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.
Now, we will look to use the power of complex exponentials to see how we may
represent arbitrary signals in terms of a set of simpler functions by
superposition of a number of complex exponentials. Below we will present the
Continuous-Time Fourier Transform (CTFT), commonly
referred to as just the Fourier Transform (FT). Because the
CTFT deals with nonperiodic signals, we must find a
way to include all real frequencies in the
For the CTFT we simply utilize integration over real numbers rather than
summation over integers in order to express the aperiodic signals.
Fourier Transform Synthesis
Fourier demonstrated that an arbitrary
can be written as a linear combination of harmonic
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; if
is continuous, then Equation 2 holds
pointwise. Also, if
meets some mild conditions (the Dirichlet
conditions), then Equation 2 holds
pointwise everywhere except at points of discontinuity.
- 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 T-periodic continuous
Now, in order to take this useful tool and apply it to arbitrary non-periodic signals, we will have to delve deeper into the use of the superposition principle. Let
be a periodic signal having period
We want to consider what happens to this signal's spectrum as the period goes to infinity. We denote the spectrum for any assumed value of the period by
We calculate the spectrum according to the Fourier formula for a periodic signal, known as the Fourier Series (for more on this derivation, see the section on Fourier Series.)
where and where we have used a symmetric placement of the integration interval about the origin for subsequent derivational convenience. We vary the frequency index proportionally as we increase the period. Define
making the corresponding Fourier Series
As the period increases, the spectral lines become closer together, becoming a continuum. Therefore,
Continuous-Time Fourier Transform
It is not uncommon to see
the above formula written slightly different. One of the
most common differences is the way
that the exponential is written. The above equations use the radial
frequency variable in the exponential, where
, but it is also common to include the more explicit expression,
, in the exponential.
Click here for an
overview of the notation used in Connexion's DSP modules.
Find the inverse Fourier transform of the ideal lowpass filter
Here we will use Equation 9 to
find the inverse FT given that
Fourier Transform Summary
Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The continuous time Fourier series synthesis formula expresses a continuous time, periodic function as the sum of continuous time, discrete frequency complex exponentials.
The continuous time Fourier series analysis formula gives the coefficients of the Fourier series expansion.
In both of these equations
is the fundamental frequency.