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Digital Communications
Lecture # 17
Vector Space and Signal Space
Sidra Shaheen Syed
[email protected]
Topical Overview
• Vector Space
• Signal Space
• Norm or Length
• Inner Product
• Orthonormal Signal Set
• Gram-Schimidth orthogonalization
Algebra: Field
• A field is a set of two or more elements
F={a,b,..} closed under two operations, +
(addition) and * (multiplication) with the
following properties
– F is an Abelian group under addition
– The distributive law is satisfied:
(a+b)*g = a*g+b*g
Immediately Following Properties
• a*b=0 implies a=0 or b=0
• For any non-zero a, a*0= ?
– a*0 + a = a*0 + a *1= a*(0 +1)= a*1=a;
therefore a*0 =0
• 0*0 =?
For a non-zero a, its additive inverse is non-zero.
0*0=(a+(- a) )*0 = a*0+(- a)*0 =0+0=0
Examples:
• The set of real numbers
• The set of complex numbers
Vector Space
• A vector space V over a given field F is a set of
elements (called vectors) closed under and operation +
called vector addition. There is also an operation *
called scalar multiplication, which operates on an
element of F (called scalar) and an element of V to
produce an element of V. The following properties are
satisfied:
– V is an Abelian group under +. Let 0 denote the additive
identity.
– For every v,w in V and every a,b in F, we have
•
•
•
•
(a*b)*v= a*(b*v)
(a+b)*v= a*v+b*v
a*( v+w)=a*v+ a *w
1*v=v
Examples of Vector Space
• Rn over R
• Cn over C
Linear Independence of Vectors
Def)
A set of vectors v1 , v2 , vn V are linearly independent iff none of the vectors
can be represented as linear combination of any other vectors in a set
Basis
Consider vector space V over F (a field).
We say that a set (finite or infinite) B  V is a basis, if
* every finite subset B0  B of vectors of linearly independent, and
* for every x  V ,
it is possible to choose a1 , ..., an  F and v1 , ..., vn  B
such that x = a1v1 + ... + an vn .
The sums in the above definition are all finite because without
additional structure the axioms of a vector space do not permit us
to meaningfully speak about an infinite sum of vectors.
Finite Dimensional Vector Space
A set of vectors v1 , v2 , vn V is said to span V if
every vector u  V is a linear combination of v1 , v2 , vn .
Example: R
n
Finite dimensional vector space
• A vector space V is finite dimensional if there
is a finite set of vectors u1, u2, …, un that span V.
Finite dimensional vector space
Let V be a finite dimensional vector space. Then
If v1 , v2 , vm are linearly independent but do not span V , then V
has a basis with n vectors (n  m) that include v1 , v2 , vm .
If v1 , v2 , vm span V and but are linearly dependent, then
a subset of v1 , v2 , vm is a basis for V with n vectors (n  m) .
Every basis of V contains the same number of vectors.
Dimension of a finiate dimensional vector space.
Example: Rn and its Basis Vectors

Example: Rn




Subspace
Let V be a vector space. Let V be a vector space and S  V .
If S is also a vector space with the same operations as V ,
then S is called a subspace of V .
S is a subspace if
v, w  S  av + bw  S
Signal Space
S(t)
S=(s1,s2,…)
•Inner Product (Correlation)
•Norm (Energy)
•Orthogonality
•Distance (Euclidean Distance)
•Orthogonal Basis
Norm - ||x(t)||
x(t )  ( x(t ), x(t ) ) =  x 2 (t )dt = Energy = Ex
T
2
0
x(t ) = Ex
2
x = x x
x
Similar to norm of vector
A
T
-A
T
x(t ) =  ( A cos
0
2 2
T
t ) dt = A
= Ex
T
2
ONLY CONSIDER SIGNALS, s(t)
s (t ) = 0 if
Energy
t0
T
t T
t
T
= Es =  s 2 (t )dt  
0
Inner Product - (x(t), y(t))
T
 x(t ), y (t )   x(t ) y (t )dt
0
y
x y

x
x  y = x y cos
Similar to Vector Dot Product
Example
A
T
t
-A
2A
A/2
t
T
A T
T
3 2
( x(t ), y(t )) = ( A)( ) + (- A)(2 A) = - A T
2 2
2
4
Orthonormal Set and Projection Theorem
Def)
A non-empty subset S of an inner product space is said to be
orthonormal iff
1) x  S ,  x, x = 1 and
2) If x, y  S and x  y, then  x, y = 0.
Orthogonality
(x(t ), y(t )) = 0
T
 x(t ) y (t )dt = 0
0
A
y
T
-A
x y = 0
x
Y(t)
B
Similar to orthogonal vectors
T
•ORTHONORMAL FUNCTIONS
X(t)
2/T
{
( x(t ), y (t ) ) = 0
T
and
x (t ) = y (t ) = 1
T
 x(t ) y(t )dt = 0
Y(t)
2/T
T
0
T
T
0
0
2
2
x
(
t
)
dt
=
y

 (t )dt = 1
( x(t ), y(t ) ) = 0
x(t ) = y (t ) = 1
y =1
x =1
Gram –Schmidt Orthonormalization
Consider linearly independent s1 , ..., sn V , and inner product space.
We can construct an orthonormal set 1 , ..., n  V so that
span{s1 , ..., sn } = span 1 , ..., n 
Gram-Schmidt Orthog. Procedure
Step 1 : Starting with s1(t)
Step 2 :
Step k :
Key Facts
Examples (1)
cont … (step 1)
cont … (step 2)
cont … (step 3)
cont … (step 4)
Signal Space summary
• Inner Product
( x(t ), y (t ) )   x(t ) y (t )dt
T
0
•Norm ||x(t)||
T
x(t ) = (x(t ), x(t )) =  x 2 (t )dt = Energy
2
0
•Orthogonality
( x(t ), y (t ) ) = 0
if
x(t ) = y (t ) = 1
35
(Orthogonal )