Radar stands for "RAdio Detection And Ranging", initially a top-secret military technology for detecting invading aircraft long before they are visible, is now making its way into our daily lives. Many modern vehicles are equipped with short-range radars as a safety feature, in adaptive cruise control and collision avoidance systems. Google's Project Soli takes this to the next level by using it as a close-range sensor for use in mobile devices.

Basic Principles of Radars

Radars work on a simple idea: send out a radio signal, wait for an echo. The time it takes for the echo to arrive is directly proportional to the distance of the reflecting object.

A basic version of this idea would be a Pulse Radar^[1]. Transmit is on for an instant, followed by a period of waiting for echoes. Mathematically, the transmitted signal is:
$$S_T = A(t)sin( 2 \pi f_c t + \phi_0 )$$

Where $A(t)$ is a constant transmit amplitude when the radar is transmitting and zero otherwise. $f_c$ is transmission frequency and $\phi_0$ is the starting phase. Without loss of generality, can assume that the starting phase is $0$ and will drop the term for clarity of notation, and only reintroduce it where the difference is significant.

In addition to estimating range from the time delay, non-zero relative velocity results in frequency shifts in a phenomenon known as the Doppler effect^[2]. As the transmitted signal is a single frequency, we can estimate the relative velocity of the reflecting object by measuring the Doppler effect that causes a change in frequency of the reflected pulse.

Although simple in terms of operating principles, due to the speed of light, pulse radars are blind at very short ranges (below 1Km). While not an issue for long-range applications (e.g. aircraft, ships), this makes them of limited use where the range is small.

FMCW Radars

In contrast with traditional pulse radars, an FMCW (Frequency Modulated Continuous Wave) radar transmits a signal who's frequency changes with time, often referred to as a chirp:

$$S_{T}(t) = A_{T} \cos\left(2 \pi (f_c + f_\tau(t) ) t \right)$$

Where $f_c$ is the starting frequency and $f_\tau(t)$ is a function describing how the frequency changes over time. One possible waveform is a sawtooth (in frequency-time) signal. i.e. (for single chirp):

$$S_{T}(t) = A_{T} \cos\left(2 \pi (f_c + B t ) t \right)$$
Where $B$ is the slope or the rate of frequency. For the rest of the discussion, we assume that we are working with a sawtooth wave.

Similar to the classical radar, we expect to receive a time-delayed and Doppler-shifted version of the transmitted signal. In contrast with the classical radar, both the transmitter and receiver are on simultaneously. Thus, there are no problems with very short ranges.

Estimating Range

The reflected waveform is a delayed version of the transmitted wave. Again, by measuring this delay, we can compute the distance the object is away from the radar. At the receiver, a mixer (multiplier) mixes the reflected signal with the transmitted signal. Next, this signal passes through a low-pass filter and is sampled by an ADC. At any instant, we can describe the signal as:

$$S_{rx} = A \cos(\alpha)\cos(\beta)$$

Where, $\alpha$ is the frequency that is being transmitted and $\beta$ is the frequency that had been reflected. Using the product to sum identity, we can see that:

$$S_{rx} = (A/2) \left( \cos(\alpha-\beta) + \cos(\alpha + \beta) \right)$$

In this form, we see that there are two frequency components in the received signal — one of much lower frequency than the transmitted waveform and one of very high frequency. After a low-pass filter, this leaves us with a signal that does not have very stringent ADC requirements, as compared to the original GHz-band signal.

Since the slope is known, we can determine the time delay (and distance) easily as follows:
$$d = \frac{ f }{2B} \cdot c_0$$
Where $c_0$ is the speed of light in free space.

Since the mixed signal gives us a frequency difference, all we have to do is to perform an FFT over the entire chirp, and the (frequency) location of the (amplitude) peaks is directly proportional to the range of the target. In FMCW radar literature, this is often referred to as the "intermediate frequency", "beat frequency" or the IF signal.

On Doppler Effect

With a sawtooth wave, there is no way to disentangle frequency shifts that is due to a non-zero relative velocity. It is treated as measurement noise for low-velocity targets. If this is not the case, a different waveform might be a more suitable choice.

Estimating Relative Velocity

While we are unable to resolve the velocity of a target from a single chirp, if we look across multiple chirps, the relative velocity can be recovered. Recall that we are assuming that the velocity of the target is small, and its range does not change significantly over several chirps. Numerically this results in FFTs with peaks at the same frequency bin. While unable to be resolved as different distances, this small displacement manifests as a phase shift.

Suppose two chirps are sent $T_c$ seconds (usually in the order of microseconds) apart. Recall that the IF signal is a sinusoid:
$$A\cos(2\pi f t + \phi_0)$$

If the object is stationary, the phase term of the first chirp will be identical to that of the second chirp. However, if there is a slight change in distance between the first and second chirp, the IF signal of the second chirp will be a phase delayed version of the first chirp. With phase delay:
$$\Delta \phi = \frac{4\pi \Delta d}{\lambda}$$

Using a $77\text{GHz}$ radar, a $1\text{mm}$ ($\lambda/4$) displacement will result in a $\pi/2$ phase shift, with only an insignificant change in frequency. (The reader is encouraged to plug in some values here to see this. A typical slope, $B$, for a 77GHz FMCW radar is $50 \text{MHz}/\mu s$.)

Rearranging and dividing by the time between chirps, $T_c$, we obtain the relationship between the phase difference and the velocity of the target:
$$v = \frac{\lambda \Delta \phi}{4 \pi T_c}$$

Numerically, the phase difference can be obtained by performing an FFT across chirps. The number of chirps and the period between the chirps determines the velocity resolution.

In a practical FMCW radar system, $N$ chirps are sent and processed as a group in order to determine the velocity of the target. We call this sequence of of chirps a frame and this is the basic unit of FMCW radar signal.

Conclusion

We have now established the basic principles behind the FMCW radar. We saw that by performing 2 FFTs, one within a chirp and another across chirps, we can estimate the range and relative velocity of a reflecting target. To design a system that operates with some desired performance parameters, we leave the following points as things to ponder about:

1. What are the limitations of an FMCW radar?
2. What determines the minimum resolvable distance (i.e. range resolution)?
3. What is the velocity resolution?
4. Is there an ambiguity in velocity estimation?
5. To measure the speed of vehicles, how long should each chirp be? What's the periodicity of the chirps?
6. What about angle estimation?

A reader with some knowledge in digital signal processing should be able to derive these limits with the information in this post. We will leave these topics as an exercise for now, and provide a detailed treatment in the next post.