Question

Show that the set {DiracDelta(t-tao): t, tao are real numbers} is a basis set for the analog complex signal space.

Answer 1

A basis set is a collection of vectors which defines a space in which a problem is solved.

We know that, (, , ) which define a "Cartesian 3D linear vector space".

In quantum chemistry, the “basis set” usually refers to the set of one-particle functions which used to build a molecular orbitals. An “orbital” is a one-electron function. Here, AO’s represented by atom-centered Gaussians in most quantum chemistry programs.

A complete basis set can represent exactly any molecular orbital. Unfortunately, a complete basis sets which tend to have an infinite number of functions and are not practical for calculations.

When messages are sent sequentially, then the transmitted signal becomes a sequence of corresponding analog signals which given as :

s (t) = s₁ (t) + s₂ (t - T) + s₁ (t - 2T) + s₁ (t - 3T)

The set of signal waveforms {s_i (t)} can be viewed as a set of signal vectors which given by -

s_i = {s_i1, s_i2, ......., s_iN}

The basic idea is to view waveforms such that a finite-energy functions as vectors in a certain vector space which is called as "signal space". Signal space representation of signals (waveforms) is a very effective and useful tool in the analysis of digitally modulated signals. We know that, any set of signals is an equivalent to a set of vectors.

Dirac delta function is a linear functional which maps every function to its value at zero. Mathematically, the delta function is not a function, because it is too singular.

It can be defined by an equation which given below as -

f(t) . (t - ) dt = f ()

where, = real numbers