*United States of America*

Estimation of Radar Linear Frequency Modulation (LFM) bandwidth at the
-40 dB point

### Introduction

Unwanted emission masks in the out-of-band domain for radiodetermination
radar systems are contained in Recommendation ITU-R SM.1541, Annex 8.
The ITU-R out-of-band (OOB) emission masks for primary radars systems
are intended to recommend an acceptable degree of electromagnetic
compatibility among radar systems, and between such systems and those of
other radio services sharing the frequency spectrum.

In the development of the out-of-band emission masks for
radiodetermination radar systems, it was the view of several
administrations that there was a need to further investigate the
state-of-the-art radar systems and their ability to suppress unwanted
emissions. This review of radar unwanted emissions in the out-of-band
domain is to be completed by 2006. See Recommendation ITU-R SM.1541,
Annex 8, Note 3.

The outstanding issue concerns the "design objectives" highlighted in
this document. This "design objective" concerns the possibility of
making more stringent the requirement for radars with regard to the
"roll-off\" of out of-band (OOB) emissions.

This document describes the derivation of a proposed equation designed
to approximate radar spectral bandwidth at the -40 dB below normalized
waveform spectrum magnitude for Linear Frequency Modulation pulse and
compares it to the theoretical FFT calculations.

### Equation Derivation Steps and Assumptions

The following paragraphs have short descriptions of the steps taken to
derive the proposed B~-40dB~ equation along with the necessary
assumptions.

[The assumptions are:]{.underline}

1.  The shapes of pulses are trapezoidal with finite rise and fall
    times.

2.  The pulse modulation is Linear Frequency Modulation (LFM).

3.  Pulse rise and fall times vary between 0.00004% and 50% of waveform
    pulse width duration.

4.  Chirp bandwidths vary between 0.375 MHz and 8 880 MHz.

5.  Compression ratios vary between 10 and 13 320 000.

6.  The minimum waveform compression ratio to a value is 10.

[The analysis steps taken are:]{.underline}

1.  Identify a broad set of radar systems from ITU recommendations, real
    operational radars and possible radar waveform designs derived from
    reference (2).

2.  Eighty (80) radar systems are used in this analysis, with varying
    pulse width, bandwidth and rise and fall times as shown in the
    assumptions list.

3.  Compute the spectrum using FFT routine.

4.  Record the FFT results for the frequency bandwidth at the 40 dB
    point below the normalized maximum.

5.  Derive equations based on curve fit to the theoretical B~-40~ dB
    values

6.  Refine equation until the resulting estimates are bounded within ±9%
    of the theoretical FFT results.

7.  Compute and plot results.

[The Proposed Equation:]{.underline}

The proposed equation attempts to compare to the theoretical FFT
results. There are no margin or implementation factors included in the
formula. The new B~-40dB~ is given by:

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

  τ                                                                                                       Pulse length (us) including rise and fall     
                                                                                                          times                                         

  t~r~                                                                                                    Pulse rise time (us)                          

  t~f~                                                                                                    Pulse fall time (us)                          

  B                                                                                                       LFM bandwidth (MHz)                           

  π                                                                                                       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.

**Results**

The result of the proposed formula is compared with the theoretical FFT
results as shown in Figure-1 below. The results track well with the FFT
with a maximum error of ±9%:

![](media/image1.wmf){width="6.501388888888889in"
height="4.441666666666666in"}

Figure 1: Results of FFT compared to proposed formula

### Recommendations

The proposed formula provides good results as compared to the FFT for
all the cases tested. It is recommended that this formula be further
investigated as a potential first step to introducing the B(-40dB). The
necessary radar waveform implementation factors, if needed, are to be
determined in future work.

### References

The following references are used:

1.  Report 837 Methods for Calculating Pulsed Radar Emission Spectrum
    Bandwidth (Document JRG-10).

2.  Skolnik, Radar Handbook 2nd Edition 1990, Table 10.2 Page 10.13