Using this and the fact that
is linear, calculating
for combinations of complex exponentials is also
The action of on an input such
as those in the two equations above is easy to explain.
scales each exponential component
by a different complex number
. As such, if we can write a function
as a combination of complex exponentials it allows us to easily calculate the output of a system.
is the fundamental frequency. For almost all
of practical interest, there exists
to make Equation 2 true. If
is finite energy (
), then the equality in Equation 2
holds in the sense of energy convergence; with discrete-time signals, there are no concerns for divergence as there are with continuous-time signals.
- called the Fourier coefficients -
tell us "how much" of the sinusoid
The formula shows
as a sum of complex exponentials, each of which is easily processed by an
LTI system (since it is an eigenfunction of
every LTI system). Mathematically,
it tells us that the set of
form a basis for the space of N-periodic discrete
DFT Synthesis Demonstration
we have the following set of numbers that describe a periodic,
discrete-time signal, where
Such a periodic, discrete-time signal (with period
) can be thought of as a
finite set of numbers. For example,
we can represent this signal as either a periodic signal or as
just a single interval as follows:
The cardinalsity of the set of discrete time signals with period
Here, we are going to form a basis
using harmonic sinusoids. Before we look into
this, it will be worth our time to look at the discrete-time,
complex sinusoids in a little more detail.
If you are familiar with the basic sinusoid signal and with complex exponentials
then you should not have any problem understanding this
section. In most texts, you will see the the discrete-time,
complex sinusoid noted as:
In the Complex Plane
The complex sinusoid can be directly mapped onto our complex plane, which
allows us to easily visualize changes to the complex
sinusoid and extract certain properties. The absolute
value of our complex sinusoid has the following
which tells that our complex sinusoid only takes values on
the unit circle. As for the angle, the following
statement holds true:
For more information, see the section on the Discrete Time Complex Exponential to learn about Aliasing , Negative Frequencies, and the formal definition of the Complex Conjugate .
Now that we have looked over the concepts of complex
sinusoids, let us turn our attention back to finding a basis
for discrete-time, periodic signals. After looking at all the
complex sinusoids, we must answer the question of which
discrete-time sinusoids do we need to represent periodic
sequences with a period .
Find a set of vectors
are a basis for
In answer to the above question, let us try the "harmonic"
sinusoids with a fundamental frequency
is periodic with period and has "cycles"
If we let
where the exponential term is a vector in
is an orthonormal basis for
First of all, we must show
is orthonormal, i.e.
then we must use the "partial summation formula" shown
where in the above equation we can say that
, and thus we can see how this is in the form
needed to utilize our partial summation formula.
is an orthonormal set.
is also a basis, since there
are vectors which are
linearly independent (orthogonality
implies linear independence).
And finally, we have shown that the harmonic sinusoids
form an orthonormal basis for
Periodic Extension to DTFS
Now that we have an understanding of the discrete-time Fourier series
(DTFS), we can consider the periodic
(the Discrete-time Fourier coefficients). Figure 7 shows a simple illustration of how we can represent
a sequence as a periodic signal mapped over an infinite number
Why does a periodic
extension to the DTFS coefficients
→ DTFS coefficients are also periodic with
Example 3: Discrete time square wave
Calculate the DTFS
Just like continuous time Fourier series, we can take the summation
over any interval, so we have
(so we can get a geometric series starting at 0)
Now, using the "partial summation formula"
Manipulate to make this look like a sinc function (distribute):
Using the steps shown above in the derivation and our
previous understanding of Hilbert Spaces and Orthogonal Expansions, the rest of the
derivation is automatic. Given a discrete-time, periodic
signal (vector in
, we can write:
Note: Most people collect both the
terms into the expression for
Discrete Time Fourier Series:
Here is the common form of the DTFS with the above note
taken into account:
This is what the fft command in MATLAB does.