Signal-Space Analysis ENSC 428 – Spring 2008 Reference: Lecture 10 of Gallager

Digital Communication System

Representation of Bandpass Signal Bandpass real signal x(t) can be written as: Note that In-phaseQuadrature-phase

Representation of Bandpass Signal (1) (2)Note that

Relation between and f x fcfc -f c f fcfc ff

Energy of s(t)

Representation of bandpass LTI System

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 non- negative 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={ , ,..} 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: (  +  + 

Immediately following properties   implies  or   For any non-zero    therefore    For a non-zero  its additive inverse is non-zero. 

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  in F, we have  (  v  v)  (  v  v  v   v+w)=  v  w  1*v=v

Examples of vector space  R n over R  C n over C  L 2 over

Subspace.

Linear independence of vectors

Basis

Finite dimensional vector space

 A vector space V is finite dimensional if there is a finite set of vectors u 1, u 2, …, u n that span V.

Finite dimensional vector space

Example: R n and its Basis Vectors

Inner product space: for length and angle

Example: R n

Orthonormal set and projection theorem

Projection onto a finite dimensional subspace Gallager Thm 5.1 Corollary: norm bound Corollary: Bessel’s inequality

Gram –Schmidt orthonormalization

Gram-Schmidt Orthog. Procedure

Step 1 : Starting with s 1 (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

L 2 ([0,T]) (is an inner product space.)

Significance? IQ-modulation and received signal in L 2

On Hilbert space over C. For special folks (e.g., mathematicians) only L 2 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 l 2 (the set of square summable sequence of complex numbers) If H is n-dimensional, then it is isomorphic to C n. 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, 3 rd 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: L 2

Orthonormal set S in Hilbert space H is complete if

Only for mathematicians (We don’t need separability.)

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.)

Signal Spaces: L 2 of complex functions

Use of orthonormal set

Examples (1)

Signal Constellation

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