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Chapter: Digital Communication - Digital Modulation Scheme

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Geometric representation of Signals

Derive Geometrical representation of signal.

Geometric representation of Signals:

 

Derive Geometrical representation of signal.



 

Basis Vectors

 

The set of basis vectors {e1, e2, …,en} of a space are chosen such that: Should be complete or span the vector space: any vector a can be expressed as a linear combination of these vectors.

 

Each basis vector should be orthogonal to all others

 

·              Each basis vector should be normalized:

 

·              A set of basis vectors satisfying these properties is also said to be a complete

orthonormal basis

 

·              In an n-dim space, we can have at most n basis vectors

 

Signal Space

 

Basic Idea: If a signal can be represented by n-tuple, then it can be treated in much the same way as a n-dim vector.

 

Let φ1(t), φ2(t),…., φn(t) be n signals

 

Consider a signal x(t) and suppose that If every signal can be written as above  ~ ~ basisfunctions and we have a n-dim signal space

 

Orthonormal Basis

 

Signal set {φk(t)}n is an orthogonal set if


Then, we can express each of these waveforms as weighted linear combination of orthonormal signals


where N ≤ M is the dimension of the signal space and are called the orthonormal basis functions

 

Let, for a convenient set of {ϕj (t)}, j = 1,2,…,N and 0 ≤ t <T,


Now, we can represent a signal si(t) as a column vector whose elements are the scalar coefficients

sij, j = 1, 2, ….., N :



These M energy signals or vectors can be viewed as a set of M points in an N – dimensional


 

Euclidean space, known as the „Signal Space’.Signal Constellation is the collection of M signals points (or messages) on the signal space




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