Digital signal processing

Digital Signal Processing
.
Mrs.R.Chitra, Assistant Professor(SS),
Department of ECE,
Faculty of Engineering,
Avinashilingam Institute for Home Science
and Higher Education for Women,
Coimbatore

SYSTEM
DEFINITION:
 Combination of set of elements.
 It will process or manipulate over input signal to
produce output signal

Classification of systems
 Continuous time and discrete time signal
 Lumped-parameter and distributed -parameter
systems
 Static and dynamic systems
 Causal and non-causal systems
 Linear and non-linear systems
 Time variant and time -invariant systems
 Stable and unstable system

Continuous time system
A Continuous time system is one which operates on a
continuous time input signal and produces a continuous
time output signal.
y(t)= Tx(t)

Discrete time system
A discrete time system is the one which operates on a
discrete- time input signal and produces a discrete -time
output signal.
y(n)=T(X(n)

Lumped and distributed parameter
 In lumped parameter system each component is
lumped at one point in space.
 In distributed parameter systems the signals are
functions of space as well as time.

Static and dynamic system
 A system is called static or memory less if its output at
any instant depends on the input at that instant.
 A system output is depends up on the past and future
values of input.

Causal and non causal system
 A causal system is one for which the output at any time
t depends on the present and past inputs.
 A non-causal system output depends on the future
values of inputs.

Linear and non linear system
A System is said to be linear if it follows super position
principle is called as linear other wise non-linear.
T[ax1(t)+bx2(t)]=aT[x1(t)]+bT[x2(t)]
Similarly for a discrete-time linear system,
T[ax1(n)+bx2(n)]=aT[x1(n)]+bT[x2(n)]

Stable and unstable system
 If the system has to be stable if and only if every
bounded input produces bounded output.
 If system has to be unstable if bounded input produces
unbounded output

Time variant and invariant system
 Time variant system have same input and output,
y(n,k)=y(n-k).
 Time invariant system has no same input and output.

CONVOLUTION
 Convolution is the process by which an input interacts
 with an LTI system to produce an output
 Convolution between of an input signal x[n] with a
 system having impulse response h[n] is given as,
 where * denotes the convolution
 x [ n ] * h [ n ] = x [ k ] h [ n - k ]

EXAMPLE
 Convolution
 We can write x[n] (a periodic function) as an infinite
sum of the function xo[n] (a non-periodic function)
shifted N units at a time

Properties of convolution
Commutative…
 x1[n]* x2[n] = x2[n]* x1[n]
Associative…
 {x1[n]* x2 [n]}* x3[n] = x1[n]*{x2 [n]* x3[n]}
Distributive…
 {x1[n] *x2[n]}* x3[n] = x1[n]x3[n] * x2[n]x3[n]

Linear convolution
)(*)()()()( 2113 nxnxmnxmxnx
m
 


)()( 11

 j
eXnx FT
)()( 22

 j
eXnx FT
)()()()(*)()( 213213

 jjj
eXeXeXnxnxnx FT

Circular convolution
 Circular convolution of of two
finite length sequences
16
       



1
0
213
N
m
Nmnxmxnx
       



1
0
123
N
m
Nmnxmxnx

Circular convolution
)()())(()()( 21
1
0
13 nxnxmnxmxnx
N
m
N  


)()( 11 kXnx  DFT
)()( 22 kXnx  DFT
)()()()()()( 213213 kXkXkXnxnxnx   DFT
both of length N

EXAMPLE
)()( 01 nnnx 
)(2 nx
)(*)( 21 nxnx
0
0 N
0 n0 N
)()( 21 nxnx 
n0=2, N=7
0

DISCRETE FOURIER TRANSFORM
 It turns out that DFT can be defined as
Note that in this case the points are spaced 2pi/N;
thus the resolution of the samples of the frequency
spectrum is 2pi/N.
 We can think of DFT as one period of discrete
Fourier series

EXAMPLE OF DFT
 Find X[k]
 We know k=1,.., 7; N=8

PROPERTIES OF DFT
Linearity
Duality
Circular shift of a sequence
Copyright (C) 2005 Güner
Arslan
   
   
       kbXkaXnbxnax
kXnx
kXnx
21
DFT
21
2
DFT
2
1
DFT
1
 
 
 
   
       mN/k2jDFT
N
DFT
ekX1-Nn0mnx
kXnx

 
 
   
     N
DFT
DFT
kNxnX
kXnx
 
 

INVERSE OF DFT
 We can obtain the inverse of DFT
 Note that

Digital signal processing

More Related Content

What's hot (20)

Similar to Digital signal processing (20)

More from chitra raju (6)

Recently uploaded (20)

Digital signal processing