Derive Geometrical representation of signal.

**Geometric representation of Signals:**

Derive Geometrical representation of signal.

The set of basis vectors {**e**1, **e**2, …,**e**n} 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

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**

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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Digital Communication : Digital Modulation Scheme : Geometric representation of Signals |

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