# FMCW Radar introduction¶

TI mmwave radars are frequency modulated continuous wave (FMCW) radars such that the frequency is swept linearly. FMCW radars have unique advantages, which cannot be presented in other radars at once. Those are 1:

• Being a mm-wave radar: the high attenuation in mmwave frequencies provides a high isolation between the co-located operating radars even if they are separated in a few meters. Indeed, tiny displacements in mm are comparable to the wavelength thus they can be detected. This high sensitivity is required to detect the chest wall movement, which is in mm order.

• Discriminating range or localizing: because the radar can distinguish the reflections from different ranges, potentially it can be used for multi-subject vital signs detection. This feature is recognized as the main advantage of an FMCW radar in 2. Indeed, high propagation attenuation reduces the possibility of having an echo signal, which is bounced off multiple reflectors. Most probably, the echo signal is reflected off a single object if the environment is not rich scattering. In that area, the received signal at particular range experienced a line of sight wireless channel. In contrast, CW radars suffer from multipath fading because they collect all reflections from all objects at all visible ranges in a one sinusoid signal.

• Being robust against thermal noise: FM signals are more robust against noise in comparison to AM signals. Also, in FMCW radars the vital sign information is encoded in the received phase similar to FM signals. Thus, FMCW radar is less affected by the noise in comparison to impulse radars.

## FMCW radar basics¶

In any radar, the electromagnetic wave is sent into the environment containing various objects. Then the echo of the wave is captured at a receiver. A simplified block diagram of such a system is shown in Fig. 1 in which both the transmitter and the receiver are at the same location. Each chirp at the output of the FMCW generator is a sinusoid signal whose frequency is swept from fmin to fmax (Fig. 2). Here the frequency is swept linearly with a positive slope of K and a duration of $$T_r$$ implying that the sweeping bandwidth is $$f_{max} − f_{min} = K T_r$$. The received signal at the output port of the receiver antenna is amplified and correlated with the transmit signal, which results in a signal called beat signal. The beat signal contains information about the objects in the scene. Particularly, the delay in the reflected signal is translated to an instantaneous frequency difference between the transmitted and the received chirps.

Fig. 1

Fig. 2

Assume that the complex chirp signal is:

$s(t) = A_t \exp \left( j(2 \pi f_{min} t + \pi K t^2) \right) , 0 < t < T_r \quad\quad (1)$

$$f_{min}$$ is the start frequency (and $$λ_{max}$$ is the corresponding wavelength) and At is the magnitude related to the transmit power. Suppose that there is only a single small object situated at the distance of R0 to the radar but it is moving around R0, which results in a time-varying distance to the radar. Let us denote this time-varying distance by $$R(t) = R0 + x(t)$$ and x(t) is a function represents the distance variations around R0. Furthermore, the reflected wave off the object at the receiver is the delayed version of s(t) with a delay of $$t_d = 2 R(t)/c$$, which is the round-trip time of the wave. c is the light speed throughout the whole paper. Consequently, the IF signal for only a single chirp duration will be:

$\begin{split}y(t)&=s(t)×s^*(t-t_d)\nonumber\\ &=A_t A_r \exp\left(j(\phi(t)-\phi(t-t_d))\right),\quad t_d<t<T_r , \quad\quad (2)\end{split}$

The thermal noise and other channel considerations are ignored for simplifications, but $$A_r$$ has a relationship to $$A_t$$ by the radar equation 3. The beat signal, $$y(t)$$, can be expressed as follows:

$\begin{split}y(t)&=A_t A_r \exp\left(j(2\pi f_{min}t_d+2\pi Kt_d t-\pi Kt_d^2)\right)\nonumber\\ &\approx A_t A_r \exp\left(j(2\pi f_{min} t_d+2\pi Kt_dt\right)\nonumber\\ &=A_t A_r \exp\left(j(\psi(t)+\omega_bt)\right) , \quad t_d<t<T_r\end{split}$
$\begin{split}y(t)&\approx A_t A_r \exp\left(j(\psi(t) + \omega_b t)\right), \quad t_d<t<T_r \quad\quad (3) \\\end{split}$
$\psi(t) =4\pi \frac{R_0+x(t)}{\lambda_{max}} , \qquad \ \ \omega_b=4\pi \frac{KR_0}{c} \label{eq4}, \quad\quad (4)$

the second approximate equality in eq3 is obtained by ignoring the third term in the phase, which is very small. The third term is negligible because K is in $$10^{12} Hz/s$$ order while $$t_d$$ is in 1ns thus the term is in the order of $$10^{-6}$$. Equation (eq4) obtained after replacing $$t_d$$ to (eq3) and ignoring the $$x(t)t$$ term because t is in $$1\mu s$$ and x(t) is almost constant for one chirp as we will see later. Furthermore, $$\psi(t)$$ varies with x(t) relative to $$\lambda_{max}$$. So, the phase variations in the scale of the maximum wavelength can greatly change the beat signal phase. For example, a radar operating at 6 GHz is 10 times less sensitive in comparison to a 60 GHz radar. In addition, x(t) is almost constant within one chirp because subjects are not moving more than 1 mm per chirp equivalent to $$1 mm/1\mu s=10^3 m/s$$.

### Radar equation¶

If the $$P_t$$ is nominal transmit power and the target is at a distance of R, then the received power is related to the transmit power of a radar by the following:

$P_r = \frac{P_t G_t G_r \sigma \lambda^2 }{(4\pi)^3 R^4}, \quad \quad (5)$

where $$G_t$$, and $$G_r$$ are the transmit and the receive antenna gains, respectively. $$\sigma$$ is the RCS of the target. Also, $$\lambda$$ is the wavelength of the travelling wave. For physiological motion, the area under which the chest moves determines the RCS. Equation eq5 is known as the radar equation. We assume that the room temperature is 300 Kelvin (or about 25 Celsius) and the transmit chirp has a sweeping bandwidth of 4 GHz, then the noise power at the output terminal of the receiver antenna is $$P_n = 10 \: log_{10} (K T b_s) \approx -103 \: dB$$ in which K is Boltzmann’s constant *. In addition, if the receiver NF is NF dB and the minimum SNR required at the based band is denoted by $$SNR_{min}$$, then the minimum received signal power at the output port of Rx antenna will be $$P_{r,min} = NF + P_n + SNR_{min}$$ in which NF is the noise figure and all the variables are in dB. By using eq5, which relates $$P_r$$ to the target range, the maximum range versus minimum required SNR at the based band can be obtained:

$R_{max} = \sqrt[\leftroot{-1}\uproot{8}4]{\frac{P_t G_t G_r \sigma \lambda^2}{P_{r,min} (4\pi)^3}}, \quad P_{r,min} = NF + P_n + {SNR}_{min}, \quad\quad (6)$

Fig. 3: Maximum range of a target versus minimum required SNR at the based band. The gain of Tx/Rx antennas are the same as G.

In figure Fig. 3, $$R_{max}$$ is plotted versus minimum SNR with values annotated. In fact, $$\sigma = 0.39 m^2$$ is an approximate value for a human as mentioned in 4. The transmit power of 12 dBm is the output power of AWR1642/AWR1443 chips (see table ref{TIParams}).

*

$$K = 4.138 \times 10^{-23} \: \: J/K$$. Do not confuse this K with the chirp slope.

## Range detection¶

From eq3 and eq4, $$\psi(t)$$ can be approximated by sampling x(t):

$\psi(t)=4\pi \frac{R_0+x(t_0)}{\lambda_{max}} ,\qquad \omega_b=4\pi \frac{KR_0}{c}, \quad\quad (7)$

where $$t_0$$ is any time in $$[t_d,T_r]$$. This equation is used to detect the range of a subject, $$R_0$$. To this end, an FFT is applied over samples of a chirp to obtain the spectrum of the beat signal, which has peaks corresponding to the subjects at different ranges. This FFT reveals range information so it is called range FFT. Each range FFT bin represents a particular distance with an associated phase similar to $$\psi(t)$$. Furthermore, as we mentioned before, there can be a very small shift in $$\omega_b$$ due to residual delays incurred by the PA and the LNA. Although the little frequency shift exists, it diminishes after the radar warms up.

## Doppler or speed detection¶

From eq3, eq4, x(t) can be any function of time depending on the moving trajectory of the target with respect to the radar. Assuming a radial movement , so $$x(t)=vt$$ where v is the radial velocity. By substituting it to eq4 then to eq3, we have:

$y(t) \approx A_t A_r \exp\left( j \left(4\pi \frac{R_0}{\lambda_{max}} + 4\pi K \frac{R_0}{c} t + 4\pi K \frac{v}{c} t^2 + 4 \pi \frac{v}{\lambda_{max}}t \right) \right) \quad\quad (8)$

the first term in the exponent is constant, the second term is for the range, and the third term is very small due to the order of t. The last term in the exponent is desired since it is linear in time and is the function of v. Though, if we take the range FFT and look to specific range, we observe a signal like eq8 with only the last term in the exponent. Thus, by taking the second FFT on a range bin across a sequence of chirps, we obtain the spectrum of the range containing peaks corresponding to the target velocities .

Radial movement is a movement along the radial axis of a spherical coordinates with the radar at the centre.

The spectrum of the range bin should be scaled by $$\lambda_{max}/2$$ to convert the frequency to the actual m/s (see eq8).

## Angle of arrival detection¶

In a typical FMCW radar, the received signal from l’th receiver antenna and the k’th target can be expressed as:

$\boldsymbol{x}_{kl}(t_f, t_s) = b_{kl} \exp \left( -j \left( 2\pi f_b t_f +\tau_{kl}+\xi_{kl}+\Delta \psi_{kl}(t_f,t_s) \right) \right) \quad\quad (9)$

where $$b_{kl}$$ relates to the received power from the l’th virtual receiver and the k’th target, which depends on the target’s RCS and range, and the antenna gain in the direction from the wave is scattered and received at the antenna. $$f_b$$ is beat frequency corresponding to the range of the target. $$\tau_{kl}$$ is phase shift due to angle of arrivals and is a function of $$\theta_k$$ and $$\phi_k$$ corresponding to the azimuth and elevation angles (see Fig. 4). $$\xi_{kl}$$ is the constant phase representing the phase difference between the virtual receivers for k’th target due to manufacturing differences of the receiver channels and the angle of arrivals (or antenna pattern in the direction of the received wave) as well. Note that $$\xi_{kl}$$ is different than $$\tau_{kl}$$ as it is not a particular function of the arrival angles, but we can say something about $$\tau_{kl}$$ in terms of $$\theta_k$$ and $$\phi_k$$. Also, $$\psi_{kl}(t_f,t_s)$$ is the residual phase noise 5. Also, we did not included the thermal additive noise in eq9. The previous equation can be written in a matrix-vector form containing all the signal received by all virtual receivers if we ignore the residual phase noise. So,

$x_k (t_f,t_s )= \Gamma_k a_k (\theta_k, \phi_k ) s_k (t_f )+e_k (t_f,t_s ) \quad\quad (10)$

in which $$a_k (\theta_k, \phi_k )$$ is known as steering vector and $$\Gamma_k$$ is a diagonal matrix containing the complex coefficients of $$b_{kl} exp⁡(-j\xi_{kl} )$$. $$s_k (t_f )$$ has only range information if the targets are stationary. In fact, if they are moving, $$s_k (t_f )$$ will be depending on the target’s range and the velocity . If we assume that $$\Gamma_k$$ is the same for all k, then the received vector from all virtual antennas and all targets are:

$\boldsymbol{x}(t_f,t_s ) = \sum_{k=1}^K (\Gamma \boldsymbol{a}_k (\theta_k, \phi_k ) s_k (t_f ) + e_k (t_f,t_s )) = \Gamma A \boldsymbol{s} (t_f ) + e(t_f,t_s ) \quad\quad (11)$

where matrix A is:

$A = [\boldsymbol{a}(\theta_1, \phi_1), \boldsymbol{a}(\theta_2, \phi_2), \cdots, \boldsymbol{a}(\theta_K, \phi_K)] \quad\quad (12)$

and $$s(t_f )$$ is:

$\boldsymbol{x}(t_f) = [s_1(t_f), s_K(t_f), \cdots, s_K(t_f)]^T \quad\quad (13)$

and K is the number of targets and the noise vector has elements which are the accumulation the noises from each antennas. If there is coupling between the antenna elements, then a matrix C should be multiplied to the left of $$\Gamma$$ to account the coupling.

As previously mentioned, the steering vector has a specific relationship to the angle of arrivals. Fig. 4 helps to derive the relationship. The antennas are shown by the solid black dots. They are separated by dx and dy along x-axis and z-axis, respectively. The red arrows are along the direction at which the wave is received by the radar. It makes an angle of $$\phi$$ with xy plane and $$\theta$$ with y-axis in xy plane. The relative phase difference between the signal received from l’th receiver to the first receiver on the origin can be expressed as below if the antenna is on the x-axis:

$-\frac{2 \pi}{\lambda_{max}} l dx \sin⁡(\theta) \cos⁡(\phi) \quad\quad (14)$

and this is the following if the antenna is shifted up in z direction:

$-\frac{2 \pi}{\lambda_{max}} l_x dx \sin⁡(\theta) \cos⁡(\phi)+ \frac{2 \pi}{\lambda_{max}} l_z dy \sin⁡(\phi) \quad\quad (15)$

where $$l_x,\: l_z$$ are the shifts to the negative x and positive z, respectively. By knowing the antenna array configuration, we can obtain the steering vector $$a_k (\theta_k, \phi_k )$$.

Fig. 4: Steering vector construction based on the array configuration

## Range, Doppler, and angle of arrival minimum and maximum bounds¶

The minimum range detection can be expressed as $$c/2B$$ where B is the sweeping bandwidth 6 and the maximum detectable range is $$f_{b,max}c/(4K)$$. For the Doppler, the chirp duration $$T_c$$ is the sampling period in the slow-time and by the Nyquist theorem the maximum detectable Doppler frequency is $$f_s/2=1/(2T_c)$$. On the other hand, the minimum Doppler frequency is determined by the time duration within which the Doppler FFT is computed. For N-chirp FFT, corresponding to an N-point Doppler FFT, the minimum Doppler is $$1/(NT_c)$$. Furthermore, the minimum and maximum detectable angles are determined by the aperture size of the whole transmit-receive arrays and the spacing of the array elements, respectively. For an effective array aperture size A and spacing of d, the minimum angular detection is $$\lambda_{max}/A$$ and the maximum unambiguous angle is $$sin^{-1}(\lambda_{max}/(2d))$$.

Table 1: Target parameters and the minimum and maximum detectable values

Parameters

Range

Doppler $$^*$$

Angle of arrival

Minimum

$$\frac{c}{2B}$$

$$\frac{1}{NT_c}$$

$$\frac{\lambda_{max}}{A}$$

Maximum

$$\frac{f_{b,max}c}{4K}$$$$^{**}$$

$$\frac{1}{2T_c}$$

$$sin^{-1}(\frac{\lambda_{max}}{2d})$$

$$^*$$ The Doppler can be scaled to represent the velocity by a factor of $$\lambda_{max}/2$$.

$$^{**} \: f_{b,max}$$ is the ADC sampling rate.

## References¶

1

M. Alizadeh, G. Shaker, and S. Safavi-Naeini, “Experimental study on the phase analysis of FMCW radar for vital signs detection,” in 2019 13th European Conference on Antennas and Propagation (EuCAP), 2019, pp. 1–4.

2

S. Wang, A. Pohl, T. Jaeschke, M. Czaplik, M. Köny, S. Leonhardt, and N. Pohl, “A novel ultra-wideband 80 ghz fmcw radar system for contactless monitoring of vital signs,” in 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Aug 2015, pp. 4978–4981.

3

C. A. Balanis, Modern Antenna Handbook. John Wiley & Sons, Incorporated, google-Books-ID: Q9OgkQEACAAJ.

4

Amy Diane Droitcour. Non-Contact Measurement of Heart and Respiration Rates with a Single-Chip Microwave Doppler Radar. 2006.

5

M. C. Budge and M. P. Burt, “Range correlation effects in radars,” in The Record of the 1993 IEEE National Radar Conference, 1993, pp. 212–216.

6

P. Pahl, T. Kayser, M. Pauli, and T. Zwick, “Evaluation of a high accuracy range detection algorithm for FMCW/phase radar systems,” in The 7th European Radar Conference, Sep. 2010, pp. 160–163.