Polar Fourier Transform and its similarity to the Z-Transform

Question

The DFT is given by

$X_{\mathcal{F}}(k) = \sum_{n=0}^{N-1} x_n e^{-j 2 \pi k n / N}$

Knowing that complex numbers can also be represented using a polar form $A \angle\theta$, I was looking for a version of FFT that would output exactly that.

So I found this Polar Fourier Transform.

I am not sure yet if it is what I am looking for. But it is related, at least to the same extent FFT is related.

---

In this paper, The Polar Fourier Transform is given by,

---

However when writing the frequency $\phi$ in polar coordinates, a term is added, the amplitude $r$, which was not originally there. Right away I started wondering if that wouldn't be exactly the same as the Z-Transform.

The Z-Transform is presented in the following way.

$X(z)=\mathcal{Z}\{x[n]\}=\sum_{n=-\infty }^{\infty }x[n]z^{-n}$

where

${z=Ae^{j\phi }=A\cdot (\cos {\phi }+j\sin {\phi })}$

---

Questions:

1. What are the main differences and similarities of between Polar Fourier Transform.
2. Is there away to optimally have a variant of the FFT and iFFT transform that represents the frequency components on their polar form, instead of adding the extra computation for computing the module and angle of these values? (At first I imagined that was exactly what the Polar Fourier Transform as doing, but now I don't think so anymore.)

Cheers,

Answer 1

Some comments which might answer your questions:

"However when writing the frequency ϕ in polar coordinates, a term is added, the amplitude r, which was not originally there."

I think you have a missunderstanding what $\phi$ is. For a two-dimensional function (i.e., function on the plane) the frequencies are also two-dimensional so $\phi \in \mathbb{R}^2$. There are several ways to encode a vector in the plane. One example is the standard way by using two axis (cartesian coordinate system) $\phi_{\text{cart}} = (\phi_1, \phi_2)$. Another one, used in the paper, is to use an angle and the distance to zero (polar coordinate system) $\phi_{\text{polar}} = (r,\theta)$. Both systems can be mapped one-to-one to each other. The direction $\phi_{\text{polar}} \to \phi_{\text{cart}}$ is already in your post, the direction $ \phi_{\text{cart}} \to \phi_{\text{polar}}$ is $$ r = \sqrt{\phi_1^2 + \phi_2^2} \\ \theta = \begin{cases} \arccos \frac{x}{r} & \text{if } y \geq 0\\ - \arccos \frac{x}{r} & \text{if } y < 0\end{cases}. $$ So the term $r$ is not added, it was always there.

I hope this also answers your second question: the polar form is only arepresentation (= way of writing your vector) of the complex numbers, so depending how you represent the complex numbers in your computer, the output is either in cartesian or polar coordinates (you can use the same FFT and iFFT algorithm in both cases).

"What are the main differences and similarities of between Polar Fourier Transform."

The main differences:

• The PolarFT is an integral transform, acting on continuous functions, while the z-transform is sum transform, acting on discrete functions
• The PolarFT is defined for two-dimensional functions, while your z-transform is defined for one-dimensional functions

The one-dimensional z-transform reduces to the discrete time Fourier transform for $|z| = 1$ though.