**Discrete signals in vector space**The concept of an N-D vector space can be generalized to an infinite dimensional space spanned by a set of basis vectors with . Moreover, the dimensions of the space can be uncountable so that the space is spanned by a set of uncountable basis vectors with .

The

*inner product*between any two vectors and in the vector space is defined as:

If , then and are*orthogonal*, if , then is*normalized*. Two vectors that are both normalized and orthogonal are called*orthonormal*.A set of countable orthonormal vectors satisfies:

Here is called the*Kronecker delta function*.A set of

*uncountable*orthonormal vectors satisfies:

Here is the*Dirac delta function*which also satisfies:

A discrete signal can be represented as a vector in a vector space. The signal can be either infinite , or finite , in which case it can be assumed to repeat itself beyond to become a periodic signal . The signal vector can be represented as a linear combination of the basis vectors:

To find the coefficients , we find the inner product of both sides with :

Now we have a generalized discrete Fourier transform pair:

In particular, it can be shown (homework solution) that

are orthonormal:

Now the transform above becomes the discrete Fourier transform of a discrete and periodic signal .If the basis is not countable, the linear combination becomes:

To find the coefficients , we find the inner product of both sides with :

Now we have a generalized Fourier transform pair:

In particular, it can be shown (homework solution) that

are orthogonal

and the transform above becomes the discrete-time Fourier transform of a discrete and non-periodic signal .**Continuous signals in function space**The concept of vector space can be generalized to a function space spanned by a set of basis functions with . The function in the space can be either finite or infinite . The dimensions of the space can also be uncountable so that the space is spanned by a set of uncountable basis vectors with .

The inner product between two functions and in the function space is defined as

Depending on the nature of the function space, the integral can be from 0 to or from to .If , then and are orthogonal. if , then is normalized. Two vectors that are both normalized and orthogonal are called

*orthonormal*.A set of countable orthonormal functions satisfies:

A set of uncountable orthonormal functions satisfies:

A continuous signal can be represented as a time function in a function space. The function can be infinite with , or finite with , in which case it can be assumed to repeat itself beyond to become a periodic signal . The signal function can be represented as a linear combination of basis functions of the space:

To find the coefficients , we find the inner product of both sides with :

Now we have a generalized continuous Fourier transform pair:

In particular, it can be shown (homework solution) that

are orthogonal

and the transform above becomes the Fourier series expansion of a periodic signal .If the basis is not countable, the linear combination becomes:

To find the coefficients , we find the inner product of both sides with :

Now we have a generalized Fourier transform pair:

In particular, it can be shown (homework solution) that

are orthogonal

and the transform above becomes the Fourier transform of a continuous and non-periodic function .