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.
The Inverse Continuous Time Fourier Transform takes us from the frequency domain reprsentation of a function
to its time domain representation
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
A big question that arises when thinking about the Fourier Transform is whether or not the integral
exists and is continuous. In addition,
approaches 0 as
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.
The generalized Fourier Transform for a periodic signal
are the coefficients of the Fourier Series of
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.
Discrete Time Fourier Transform
The Discrete Time Fourier Transform converts aperiodic discrete signals into the frequency domain.
The intution for the Discrete Time Fourier Transform is more or less the same as that of the Continuous Time Fourier Transform.
The Inverse Discrete Time Fourier Transform converts the frequency domain representation of a signal back into its time domain representation.
Properties of the DTFT
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.
Convergence of the DTFT
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.
exists and is continuous.
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
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.
For a perioidic signal
, the generalized Discrete Time Fourier Transform is
are the Fourier Series coefficients of
Discrete 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.
For a length
, the Discrete Fourier Transform of the signal is a length N finite sequence
One way to interpret the DFT is in terms of the Fourier series for a disrete periodic signal
x~[n]=x[n mod N]
. 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.
For a length N finite sequence
representing the DFT of a finite perioidc signal
, the inverse DFT is given by
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.
2D Fourier Transforms
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.