Every
periodic signal can be written as a summation of sinusoidal
functions of frequencies which are multiples of a constant
frequency (known as fundamental frequency). This
representation of a periodic signal is called the
Fourier Series. An aperiodic signal can always be
treated as a periodic signal with an infinite period.
The frequencies of two consecutive terms are
infinitesimally close and summation gets converted to
integration. The resulting pattern of this representation
of an aperiodic signal is called the Fourier Transform.
Signals Treated
as Vectors
Any vector in
N-dimensional space can be fully specified by a set
of N numbers (i.e. its components in various directions).
Similarly we can also treat signals in continuous and
discrete times as special cases of vectors with infinite
dimensions.
Why do we
need signals to be treated as vectors?
The mathematical analysis of vectors is highly
advanced compared to signals. Treating signals as vectors helps us to attribute many additional properties to them. Moreover we do feel comfortable taking signals
as vectors in a problem involving number of signals.
Countable
Infinity: A set is called countably infinite if and only
if its all elements have 1-1
correspondence with set of natural numbers or any other
countable infinite set. We can easily see set of integers
satisfying this property. Now we can call a set countably
infinite if its elements have 1-1 correspondence with
integers (it will ensure automatically that the condition
be satisfied for natural numbers). Every rational number
can be taken as a tuple of two integers (numerator and
denominator) making the set of rational numbers also countably
infinite.
Exercise: Prove that the set of real numbers is not
countably infinite.
Note:
A Discrete Signal x[n] can be thought of
as a " Vector " with countably infinite dimensions.
A Continuous Signal x(t) can be thought of
as a vector with uncountably infinite dimensions.
