SignalSpace Analysis

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Signal-Space Analysis

• ENSC 428 Spring 2008
• Reference Lecture 10 of Gallager

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Digital Communication System
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Representation of Bandpass Signal
Bandpass real signal x(t) can be written as
Note that
In-phase
Quadrature-phase
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Representation of Bandpass Signal
(1)
(2)
Note that
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Relation between and
x
f
f
f
f
fc
-fc
fc
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Energy of s(t)
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Representation of bandpass LTI System
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Key Ideas
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Examples (1) BPSK
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Examples (2) QPSK
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Examples (3) QAM
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Geometric Interpretation (I)
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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!

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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, ab is in the
  set.
• The associative law is satisfied that is for
  a,b,c in the set (ab)c a(bc)
• There is an identity element, e, in the set such
  that ae eaa 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 ae.

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Group example

• A set of non-singular nn 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 non-negative
  integers, with

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Unique identity? Unique inverse fro each element?

• axa. Then, a-1axa-1ae, so xe.
• xaa
• axe. Then, a-1axa-1ea-1, so xa-1.

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Abelian group

• If the operation is commutative, the group is an
  Abelian group.
• The set of mn real matrices, with .
• The set of integers, with .

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Application?

• Later in channel coding (for error correction or
  error detection).

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Algebra field

• A field is a set of two or more elements
  Fa,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
• (ab)g agbg

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Immediately following properties

• ab0 implies a0 or b0
• For any non-zero ?, ?0 ?
• ?0 ? ?0 ? 1 ?(0 1) ?1? therefore
  ?0 0
• 00 ?
• For a non-zero ?, its additive inverse is
  non-zero. 00(?(- ?) )0 ?0(- ?)0 000

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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)

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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
• (ab)v a(bv)
• (ab)v avbv
• a( vw)av a w
• 1vv

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Examples of vector space

• Rn over R
• Cn over C
• L2 over

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Subspace.
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Linear independence of vectors
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Basis
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Finite dimensional vector space
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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.

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Finite dimensional vector space
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Example Rn and its Basis Vectors
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Inner product space for length and angle
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Example Rn
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Orthonormal set and projection theorem
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Projection onto a finite dimensional subspace
Gallager Thm 5.1 Corollary
norm bound Corollary Bessels
inequality
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Gram Schmidt orthonormalization
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Gram-Schmidt Orthog. Procedure
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Step 1 Starting with s1(t)
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Step 2
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Step k
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Key Facts
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Examples (1)
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cont (step 1)
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cont (step 2)
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cont (step 3)
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cont (step 4)
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Example application of projection theorem

• Linear estimation

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L2(0,T)(is an inner product space.)
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Significance? IQ-modulation and received signal
in L2
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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.

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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

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Orthonormal set S in Hilbert space H is complete
if
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Only for mathematicians (We dont need
separability.)
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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.

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Only for mathematicians. (Separability is nice.)
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Signal Spaces L2 of complex functions
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Use of orthonormal set
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Examples (1)
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Signal Constellation
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cont
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cont
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cont
QPSK
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Examples (2)
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Example Use of orthonormal set and basis

• Two square functions

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Signal Constellation
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Geometric Interpretation (III)
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Key Observations
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Vector XTMR/RCVR Model