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Impulse Function
Discrete impulse
1, n  0
0, n  0
 [ n]  
Continuous time impulse
 (t )  0 for t  0

 (t )dt  1
 (t ) 
d
u (t )
dt
t
u (t )    ( )d

Ramp Function
t, t  0
r (t )  
0, t  0
Discrete Time Function
n, n  0
r[n]  
0, n  0
x (t )
 /2
/2
 /2
/2
Systems viewed as interconnections of Operations
is the collection of all real-valued continuous
functions defined on some interval .
is the
collection of all functions
with continuous th
derivatives. A function space is a topological vector
space whose "points" are functions.
http://mathworld.wolfram.com/FunctionSpace.html
A vector space with a Hausdorff topology such that
the operations of vector addition and scalar
multiplication are continuous.
An
operator
assigns
to
every
function
a function
. It is therefore a mapping
between two function spaces. If the range is on the real line
or in the complex plane, the mapping is usually called a
functional instead.
http://mathworld.wolfram.com/Operator.html
1. A vector space is a set that is closed under finite
vector addition and scalar multiplication.
2. A topological space fulfilling the -axiom. In the
terminology of Alexandroff and Hopf (1972), a space is called a Hausdorff space.
a. Given any two distinct points
, there
exist neighborhoods and of and ,
respectively, with
.
y (t )  H {x(t )},
y[n]  H {x[n]}
Moving Average System
(a) cascade form of implementation and
(b) parallel form of implementation.
1
y[n]  ( x[n]  x[n  1]  x[n  2])
3
1
H  (1  S  S 2 )
3
Properties of Systems
Stability BIBO
y (t )  M y   for x(t )  M x  
Memory
A system is said to possess memory if its output depends on past or future values of the input signal.
Causality
A system is said to be causal if the present value of the output signal depends only on the present or past values
of the input signal
Invertibility
A system is said to be invertible if the input of the system can be recovered from the output.
H inv{ y(t )}  H inv{H {x(t )}}  H inv H {x(t )}
H inv H  I
Time Invariance
A system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time
shift of the output signal.
Linearity
A system is said to be linear in terms of the system input (excitation) and the system output (response) if it
satisfies the properties of:
1. Superposition
2. Homogenity