The Fourier Transform
The intuition for this transform comes from the Fourier Series. Only periodic signals can be represented by the Fourier Series. If we start with a finite signal
, then we can just make it periodic by copying the domain over which it is nonzero so it repeats over a period
. Call this signal
is periodic, we can find its fourier series coefficients.
These steps are possible because
over a single period, and
is zero outside that period.
Notice that if we let
approach infinity, then
becomes very small, so the
can almost be thought of as samples of some continuous time function. What this means is for a general aperiodic signal, regardless of if it is finite or not, we can think of it as having "infinite period" and thus made up of a continuous set of frequencies. This is what motivates the CTFT.
We can arrive at this equation by starting from the Fourier series again. Our faux signal
which was the periodic function we constructed out of our aperiodic one is represented by its Fourier Series
Notice this is just rewriting
as the samples of the Fourier Transform
For all these properties, assume that
A big question that arises when thinking about the Fourier Transform is whether or not the integral
Conceptually, this theorem makes sense because
So if one converges, the other must converge. However, this means that
don’t have a "strict" Fourier Series because the integral doesn’t converge for these periodic signals. In order to get around this, we can define a "generalized" Fourier Transform which operates on periodic signals.
, we know that in the frequency domain, the only consituent frequency is
. This means that
is some scalar. Using the Inverse Fourier Transform,
Now if we apply the frequency shift property, we see that
With this, we can define our generalized Fourier Transform for periodic signals.
This definition works because any periodic signal can be represented by its Fourier Series. The rational behind using the Dirac Delta in this generalized Fourier Transform is exlained by the Theory of Distributions which can be found in Appendix.
The intution for the Discrete Time Fourier Transform is more or less the same as that of the Continuous Time Fourier Transform.
For all these properties, assume that
Time Expansion: In discrete time, compression of a signal doesn’t make sense because we can’t have partial steps (i.e n must be an integer). However, we can stretch a signal.
Just like in continuous time, it was unclear whether or not the integral would converge, in discrete time, it is unclear if the infinite sum will converge. The convergence theorem for both are essentially the same.
Just like in continuous time, periodic signals like
are problematic because they don’t converge under the "strict" transform, so they require a generalized transform. In the frequency domain, a constant signal like
will be the sum of all frequencies. This will look like a sum of Dirac Deltas
Applying the synthesis equation to this, we get
and we can apply the frequency shift property to get
Once again using the Fourier Series representation of
, we can define the generalized Discrete Time Fourier Transform.
Whereas the CTFT takes a continuous signal and outputs a continuous frequency spectrum and the DTFT takes a discrete signal and outputs a continuous, periodic frequecy spectrum, the Discrete Fourier Transform takes a discrete periodic signal and outputs a discrete frequency spectrum.
One way to interpret the DFT is in terms of the Fourier series for a disrete periodic signal
. Recall that the coefficient of the kth term of the Fourier Series is
Notice that the
of the Fourier Series are the DFT values except scaled by a factor of
. This gives an intuitive inverse DFT.
One important property of the DFT is its complex conjugacy. When
is a real valued signal, then
. This can easily be shown by substituting
into the DFT formula. Further intuition for the DFT comes from relating it to the DTFT. Suppose we have a finite signal
. The DTFT of this signal is
Suppose we sample the DTFT at intervals of
Thus we can think of the DFT as a
point sample of the DTFT.
So far, our Fourier Transforms have been limited to signals of a single dimension. However, in the real world, signals might be multidimensional (think images). Thankfully, each of the Fourier Transforms generalizes easily into higher dimensions.\
Just like in 1 dimension, absolute summability/integrability guarantee the convergence of these transforms. It turns out that when a 2D signal is simply a multiplication of two 1D signals, the Fourier Transforms are very easy to compute.