Understanding Fourier Transform from Vehicle-borne Radar

Looking back, my first encounter with Fourier Transform was when I was studying signals and systems. I thought its mathematical expression was beautiful and wanted to learn more about it. However, I could only observe it from a distance and could not comprehend its profound meaning. My understanding of it was limited to a kind of mathematical transformation. During the exam, I would calculate the Fourier transform of a rectangular window and perform the calculations step by step. That was all. I could not master it and we became strangers from then on.

Figure 1 Fourier transform, simple and elegant, beautiful as a flower

The turning point came when I was learning OFDM (Orthogonal Frequency Division Multiplexing). Modulation and demodulation are one of the core parts of communication. OFDM modulation and demodulation can be expressed so concisely and profoundly using Fourier transform. I was amazed. For the first time, Fourier gave me such a clear and preliminary physical concept. I always think that mathematicians are very powerful, because any mathematical deduction without actual physical concepts is extremely abstract, and it is not easy to think clearly. The abstraction and rigor of mathematics without the intuition of additional physical concepts is an important reason why we are afraid of mathematics, isn't it? The cold mathematical formulas are given clear physical concepts and become so vivid, just like general relativity to Riemann geometry and Yang-Mills gauge field to fiber bundle theory. So, the key to understanding Fourier transform is physical concepts.

As the saying goes, all roads lead to Rome. Different fields, different physical or engineering perspectives provide more possibilities for interpreting Fourier transform. The interpretation from the perspective of radar, especially vehicle-mounted radar, is very interesting. Therefore, this article attempts to fully explain the physical concept and extension of Fourier transform from the perspective of millimeter-wave radar. I hope it will be useful to you and deepen your understanding of it.

In automotive radar applications, I think many people know that 1D FFT is performed on ADC Raw Data, 2D FFT is performed on one-dimensional data, and DoA is 3D FFT, just like we know that the FT of rect(t/T) is sinc(w), but we may not be clear about the physical concepts and extensions behind it, such as how to understand that FFT can be performed in three dimensions, why FFT is needed, what to do if FFT does not meet the requirements, is it okay not to perform FFT, etc. The formula shown in Figure 2 is the single-target MIMO radar echo expression, and is also one of the most commonly used echo models for automotive radars [1]:

Figure 2. Automotive MIMO Radar echo expression (single target)

It has three dimensions, l, n, p, representing angle bin, distance bin, and speed bin respectively. So within the value range of each dimension, d(l,n,p) represents a data cube, also called radar cube, as shown in Figure 3. We perform FFT on the cube along the range axis, Doppler axis, and Space axis to extract range, Doppler, and Azimuth information. This is the first level of understanding FT as a radar algorithm engineer - knowing the operation process, that is, how to do it.

Figure 3 Radar Cube

Next, we set fixed values ​​for two of the three dimensions in turn, and continue to decompose d(l,n,p) into three dimensions:

The Gaussian white noise expression is simplified, and the above three equations can be further summarized as follows:

We find that each formula actually starts with a constant phase, so we can simplify and organize it:

It can be seen that the qualities of the above three equations are the same, and they all have the following form:

This is why FFT can be performed in all three dimensions. On the other hand, the distance, velocity, and azimuth information we care about is contained in a corresponding frequency. Information and frequency are one-to-one corresponding, so the acquisition of information is converted into frequency extraction, and FFT is an important tool for extracting frequency.

Then the question arises again, why is FFT used instead of other tools to extract frequency?

Continuing the analysis, in actual engineering, we get d, and d contains noise, which is w in the above formula. Usually, we assume that the noise is Gaussian white noise, so the problem is transformed into the frequency estimation problem under Gaussian white noise conditions. Let's sort it out again.

1. What we have is measurements, i.e. noisy data

2 What do we assume? The noise is Gaussian white noise, that is, we know the distribution

3 What do we want to do, estimate the frequency

Based on 1, 2, and 3 above, we should think of the likelihood function. The likelihood function of the above problem is:

The specific derivation can be found in reference [2], which will not be repeated here. From the derivation results, the maximum likelihood result of frequency estimation under Gaussian white noise conditions is FFT:

Where L is the likelihood function and A(w) is FFT. This is the second level of understanding FT - tracing back to the source and knowing why FFT is used. It is because it is the best way to estimate frequency under the likelihood criterion.

As mentioned before, FFT is ultimately a tool for frequency estimation. Can it be replaced by other methods? The answer to this question depends on your application. Radar outputs three quantities: distance, speed, and angle, and the important indicators for evaluating these three quantities are resolution and accuracy. Depending on the application, our requirements for resolution and accuracy are different. Of course, if conditions permit, we hope that the resolution and accuracy are as high as possible. The Fourier transform is already excellent, and with the support of fast algorithms, it has been leading the way in embedded platforms such as automotive radars for decades. But we must always look at problems with a developmental, changing, and critical eye. Yes, then the excellent FFT also has flaws:

• There is a Rayleigh limit and the resolution is not high

• Sidelobes can easily cause false alarms (High False Alarms)

• Frequency is the average frequency of time statistics, lacking local frequency information

• Under multi-target conditions, the frequency estimation accuracy is poor

The defects of FFT make the resolution and accuracy of the three quantities estimated by radar poor, which may not meet your application requirements. Based on this, in the past few decades, various heroes have emerged in the battle for the power of frequency estimation, which is very exciting!

Fourier is the originator of analytical signal processing, and the masters of each era have gradually developed the Bayesian estimation framework, state filtering, cognitive signal processing, and machine learning or neural network methods based on the work of their predecessors and their own unique insights. We hope to gradually use these methods for better estimation of the three quantities, especially the angle, under the conditions of hardware computing power and memory. This is the third level of understanding Fourier - carrying forward the past and opening up the future. This is not a denial or abandonment of FFT, but inheritance and development. These methods are also collectively referred to as modern digital signal processing methods, which will be introduced and shared in future articles!

FFT is still an endless treasure. Looking back at it from time to time, I still feel awe. Just like us who are wandering in a foreign land, no matter how successful we are, every time we step on our homeland, we will also be filled with emotion:

Picture 4 Let me take another look at you, I hope you are okay.