) from its samples is
and equivalently
The two thresholds,
and
are respectively called the
Nyquist rate
and
Nyquist frequency
.
1.A.7)
Define LTI systems
(Nov/Dec 2010)
These systems are also called
linear translation

invariant
to give the theory the most general reach. In the case
of g
eneric
discrete

time
(i.e.,
sampled
) systems,
linear shift

invariant
is the corresponding te
rm. A good example of
LTI systems are electrical circuits that can be made up of resistors, capacitors, and inductors.
1.A.8)
Differentiate linear and circular convolution(Nov/Dec 2010)
1.A.9)
State sampling theorem and find Nyquist rate of the signal x(t)=5sin250
π
t+6cos600
π
t
(APRIL/MAY 2008)
Take T=.004 ; fs=1/T = 250 Hz.
Ω
s=2
πfs=2π*25 Hz=50π rad/sec
Ωm=2πfm=600π
fm=300
2fm=600 So the nyquist rate is = 600 Hz.
1.A.10)
State and prove the convolution property of Z transform(APRIL/MAY 2008)
1.A.11)
Find convolution of {
5,4,3,2} and {1,0,3,2} (APRIL/MAY 2008)
Yz)=H(z)X(z)
H(z)=5+4Z

1
+3Z

2
+2Z

3
and X(z)=(1+3Z

2
+2Z

3
)
Y(z)=(
5+4Z

1
+3Z

2
+2Z

3
)
(
1+3Z

2
+2Z

2
)
=5+4z

1
+18z

2
+39z

3
+9z

4
+6z

5
+10z

6
y
(n)={5,4,18,39,9,6,10}
1.A.12)
Find r
xy
and r
yx
for x={1,0,2,3} and y={4,0,1,2} (APRIL/MAY 2008)
X(z)=(1+2Z

2
+3Z

3
); Y(z)=(4+Z

2
+3Z

3
)
Correlation
(r
xy
)
=(1+2Z

2
+3Z

3
)(4+Z
2
+3Z
3
)=
15+6Z

1
+8Z

2
+12Z

3
+6Z+Z
2
+3Z
3
Correlation
(r
yx
)
=
(1+2Z
2
+3Z
3
)(4+Z

2
+3Z

3
)=15+z

2
+3z

3
+8z
2
+6z

1
+12z
3
+3z
1.A.13)
Determine Z transform
for X(n)=

na
n
u(

n

1)
(APRIL/MAY 2008)
1.A.14)
Find whether the signal Y(n)=

n
2
X(n) is linear.
(APRIL/MAY 2008)
Y1(n)=

n
2
X
1
(n) Y2(n)=

n
2
x
2
(n)
Y(n)=

n
2
X
1
(n)

n
2
x
2
(n)

1
Y(n)=

n
2
(
X
1
(n)+
x
2
(n))

2
1
≠
2 so the system is linear
1.A.15)
What is causal
system?(May/June 2013)
The system is said to be causal if the output of the system at any time ‘n’ depends
only on present and past inputs but does not depend on the future inputs.
e.g.:

y (n) =x (n)

x (n

1)
A system is said to be non

causal if a system d
oes not satisfy the above definition.
1.A.16)
Define impulse response of the system?
(May/June 201
2
)
In
signal processing
, the
impulse response
, or
impulse response function (IR
F)
, of a
dynamic system
is its
output when presented with a brief input signal, called an
impulse
. More generally, an impulse response refers
to the reaction of any dynamic system in response to some external change. In both cases, the impulse response
describes the reaction of the system as a
function
of time (or possibly as a function of some other
independent
variable
that parameterizes the dynamic
behavior of the system).
1.A.17)
Check for causality and linearity y(n)=x(n)

x(n
2

n)
(May/June 2013)
The output of the system depends on the future values of the input. so the system is called causal system.
Y1(n)=x1(n)

x1(n
2

n) ; y2(n)=
x2(n)

x2(n
2

n)
Y(n)=
x1(n)

x1(n
2

n)+
x2(n)

x2(n
2

n)
Y’(n)=
x1(n)

x2(n
2

n)+
x1(n)

x2(n
2

n)
Y(n)
≠
Y’(n)
So the system is called nonlinear systems.
1.A.18)
What are the advantages of DSP?(
(Nov/Dec 2009)
+ Linear and nonlinear math operations work over a wide dynamic ra
nge of signal, 2^31 to 2^

31 for standard floating
point. Also a suite of operations, like cos(), atan(), sqrt(), log() are available.
+ Higher order filters can be implemented with a relatively low incremental cost. Additional memory and computations
onl
y.
+ Filter design techniques provide a relatively high degree of freedom in spectral shaping, as in the Frequency Sampling
method, for example.
+ No tuning of analog components (R,L,C) during production or during maintenance.
+ Good version control. Bu
rn filter coefficients into memory and these will never change from one unit to the next.
+ Software

based implementations require no custom hardware

just use standard signal I/O boards and write custom
software.
+ Small and rugged implementation using
mixed

type VLSI, combining both DSP and analog I/O on a single chip.
1.A.19)
Define impulse signal
(NOV/DEC 2009)
An impulse function is not realizable, in that by definition the output of an impulse function is infinity at certain
values. An impulse functio
n is also known as a "delta function", although there are different types of delta
functions that each have slightly different properties. Specifically, this
unit

impulse function
is known as the
Dirac delta function. The term "Impulse Function" is unambig
uous, because there is only one definition of the
term "Impulse".
1.A.20)
What is meant by aliasing? How can it be avoided?
(May/June 2006)
Given a power spectrum (a plot of power vs. frequency), aliasing is a false translation of power falling in
some frequency
range
outside the range. Aliasing can be caused by discrete sampling below the
Nyquist
frequency
. The sidelobes of any
instrument function
(including the simple
sinc
squared
function obtained simply
from
finite
sampling) are also a form of aliasing. Although sidelobe contribution at large offsets can be
minimized with the use of an
apodization function
, the tradeoff is a widening
of the response (i.e., a lowering of
the resolution).
1.A.21)
Is the system Y(n)=ln[x(n)] is linear and time invariant
.
(May/June 2006)
Here the Output of the systems depends on present input so the systems is called linear systems.
This systems did not satisfy
the BIBO condition . so this system is said to be linear system.
1.A.22)
Determine the energy of the sequence x(n)=(1/2)
n
n
≥
0.
(May/June 2011)
E=
1/1

1/2 = 2 joules.
1.A.23)
Find final value of x(n) if x(z)=(1+z

1
)/(1

0.25 z

2
)
(May/June 2011)
Final value:
X(Z)=(1+Z

1
)(
1

0.25Z

2
)
X(Z)=Z=0; X(Z)=z=0;
=(1+0)(1

.025)
=(1*0.75)
X(Z)
=0.75
1.A.24)
What is zero padding ? What are its uses.(April/May 2010)
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.
The uses of zero padding are
1) We can get better display of the frequency spectrum.
2)With zero padding the DFT can be used in linear filtering.
1.A.25)
Define sy
mmetric and anti

symmetric signals (May/June 2007)
Even signal
: continuous time signal x(t) is said to be even if it satisfies the condition
x(t)=x(

t) for all values of t.
Odd signal
: he signal x(t) is said to be odd if it satisfies the condition
x(

t)=

x(t) for all t.
In other words even signal is symmetric about the time origin or the vertical axis, but odd
signals are anti

symmetric about the vertical axis.
1.A.26)
What are the properties of ROC?
(May/June 2007)
The ROC does not contain any poles.
Wh
en x(n) is of finite duration then ROC is entire Z

plane except Z=0 or Z=._
If X(Z) is causal,then ROC includes Z=
∞
If X(Z) is anticasual,then ROC includes Z=0.
Part

B
1.B.1)
(i) explain
the concept of energy and power signals. Also checks whether the following signals are energy or
power signal.
(12)
(1)x(n)=(1/3)
n
u(n)
(2)x(n)=sin (p*/4)n
(12)
(APRIAL/MAY 2008)
∞
∞
Energy =
∑ x(n)
2
Power=1/(2N+1)
∑ x(n)
2
n=

∞
n=

∞
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.1.34

1.36)
(ii)briefly explain Quantization.(4)
(APRIAL/MAY 2008)
Quantization
, in mathematics and
digital signal processing
, is the process of mapping a
large set of input values to a smaller set
–
such as
rounding
values to some unit of precision. A
device or
algorithmic function
that performs quantization is called a
quantizer
. The
round

off
error
introduced by quantization is referred to as
quantization error
.
In
analog

to

digital conversion
, the difference between the actual analog value and
quantized digital value is called
quantization error
or
quantization distortion
. This error is either
due to rounding or truncation. The error signal is sometimes conside
red as an additional random
signal called
quantization noise
because of its
stochastic
behaviour. Quantization is involved to
some degree in nearly all digital signal processing, as t
he process of representing a signal in
digital form ordinarily involves rounding. Quantization also forms the core of essentially all
lossy compression
algorithms.
(P.Ra
mesh babu,Digital signal processing,fourth edition,2011.Page no.1.
173

1.
74
)
1.B.2)
check the following system for linearity, time invariance , causality and stability .
(APRIAL/MAY 2008)
(i) y(n) = e^x(n)
(ii)y(n) = x(

n+2).
(16)
Causal :
The
output of the systems depends on past and present values of the input and does not
depend on future values of input.
Static:
The output of the systems depends only on present value of the input and does not depend past
and future value of input.
Time inv
ariance:
The output of the systems will not vary with respect to time period.
Y’(n

k)=y(n,k)
Linearity:
The systems will satisfy the BIBO coditions.
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.1.
53

1.
57
)
1.B.3)
(i)
D
etermine the Z

transform of x(n)=cos wn u(n). (6) (APRIAL/MAY 2008)
∞
X(Z) =
∑ x(n)Z

n

∞
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2
.
16

2
.
17
)
(ii) state and prove the following properties of Z

transforms: (APRIAL/MAY 2008)
(1)time shifting
(2)time reversal
(3)differentiation
(4)scaling in Z domain. (10)
Properties of Z

Transform
The z

transform has a set of properties in parallel w
ith that of the Fourier transform (and Laplace transform).
The difference is that we need to pay special attention to the ROCs. In the following, we always assume
and
Time Shifting
Proof:
Define
, we have
and
The new ROC is the same as the old one except the possible addition/deletion of the origin or infinity as
the shift may change the duration of the signal.
Time Expansion (Scaling)
The discrete signal
cannot be continuously scaled in time as
has to be an integer (for a non

integer
is zero). Therefore
is defined as
Example:
If
is ramp
1
2
3
4
5
6
1
2
3
4
5
6
then the expanded version
is
1
2
3
4
5
6
0.5
1
1.5
2
2.5
3
1
2
3
0
1
0
2
0
3
where
is the integer part of
.
Proof:
The z

transform of such an expanded signal is
Note that the change of the summation index from
to
has no effect as the terms skipped are all
zeros.
Time Reversal
Proof:
where
Scaling in Z

domain
Proof:
In particular, if
, the above becomes
The multiplication by
to
corresponds to a rotation by angle
in the z

plane, i.e., a frequency
shift by
. The rotation is either clockwise (
) or counter clockwise (
) corresponding
to, respectively, either a
left

shift or a right shift in frequency domain. The property is essentially the
same as the frequency shifting property of discrete Fourier transform.
Differentiation in z

Domain
Proof:
i.e.,
(P.Ramesh babu,Digital signal processing,fourth editi
on,2011.Page no.
2
.
8

2
.
15
)
1.B.4)
(i)
Determine
the inverse Z transform of X(z)=(1+3z ^

1)/(1+3z^

1 +2z ^

2) for z >2. (8) (APRIAL/MAY 2008)
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2
.
34

2
.
38
)
(ii) compute the
response of the system y(n)=0.7y(n

1)

0.12y(n

2)+x(n

1)+x(n

2) (APRIAL/MAY 2008)
to input x(n)= n u(n). (8)
Step 1:Find Y(Z) using Z transform
Step 2 :Find X(z) using Z transform
Step 3: find h(n) , first find H(z) using formula H(z
)=Y(z)/X(z)
Step 4:Find the h(n) using inverse Z transform.
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2
.
56

2
.
57
)
1.B.5)
i.Find the response of the system for the input signal x(n)={1,2,2,3}&h(n)={1,0,3,2}
(APRIL/MAY 2007)
Step 1:Find
Y(Z) using Z transform
Step 2 :Find X(z) using Z transform
Step 3: find h(n) , first find H(z) using formula H(z)=Y(z)/X(z)
Step 4:Find the h(n) using inverse Z transform.
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2
.
59

2
.
60
)
1.B.6)
ii.Find the inverse Z transform of the system H(z)= 1/(1

1/2Z

1
)( 1

1/4Z

1
)
(APRIL/MAY 2007)
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2
.
34
)
1.B.7)
Find the convolution and correlation of the for x(n)={0,1

2,3,

4} and
h(n)={0.5,1,2,1,.0.5}
(APRIL/MAY 2008)
Step 1:Find X
(Z) using Z transform
Step 2 :Find
H
(z) using Z transform
Step 3:
Using formula find Y(z)=X(z)H(z)
Step 4: To find convolution y(n) using inverse Z transform
Step 5: to find correlation Y(z)=X(z)X(Z

1
)
St
ep 6: find y(n) usinf Z transform.
1.B.8)
Determine the impulse of the difference equation y(n)+3y(n

1)+2y(n

2)=2x(n)

x(n

1)
(APRIL/MAY 2008)
Step 1:Find Y(Z) using Z transform
Step 2 :Find X(z) using Z transform
Step 3: find h(
n) , first find H(z) using formula H(z)=Y(z)/X(z)
Step 4:Find the h(n) using inverse Z transform.
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2
.
65
)
1.B.9)
A linear shift invariant system has a unit impulse response h (n) =u (

n). find the output if the input is
X(n)=(1/3)
n
u(n)
(May/June 2013)
Step 1:Find
H
(Z) using Z transform
Step 2 :Find X(z) using Z transform
Step 3: find h(n) , first f
ind H(z) using formula
y(z)=
X(z)
H(z)
Step 4:Find the h(n) using inverse Z transform.
(Dr.S.Palani, Digital signal processing,first edition,Ane Books Pvt.,Ltd.,2010.Page.No.4.66)
1.B.10)
Classify, express
and explain the various types of signals.
(May/June 2013)
Classifications of Signals
Along with the classification of signals below, it is also important to understand the Classification
of Systems.
Continuous

Time vs. Discrete

Time
As the names suggest, this classification is determined by whether or not the
time axis
(x

axis) is discrete (countable) or continuous . A continuous

time signal will contain a
value for all real numbers along the time axis. In contrast to this, a discrete

time signal is
often created by using the sampling theorem to sample a conti
nuous signal, so it will only
have values at equally spaced intervals along the time axis.
Analog vs. Digital
The difference between analog and digital is similar to the difference between
continuous time
and discrete

time. In this case, however, the dif
ference is with respect to the value of
the function (y

axis). Analog corresponds to a continuous y

axis, while digital corresponds
to a discrete y

axis. An easy example of a digital signal is a binary sequence, where the
values of the function can only be
one or zero.
Periodic vs. Aperiodic
Periodic signals repeat with some period T, while aperiodic, or nonperiodic, signals do not.
We can define a periodic function through the following mathematical expression, where t
can be any number and T is a positi
ve constant:
f (t) = f (T + t) (2.7)
The fundamental period of our function, f (t), is the smallest value of T that the still
allows the above equation, Equation 2.7, to be true.
Causal vs. Anticausal vs. Noncausal
Causal signals are signals that are zero for all negative time, while anitcausal are signals
that are zero for all positive time. Noncausal signals are signals that have nonzero values
in both positive and negative time.
Even vs Odd
An even signal is any
signal f such that f (t) = f (−t). Even signals can be easily spotted
as they are symmetric around the vertical axis. An odd signal , on the other hand, is a
signal f such that f (t) = −(f (−t)).
Using the definitions of even and odd signals, we can show
that any signal can be
written as a combination of an even and odd signal. That is, every signal has an odd

even
decomposition. To demonstrate this, we have to look no further than a single equation.
f (t) = (f (t) + f (−t)) +(f (t) − f (−t))
By multipl
ying and adding this expression out, it can be shown to be true. Also, it can be
shown that f (t) + f (−t) fulfills the requirement of an even function, while f (t) − f (−t)
fulfills the requirement of an odd function.
Deterministic vs Random
A
deterministic signal is a signal in which each value of the signal is fixed and can be
determined by a mathematical expression, rule, or table. Because of this the future values
(P.Ramesh babu
, Digital
signal
processing, fourth
edition
, 2011.Page
no.1.22

1.34)
1.B.11)
Obtain the Z transform and ROC for
i).x(n)=

b
n
u(

n

1)
ii). X(n)=a
n
u(n)
Z transform of the sequence
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.4

2.5
)
1.B.12)
Find x(n) if the given
X(Z)=
(1
+3
Z

1
)
/
( 1
+3Z

1
+2Z

2
)
for ROC Z
2
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.34

2.35
)
1.B.13)
Explain the following with suitable examples
(May/June 2012)
I. Linear systems ii.
Time invari
ant systems
iii.Causal systems
Memoryless
systems
A system is memoryless if the output y[n] depends only on x[n] at the
same n. For example, y[n] = (x[n])2 is
memoryless, but the ideal delay
Linear systems
A system is linear if the principle of superposition applies. Thus if y1[n]
is the
response of the system to the
inpu
t x1[n], and y2[n] the response
to x2[n], then linearity implies
Time

invariant systems
A system is time invariant if a time shift or delay of the input sequence
causes a corresponding shift in
the output sequence. That
is, if y[n] is the response to x[n],
then y[n

n0] is the response to x[n

n0].
For example, the accumulator system
is time invariant, but the compressor system
for M a positive integer
(which selects every Mth sample from a sequence) is not.
Causality
A system is causal if the output at n depends only on the input at n
and earlier inputs. For example, the
backward difference system
is causal, but the forward difference system
Stability
A system is stable if every bounded input sequence produces a bou
nded
output sequence:
x[n] is an example of an unbounded system, since its response to the unit
which has no upper bound.
Linear time

invariant systems
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.52

2.63
)
1.B.14)
Derive
the relationship between input and output of LTI system in Z transform.
(May/June 2012)
Step 1:Find Y(Z) using Z transform
Step 2 :Find X(z) using Z transform
Step 3: find h(n) , first find H(z) using formula H(z)=Y(z)/X(z)
Step 4:Find the h(n) using
inverse Z transform.
(Dr.S.Palani, Digital signal processing,first edition,Ane Books Pvt.,Ltd.,2010.Page.No.4.
2

4.3
)
1.B.15)
Test the stability and causality
of the following system (Nov/Dec 2009)
i.Y(n)=Cosx(n)
ii.Y(n)=X(

n

2)
Causal :
The output of the syste
ms depends on past and present values of the input and does not
depend on future values of input.
Stable:
for any value of n the output of the systems will produce the constant output with respect to the
input.
(P.Ramesh babu,Digital signal
processing,fourth edition,2011.Page no.
1.53

1.63
)
1.B.16)
Find the one sided Z transform of discrete sequences generated by the mathematically sampling of the following
continuous time function
(Nov/Dec 2009)
i.X(t)=sinwt
ii.x(t)=coswt
To find Z transform of the sequence
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.16

2.17
)
1.B.17)
Define correlation and bring out the difference between convolution and correlation. (May/June 2006)
A
convolution
is an integral that
expresses the amount of overlap of one function
as it is shifted over another function
.
You can use
correlation
to compare the
similarity
of
two
sets of
data. Correlation computes a measure
of
similarity
of
two
input signals
as they are
shifted by one
another
.
The
correlation result reaches
a
maximum
at the
time when the two signals match best
The difference between convolution and correlation is that
convolution is a filtering operation
and
correlation is a
measure of relatedness of two signals
Yo
u can use convolution to compute the response of a linear system to an input signal. Convolution is also the time

domain
equivalent of filtering in the frequency domain
.
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.11,2.15
)
1.B.18)
Determine the Z transform of the signal x(n)=a
n
u(n)

b
n
u(

n

1) and plot the ROC.
(May/June 2006)
To find Z transform of the sequence
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.5

2.6
)
1.B.19)
Find the system function of the system
described by y(n
)
={0.5/[(1

.75z

1
)(1

Z

1
)]}
x(n)
(May/June 2006)
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.38
)
1.B.20)
Find the system function of the system described by y(n

0.75y(n

1)

+0.125y(n

2)=x(n)

x(n

1)
and plot the poles
and zeros of H(z).
(May/June 2006)
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.63

2.65
)
1.B.21)
State and prove sampling theorem. (May/June 2011)
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
1.168

1
.172
)
1.B.22)
Find the Z transform of
i.x(n)=a
n
u(n)
ii.y(n)=a
n
u(n)

b
n
u(

n

1)
iii.w(n)=a
n
cos
ω
onu(n)
Wrte the ROC for all the above cases.
(May/June 2011)
To find Z transform of the sequence
(P.Ramesh babu,Digital signal processing,fourth edition,2011.Page no.
2.4

2.2.6
)
1.B.23)
Find the inverse Z transform of x(z)=z
2
+z/(z

1/3)
3
(z

1/4) for the ROC z
>
1/2
(May/June 2011)
(P.Ramesh babu,Digital signal
processing,fourth edition,2011.Page no.
2.50

2.51
)
UNIT II
FREQUENCY TRANSFORMATIONS
9
Part

A
2.A.1)
Define DFT pair. (APRIAL/MAY 2008)
2.A.2)
Draw the basic butterfly structure for DITFFT and DIF FFT Algorithms. (APRIAL/MAY 2008)
2.A.3)
Draw the
Radix 4 FFT DIF Butterfly diagram.(MAY/JUNE 2007)
2.A.4)
Differentiate DFT and DTFT(Nov/Dec 2010)
2.A.5)
Find DFT for {1,0,0,1}(APRIL/MAY 2008)
2.A.6)
How DFT is computed using FFT algorithm.
(May/June 2013)
2.A.7)
State the advantages of FFT over DFTs
(May/June 2012)
2.A.8)
Find the DFT f
or x(n)=a
n
u(n)
.
(NOV/DEC 2011)
2.A.9)
Calculate the number of multiplications needed for the DFT computation of N=32 point sequence using FFT
algorithm.(NOV/DEC 2011)
2.A.10)
List any four properties of DFT.(Nov/dec 2009)
2.A.11)
Define DFT pair.(May /June 2006)
2.A.12)
Differentiate between DIT and DIF FFT algorithm.
(May /June 2006)
2.A.13)
Give relation between Z

domain and frequency domain(May /June 2012)
2.A.14)
Calculate the DFT of the sequence x(n)={1,1,

2,2}(Nov/Dec 2009)
2.A.15)
Mention the applications of FFT algorithm.(Nov/Dec 2011)
2.A.16)
D
istinguish circular shifting and shifting in DFT (May/June 2011)
2.A.17)
Find the DFT OF y(n)=
δ
(n)

δ
(n

n
0
)+
δ
(n
+n0
)
?
(May/June 2012)
2.A.18)
What is Radix

2 FFT
. (May/June 2012)
2.A.19)
Draw the butterfly structure of DIT algorithm.
(May/June 2007)
Part

B
2.B.1)
D
erive and draw the flow graph of the Radix

2 DIF FFT algorithm for the computation of 8

ponit DFT. (10)
(APRIAL/MAY 2008)
2.B.2)
what are differences and similarities between DIT and DIF FFT algorithm? (6) (APRIAL/MAY 2008)
2.B.3)
compute the 8

point DFT of the
sequence x(n) ={1,2,3,4,4,3,2,1} (10) (APRIAL/MAY 2008)
2.B.4)
illustrate the concept of circular convolution property of DFT. (6) (APRIAL/MAY 2008)
2.B.5)
Compare the computational complexity of direct computation of DFT Vs FFT algorithm (May/June 2013)
2.B.6)
Write a de
tailed technical note on the use of FFT algorithms in linear filtering and correlation with an example.
(May/June 2013)
2.B.7)
Compute the DFT of the signal x(n)=Cosn
π
/2 for N=4 using DIT

FFT algorithm.Draw the flow diagram. Show all the
Computati
ons
. (May/June 2013)
2.B.8)
Compute the DFT for the sequence {1,2,0,0,0,2,1,1} using Radix

2 DIF

FFT Algorithm.(16) (APRIL/MAY 2007)
2.B.9)
Explain the properties of DFT
1.Time shifting
2.Convolution
3.Time reversal
4.Complex conjugate poerty.
(Nov/Dec 2011)
2.B.10)
Obtain
the circular convolution of the sequence
if X1(n)={1,0,1,0} and x2(n)={1,1,1,1}
(Nov/Dec 2011)
2.B.11)
Satating the relevant equations obtain the eight point DIT

FFT algorithms (May/June 2012)
2.B.12)
S
tate
and prove the time shifting property of DFT .
(May/June 2012)
2.B.13)
A
n 8 point sequence is given by x(n)={2,2,2,2,1,1,1,1} compute 8 point DFT of x(n) by
i.Radix

2 DIT FFT
ii.RADIX DIF

FFT
(Nov/Dec 2009)
2.B.14)
Using DFT

IDFT method , perform circular convolution of the sequences x(n)={1,2,0,1} and h(n)={2,2,1,1}
2.B.15)
State and
prove the circular convolution property of DFT.
2.B.16)
Find DFT of X(k) given below
X(k)={2,1,3,0,4}
2.B.17)
State and prove the multiplication in time property of DFT.
(May/June 2012)
2.B.18)
Using FFT algorithm find the response of an LSI system with impulse response h(n)={1,2
,

1} for the input x(n)={2,

4}
(April/May 2010)
2.B.19)
Derive the frequency response of linear phase FIR filter of order N(odd) with symmetric condition.
(April/May 2010)
2.B.20)
Find the linear convolution of x(n)={1,2,3} with h(n)={2,4} using DFT and IDFT.(Nov/Dec 2010
)
UNIT III
IIR FILTER DESIGN
9
Part

A
3.A.1)
State the condition for linear phase in FIR filters for symmetric and anti

symmetric response.
(APRIAL/MAY 2008)
3.A.2)
What is called pre warping? (APRIL/MAY 2008)
3.A.3)
Compare IIR and FIR digital filters.
(Nov/Dec 2010)
3.A.4)
What are the various design methods available for IIR filters (May/June 2013)
3.A.5)
State any two properties of chebychev filters.
(May/June 2012)
3.A.6)
What is the principle of impulse invariance method.(NOV/DEC 2011)
3.A.7)
Compare analog and digital filter
(Nov/Dec 2009)
3.A.8)
Sketch the mapping of s

plane and z

plane in bilinear transformation (Nov/Dec 2009)
3.A.9)
Find the transfer function for normalized Butterworth filter of order 1 by determining the pole values
.
(May/June 2006)
3.A.10)
Why we go for analog approximati
on to design a digital filter?
(May/June 2012)
3.A.11)
Why impulse invariant method is not preferred in the design of IIR filter other than low pas filter?
(Nov/Dec 2011)
3.A.12)
What is the main objective of impulse
invariant
transformation. (May/June 2011)
3.A.13)
Mention two i
mportant features of Butterworth filters. (May/June 2012)
3.A.14)
What is wraping effect? What is its effect on magnitude and phase response?
3.A.15)
What is dead band?
Part

B
3.B.1)
D
esign a digital Butterworth filter satisfying the following constraints with T=1sec.using Bilinear transformation.
(12)
(APRIL/MAY 2008)
0.707=< H(ej
ω
)=< 1 for 0 = < p /2
H(ej
ω
)=<0.2 for 3p/4
3.B.2)
Design and realize a Butterworth digital filter for the following specifications:
(APRIL/MAY 2008)
Pass band gain required:

2dB
Frequency upto which pass

banbd gain must remain more or less steady f1=300 Hz.
Amount of attenuatui
on required:

30 dB
Frequency from which the attenuation must start f2=650 Hz.
3.B.3)
Derive the impulse invariant transformation and obtain the frequency relationship
3.B.4)
Design a chebychev analog
low passfilter that have

3 dB cutoff frequency of 100 rad/sec and
a stop band
attenuation 25 dB or greater for all radian frequencies past 250 rad/sec. plot 20logH(j
Ω
) for your filter and show that
you satisfy the requirements at the critical frequencies.
(May/June 2013)
3.B.5)
Convert the analog filter with system functio
n Ha(S) into digital filter using bilinear transformation
Ha(S)=S+03/(S

0.3)
2
+16 (May/June 2013)
(May/June 2012)
3.B.6)
Obtain direct form

I canonical and parallel form realization structures for the system given by the difference
equation
y(n)=

0.1y
(n

1)+0.72y(n

2)+0.7x”(n)

.252x(n

2)
(May/June 2012)
3.B.7)
Enumerate the various steps involved in the design of low pass digital Butterworth
IIR
filter
(May/June 201
2
)
3.B.8)
Explain the concept of IIR filter design using approximation derivative.
(Nov/Dec
2011)
3.B.9)
Compare Bilinear transformation and impulse invariant method.
(Nov/Dec 2011)
3.B.10)
Compare the impulse invariant and bilinear transformation (Nov/Dec 2009)
3.B.11)
Apply the bilinear transformation for the following
(Nov/Dec 2009)
i.H
a
(S)=2/(S+1)(S+2) with T=1
and find H(z)
ii.H
a
(S)=2S/(S2+0.2S+1) WITH T=1 and find out H(z)
3.B.12)
Explain the design procedure for lowpass digital butterworth IIR filter
(Nov/Dec 2009)
3.B.13)
Find the output response of the system given the input signal x(n)={1,

2,3,

2} and impulse response
h(n)={2,

3,4}
(May/june 2006)
3.B.14)
Obtain the direct I, canonic form and parallel form realization structures for the system given by the difference
equation.
Y(n)=

0.1y(n

1)+0.72y(n

2)+0.7x(n)

0.252x(n

2)
(May/June 2006)
3.B.15)
Obtain a parallel realization for
the following H(z)=(8z
3

4z
2
+11z

2)/(z

1/4)(z
2

z+1/2) use direct form II realization
for each section.
(May/June 20
11
)
3.B.16)
Explain the different windows used in FIR filter design.
(May/June 2011)
3.B.17)
A Butterworth low pass has to meet the following speci
fications :
Pass band gain of

1db at 4rad/
sec, stop
band attenuation greater than20db at 8 rad/sec. determine the transfer
function of the lowest order to meet the above specifications.
(May/June 2012)
3.B.18)
Explain the mapping between S plane and Z plane is done using
bilinear
transformation method.
(May/June 2012)
3.B.19)
Given the analog transfer function H(s)=[(S+0.1)/(S+0.1)
2
+9]
find H(z) using
(Nov/De 2010)
i. Impulse invariant method
ii.Bilinear transform.
3.B.20)
Determine the pole zero for the system described by difference equation
y(n)

3/4y(n

1)+1/8 y(n

2)=x(n)

x(n

1)
(May/June 2007)
3.B.21)
Apply bilinear transformation to H(s)= 2/(S+1)(s+2) with T=1 sec and find H(z). (Nov/Dec 2011)
3.B.22)
Obtain the Direct form

I, cas
cade and parallel form realization for the following system.
Y(n) =

0.2y(n

1) + 0.5 y(n

2) + 6 x(n) + 4.5 x(n

1) + 0.8 x(n

2).
UNIT IV
FIR FILTER DESIGN
9
Part

A
4.A.1)
Write procedure for designing FIR filters
using windows. (APRIL/MAY 2008)
4.A.2)
What are the features of FIR filters?(May/June 2013)
4.A.3)
Obtain cascade for realization of system function 1+5/2Z

1
+2 Z

2
+2 Z

3
(May/June 2013)
4.A.4)
State Condition for FIR filters have linear phase.
(may/june 2012)
4.A.5)
Write the steps
involved in FIR filter design (Nov/Dec 2009)
4.A.6)
Write the expression for Kaiser
Window
function (Nov/Dec 2009)
4.A.7)
State the advantages ad disadvantages of FIR filter over II filter (May/June 2006)
4.A.8)
List out the different forms of structural realizations available
. (May/June 2006)
4.A.9)
What does frequency wraping? (May/June 2006)
4.A.10)
Write the equation o
f Bartlett and hamming window. (May/June 2007)
4.A.11)
Define symmetric and
ant symmetric
FIR Filters
4.A.12)
What is overflow limit cycle oscillations.(April/May 2010)
4.A.13)
Define limit cy
cle.(May/June 2011)
4.A.14)
Define power spectral density?
(Nov/Dec 20
11
)
4.A.15)
How overflow limit cycles can be eliminated.
(Nov/Dec 2009)
4.A.16)
How will you avoid limit cycle oscillations due to overflow in addition?
(
May/June 2006)
4.A.17)
Define gibbs phenomenon.(Nov/Dec 2011)
4.A.18)
Ho
w rounding is preferred over truncation in realizing digital filters.
(
May/June 2012
)
Part

B
4.B.1)
obtain the cascade and parallel realization of the system described by
(APRIAL/MAY 2008)
y(n)=

0.1y(n

1)+0.2y(n

2)+3x(n)+3.6x(n

1)+0.6x(n

2). (10)
4.B.2)
discuss
about any three window function used in the design of FIR filters. (6)
(APRIAL/MAY 2008)
4.B.3)
Design a low pass FIR
filter for
the following specifications using rectangular window.
(May/June 2013)
Frequency of pass band edge: 3000 Hz ; gain in pass band:

2d
B
Frequency from which stop band begins: 3000 Hz
Gain in stop band :

50 dB ; sampling frequency 13 kHz
4.B.4)
Write a detailed technical note on the frequency sampling techniques.
(May/June 2013)
4.B.5)
Design an FIR linear filter approximating the ideal frequency re
sponse
(May/June 2013)
H
d
(ei
ω
)
= 1
for


≤
π
/6
=0
for
π
/6
≤


<π/3
=1
for
π
/
≤


< π
Determine the coefficients of a 9 filter with Hanning window.
4.B.6)
Explain in detail about frequency sampling method of designing an FIR filter (May/June 2013)
4.B.7)
State
the issues in designing FIR filter using window method.
(May/June 2012
)
4.B.8)
Explain the type

1 and type

2 design of FIR filter using frequency sampling technique.
(May/June 2013)
4.B.9)
Explain in detail the FIR filter design using windowing method.(NOV/DEC 2011)
4.B.10)
obtain the direct form structure of FIR filter for 1+2Z

1

3Z

2

4Z

3
+5Z

4
(NOV/DEC 2011)
4.B.11)
Explain the design of FIR filter using Kaiser windows.
(NOV/DEC 2011)
4.B.12)
Explain anti symmetric
and symmetric
linear phase FIR filters.
(NOV/DEC 2011)
4.B.13)
Design a lowpass f
ilter using rectangular window by taking 9 samples of
ω
(n) and with cutoff frequency of 1.2
rad/sec.
(Nov/Dec 2009)
4.B.14)
Design a linear phase lowpass filter with cutoff frequency of
π
/2 rad/sec using frequency sampling techniques
(Take N=17)
(Nov
/Dec 2009)
4.B.15)
Obtain the transversal structure and linear phase realization structure for a filter by (n)={0.5,2.88,3.404,2.88,0.5}
(May/June 2006)
4.B.16)
Design a digital filter with
H
d
(e
j
ω
)
=
1
for
k=0,1,2,3,4
0.4
for
k=5
0
for
k=6,7
using Hamming window. Draw the frequency response.
(May/June 2006)
4.B.17)
Design an FIR filter satisfying the following specifications
α
p
≤
0.1db
;
α
p
≥
0
.1db
ω
p
= 20 rad/sec ω
s
=30 rad/sec
ω
sf
=100 rad/sec. (Nov/Dec 2011)
4.B.18)
Determine the frequency response of FIR filter defined by y(n)=0.25x(n)+x(n

1)+0.25x(n

2) .
(Nov/Dec 2011)
4.B.19)
Describe the characteristics of the various windows used for design of FIR filters. (May/June 2012 )
4.B.20)
Explain frequency sampling method of designing F
IR filter. Derive the frequency response and show that it has
linear phase
(Nov/Dec 2010)
4.B.21)
Find the system function and the impulse response of the system described by the difference equation
y(n)=x(n)+2x(n

1)

4x(n

2)+x(n

3)
(May/June 2007)
4.B.22)
Design a filter with
(May/June 2007)
Hd(e

j
ω
)
=e

j3
ω
for

π/4 ≤ω≤ π/4
=0
for π/4 ≤ω≤ π
Using a Hanning window with N=7.
4.B.23)
Consider the transfer function H(z)=H1(z) where H1(z)=1/(1

a
1
z

1
) and H2(z)=(1

a
2
Z

1
) assume a
1
=0.5 and a
2
=06
and find the output round off noise power.
(Dec 2009)
4.B.24)
Derive the quantization input noise power and determine the signal to noise ratio.
(
NOV / DEC 2011
)
4.B.25)
Limit cycle oscillations with an example.
(
NOV / DEC 2011
)
4.B.26)
Explain shortly
about signal scaling.
(April/May 2010)
UNIT V
APPLICATIONS
9
Part

A
5.A.1)
What are the factors that influence the selection of DSPs?
(May/June 2012)
5.A.2)
What are the different buses of TMS 32
0C54x processor and list their
functions
.(May/June 2012)
5.A.3)
What are the shift instructions in TMS 320 C54x?
(April/May 2010)
5.A.4)
List the on

chip peripherals of C54x processor.
(Nov/Dec 2010)
5.A.5)
List the various registers used with ARAU.
(April/May 2010)
5.A.6)
What are
the advantages and disadvantages of C54x architecture?
(Nov/Dec 2010)
5.A.7)
What is meant by vocoder
.(Nov/Dec 2011)
5.A.8)
Define quantization noise.
(Nov/Dec 2011)
5.A.9)
Distinguish Von Neumann and Harvard architectures
.(Nov/Dec 2011)
5.A.10)
Write about the accumulators in TMS320
C54X
.
(May/June 2011)
5.A.11)
Mention two advantages of Harvard
architectures
.(May/June 2012)
5.A.12)
What is stack addressing?
(May/June 2012)
5.A.13)
What are the different stages of pipelining?
(April/May 2010)
5.A.14)
What are the features of TMS320C54 dsp processor.
(April/May 2010)
5.A.15)
What are the different buses of TMS320C5X
.(Nov/Dec 2010)
5.A.16)
Mention any two processors that support fixed point operation in TMS320c5X series.
(Nov/Dec 2011)
5.A.17)
Define multirate signal processing?
Part

B
5.B.1)
Explain briefly about
(May/June 2012)
i.
Multirate signal processing
ii.
Vocoder
5.B.2)
Explain the application of DSP in Speech processing?
(Nov/Dec 2011)
5.B.3)
What is a vocoder? Explain with a block diagram?
(May/June 2012)
5.B.4)
Describe
the function of on chip peripherals of TMS 320 C 54 DSP processor.
5.B.5)
What
are the different buses of TMS 320 C 54 and their functions? (April/May 2008)
5.B.6)
Discuss in detail the various quantization
effects
in the design of digital filters. (April/May 2008)
5.B.7)
Explain in detail abo
ut the applications of PDSP
(Nov/Dec 2011)
5.B.8)
Explain bri
efly:
(Nov/Dec 2011)
i.
Von Neumann architecture
ii.
Harvard architecture
iii.
VLIW architecture
5.B.9)
Explain in detail about
(May/June 2012)
i.
MAC unit
ii.
Pipelining
5.B.10)
Draw and explain the architectu
re of TMS 320C54x processor
(Nov/Dec 2010)
5.B.11)
Explain in detail about the Add
r
essing modes of TMS 320C54X
(Nov/Dec 2010)
5.B.12)
Draw and explain the arch
itecture of TMS320C50 processor
(Nov/Dec 2010)
5.B.13)
Explain in detail about MAC unit and pipelin
ing with reference to TMS320C50
.
(April/May 2010)
5.B.14)
Write an assembly language program for linear
convolution using TMS320C50.
(Nov/Dec 2011)
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