Digital Signal Processing Fundamentals

10/7/2009
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Chapt 01
Signal and Systems
Digital Signal Processing
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Signal
• Definition: It is the function that describes the variation of a
physical variable with respect to independent variable( time,
space etc.) in a physical process.
Mathematically,
S=f(x)
Where s is signal /function and x independent variable
Examples:
1) Change in temperature sensed by the sensor placed in boiler.
T=f(t)
Here, time t is independent variable.
t(msec) 1 2 3 4 5
Temp(0C) 29 31 35 45 45

10/7/2009
Contd..
2) The digital data to be sent over a transmission channel
t(msec) 0 1 2 3 4 5
Digital
data
0 1 1 0 1 0
3) Pixel intensities in an image
2 5 7 4 12 1 6 2 4 9
10 7 4 4 4 2 9 3 1 2
8 11 14 4 1 9 7 5 11 12
1 2 3 2 3 6 8 13 4 5
6 7 8 15 0 10 7 9 6 5
6 6 7 8 4 3 11 8 9 0
7 8 5 6 4 5 13 4 4 4
7 8 9 8 9 8 7 5 7 7
7 5 4 3 3 4 5 6 14 1
14 12 12 1 5 7 2 1 9 9
Image, I=f(x,y)
In this example intensity in image is the
function / or signal which is dependent on
independent variable spatial coordinate x
and y.

10/7/2009
Contd..
4) The voltage across capacitor
RCt
eV /−
=
5) share value fluctuation in stock market during year ( month wise)
Month Jan Feb Mar Apr May Jun Jul Aug Sep
Share
value(
Rs)
200 214 214 245 198 200 210 200 234

10/7/2009
Common Discrete time signals
It is general practice to represent complex signals by using common
or standards signals.In descrete time signals independent variable
,t, can be equivalently seen as nT where T is samling period and
n is sample number. Thuis , n becomes independent variable in
descrete time signal. Some of the common signals are given here.
a)Unit impulse function
0
1
;0
;0
≠
=
n
n=)(nδ
b) Unit step function u(n):
0≥n
0<n
It is defined as u(n) = 1
0
0≥n
0<n
c)Unit ramp function ( r(n)):
r(n) = n
0

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Contd..
d) Exponential function:
It is expressed as as
an
Aenf =)(
When a ia negative when a is positive
)sin()( φω += nanf
)cos()( φω += nanf
fπω 2=
φ
e) Sinusoidal function
sine function ,
cosine function
is the angular frequency of function
where A is the amplitude of function
is the phase of the function.
f) Sgn function
It is defined as
1, n>0
Sgn(n)= 0,
n=0
-1,
n<0
g) Decaying/growing sinusoidal

10/7/2009
Signal Representation
Discrete time signals can be represented as follows
Graphical Representation
0
x(n)
Sequence Representation
{ }3021)( =nx ;......2)1(;1)0( == xx
{ }25212310)( −−
↑
nx
....2)2(,1)1(,2)0(,3)1(,1)2(,0)3(,0)4( ==−==−=−=−=− xxxxxxx
Functional representation





=
0
2/
2
)( n
n
nx
elsewhere
n
n
85
40
≤≤
≤≤



=
n
n
nx
3
)(
2
evenn
oddn
=
=

10/7/2009
Problems
Sketch following signals





=
0
2/
2
)( n
n
nx
elsewhere
n
n
85
40
≤≤
≤≤



=
n
n
nx
3
)(
2
evenn
oddn
=
=





−
−=
n
nnx
5
32
0
)(
4
42
2
>
≤≤
≤
n
n
n
1
2
3

10/7/2009
Signal Operations
• Addition/Subtraction – corresponding samples from both signals would be
added, subtracted
• Amplitude Scaling- each sample of the signal would be scaled by scaling
factor
• Delaying
• Advancing
•Time reversing
•Rate Changing

10/7/2009
x(n)
Delaying- Advancing
Original signal
x(n)
1
2
3
4
n0
Delaying
1
2
3
4
n0
x(n)
x(n-1)
x(n)
1
2
3
4
n0
x(n+1)
Advancing

10/7/2009
Time Reversing
Original signal x(n)
1
2
3
4
n0
TR & Delaying
x(n)
x(-n+1)
x(-n-1)TR & Advancing
Time Reversed x(n)
1
2
3
4
n0
x(-n)
x(n)
1
2
3
4
n0
x(n)
1
2
3
4
n0

10/7/2009
Rate Changing
Rate changing – Multirate Signal Processing
– changing the sampling rate of signal
Up sampling
Down sampling
Original signal
x(n)
1
2
3
4
n0
x(n)
x(n)
1
2
3
4
n0
x(n/2)
x(n)
1
3
n0
x(2n)

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Signal Classification
• Periodic and Aperiodic Signals
• Even and Odd signals
• Energy and Power signal
• Deterministic and stochastic signal

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Periodic and Aperiodic Signals
•A signal is periodic if it satisfies periodicity property x(n+kN)=x(n)
where N is a fundamental period and k is any integer
•If signal is periodic with fundamental period N, it sis also periodic for any
integer multiple of N
tTp
t
A signal which doesn't satisfy periodicity property is called aperiodic signal

10/7/2009
Calculation of Periodicity
• Sum of two or more periodic signals is also a periodic signal
• If xa[n] and xb[n] are two periodic singals with funfadmental
period Na and Nb respectively, then singal y[n]=xa[n]+xb[n] is a
periodic singal with fundamental period N given by
N= Na*Nb/(GCD(Na,Nb)
where GCD greatest common divisor of Na & Nb

10/7/2009
Even and Odd signals
• Signal which satisfies
x(n)=x(-n) for all values for n
is called as even signal
n
0
• Signal which satisfies
x(n)=-x(-n) for all values for n
is called as odd signal n0
Any signal can be expressed in terms
of odd and even signal
)()()( nxnxnx oddeven +=
2
)()( nxnx
xeven
−+
=
2
)()( nxnx
xodd
−−
=where and

10/7/2009
Energy and Power Signal
•The signal x(n) is said to be an energy signal if its energy, as
calculated by following equation , is finite and non-zero.
∑−=
∞→
=
N
Nn
N
nxEnergy
2
)(lim
•The signal x(n) is said to be an power signal if its power, as
calculated by following equation , is finite and non-zero.
∑−=
∞→ +
=
N
Nn
N
nx
N
Power
2
)(
12
1
lim

10/7/2009
Problem(1)
• identify signals if it is power or energy signal
a) u(n) b) 0.5
n
u(n)
∑∑
∞∞
∞=−
∞===
0
2
2
1)(
n
nxE
2
1
1
1
2
2
2
1
lim
1
12
1
lim
1
12
1
lim
)(
12
1
lim
=
+
+
=
+
+
=
+
=
+
=
∞→
∞→
−=
∞→
−=
∞→
∑
∑
N
N
N
N
N
Nn
N
N
Nn
N
N
N
N
nx
N
P
a)
∑=
+
−
−
=
2
1
21
1
1N
Nn
NN
n
a
aa
a 1≠a
We know ,
where
Power signal

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Solution
3
4
75.0
1
25.01
01
25.05.0)(
0
2
0
2
2
==
−
−
=
=−== ∑∑∑
∞∞∞
∞=−n
nxE
Energy signal
∑=
+
−
−
=
2
1
21
1
1N
Nn
NN
n
a
aa
a 1≠a
We know ,
where

10/7/2009
Quiz
1. Sketch
∑
∞
=
−=
0
)()(
k
knnx δ
2. Can step sequence be represented in terms of impulses? If yes how?
3. Signal is to be down sampled by a factor of 2.5. Is it possible? If yes ,how?
4. Verify odd-even signal equation for the signal x(n)={1 2 3 4}
5 Given DTS as { }23221231)( −−−=
↑
nx
Sketch a) x(n-3) , b)x(3-n), c) x(-n-1) , d)x(2n) u(2-n) f)x(n-2)∂(n+2)

10/7/2009
System
•Definition
System is device that follows unique relationship between
excitation and response( input and output)
A discrete-time system is essentially an algorithm for
converting one sequence (called the input) into another sequence
(called the output)
∫
x(n) y(n)
y(n)=f[x(n)]
f[.] denotes the specific
system

10/7/2009
Examples of system
• Differentiator y(n)= x(n)-x(n-1)
• Square law modulator y(n)= [ x(n)]
2
• Interpolator/ upsampler



=
0
)(
)(
m
nx
ny
otherwise
fMmultipleson =

10/7/2009
System Representation
• Impulse response (in time domain)
h[n]={1 -1} , h[n]= { 1 0.8 0.54 0.24 0.006}
•Relation between input and output
as seen in system examples
•Difference equation
y[n]= x[n]- x[n-1] , y[n]= y[n-1] + ax[n] +bx[n-2]
•Transfer function( in z-domain)
H(z)= 1/(z+1)

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System Classification
• Continuous /discrete time systems
• Time variant/invariant systems
• Memory less/memory systems( Static/dynamic system)
• Causal/anti-causal/non-causal system
• Linear/non-linear systems
• Lumped/distributed-parameter systems
• Stable/unstable systems
• Invertible and non-invertible systems

10/7/2009
Continuous /discrete time systems
Is there any application
which exist only in digital
and not in analog ?
If the system process CTS, then system is said to be continuous
time system.
e.g. R-C circuit, transmitting antenna
If the system process DTS, then system is said to be discrete time
system.
e.g. Digital adder

10/7/2009
Lumped/distributed-parameter systems
• If the component used in system has identical
values of physical parameters( current, voltages etc)
throughout its area and can be considered as a
single point (node) in the system , it is called as
lumped parameter system
• e.g normal components like resistor, capacitor in
low frequency applications etc
V1
V1
V1
V1
• If the component used in system has different
values of physical parameters ( current, voltages etc)
throughout its area and cannot be considered as a
single point (node) in the system , it is called as
lumped parameter system
•E.g. transmission lines, microwave tubes which
normally used in high frequency
V1
V2
V3
V4

10/7/2009
Memory less/memory systems( Static/dynamic
system)
A system for which the output depends only on a present input
and thus , not requires memory, is called as memory less (static )
system.
e.g. y(n)= a*x(n) , y(n)=x
2
(n)
A system for which the output depends on past and/or future
values of the input in addition to the present values of input ,
hence it needs memory, delay elements, is called as
memory system
e.g. y(n)=x(n)+x(n+2) , y(n)=x(n-2)

10/7/2009
Causal/anti-causal/non-causal system
A system for which the output at any instatnt depends only on the
past or present values of the input( not on future samples) is
called as causal system
y(n)= n*x(n) , y(n)=x(n) +x(n-1)
A system for which the output at any instatnt depends also on
future values of the iput , is called as non-causal system
e.g. y(n)=x(n2
) , y(n)=x(-n) , y(n)=x(n)+x(n+1)

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Stable/unstable systems
The system is said to be stable if any bounded input signal
results in bounded output signal
bounded signals u(n) , e
-an
The system is said to be unstable if the system gives
unbounded output signal in response to bounded input
signal
unbounded signals r(n) , n*u(n)

10/7/2009
Invertible/non-invertible systems
The system whose output can be used to determine
input uniquely and exactly, is called as invertible system.
e.g. y=2x
but y=x2 is not an invertible system as it would give two
possible inputs ( +ve and –ve)
Hence , system defined by y=x2 is non-invertible system

10/7/2009
Linear /non-linear Systems
If system holds superposition property , it is called as linear system
if system violates superposition property, it is called as nonlinear system
H
X1(n)
X2(n)
+
a
b
Y(n)
H
H
Superposition property of a system with any two inputs x1(n) and x2(n) is
defined as
H{a x1(n) +b x2 (n) }= a H{x1(n)} +b H{x2(n) }
=a y1(n) +b y2(n)
X1(n)
X2(n)
a
b
+ Y(n)

10/7/2009
Problems
1) y(n)=nx(n)
y1(n)=nx1(n)
y2(n)= nx2(n)
ay1(n)+by2(n)= nax1(n)+ nbx2(n)
=n[ax1(n)+bx2(n)] …..A
H[ax1(n)+bx2(n)]
= n[ax1(n)+bx2(n)] …..B
A=B Linear system
2) y(n)=x2
(n)
y1(n)=x12(n)
y2(n)=x22(n)
ay1(n)+by2(n)=ax12(n)+bx22(n) ….A
H[ax1(n)+bx2(n)]= [ax1(n)+bx2(n)]2
= a2x12(n)+b2x22(n)+2abx1(n)x2(n)
………B
A != B
Non-linear system

10/7/2009
Time Variant/Invariant Systems
A system is said to be time-invariant if its input-output
relationship does not change with time
A system is said to be time-variant if its input-output
relationship changes with time
In other words if a time shift or delay at the input
produces identical time shift at the output , then system
is said to be time invariant system.
i.e. H{x(n-a)}=y(n-a)
Other wise it is said to be time variant

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Contd..
HShift by a
H Shift by a
X(n) y(n-a)
X(n) y(n-a)
Output of shifted input , y(n,k)
Shifted output, y(n-k)

10/7/2009
Problems
1) Y(n)=x(n)+ x(n-1)
o/p of delayed input by k
i.e. y(n,k)=x(n-k)+x(n-1-k)
Delayed o/p by k
Y(n-k)= x(n-k)+x(n-k-1)
Y(n,k)=y(n-k)
System is time-invariant
2)Y(n)=nx(n)
i.e. y(n,k)=nx(n-k)
Delayed o/p by k
Y(n-k)= (n-k)x(n-k)
Y(n,k) !=y(n-k)
System is time-variant
3) Y(n)=x(-n)
i.e. y(n,k)=x(-n-k) as x(n) x(n-k) x(-n-k)
Delayed o/p by k
Y(n-k)= x(-(n-k))=x(-n+k)
Y(n,k)=y(n-k) System is time-variant

10/7/2009
DT System with Differential equations
System relationship between input and output
Output is a function of input as well as outputs and can be well described
by differential equation
∑ ∑= =
−+−−=
N
k
M
k
kk knxbknyany
1 0
)()()(
where {ak} and {bk} are constant parameters that specify the
system and are independent of x(n) and y(n)

10/7/2009
IIR and FIR systems
IIR is recursive structure
FIR is non-recursive structure

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Convolution
Definition: It is the tool or operation to determine the response of
the LTI system
Convolution between two DT signals x(n) and h(n) is expressed
as
∑
∑
∞
−∞=
∞
−∞=
−=
−=
=
k
k
knxkhor
kxknh
nxnhny
)()(
)()(
)(*)()(
Example: x(n) is input , h(n) = [ 1 0.8 0.4 0.01]
x(n)= { 1 3 2 1 2 2 1 1 3 2} y(n)= {1 }

10/7/2009
Properties of Convolution
• h(n) * x(n) = x(n)*h(n)
• h(n)* [ax(n)] = a [h(n)*x(n)] where a is constant
• h(n)*[x1(n)+x2(n)]=h(n)*x1(n)+h(n)*x2(n)
• h(n)*[x1(n)*x2(n)]=[h(n)*x1(n)]*x2(n)

10/7/2009
Convolution: Graphical Method
Steps :
1) Reverse one of the signal to get h(-k)
2) Shift right above signal by n to get h(n-k)
3) Multiply (dot product) h(n-k) with x(k) to get sample y(n)
4) Repeat step 2 and 3 to get sample y(n) for all values of n

10/7/2009
Contd..
]30112[)( −−=
↑
nx ]121[)( −=
↑
nh
h(k)
k
h(-k)
k
x(k)
k0 0
0

10/7/2009
Contd..
h(-k)
k0
k0
x(k)
Shift by 0 to get y(0)
0 4 1 0 0 0 = 5 = y(0)

10/7/2009
Contd..
h(-1-k)
k0
k0
x(k)
Shift by -1 to get y(-1)
0 0 2 0 0 0 0 = 2 = y(-1)

10/7/2009
Contd..
h(-2-k)
k0
k0
x(k)
Shift by -2 to get y(-2)
0 0 0 0 0 0 0 0 = 0 = y(n) for n < -1

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Contd..
h(1-k)
k0
k0
x(k)
-2 2 -1 0 0 = -1 = y(1)

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Contd..
h(2-k)
k0
k0
x(k)
0 -1 -2 0 0 = -3 = y(2)

10/7/2009
Contd..
h(3-k)
k0
k0
x(k)
0 0 1 0 -3 = -2 = y(3)

10/7/2009
Contd..
h(4-k)
k0
k0
x(k)
0 0 0 0 -6 = -6 = y(4)

10/7/2009
Contd..
h(5-k)
k0
k0
x(k)
0 0 0 0 3 = 3 = y(5)

10/7/2009
Contd..
h(6-k)
k0
k0
x(k)
0 0 0 0 0 0 0 = 0 = y(n) for n>5

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Contd..
]30112[)( −−=
↑
nx ]121[)( −=
↑
nh
h(k)
k
x(k)
k0 0
*
= 0
]3623152[)( −−−−=
↑
ny
length (x)= N1=3
length (h)=
N2=5
Thus,
length (y)=
N1+N2-1=7

10/7/2009
30112
60224
30112
−−
−−
−−
Convolution: Tabular method
]30112[)( −−=
↑
nx ]121[)( −=
↑
nh
h(n)
x(n)
30112 −−
1
2
1
−
]3623152[)( −−−−=
↑
ny

10/7/2009
Problems
• Compute convolution for the following signals:
– A)
– B)
– C)
]1231[)(
↑
=nx ]11[)( =nh
]54321[)(
↑
=nx ]11[)( −=nh
]12312[)( −=nx ]1234[)( =nh

10/7/2009
Correlation
The cross-correlation of x(n) and y(n) is given by
∑
∞
−∞=
−=
n
xy lnynxlr )()()( ∑
∞
−∞=
+=
n
xy nylnxlr )()()(or
....3,2,1,0 ±±±=lfor
If signal x(n) and y(n) are same i.e. y(n)=x(n), then auto-correlation is given
by
∑
∞
−∞=
−=
n
xx lnxnxlr )()()(
....3,2,1,0 ±±±=lfor
)()( lrlr yxxy −=

10/7/2009
Contd..
Computation of correlation is same as computation of convolution
without folding operation e.g.
]3217312[)( −−=
↑
nx ]52142211[)( −−−=
↑
ny
3217312 −−
↑x
y
1
1
2
2
4
1
2
5
−
−
−
10 -5 15 35 5 10 -15
-4 2 -6 -14 -2 -4 6
2 -1 3 7 1 2 -3
8 -4 12 28 4 8 -12
-4 2 -6 -14 -2 -4 6
4 -2 -6 14 2 4 -6
-2 1 -3 -7 -1 -2 3
2 -1 3 7 1 2 -3
rxy=[ 10 -9 19 36 -14 33 0 7 13 -18 16 -7 5 -3]

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Quiz
• Find the output of the system if input x(n) and impulse response
h(n) are given by ( May 2003, 4 marks)
•Determine the autocorrelation of the following signals (Dec 97 , 5
marks)
– i) x(n)={ 1 2 1 1} ii) y(n)= { 1 1 2 1}
– What is your conclusion?
0
2
1)(
=
=
=nx
otherwise
n
n
1
1,0,2
−=
−= )3()2()1()()( −−−+−−= nnnnnh δδδδ

10/7/2009
End of Chapter 01
Queries ???

Digital Signal Processing Fundamentals

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