Download Discrete-Time Signals and Systems

Discrete-Time Signals and Systems
Topics:
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Discrete-time signal and its classification
 What is discrete-time signal?
 Basic operations
Discrete-time systems and properties
 Properties of discrete-time systems
Convolution sum and methods for performing convolution
LCCDE Linear Constant Coefficient Difference Equation.
1. Discrete time signals
The Definition
A discrete-time signal is an indexed sequence of real or complex numbers.
It is a functions of an integer-valued variable, n, that is, often, denoted by x(n).
Graphical representation :
Three Fundamental Sequences
1. Unit sample
1 n  0
0 otherwise
 ( n)  
2. Unit step
1 n  0
u ( n)  
0 otherwise
Relationship between u(n) and δ(n):
u ( n) 
n
  (k )
n  
δ(n) = u(n) - u(n-1)
3. Exponential sequences
x(n) = an
a may be real or complex number.
a  e
jw0
is quite useful.
Classifications of Discrete Time Signals
1. Signal Duration
Finite length sequence – equals zero beyond [N1,N2]
Infinite length sequence – ex. Unit step
 Right sided sequence: zero for n less than n0
 Left sided sequence: zero for n bigger than n0
 Two sided sequence
2. Periodic and Aperiodic sequences
A signal x(n) is said to be periodic if, for some positive real integer N:
x(n) = x(n+N)
Fundamental period – N is smallest integer of the last equation.
3. Symmetric Sequences
A real valued signal is said to be even if, for all n:
x(n) = x(-n)
Whereas a signal is said to be odd if, for all n:
x(n) =- x(-n)
Any signal can be decomposed as a combination of even and odd signal:
x(n) = xe(n) + xo(n)
xe(n) = ½ [(x(n) + x(-n) ]
xo(n) = ½ [(x(n) - x(-n) ]
Complex value sequence:
It is said to be conjugate symmetric if, for all n
x(n) = x*(-n)
It is said to be conjugate asymmetric if, for all n
x(n) = - x*(-n)
4. Signal Decomposition:
x ( n) 

 x(k ) (n  k )
k  
Discrete-time Systems and properties
A discrete-time system is a mathematical operator or mapping that transforms
one signal ( the input) into another signal ( the output) by means of a fixed set of rules
or operation.
y(n)=T[x(n)]
x(n)
T[.]
1. Memory-less system
Definition: A system is said to be memoryless if the output at any time n=n0
depends only on the input at time n=n0.
Ex:
y(n) = x2(n)
Y(n) = x(n)+x(n-1)
2. Additive systems:
T[x1(n) + x2(n)] = T[x1(n)] + T[x2(n)]
3. Homogeneity:
T[cx(n)] =c T[x(n)]
4. Linear system:
T[a1x1(n) + a2x2(n)] =a1 T[x1(n)] + a2T[x2(n)]
h(n) = T[δ(n)]
hk(n) = T[δ(n-k)]
y ( n) 

 x(k )h
k
k  
( n)
5. Shift Invariant System:
For
y(n)=T[x(n), the system is said to be shift invariant if, for any delay n0,
the response to x(n-n0) is y(n-n0).
6. LSI ( Linear Shift Invariant) System:
For LSI :
hk(n) = h(n-k)
For LSI system, any input x(n) will have output:
y ( n) 

 x(k )h(n  k ) = x(n)*h(n)
k  
7. Causality
A system is said to be causal if, for any n0 the response of the system at time
n0 depends only on the input to time n= n0.
8. Stability
A sytem is said to be stable in the bounded input-bounded output sense if, for
any input that is bounded x(n)  A   , the output will be bounded,
y(n)  B  
Convolution Sums

 x(k )h(n  k ) = x(n)*h(n)
k  
Properties:
Commutative Property:
x(n)*h(n) = h(n)*x(n)
Associative Property:
{x(n)*h1(n)}*h2(n) =x(n)*{h1(n)*h2(n) }
Distributive Property:
x(n)*{h1(n)+h2(n)} =x(n)*h1(n)+x(n)*h2(n)
Performing Convolution
Direct Evaluation
Graphic Approach
Sliding Rule Method.
Difference Equations
LCCDE – Linear Constant Coefficient Difference Equation:
q
p
k 0
k 1
y ( n)   b( k ) x ( n  k )   a ( k ) y ( n  k )
If the system have one or more a(k) that are nonzero, the difference equation is
said to be recursive. If all of the coefficients a(k) are equal to zero, the
difference equation is said to be non recursive.
Difference equation provide a method for computing the response of a system,
y(n), to an arbitrary input x(n).
Approaches to solve LCCDE:
 Classical approach of finding homogeneous and particular solution.
 Using z-transform.