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Signal-Space Analysis
ENSC 428 – Spring 2008
Reference: Lecture 10 of Gallager
Digital Communication System
Representation of Bandpass
Signal
x  t   s t  cos  2 fct 
Bandpass real signal x(t) can be written as:
x  t   2 Re  x  t  e j 2 fct  where x  t  is complex envelop
Note that x  t   xI  t   j  xQ  t 
In-phase
Quadrature-phase
Representation of Bandpass
Signal
(1)
x  t   2 Re  x  t  e j 2 fct 
 2 Re  xI  t   j  xQ  t   cos  2 f ct   j sin  2 f ct  
 xI  t  2 cos  2 f ct   xQ  t    2 sin  2 f ct  
(2)
Note that
x  t   x  t  e j t 
x  t   2 Re  x  t  e j 2 fct   2 Re  x  t  e j t   e j 2 fct 
 x  t  2 cos  2 f ct    t  
Relation between
x t 
and
x t 
e  j 2 f c t
x t 
x t 
2
x
f
2
2
-fc
fc
f
1
 X  f  f c   X *    f  f c   
2
 X ( f ), f  0
X  f   
, X  f   X   f  fc 
0,
f

0

Xf 
fc
f
f
Energy of s(t)

E   s 2  t  dt



S  f  df
2


 2  S  f  df
2
0

  S  f  df
0
2
(Rayleigh's energy theorem)
(Conjugate symmetry of real s(t ) )
Representation of bandpass LTI
System
s t 
h t 
r t 
s t 
h t 
r t 
r t   s t   h t 
R f   S  f H  f 
 S  f  H  f  f c  because s(t ) is band-limited.
H  f    H  f  f c   H *    f  f c   
 H ( f ), f  0
H  f   
f 0
0,
H  f   H   f  fc 
Key Ideas
Examples (1): BPSK
Examples (2): QPSK
Examples (3): QAM
Geometric Interpretation (I)
Geometric Interpretation (II)
I/Q representation is very convenient for some
modulation types.
 We will examine an even more general way of
looking at modulations, using signal space concept,
which facilitates





Designing a modulation scheme with certain desired
properties
Constructing optimal receivers for a given modulation
Analyzing the performance of a modulation.
View the set of signals as a vector space!
Basic Algebra: Group
 A group
is defined as a set of elements G and a
binary operation, denoted by · for which the
following properties are satisfied
For any element a, b, in the set, a·b is in the set.
 The associative law is satisfied; that is for a,b,c in
the set (a·b)·c= a·(b·c)
 There is an identity element, e, in the set such that
a·e= e·a=a for all a in the set.
 For each element a in the set, there is an inverse
element a-1 in the set satisfying a· a-1 = a-1 ·a=e.

Group: example
 A set
of non-singular n×n matrices of
real numbers, with matrix multiplication
 Note; the operation does not have to be
commutative to be a Group.
 Example of non-group: a set of nonnegative integers, with +
Unique identity? Unique inverse fro
each element?
 a·x=a.
Then, a-1·a·x=a-1·a=e, so x=e.
 x·a=a
 a·x=e.
Then, a-1·a·x=a-1·e=a-1, so x=a-1.
Abelian group
 If
the operation is commutative, the group is
an Abelian group.
The set of m×n real matrices, with + .
 The set of integers, with + .

Application?
 Later
in channel coding (for error correction or
error detection).
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 set F−{0} is an Abelian group under
multiplication, where 0 denotes the identity
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
 Later, finite fields (Galois fields) will be
studied for channel coding

E.g., {0,1} with + (exclusive OR), * (AND)
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
 L2 over
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
Linear independence of vectors
Def)
A set of vectors v1 , v2 , vn V are linearly independent iff
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

Inner product space: for length and
angle
Example: Rn




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.
Projection onto a finite dimensional
subspace
Gallager Thm 5.1
Corollary: norm bound
Corollary: Bessel’s inequality
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)
Example application of projection
theorem
Linear estimation
L2([0,T])
(is an inner product space.)
Consider an orthonormal set


1
 2 kt 
exp  j
k  t  
 k  0, 1, 2,... .
T
T




Any function u (t ) in L2  0, T  is u   k  u , k k . Fourier series.

For this reason, this orthonormal set is called complete.
Thm: Every orthonormal set in L2 is contained in some
complete orthonormal set.
Note that the complete orthonormal set above is not unique.
Significance? IQ-modulation and
received signal in L2
r  t ,    s  t   N  t ,    L2  0, T 
s  t   span

2 T cos 2 f ct ,  2 T sin 2 f ct
Any signal in L2 can be represented as
There exist a complete orthonormal set



r  (t ).
i i i

2 cos 2 f c t ,  2 sin 2 f ct , 3 (t ), 4 (t ),...
On Hilbert space over C. For special
folks (e.g., mathematicians) only
L2 is a separable Hilbert space. We have very useful
results on
1) isomorphism 2)countable complete orthonormal set
Thm
If H is separable and infinite dimensional, then it is
isomorphic to l2 (the set of square summable sequence
of complex numbers)
If H is n-dimensional, then it is isomorphic to Cn.
The same story with Hilbert space over R. In some sense there is only one real and one
complex infinite dimensional separable Hilbert space.
L. Debnath and P. Mikusinski, Hilbert Spaces with Applications, 3rd Ed., Elsevier, 2005.
Hilbert space
Def)
A complete inner product space.
Def) A space is complete if every Cauchy
sequence converges to a point in the space.
Example: L2
Orthonormal set S in Hilbert space
H is complete if
Equivalent definitions
1) There is no other orthonormal set strictly containing S . (maximal)
2)  x  H , x   x, ei ei
3) x, e , e  S implies x  0
4)  x  H , x  
2
x, ei
2
Here, we do not need to assume H is separable.
Summations in 2) and 4) make sense because we can prove the following:
Only for mathematicians (We don’t
need separability.)
Let O be an orthonormal set in a Hilbert space H .


For each vector x  H , set S  e  O x, e  0 is
either empty or countable.

Proof: Let Sn  e  O x, e
2
 x
2

n .
Then, Sn  n (finite)
Also, any element e in S (however small x, e is)
is in S n for some n (sufficiently large).
Therefore, S 

n 1
Sn . Countable.
Theorem
Every orothonormal set in a Hilbert space is
contained in some complete orthonormal set.
 Every non-zero Hilbert space contains a complete
orthonormal set.


(Trivially follows from the above.)
( “non-zero” Hilbert space means that the space has a non-zero element.
We do not have to assume separable Hilbert space.)

Reference: D. Somasundaram, A first course in functional analysis, Oxford, U.K.: Alpha Science, 2006.
Only for mathematicians.
(Separability is nice.)
Euivalent definitions
Def) H is separable iff there exists a countable subset D
which is dense in H , that is, D  H .
Def) H is separable iff there exists a countable subset D such that
x  H , there exists a sequence in D convergeing to x.
Thm: If H has a countable complete orthonormal set, then H is separable.
proof: set of linear combinations (loosely speaking)
with ratioanl real and imaginary parts. This set is dense (show sequence)
Thm: If H is separable, then every orthogonal set is countable.
proof: normalize it. Distance between two orthonormal elements is 2. .....
Signal Spaces:
L2 of complex functions
Use of orthonormal set
M-ary modulation {s1 (t ), s2 (t ),..., sM (t )}
Find orthonormal functions f1 (t ), f 2 (t ),.., f K (t ) so that
{s1 (t ), s2 (t ),..., sM (t )}  span{ f1 (t ), f 2 (t ),.., f K (t )}
Examples (1)
T
2
T
2
Signal Constellation
cont …
cont …
cont …
QPSK
Examples (2)
Example: Use of orthonormal set
and basis
 Two
square functions
Signal Constellation
Geometric Interpretation (III)
Key Observations
Vector XTMR/RCVR Model
N
s(t) =  s i i t ,
r(t) = s(t) + n(t)
i 1

Waveform channel / Correlation
Receiver
1 t
Vector
XTMR

.
.
.
sN
i=j
i 1

s2
j
n(t) =  ni i t
n(t)
s1
i
A

s(t)
 t
.
.
.

s(t)

n(t)

1 t
r(t)
.
.
.

z
r1 = s 1 + n1
z
r2 = s 2 + n2
T
0
T
0
.
.
.
 t


  t 
  t 
z
T
0
rN = sN + nN
}
Vector
RCVR