FMCW Range Radar: working
principle and application
By Jackson Z
AIM
This essay aims to display the progress I made when investigating with
Multidimensional photonics laboratory in Wuhan Optoelectronics National
Research Center on using the FMCW radar to measure the distance. During the
research, I understood the working principle of the radar and use this technology to
carry out an experiment. The laboratory also show me the present application and
future improvement about the FMCW range radar.
INTRODUCTION
FMCW stands for Frequency Modulated Continuous Wave. FMCW range radar
belongs to laser ranging radar, which was first been invented by American scientist
in 20 century. Unlike most of the laser radar depend on measuring the time take for
the laser to reach the object to be measured and reflect back, FMCW range radar
will record the difference between the frequency of the emergent light and the
reflect light to get a beat note signal. Then get the distance of the object from the
signal and the phase difference of the 2 waves.
RELATED TERMS
Since the FMCW range radar is not included in the IB syllables, many unfamiliar
terms can appear frequently during my investigation. So I list these terms below
and give definition base on my own understand.
1. Pulse width：it is the width of the pulse signal. Can be seen as the time period
from the start of the signal to it’s end. ( Unit s,ns )
2. Pulse width period：it is the time period between 2 pulse signal.
3. Frequency band width：it is the gap between the highest and lowest frequency of
a signal. （ Unit HZ )
4. Optical frequency：it is the frequency of the vibration of the light wave.
5. Fourier transform: under certain condition, we can represent a function by the
combination of several trigonometric function or their integration function.
6. Time domain: is to describe the change in signal as time passing.
7. Frequency domain: waveform in time domain get through Fourier transform will
become waveform in frequency domain. This domain shows the relationship
between the frequency and amplitude of the signal.
WORKING PRINCIPLE
The structure of the FMCW range radar can be divided to 3 parts: launch sector,
receive sector and signal processing sector. The general radar system is shown
below:
Wave source
Light guide
system
Divider
Collimator
Light path
Reflected
light
Emergent
light
Mixer
Light
detector
Signal
processing
Object
As the system shows, the wave source(usually a chip or an oscillator) will send out
continuous electric or light wave which has specially controlled launch frequency.
This wave will be divided by the divider in to 2 parts. One part will get in to the
mixer directly while the other part will get through the light guide and collimator
then shot toward the object. The wave will reflect back by the object and received
by the radar and get into the mixer from the light guide system. The frequency of
the emergent and reflected wave will be mixed to produce 2 new sine waves. One
of the sine wave has the frequency of the difference between the frequency of the
emergent and reflected wave, which can be seen as the beat note signal for signal
processing system to produce relevant graph for distance determine.
Some of the instructions above may seem confusing but actually essential, so I list
them out to give detailed explanation.
1. Why should the frequency of the emergent and reflected wave be mixed？
Because during the process of reaching the object and reflect back to the radar, the
energy of the wave can descend a lot. By mixing the strong emergent wave with it,
the
2. Why take the sine wave that has frequency of the difference between the
frequency of the emergent and reflected wave as the beat note signal?
Because the FMCW range radar can’t process the signal with too high frequency,
this choice will make the result more accurate.
CACULATION PROCESS
When controlling frequency of the radar signal, we usually use another wave to
combine with the signal that we want to transcend, the frequency of these 2 waves
will be mixed to produce a new wave. In this process the sawtooth wave is the
common choice since the application of this wave can make the formation of the
waveform easier.
The frequency of this compound wave send by the FMCW range radar will increase
as the time passing on, the amplitude and time graph of the wave is shown below
to make this increase visible.
And the math formula of a wave signal is:
j  2ft  
j t  
S

Ae

Ae
E
E1.
Where A is the amplitude of the signal and 2ft    is the phase of the signal. To
get the formula of the sawtooth wave, we need to introduce the frequency time
graph of it:
(the graph is provided by Multidimensional photonics laboratory）
As the graph shows, the formula of the frequency f of the wave vary with the time t
is
f  kt  f 0 where k is the gradient of the graph. What we need to do here is to
find the change in
 otherwise is the change in frequency of the wave in time
period. This require us to calculate the integration of the equation
t
E2.
 kt  f 0 dt 
0
f  kt  f 0
1 2
kt  f 0t
2
And we then take the change in frequency in to the equation of the
S E to get
the formula of the wave emerge from the radar:

S

Ae
E
E3.
j 2f 0t kt 2 

We assume the time taken for the wave to reflect and receive by the radar is �1 , the
distance between the radar and the object is d. So the signal of the reflected wave
can be seen as the emergent wave delay time t1:
E4.
S R t   S E t  t1 
So we can quickly write down the formula for the reflected wave, just need to
replace the t to t  t1  :

S

Ae
R
E5.
j 2f 0 t t1 k t t1 2 

Like we said above, the emergent and reflected light frequency will be mixed
together. This process can actually be seen as the product of these 2 wave signal,
which requires the introduce of the product to sum formula to work out the answer.
E6.
cos  cos  
1
 cos     cos   
2
Taking E6 to the emergent and reflect waves we need to mix, the new equation is
shown below:
cos1t  1  cos2t   2  
1
 cos1t  2t  1   2   cos1t  2t  1   2 
E7. 2
We can see that through the formula, our mixed signal is turned to 2 new signals
with different frequency. As the explanation above, we will choose the lower
frequency one:
E8.
E9.
S IF 
cos    cos1t  2t  1   2 

2
2
S IF  S E S R
So we can use the formula of the emergent and reflected wave to represent
S IF
S IF  Ae j 2f 0t1  2ktt1 kt1  
2
E10.
And this signal will be our beat note signal.
So the phase of the signal will be :
2f t  2ktt kt  
01
1
2
1
E11.
If we take the differentiation of the equation above, we can get the frequency of the
beat note signal vary with time:
E12.
f IF  2kt1 
1
 kt1
2
Then what we need is take this signal into a equation to calculate the distance. I will
show the derivation of the equation here but actually all the calculation is finished
by the computer when conducting the experiment. We can first have the equation
below:
E13. t1 
2d
c
Where c is the velocity of the wave. If we take the E13 into E12, what we get is:
E14. f IF 
2dk
c
E15. d 
cf IF
2k
We should remember in all the equations above, k is the gradient of the sawtooth
wave in graph1, which is a ft graph. So the gradient k can also represent by dividing
the pulse width of the wave by the band width:
E16. k 
B
T
Where T is the pulse width of the wave, unit s while B is the band width of the
wave,unit Hz. So the final equation for the distance calculation is :
E17. d 
cf IF T
2B
EXPERIMENT APPARATUS
All the apparatus are provided by Multidimensional photonics laboratory
1. Wave source: TPS-400 radar source with wave frequency 193.4THz, wave length
1550 nm, power 10.0dBm
2. Divider: it will divide the emergent wave into 2 parts, one with 1% strength of the
original wave and waited to be mixed with reflected wave, one with 99% strength of
the original wave and will be emit towards the object.
3. 3 port fiber circulator: act as light guide system, it guide the reflected wave into
the mixer.
4. Collimator: to adjust and get the divergent wave together and shot toward the
object accurately.
5. Mixer: to mix the emergent and reflected signal together to produce beat note
signal.
6. Oscilloscope: it will show the frequency of the beat note frequency against dB.
7. Photo diode: act as detector to receive the reflected light wave.
8. Diastimeter: it will help us to measure the actual distance from the object to the
collimator of the radar.
9. Mirror: act as object.
So we set up the apparatus, connect the oscilloscope with the mixer through wire,
make sure the collimator is in a straight line with the center of the mirror. Us
diastimeter to measure the actual distance from object to the collimator of the
radar, record as distance 1. Then we can open the radar, set up the band width and
pulse width of the wave on the radar source. Using the frequency of the beat note
signal shows on the oscilloscope to calculate the measured distance from object to
collimator of the radar, record as distance 2.
Variables
Controlled variables:
controlled
variable
method of control
possible effect on result
object
humidity
Wave source
Use the same lens as the
Some object or lens can reflect the light wave and
object
in
every
weakening the signal.
experiment
Use air conditioning to water particles in the environment can prevent the
control
light wave from traveling and can weakening the signal.
Use the same radar
Different wave source can emit light wave with different
source
in
every
power, make the final not beat signal different.
experiment
Table1: controlled variables in the experiments
DATA ANALYZING
For the graph below, distance 1 is the actual distance between the object and the
collimator of the radar while the distance 2 is the distance we get by using the
FMCW range radar to measure. During the experiment the velocity of the wave is
c  3  108 m / s , band width is B  7.9  1012 Hz and pulse width is T  130 s .
This is a screenshot from the oscilloscope during one of our experiment. We can
see in the picture above, the yellow graph is the beat note signal we send with
constant frequency, but how dose the purple line come? Here we need to introduce
the Fourier transform. That is for any signal which can be integral in time domain
will change to frequency domain through the equation below:
F  


f t e it dt


-

  f t cost dt  i 
f t sin t dt
Where the �� is the signal in frequency domain, �� is the signal in time domain
and ω is the frequency. So what we do to the yellow line is we choose different ω
and take it into the equation above to get the amplitude of the signal in frequency
domain. Then we can plot the amplitude against the frequency to get the purple
line. We should find the largest value of amplitude on the purple line which means
the frequency of the signal gives this amplitude takes up the most in all frequency.
So this frequency will be our beat note signal frequency ���
experiment
s
1
2
3
4
5
6
7
8
9
object
distance 1/m
f IF /kHz
mirror1
mirror1
mirror1
mirror1
mirror1
mirror1
mirror1
mirror1
mirror1
2.394
2.394
2.394
2.419
2.419
2.419
2.444
2.444
2.444
1.660
1.630
1.621
1.767
1.718
1.718
1.650
1.640
1.679
Table 2: experiment factors and collected data.
Take experiment 1 as example:
dis tan ce1 
cf IF T 3  108 m / s  1.660 Hz  130 s

 2.049m
2B
2  7.9  1012 Hz
distance
2/cm
2.049
2.012
2.000
2.181
2.120
2.120
2.036
2.024
2.072
uncertainty
%
14.41
15.96
16.46
9.839
12.36
12.36
16.69
17.18
15.22
uncerta int y 
d1  d 2
d1

2.394m  2.049m
 100%  14.41%
2.394m
REFLECTION
We can see from the table 2 that there still has uncertainties in our measured data.
Such uncertainties can be a result of the several factors below:
1. Even if we set up a constant power for the emergent light wave on the radar
source, the power of the wave can still vibrate during the long experiment, which
can make the
2. The beat note signal shown on the graph is hard to read a accurate value, so
most of the time we can only take average to make the calculation, which introduce
the uncertainty.
3. When using the equation to calculate the frequency, we often preserve to 3
significant figure for the result so there can have difference between measured
distance 1 and distance 2.
DEVELOPMENT
Comparing to other laser radar, FMCW range radar has the advantage of burst
transmission, hard to intercept signal and require no physical contact with the
object. This technology has been used widely in military radar, pilotless automobile
and automatic production. During the investigation in the Multidimensional
photonics laboratory, I also leaned that the researchers there begin to design a chip
to replace the electric signal send by the FMCW radar by light signal and try to
concentrate the whole system except the wave source and oscilloscope in to this
chip which no larger than a stamp. In this case the radar system will be more
portable and convenient to set up. The antijamming capacity of the radar will also
improve since many machines are no longer expose in the air.
REFERENCE
1. 杨牧，《FMCW 雷达交通目标检测与识别方法研究》
2. 聊冉，《基于 FMCW 的高分辨率激光雷达测距技术研究与实现》
3. 邱浩铖，《基于连续调频波的激光雷达测距测速性能研究》