Linear FM Pulsed Radar Waveform

Radar Simulation

And Measurement Results

[Table of Contents]{.underline}

[1. Introduction 3](#introduction)

[1.1. Background 3](#background)

[2. Pulsed FM Chirped 4](#pulsed-linear-fm-chirped)

[2.1.1. FM Pulse Waveforms for Generation and Measurement
5](#fm-pulse-waveforms-for-generation-and-measurement)

[2.2. Proposed -40 dB Bandwidth Equation
8](#proposed--40-db-bandwidth-equation)

[2.3. Proposed -20 dB Bandwidth Equation
8](#proposed-lfm--20-db-bandwidth-equation)

[3. Chirped Simulation and Measurement Results
9](#lfm-chirped-simulation-and-measurement-results)

[3.1. System Measurements 10](#system-measurements)

[3.2. VSG Measurements 13](#vsg-measurements)

[3.3. Implementation Factor 17](#implementation-factor)

[4. Summary Results 17](#summary-results)

[5. References 18](#references)

Table of Figures:

[Figure 1 - Trapezoidal Pulse Model. 5](#__RefHeading___Toc148753543)

[Figure 2 - Linear FM chirp 5](#__RefHeading___Toc148753544)

[Figure 3 -- Plot of System 1, VSG Measurement, FFT Theory, and Proposed
Equation -40 dB Bandwidths. 13](#__RefHeading___Toc148753545)

[Figure 4 - Plot of System-2, VSG Measurement, FFT Theory, and Proposed
Equation -40 dB Bandwidths. 13](#__RefHeading___Toc148753546)

[Figure 5 - Plot of System 3, VSG Measurement, FFT Theory, and Proposed
Equation -40 dB Bandwidths. 14](#__RefHeading___Toc148753547)

[Figure 6 - Plot of System 4, VSG Measurement, FFT Theory, and Proposed
Equation -40 dB Bandwidths. 14](#__RefHeading___Toc148753548)

[Figure 7 -- VSG Measurement, FFT Theoretical, and Proposed -40 dB
Bandwidth of CP23. 18](#_Ref148323994)

[Figure 8 - VSG Measurement, FFT Theoretical, and Proposed -40 dB
Bandwidth of CP11. 18](#__RefHeading___Toc148753550)

[Figure 9 - VSG Measurement, FFT Theoretical, and Proposed -40 dB
Bandwidth of CP12. 19](#__RefHeading___Toc148753551)

Table of Tables:

[Table 1 - LFM Pulse Parameters for Waveform Simulations and
Measurements. 6](#_Ref147037436)

[Table 2 -- Measured System, VSG Measurement, FFT Theory, and Proposed
Equation -40 dB Bandwidths. 10](#__RefHeading___Toc148324187)

[Table 3 - Measured System, VSG Measurement, FFT Theory, and Proposed
Equation -20 dB Bandwidths. 11](#__RefHeading___Toc148324188)

[Table 4 -- VSG Measurements, Proposed -40 dB Equation (EQ-1), and FFT
Theory Bandwidth Comparison 15](#__RefHeading___Toc148324189)

[Table 5 - VSG Measurements, Proposed -20 dB Equation (EQ-5), and FFT
Theory Bandwidth Comparison 15](#__RefHeading___Toc148324190)

#  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 pulsed Linear FM chirped radars. Since access to actual radars must
be arranged months in advance as well as expenses incurred to measure
their emission spectrum, the United States took on the task to generate
and measure various types of modulated radar waveforms to help progress
the work in the JRG. These radar waveforms were generated with an
Agilent Vector Signal Generator (VSG) at the Radio Frequency (RF) level,
and measured with a system described in ITU-R M.1177 to produce emission
bandwidth plots of frequency versus power.

It is important to note that, in order to provide an accepted difference
between theory and practical implementation, the JRG has determined that
it is necessary to include an implementation factor in all bandwidth
formulas under consideration that are based on theory alone. Thus, in
order to determine the necessary implementation factor, it was decided
that various measurements of real systems need to be taken. The United
States has a small library of emission measurements of various radar
systems that use either chirped or phase coded modulation schemes. These
measurements were performed over the years to support other programs and
have a large enough dynamic range tuned away from the fundamental to
"see" artifacts in the emissions. The measurements are for systems
employing various output devices. Additional emission measurements will
be done when systems are available.

By comparing available measured radar emissions with the waveforms
generated by the VSG (programmed to emulate the real radar as closely as
possible) and the proposed equations, the JRG will be able to
investigate how the proposed -20 and -40 dB bandwidth equations for
these systems compare to real systems and how the implementation factors
also affect the emissions. Note that the VSG does have limitations for
the rise and fall times (30 nanoseconds), chirp rate (80 MHz/µs), and
chirp length (80 MHz). Also, note that with measured data, the plots
have been normalized to show the frequency response referenced to 0 MHz.
Finally, the reader should recognize the VSG operating at a relatively
low RF output level does not necessarily represent the performance of
radar transmitters operating at very high power levels.

## Background

The ITU tasked members of Study Groups 1A, 1C, and 8B to investigate the
emission roll-offs of radars. However, once the JRG began their task,
the members realized that the -20 and -40 dB bandwidth equations are
closely linked to the emission roll-offs. The current -20 and -40 dB
bandwidth formulas in the Recommendation ITU-R SM.1541 overestimate the
spectrum for high time-bandwidth products, therefore it was determined
that these equations need to be addressed before the investigation of
the emission roll-offs from the -40 dB bandwidth.

# Pulsed Linear FM Chirped

**Error: Reference source not found** defines the trapezoidal pulse
envelope under consideration. Shown in the figure are the 50% pulse
width, the 100% rise and fall times, and total pulse width. Figure 2
shows that the chirp bandwidth is B~c~, that the chirp is linear
(up-chirp), and occurs over the full base of the pulse width. These were
the parameters that were used to program the chirp into the VSG.

![](media/image1.png){width="4.622916666666667in"
height="3.467361111111111in"}

[]{#__RefHeading___Toc148753543 .anchor}Figure 1 - Trapezoidal Pulse
Model.

![](media/image2.png){width="4.810416666666667in"
height="2.8381944444444445in"}

[]{#__RefHeading___Toc148753544 .anchor}Figure 2 - Linear FM chirp

## FM Pulse Waveforms for Generation and Measurement

The following FM pulse signals were generated at the RF level and
measured in the time and frequency domain with a measurement system
described in ITU-R M.1177. The data in, ASCII file format, was then
plotted.

Table 1 also lists a matrix of parameters for the different chirped
signals. consists of systems described by various combinations of
parameters (pulse width, chirp length, chirp rate, rise time, fall
time). The systems that were measured in the field are denoted by an \*
in the file name. The list is not exhaustive, but does contain systems
with a wide variety of time bandwidth products and other system
parameters. Many of the pulses that did not come from real systems came
from References 2 and 3, the texts by Merrill Skolnik and Eli Brookner.

[]{#_Ref147037436 .anchor}

**Table 1 - LFM Pulse Parameters for Waveform Simulations and
Measurements.**

  -------------------------------------------------------------------------------------------------------------------------------
  **Number**   **Nomenclature**   **50%    **0-100%**   **0-100%**   **Chirp     **Chirp Rate **Compression   **Pulse   **Pulse
                                  Pulse                              Deviation   (MHz/us)**   Ratio**         Rise Time Fall Time
                                  Width                              (Chirp                                   to Pulse  to Pulse
                                  (us)**                             Bandwidth                                Width**   Width**
                                                                     in MHz)**                                          
  ------------ ------------------ -------- ------------ ------------ ----------- ------------ --------------- --------- ---------
                                           **Pulse Rise **Pulse Fall                                                    
                                           Time (us)**  Time (us)**                                                     

  1            CP1                1500     12.5         37.5         0.375       0.00025      562.5           0.67%     2.00%

  2            CP2                35       0.25         0.75         1           0.0286       35              0.57%     1.71%

  3            CP3[^1]            80       0.0375       0.0375       1           0.0125       80              0.04%     0.04%

  4            CP4[^2]            200      0.875        1.25         5           0.025        1000            0.35%     0.50%

  5            CP5\*              51.2     0.1875       0.1875       1.25        0.0244       64              0.29%     0.29%

  6            CP6                24.8     0.1875       0.1875       2           0.0806       49.6            0.60%     0.60%

  7            CP7\*              245      2            2            2           0.0082       490             0.65%     0.65%

  8            CP8                5        0.125        0.125        2.04        0.408        10.2            2.00%     2.00%

  9            CP9\*              25.6     0.1875       0.1875       2.89        0.1129       73.98           0.59%     0.59%

  10           CP10^2^            10       0.125        0.125        3           0.3          30              1.00%     1.00%

  11           CP11^2^            100      0.0625       2.5          3           0.03         300             0.05%     2.00%

  12           CP12               89       0.875        1.25         4           0.0449       356             0.79%     1.12%

  13           CP13               12.48    0.125        0.125        4.2         0.3365       52.42           0.80%     0.80%

  14           CP14\*             150      0.1875       0.1875       1.3         0.00867      195             0.07%     0.07%

  15           CP15^2^            17       0.125        0.125        10          0.5882       170             0.59%     0.59%

  16           CP16               5.4      0.125        0.125        11.2        2.0741       60.48           1.85%     1.85%

  17           CP17^2^            70       0.1875       0.1875       14          0.2          980             0.21%     0.21%

  **Number**   **Nomenclature**   **50%    **0-100%**   **0-100%**   **Chirp     **Chirp Rate **Compression   **Pulse   **Pulse
                                  Pulse                              Deviation   (MHz/us)**   Ratio**         Rise Time Fall Time
                                  Width                              (Chirp                                   to Pulse  to Pulse
                                  (us)**                             Bandwidth                                Width**   Width**
                                                                     in MHz)**                                          

  18           CP18[^3]           7.5      0.125        0.125        40          5.3333       300             1.33%     1.33%

  19           CP19               1        0.625        0.625        50          50           50              50.00%    50.00%

  20           CP20^3^            30       0.125        0.125        50          1.6667       1500            0.33%     0.33%

  21           CP21^3^            33       0.0375       0.0375       60          1.8182       1980            0.09%     0.09%

  22           CP22[^4]           1        0.0375       0.0375       80          80           80              3.00%     3.00%

  23           CP23               5        0.0625       0.0625       80          16           400             1.00%     0.90%

  24           CP24^4^            125      1.25         1.25         80          0.64         10000           0.80%     0.80%

  25           CP25^3^            20       0.025        0.025        100         5            2000            0.10%     0.10%

  26           CP26               25       0.025        0.0075       640         25.6         16000           0.1%      0.03%

  27           CP27               5.5      0.0125       0.0125       0.75        0.136        4.125           0.227%    0.227%

  28           CP28[^5]           5        1            1            1.0         0.2          5               20%       20%
  -------------------------------------------------------------------------------------------------------------------------------

## Proposed -40 dB Bandwidth Equation

When the following proposed LFM -40 dB equation (**Equation 1**) is
compared to the theoretical FFT, the result is within an error margin of
9.0%. It is important to note that there is no added margin or
implementation factor included in the formula. However, once the JRG
investigates and determines the appropriate value for the margin or
implementation factor, that value needs to be added by utilizing the
constant parameters R and N. The proposed B~-40dB~ is given by:

$$B_{- \text{40}\text{dB}_{\text{New}}} = R \ast B + N \ast \sqrt{\pi} \ast \lbrack\text{ln}(B \ast \tau)\rbrack^{0\text{.}\text{53}} \ast \lbrack\text{Min}(B_{\text{rise}},B_{\text{fall}},B_{\text{rise} \land \text{fall}}) + \text{Max}(B_{\text{rise}},B_{\text{fall}},B_{\text{rise} \land \text{fall}})\rbrack$$

Equation 1

Where,

  ------------------------------------------------------------------------------------------------------- --------------------------------------------- ------
  $$B_{\text{rise}} = \frac{1}{\sqrt{\tau \ast t_{r}}}$$                                                  To account for the rise time                  EQ-2

  $$B_{\text{fall}} = \frac{1}{\sqrt{\tau \ast t_{f}}}$$                                                  To account for the fall time                  EQ-3

  $$B_{\text{rise}_{\text{fall}}} = \frac{1}{\left( \tau \ast t_{r} \ast t_{f} \right)^{\frac{1}{3}}}$$   To account for both the rise and fall times   EQ-4
                                                                                                          combination                                   

                                                                                                          Total Pulse length (us) including rise and    
                                                                                                          fall times                                    

  t~r~                                                                                                    100% Pulse rise time (us)                     

  t~f~                                                                                                    100% Pulse fall time (us)                     

  B                                                                                                       LFM bandwidth (MHz)                           

                                                                                                          3.141592654.....                              

  R                                                                                                       1 (Constant reserved for possible future use  
                                                                                                          or more refinement)                           

  N                                                                                                       1 (Constant reserved for possible future use  
                                                                                                          or more refinement)                           
  ------------------------------------------------------------------------------------------------------- --------------------------------------------- ------

From the calculated results it was determined that the condition of
$B \ast \text{MIN}(t_{r},t_{f}) > 0\text{.}1$ is necessary to obtain
required results.

It is important for the reader to note that there is a condition imposed
on the formula where the chirp bandwidth multiplied by either the rise
or fall time of the pulse must exceed 0.10. For smaller values than
0.10, the formula will not provide good results. Also for Bτ product
(compression ratio) less than 10 the formula is not valid.

## Proposed LFM -20 dB Bandwidth Equation

The proposed LFM -20 dB bandwidth equation is shown in Equation 5. It
was also compared to the theoretical FFT results. As in the case of the
LFM --40 dB bandwidth equation, there are no margin or implementation
factors included in the formula. However, once the JRG investigates and
determines the appropriate value for the margin or implementation
factor, those values need to be added to the equation by utilizing the
constant parameter W and V. The minimum Bτ product for this formula to
be valid is 10. This equation has been shown to be within an error
margin of +/- 12% when compared with the FFT theoretical -20 dB
bandwidth. The proposed new B~-20dB~ is given by:

$$B_{- \text{20}\text{dB}_{\text{New}}} = W \ast B + V \ast \lbrack\text{ln}(B \ast \tau)\rbrack^{0\text{.}\text{37}} \ast \text{Min}(B_{\text{rise}},B_{\text{fall}},B_{\text{rise} \land \text{fall}}) + \frac{\text{Max}(B_{\text{rise}},B_{\text{fall}},B_{\text{rise} \land \text{fall}})}{(B \ast \tau)^{1\text{.}5}}$$

Equation 5

Where,

  ------------------------------------------------------------------------------------------------------- --------------------------------------------- ------
  $$B_{\text{rise}} = \frac{1}{\sqrt{\tau \ast t_{r}}}$$                                                  To account for the rise time                  EQ-6

  $$B_{\text{fall}} = \frac{1}{\sqrt{\tau \ast t_{f}}}$$                                                  To account for the fall time                  EQ-7

  $$B_{\text{rise}_{\text{fall}}} = \frac{1}{\left( \tau \ast t_{r} \ast t_{f} \right)^{\frac{1}{3}}}$$   To account for both the rise and fall times   EQ-8
                                                                                                          combination                                   

                                                                                                          Pulse length (us) including rise and fall     
                                                                                                          times                                         

  t~r~                                                                                                    Pulse rise time (us)                          

  t~f~                                                                                                    Pulse fall time (us)                          

  B                                                                                                       LFM bandwidth (MHz)                           

  W                                                                                                       1 (Constant reserved for possible future use  
                                                                                                          or more refinement)                           

  V                                                                                                       1 (Constant reserved for possible future use  
                                                                                                          or more refinement)                           
  ------------------------------------------------------------------------------------------------------- --------------------------------------------- ------

# LFM Chirped Simulation and Measurement Results

Results of the chirped pulse waveform measurements are detailed in this
section. Data was taken for all 28 LFM waveforms, although in the first
round of tests a non-optimum resolution bandwidth was used for the
measurements. Ten waveforms were then chosen as representing a wide
variety of pulse parameters to be re-tested using the correct
measurement resolution bandwidth. These ten pulses were: CP1, CP5, CP7,
CP9, CP11, CP12, CP14, CP16, CP19, and CP23. Of these ten, four (CP5,
CP7, CP9, and CP14) correspond to actual systems with field measured
data, and results of these four pulses will be shown in detail in
Section 3.1. Then a brief discussion will follow in Section 3.2 on a
comparison of all ten waveforms with the VSG measurements only. To
understand the results from Table 2 through Table 5, the column
comparisons must be explained. For example, in Table 2 column 7, the
difference in percent of Equation 1 to Measured is mathematically
represented as:

$\text{Diff} = \frac{B_{\text{eq}1} - B_{\text{meas}}}{B_{\text{meas}}} \times \text{100}\text{\%}$,
EQ-9

with B~eq1~ the -40 dB bandwidth from Equation 1, and B~meas~ the -40 dB
bandwidth from the radar measurement. The negative represents the fact
that the measured bandwidth was greater than the bandwidth from the
equation (i.e. Equation 1 underestimated the radar measured bandwidth).
The rest of the comparisons in each table are computed in a similar
manner.

## System Measurements

Shown in Figure 3 through Figure 6, are the measured systems, FFT
theoretical spectrum, the spectrum generated by the VSG (Vector Signal
Generator) measured by the M.1177 compliant system (henceforth called
the VSG measurement system), and the -40 dB proposed bandwidth equation.
These results are also summarized in Table 2 and Table 3. Two of the
systems were measured in a frequency hopping mode, and therefore no
bandwidth information for single frequency channel operation could be
obtained. Plots were still included to give an indication of how the
band edge roll-off compared to the theoretical and VSG measurements. For
the two systems for which bandwidth could be calculated, System 2 (CP7)
and System 4 (CP14), the results were very different. In the first case,
System 2, the FFT theory, the VSG measurement, and the proposed equation
underestimated the actual system bandwidth by about 45%. In the second
case, System 4, the FFT theory, VSG measurement, and the proposed
equation overestimated the system bandwidth (-40 dB) by 10% to 25%. It
must be noted that this system actually transmits a non-linear FM (NLFM)
pulse, and this was compared to a linear FM pulse (that was set to have
the same rise time, fall time, bandwidth, and pulse width). It was shown
in Holloway (4), that NLFM will have bandwidths equal to or less than
equivalent LFM waveforms. (The JRG also decided that NLFM would not have
separate bandwidth equations, and has accepted the slight margin the LFM
based equations would provide.) For the -20 dB bandwidth, summarized in
Table 3, the FFT theory, VSG measurements, and the proposed equation
overestimated in both cases, although by differing amounts. For System
2, they overestimated by about 40-45%. For System 4, they overestimated
by 10-15%, but again this system actually transmits a non-linear FM
pulse, not the linear FM that it was compared with.

[]{#__RefHeading___Toc148324187 .anchor}Table 2 -- Measured System, VSG
Measurement, FFT Theory, and Proposed Equation -40 dB Bandwidths.

+-------+-------+--------+--------+--------+--------+--------+--------+
| S     | Mea   | VSG    | -40 dB | FFT    | Diff   | Diff   | Diff   |
| ystem | sured | -40 dB | Ban    | -40 dB | erence | erence | erence |
|       | -40   | Ban    | dwidth | Ban    | (pe    | (pe    | (pe    |
|       | dB    | dwidth | Eq     | dwidth | rcent) | rcent) | rcent) |
|       |       | (MHz)  | uation | (MHz)  | VSG to | Eqn. 1 | FFT to |
|       | Band  |        | (MHz)  |        | Me     | to     | Me     |
|       | width |        |        |        | asured | Me     | asured |
|       | (MHz) |        |        |        |        | asured |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | NA    | 5.35   | 5.56   | 5.53   | NA     | NA     | NA     |
| tem-1 |       |        |        |        |        |        |        |
| (CP5) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | 5.04  | 2.76   | 2.68   | 2.73   | -45.31 | -46.90 | -45.84 |
| tem-2 |       |        |        |        | %      | %      | %      |
| (CP7) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | NA    | 10.43  | 8.61   | 8.46   | NA     | NA     | NA     |
| tem-3 |       |        |        |        |        |        |        |
| (CP9) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | 3.64  | 4.00   | 5.41   | 4.22   | 9.95 % | 25.54  | 16.31  |
| tem-4 |       |        |        |        |        | %      | %      |
| (     |       |        |        |        |        |        |        |
| CP14) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+

[]{#__RefHeading___Toc148324188 .anchor}Table 3 - Measured System, VSG
Measurement, FFT Theory, and Proposed Equation -20 dB Bandwidths.

+-------+-------+--------+--------+--------+--------+--------+--------+
| S     | Measu | VSG -  | \-     | FFT -  | Diff   | Diff   | Diff   |
| ystem | red - | 20dB   | 20dB   | 20dB   | erence | erence | erence |
|       | 20dB  | Ban    | Ban    | Ban    | (pe    | (pe    | (pe    |
|       |       | dwidth | dwidth | dwidth | rcent) | rcent) | rcent) |
|       | Band  | (MHz)  | Eq     | (MHz)  | VSG to | Eqn. 5 | FFT to |
|       | width |        | uation |        | Me     | to     | Me     |
|       | (MHz) |        | (MHz)  |        | asured | Me     | asured |
|       |       |        |        |        |        | asured |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | NA    | 1.74   | 1.80   | 1.66   | NA     | NA     | NA     |
| tem-1 |       |        |        |        |        |        |        |
| (CP5) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | 3.89  | 2.22   | 2.09   | 2.21   | -42.89 | -46.30 | -43.28 |
| tem-2 |       |        |        |        | %      | %      | %      |
| (CP7) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | NA    | 3.84   | 3.67   | 3.75   | NA     | NA     | NA     |
| tem-3 |       |        |        |        |        |        |        |
| (CP9) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+
| Sys   | 1.82  | 1.59   | 1.65   | 1.55   | -12.64 | -9.28  | -14.56 |
| tem-4 |       |        |        |        | %      | %      | %      |
| (     |       |        |        |        |        |        |        |
| CP14) |       |        |        |        |        |        |        |
+-------+-------+--------+--------+--------+--------+--------+--------+

![](media/image3.png){width="5.503472222222222in"
height="4.127083333333333in"}

[]{#__RefHeading___Toc148753545 .anchor}Figure 3 -- Plot of System 1,
VSG Measurement, FFT Theory, and Proposed Equation -40 dB Bandwidths.

![](media/image4.png){width="5.503472222222222in"
height="4.127083333333333in"}

[]{#__RefHeading___Toc148753546 .anchor}Figure 4 - Plot of System-2, VSG
Measurement, FFT Theory, and Proposed Equation -40 dB Bandwidths.

![](media/image5.png){width="5.127083333333333in"
height="3.845138888888889in"}

[]{#__RefHeading___Toc148753547 .anchor}Figure 5 - Plot of System 3, VSG
Measurement, FFT Theory, and Proposed Equation -40 dB Bandwidths.

![](media/image6.png){width="5.111111111111111in"
height="3.838888888888889in"}

[]{#__RefHeading___Toc148753548 .anchor}Figure 6 - Plot of System 4, VSG
Measurement, FFT Theory, and Proposed Equation -40 dB Bandwidths.

## VSG Measurements

For the other six systems that had no actual system measurements, all
that could be done was a comparison between the bandwidths (-20 and -40)
measured by the VSG measurement system, the FFT theoretical bandwidths,
and the bandwidths proposed by Equation 1 and Equation 5. This gives an
idea of how a simulated radar waveform spectrum compared to the FFT
theoretical spectrum, and how well the proposed equations fit these
measurements. The -40 dB bandwidth results are summarized in , and the
-20 dB results are summarized in Table 5.

After the measurements were taken, it was noticed that there was an
error with the VSG software that was causing the pulses to be truncated
in time. This led to a loss of high frequency components of each
waveform, but this loss was only discovered by examining CP16, CP19, and
CP23, as these waveforms had the highest bandwidths and chirp rates
(bandwidths greater than 11 MHz, and chirp rates greater than 2 MHz/us).
The waveforms (CP1, CP5, CP7, CP9, CP12, and CP14) are thought to also
contain this problem, but since their chirp rate is so low (less than
0.12 MHz/us), the truncated portion of those waveforms contains only
minimal frequency content. The software problem has been corrected by
the VSG manufacturer, and CP16, CP19 and CP23 have been re-measured at
this time.

For the -40 dB bandwidth, the first instance where the proposed equation
differed significantly with the FFT theoretical bandwidth was for
waveforms CP11. When this waveform was examined more closely, it was
revealed that$B \ast \text{MIN}(t_{r},t_{f}) \simeq 0\text{.}2$. This
shows that the proposed equation is only valid when the condition is
sufficiently met.

The second instance where the proposed -40 dB bandwidth equation
differed significantly is waveform CP23. A plot for CP23 is shown in
Figure 7. It matches the FFT theory well to about the -20 dB points.
There is a difference in fall-off on the higher frequency side versus
the lower frequency side. Even though this waveform has been re-measured
without the VSG truncation, there are still artifacts present in the VSG
measurement that show a wider -40 dB bandwidth than the FFT theory or
proposed equation predict (these artifacts are clearly seen in Figure 7,
at -75 MHz, and on the higher frequency roll-off). This is due mainly to
the fact that this waveform is at the limits of the VSG capabilities (80
MHz bandwidth), and it is theorized that this is causing the errors.

For the other eight waveforms (CP1, CP5, CP7, CP9, CP12, CP14, CP16, and
CP19) the proposed -40 dB equation fit the VSG within 21%. (Example
plots of two of these waveforms, CP11 and CP12, are shown in Figure and
Figure , respectively.) The proposed -20 dB bandwidth equation fit the
VSG measurements within +/- 14.2% (including CP11, since this equation
has no rise/fall-time limit, and CP23, as the -20 dB bandwidth was not
affected by the VSG limitations). These numbers are a good starting
point for discussions and study of the necessary implementation factors,
but they do not take into account real systems. As shown in Section 3.1,
the VSG measurements can differ for a real system by as much as 45% for
both the -40 dB and -20 dB bandwidths.

![](media/image7.png){width="5.375in" height="4.03125in"}

Figure 7 - VSG Measurement, FFT Theory, and Proposed -40 dB Bandwidth of
CP23.

[]{#__RefHeading___Toc148324189 .anchor}

Table 4 -- VSG Measurements, Proposed -40 dB Equation (EQ-1), and FFT
Theory Bandwidth Comparison

+-----------+-------+--------+--------+--------+--------+-----------+
| Waveform  | VSG   | Eqn. 1 | FFT    | Diff   | Diff   | D         |
|           | wav   |        | Theory | erence | erence | ifference |
|           | eform | -40 dB | -40 dB | in     | in     | in        |
|           |       | Ban    | BW     | p      | p      | percent   |
|           | -40   | dwidth | (MHz)  | ercent | ercent |           |
|           | dB    | (MHz)  |        |        |        | FFT to    |
|           | Band  |        |        | FFT to | Eqn. 1 | VSG       |
|           | width |        |        |        | to VSG |           |
|           |       |        |        | Eqn. 1 |        |           |
|           | (MHz) |        |        |        |        |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP1       | 0.57  | 0.45   | 0.48   | 6.05 % | -20.92 | -16.63 %  |
|           |       |        |        |        | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP5       | 5.35  | 5.56   | 5.53   | -0.63  | 4.06 % | 3.35 %    |
|           |       |        |        | %      |        |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP7       | 2.76  | 2.68   | 2.73   | 1.94 % | -2.91  | -0.98 %   |
|           |       |        |        |        | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP9       | 10.43 | 8.61   | 8.46   | -1.76  | -17.46 | -18.92 %  |
|           |       |        |        | %      | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP11      | 7.26  | 5.06   | 5.92   | 17.01  | -30.31 | -18.44 %  |
|           |       |        |        | %      | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP12      | 5.69  | 5.41   | 5.46   | 0.93 % | -4.87  | -4.01 %   |
|           |       |        |        |        | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP14      | 4.00  | 4.56   | 4.22   | -7.53  | 14.18  | 5.62 %    |
|           |       |        |        | %      | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP16      | 23.84 | 24.22  | 22.85  | -5.65  | 1.58 % | -4.16 %   |
|           |       |        |        | %      |        |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP19      | 59.37 | 58.38  | 58.23  | -0.25  | -1.68  | -1.92 %   |
|           |       |        |        | %      | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+
| CP23      | 1     | 105.09 | 104.17 | -0.88  | -29.97 | -30.59 %  |
|           | 50.09 |        |        | %      | %      |           |
+-----------+-------+--------+--------+--------+--------+-----------+

[]{#__RefHeading___Toc148324190 .anchor}Table 5 - VSG Measurements,
Proposed -20 dB Equation (EQ-5), and FFT Theory Bandwidth Comparison

+-----------+--------+--------+--------+--------+---------+----------+
| Wavefo    | VSG    | Eqn. 5 | FFT    | Diff   | A       | Absolute |
| rm/system | wa     |        | Theory | erence | bsolute | Di       |
|           | veform | -20 dB | -20 dB | in     | Dif     | fference |
|           |        | Ban    | BW     | p      | ference | in       |
|           | -20 dB | dwidth | (MHz)  | ercent | in      | percent  |
|           | Ban    | (MHz)  |        |        | percent |          |
|           | dwidth |        |        | FFT to |         | FFT to   |
|           |        |        |        |        | Eqn. 5  | VSG      |
|           | (MHz)  |        |        | Eqn. 5 | to VSG  |          |
+-----------+--------+--------+--------+--------+---------+----------+
| CP1       | 0.45   | 0.39   | 0.41   | 5.59 % | -12.94  | -8.99 %  |
|           |        |        |        |        | %       |          |
+-----------+--------+--------+--------+--------+---------+----------+
| CP5       | 1.74   | 1.80   | 1.66   | -7.66  | 3.29 %  | -4.50 %  |
|           |        |        |        | %      |         |          |
+-----------+--------+--------+--------+--------+---------+----------+
| CP7       | 2.22   | 2.09   | 2.21   | 5.82 % | -5.96 % | -0.68 %  |
+-----------+--------+--------+--------+--------+---------+----------+
| CP9       | 3.84   | 3.67   | 3.75   | 2.11 % | -4.37 % | -2.43 %  |
+-----------+--------+--------+--------+--------+---------+----------+
| CP11      | 3.61   | 3.12   | 3.36   | 7.70 % | -13.47  | -6.86 %  |
|           |        |        |        |        | %       |          |
+-----------+--------+--------+--------+--------+---------+----------+
| CP12      | 4.67   | 4.18   | 4.45   | 6.42 % | -10.47  | -4.70 %  |
|           |        |        |        |        | %       |          |
+-----------+--------+--------+--------+--------+---------+----------+
| CP14      | 1.59   | 1.65   | 1.55   | -6.00  | 3.84 %  | -2.21 %  |
+-----------+--------+--------+--------+--------+---------+----------+
| CP16      | 15.48  | 13.24  | 14.51  | 9.61 % | -14.46  | -6.23 %  |
|           |        |        |        |        | %       |          |
+-----------+--------+--------+--------+--------+---------+----------+
| CP19      | 46.73  | 51.72  | 45.95  | -11.15 | 10.67 % | -1.68 %  |
|           |        |        |        | %      |         |          |
+-----------+--------+--------+--------+--------+---------+----------+
| CP23      | 90.18  | 83.45  | 88.19  | 5.68 % | -7.26 % | -2.21 %  |
+-----------+--------+--------+--------+--------+---------+----------+

![](media/image8.png){width="5.375in" height="4.03125in"}

[]{#_Ref148323994 .anchor}Figure 8 -- VSG Measurement, FFT Theory, and
Proposed -40 dB Bandwidth of CP11.

![](media/image9.png){width="5.375in" height="4.03125in"}

[]{#__RefHeading___Toc148753550 .anchor}Figure 9 - VSG Measurement, FFT
Theory, and Proposed -40 dB Bandwidth of CP12.

## Implementation Factor

Experience and measurements show us that emission data from real systems
almost never exactly matches theory. That being the case, the bandwidth
equations must have a term to include implementation factors to account
for features of the real systems (noise, filtering, output device
characteristics, etc,). In this work, only two of the measured systems
have measurable system bandwidth, as the other two measured systems were
in a hopping mode during data collection and this made bandwidth
calculations (based on the measurements) impossible for a single
channel. This was not enough data to calculate an implementation factor
at this time. As more system measurements become available, the
implementation factor can be further investigated. The appropriate
implementation factor can be added to EQ-1 and EQ-5, by adjusting the
constant parameters of each equation. Other administrations are
encouraged to provide data on implementation factors.

# Summary Results

The goal of the measurements was to verify the proposed --20 dB and --40
dB bandwidth equations, EQ-1 and EQ-5, and to capture data for the
purpose of gaining insight into the proper way to model the ideal
spectrum of Pulsed FM radars. Analyses of the captured data included
testing the validity and appropriate use of FFT and associated window
functions and comparing those measurements to field measurements of
actual systems as well as VSG simulated waveform measurements. This
comparison allows one to view the difference between theory, simulated,
and actual radar waveforms.

# References

1.  "Techniques for Measurements of Unwanted Emissions of Radar
    Systems," ITU-R M.1177. June 2003.

2.  Merrill Skolnik, Radar Handbook Second Edition, McGraw Hill 1990.

3.  Eli Brookner, Radar Technology, Lex Book, 1996.

4.  John Holloway, "Spectrum of Chirped Radar Pulses using Non-Linear
    FM," JRG-13. July 2004.

[^1]: Brookner Radar Technology, Page 173, Table 1.

[^2]: Skolnik Radar Handbook 2^nd^ Edition, Table 10.2

    \* Measurement data on this chirp is provided.

[^3]: Skolnik Radar Handbook 2^nd^ Edition, Table 10.2

[^4]: Brookner Radar Technology, Page 173, Table 1.

[^5]: Hypothetical Pulsed Waveform.