# Results of FMCW Simulation and Measurements

## Introduction

There is work going on in the ITU-R Joint Rapporteurs Group (JRG)
1A-1C-8B toward reviewing and modifying the -20 dB and -40 dB bandwidth
formulas for frequency modulated (FM) chirped radars, both pulsed and
continuous wave (CW). Bandwidth formulas have been deemed to
underestimate the bandwidths for frequency modulated continuous wave
(FMCW) radar. Ref. [^1] proposed a set of new formulas based on
theory.[^2] In order to provide an accepted difference between theory
and practical implementation the JRG determined that it is necessary to
include an implementation factor in all bandwidth formulas under
consideration. In order to determine the necessary implementation
factor, it was decided that various measurements of real systems need to
be taken. Since access to actual radars must be arranged months in
advance as well as expenses incurred to measure their emission spectrum,
as a starting point, the U.S. has undertaken a task to generate and
measure various types of modulated radar waveforms using an Agilent
Vector Signal Generator (VSG) and a RF measurement system described in
ITU-R M.1177. While most of the simulations were devoted to FM pulse,
there were a few FMCW simulations performed.

Specifically, for FMCW, it was determined to generate two sets of 3
different FMCW signals at the RF level and then measure them in the time
and frequency domain with a spectrum analyzer. The first set of signals
ensure phase continuity across all periods of the waveform. The second
set of signals ensure that the phase is not continuous across the
waveform.

The 3 different FMCW signals are the following:

1.  Triangular using realistic chirp bandwidth (B~c~) and forward chirp
    time (τ).

2.  Sawtooth using zero flyback (or as near zero as possible)

3.  Sawtooth using a realistic flyback.

The following table lists a matrix of parameters for the different
signals.

Table 1: FMCW Test Parameters

  -------------- -------------- -------------- -------------- ------------
  Signal \#      Bc (MHz)       τ (msec)       tfb (msec)     Continuous
                                                              Phase?

  FMCW1          1              10             10.0           YES

  FMCW2          1              10             0.01           YES

  FMCW3          1              10             0.10           YES

  FMCW4          1              10             10.0           NO

  FMCW5          1              10             0.01           NO

  FMCW6          1              10             0.10           NO
  -------------- -------------- -------------- -------------- ------------

## Results

The results of six VSG generated FMCW simulations and measurements are
shown in Figure 1 through Figure 6. Each figure contains a blue curve,
which is the Fast Fourier Transform (FFT) of the waveform, and a black
curve, which is the measured VSG output. The measurement bandwidth for
the simulations was 18 kHz, except for FMCW3 and FMCW6, which used 47
kHz. The title of each plot lists center frequency (Fo), chirp bandwidth
(B~c~), the forward chirp time (τ), and the return chirp time
(t~fb~).[^3] These are followed by the sampling frequency for the FFT
(F~s~ = 100 MHz for all simulations), followed by the 3, 20, 40, and 60
dB bandwidths of the VSG output. The second line of the title contains
3, 20, 40, and 60 dB bandwidths of the FFT. In addition to plotting
measured VSG data, Figure 1 through Figure 3 illustrate the FFT of the
FMCW waveform with continuous phase (CP) across the FM bandwidth. Figure
4 through Figure 6 illustrate the FFT of the FMCW waveform with
non-continuous phase (NCP) across the FM bandwidth.

![](media/image1.png){width="6.499305555555556in" height="2.5in"}

Figure 1. FMCW1

![](media/image2.png){width="6.499305555555556in" height="2.5in"}

Figure 2. FMCW2

![](media/image3.png){width="6.499305555555556in" height="2.5in"}

Figure 3. FMCW3

![](media/image4.png){width="6.499305555555556in" height="2.5in"}

Figure 4. FMCW4

![](media/image5.png){width="6.499305555555556in" height="2.5in"}

Figure 5. FMCW5

![](media/image6.png){width="6.499305555555556in" height="2.5in"}

Figure 6. FMCW6

One most important finding is that the VSG generates the signal in a
different way than described in Ref. Error: Reference source not found,
which results in continuous phase across all periods of the waveform,
regardless of the relationship between waveform period and center
frequency. This finding was deduced from the fact that measured
bandwidths were essentially identical between FMCW1 and FMCW4, FMCW2 and
FMCW5, and FMCW3 and FMCW6. In each case, the only difference between
these signal pairs is that one was supposed to be continuous-phase, and
the other was not. The simulation was set up attempting to achieve a
phase discontinuity of approximately -π/2 radians in FMCW4, FMCW5, and
FMCW6, by deviating slightly from the nominal center frequency of 4 GHz,
as shown in Table 2.[^4] The blue curves in Figure 4 through Figure 6
show what the FFT would look like if it had the expected
phase-discontinuity.

Table 2. Center frequencies used in an attempt to achieve a phase
discontinuity of --π/2

  ------- ----------- ----------- ------------- ------- ------- ----------------
          \'=/(2π)    n           Fo (MHz)      τ       t~fb~   Fo(τ+t~fb~)

  FMCW4   -0.26       80000000    4000.000013   10000   10000   80000000.26

  FMCW5   -0.25025    40040000    4000.000025   10000   10      40040000.25025

  FMCW6   -0.2525     40400000    4000.000025   10000   100     40400000.2525
  ------- ----------- ----------- ------------- ------- ------- ----------------

It is speculated that the phase-discontinuity could not be achieved
because the VSG used a voltage controlled oscillator (VCO) to generate
the FMCW. In this case the frequency deviation, as depicted in Figure 5
of Ref. Error: Reference source not found, is the input to a VCO whose
oscillator frequency is directly proportional to the input. In this case
a given period's starting phase is allowed to drift from period to
period as needed in order to preserve phase continuity.

Table 3 and Table 4 summarize the -20 dB and -40 dB bandwidth results of
this section, respectively. Data for these tables are taken primarily
from Figure 1 through Figure 6, except that the FFT data for FMCW4
through FMCW6, which assumed a phase discontinuity, was replaced by the
continuous-phase FFT data for FMCW1 through FMCW3.

The first column of the tables lists bandwidths based on the FFT. The
next column lists bandwidths based on equations taken from Ref. Error:
Reference source not found (these equations will be described shortly).
The next column lists bandwidths measured in the simulation. The last
three columns of the tables compare the VSG and FFT bandwidths, stated
in terms of percentages. As an example, "VSG to FFT" calculates (VSG --
FFT)/FFT.

Table 3. Various -20 dB Bandwidth Comparisons [^5]

  ----------------- -------- ---------- --------- ---------- -------- ----------
  Waveform/system   -20 dB   B(-20dB)   VSG (-20  VSG to     VSG to   B(-20dB)
                    based on with       dB)       B(-20dB)   FFT      to FFT
                    FFT      k~20~=1                                  

  FMCW1             1.0083   1.0541     1.0409    -1.25%     3.23%    4.54%

  FMCW2             1.0244   1.0541     1.0422    -1.13%     1.74%    2.90%

  FMCW3             1.0207   1.0541     1.1077    5.09%      8.52%    3.27%

  FMCW4             1.0083   1.0541     1.0411    -1.23%     3.25%    4.54%

  FMCW5             1.0244   1.0541     1.0431    -1.04%     1.83%    2.90%

  FMCW6             1.0207   1.0541     1.1081    5.13%      8.56%    3.27%

                                        mean:     0.93%      4.52%    3.57%

                                        stddev:   3.24%      3.18%    0.77%
  ----------------- -------- ---------- --------- ---------- -------- ----------

Table 4. Various -40 dB Bandwidth Comparisons [^6]

  ----------------- -------- ---------- --------- ---------- -------- ----------
  Waveform/system   -40 dB   B(-40dB)   VSG (-40  VSG to     VSG to   B(-40dB)
                    based on with       dB)       B(-40dB)   FFT      to FFT
                    FFT      k~40~=1                                  

  FMCW1             1.0263   1.1021     1.0611    -3.72%     3.39%    7.39%

  FMCW2             1.1563   1.1021     1.2344    12.00%     6.75%    -4.69%

  FMCW3             1.0954   1.1021     1.1972    8.63%      9.29%    0.61%

  FMCW4             1.0263   1.1021     1.0613    -3.70%     3.41%    7.39%

  FMCW5             1.1563   1.1021     1.2337    11.94%     6.69%    -4.69%

  FMCW6             1.0954   1.1021     1.2001    8.89%      9.56%    0.61%

                                        mean:     5.67%      6.52%    1.10%

                                        stddev:   7.41%      2.70%    5.41%
  ----------------- -------- ---------- --------- ---------- -------- ----------

Ref. Error: Reference source not found sets forth a methodology for
determining the -20 and -40 dB bandwidth formulas for FMCW. Initially,
there were three sets of bandwidth formulas proposed depending on the
phase discontinuity. Since this simulation has shown that phase is
always continuous, for this exercise, the bandwidth pair in that paper
corresponding to a continuous phase signal is used in the tables:

  ----------------------------------------------------------------- -----
  ![](media/image7.wmf)                                             ()

  ----------------------------------------------------------------- -----

  ----------------------------------------------------------------- -----
  ![](media/image8.wmf)                                             ()

  ----------------------------------------------------------------- -----

where B~c~ is the total frequency deviation, τ is the time over which
the main chirp occurs, and k~20~ and k~40~ are the implementation
factors to account for accepted differences between theory and real
system output.

Neither () nor () are functions of the flyback time t~fb~. In fact a
smaller t~fb~ will produce a larger bandwidth, but the formulas already
assume a small t~fb~. Thus one would expect the formulas to overestimate
the bandwidth somewhat for larger t~fb~/τ ratios.

## Discussion

The first thing to note is that the resolution of VSG measurements is
not small enough. For example, the difference between VSG and FFT
bandwidths is only 1 to 4 measurement bandwidths. For this reason the
contributors of this document hope to redo the measurements for FMCW1,
FMCW2, and FMCW3 in the recommended measurement bandwidth of 1 kHz (see
Ref. Error: Reference source not found). The following discussion points
will be revised if warranted by the new findings.

As expected, B(-20dB) to FFT in Table 3 showed the largest
overestimation (4.54%) for the signal which has the largest t~fb~/τ
ratio (unity), namely, FMCW1. FMCW3 has the next largest t~fb~/τ (0.01),
and the next largest overestimation (3.27%) of B(20dB) to FFT. This same
finding holds for B(-40dB) to FFT in Table 4. This simply confirms that
B(-20dB) and B(-40dB) are based on the FFT.

The tables also show that comparison of VSG to FFT is positive in every
case, suggesting that there may be some sort of spectrum re-growth
occurring in the VSG implementation. It is suspicious, though, that VSG
to FFT is highest with the VSG signal that used the larger measurement
bandwidth, namely FMCW3 and FMCW6. Since this measurement bandwidth is
47 times larger than that recommended in Ref. [^7], the VSG measurements
are likely larger than true, and if that is the case, that would make
VSG to FFT comparisons in the table larger than true, especially for
FMCW3 and FMCW6. If we discard the value for those two signals, we are
looking at a VSG to FFT of up to 3% for the -20 dB bandwidth, and 6-7%
for the -40 dB.

The main purpose of this study has to do with evaluating bandwidth
formulas to make sure the proposed formulas are a good approximation of
theory (FFT). It is not the purpose of this paper to propose
implementation factors, but rather to show one possible way of how they
might be derived. The VSG to B(-20dB) column adds valuable input for
this purpose because, if B(-20dB) underestimates VSG, it plus one is the
factor by which B(-20dB) would be multiplied to equal VSG. In this case
all values are slightly negative (\~ -1%) indicating no underestimation,
with the exception of FMCW3 and FMCW6, whose values may be erroneously
positive due to enlarged VSG as described in the previous paragraph.

The other two bandwidth comparisons can be useful in making the final
determination of the implementation factor. In each case the theory
(FFT) is in the denominator, because it must be the baseline. Ultimately
the implementation factor must be tied to theory, because logically it
describes the accepted amount by which real system implementations can
exceed theory. But to arrive at an implementation factor to be applied
to a bandwidth formula, which is based on an accepted implementation
above and beyond theory, will require some compromise, and more thought,
hopefully articulated in another future paper.

At this point we remind the reader that the implementation factor is
heavily dependent on the output device, amongst other things. Noting
that the VSG is a solid-state device, as well as low power, and that
other output devices (tube type for example) have their own
characteristics, the final implementation factor may differ from those
indicated in this report. Actual measured data of real systems could
also shed more light on what is a realistic implementation factor. Other
administrations are encouraged to review these equations and present
data based on different output devices in tables similar to Table 3 and
Table 4.

For the -40 dB bandwidth, the largest VSG to B(-40dB) is 12% for FMCW2
and FMCW5, which both have the smallest t~fb~/τ ratio. This is expected
since the B(-40dB) formula, which is equation (), is independent of
t~fb~, whereas the actual signal (VSG) would be expected to be largest
when t~fb~ is smallest.

There are many factors that could influence the implementation factor.
Output device is just one. Another factor is the quality of the
electronics that go into shaping the signal. If these were the only
factors, we could speculate that the implementation factors would be
much larger than those indicated by the VSG. But there are other
hardware aspects of real radar systems, such as filtering, and others,
which may allow real systems to use the spectrum more efficiently than
the VSG. In either case, additional input of real system data from
administrations could give a better estimate of the implementation
factor.

The following figures show the measured frequency deviation for each
signal.

![](media/image9.png){width="6.499305555555556in" height="2.5in"}

Figure 7. FMCW1 Frequency deviation

![](media/image10.png){width="6.499305555555556in" height="2.5in"}

Figure 8. FMCW2 Frequency deviation

![](media/image11.png){width="6.499305555555556in" height="2.5in"}

Figure 9. FMCW3 Frequency deviation

![](media/image12.png){width="6.499305555555556in" height="2.5in"}

Figure 10. FMCW4 Frequency deviation

![](media/image13.png){width="6.499305555555556in" height="2.5in"}

Figure 11. FMCW5 Frequency deviation

![](media/image14.png){width="6.499305555555556in" height="2.5in"}

Figure 12. FMCW6 Frequency deviation

## Summary

This paper proposes a format whereby bandwidth formulas and associated
implementation factors for VSG generated FMCW waveforms can be
evaluated. It compares the output of a particular device with specific
FMCW -20 dB and -40 dB bandwidths. Other administrations are invited to
review this paper, and to compare their own measured data with the FMCW
bandwidth formulas proposed in this paper. More work also needs to be
done to determine what the implementation factors are, and how to apply
them.

[^1]: References:

    ^?^ "Theory of FMCW Radar Waveforms"

[^2]: This paper assumes a familiarity with Ref. Error: Reference source
    not found, in which various terms have already been defined.

[^3]: All times are in microseconds. All frequencies are in Megahertz

[^4]: See equation (12) in Ref. Error: Reference source not found for
    more understanding.

[^5]: For the equation () calculation k~20~ was set to one to
    essentially remove it from the equation. This was done because the
    first step in determining the necessary implementation factor is to
    see how the formula, without an implementation factor, compares with
    bandwidths derived by other means.

[^6]: For the equation () calculation k~40~ was set to one to
    essentially remove it from the equation. This was done because the
    first step in determining the implementation factor is to see how
    the formula, without an implementation factor, compares with
    bandwidths derived by other means.

[^7]: Table 1 of NTIA Report TR-05-420