*Title: Spectrum of chirped radar pulses using non-linear FM*

*Written By: John Holloway*

*Date: Saturday, 03 July 2004*

*This note is produced as an input to the 2004 July Meeting of the ITU
Radar JRC and is intended to contribute to the current discussion on the
revision of the ITU-R 1541 recommendation of Out of Band Emissions.*

***Summary: This note compares the bandwidths calculated for the 40 dB
to 60 dB regions of a pulse spectrum for a trapezoidal pulse modulated
by linear and non-linear FM.***

***It shows for the theoretical cases considered the non-linear FM
modulation gives a narrower bandwidth than the equivalent linear FM
modulation. Apart from one case the modulations used produce theoretical
bandwidths less than the ITU design objective.***

***The practical implementation of these theoretical waveforms however
needs to be considered, as the current indications are that distortions
in the transmitter can result in a non-compliant spectrum. This is
particularly the case for frequency agile systems. More work on
practical spectra is required to confirm what design objective can be
achieved.***

*John Holloway*

*Systems Engineering Department*

## AMS Radar Systems Division

*Newport Road*

*Cowes*

*IoW.*

***John.r.Holloway@amsjv.com***

**1) Introduction**

In order to achieve the stability needed to give the level of
sub-clutter-visibility required, modern radar systems make use of driven
transmitters. These transmitters are also capable of operating over wide
agile bandwidths.

These types of transmitters generally produce a much lower peak power
than was used in the early Magnetron/Klystron based systems that had
limited agility. In order produce the required energy within the pulse
to achieve target detection; a much longer pulse must be used. In order
to recover the range resolution from these long pulses, pulse
compression coding has to be used. In these systems a long expanded
pulse many range cells in length is transmitted, on reception the signal
is compressed by the receiver to recover the range resolution.

One commonly used pulse compression waveform is a FM modulated or
chirped pulse. Linear FM modulation when compressed in the receiver
leads to a main lobe in the time domain equal to the required range cell
but also produces subsidiary lobes in the time domain, the first lobes
being at a level --13 db wrt to the peak of the main lobe. These, so
called time/range side lobes, can produce false target detections or
multiple detections in range. If a target is large it can also cause
small targets in adjacent range cells to be suppressed. A representation
of a linear chirp is shown in Figure 1 below.

# **Figure 1 linear-chirp**

![](media/image1.wmf)

In order to reduce the range side lobes it is possible to apply a
weighting function to the waveform, this can be applied to the received
signal by the receiver or by weighting the expanded waveform on
transmit.

The use of weighting on receive has a disadvantage in that it gives a
non-optimum matched filter which leads to a loss of S/N. Pre-distortion
of the expended pulse by phase weighting when combined with a matched
filter receiver can minimise the system losses and maximises the S/N.

The use of non-linear FM has a further benefit in that it produces a
more compact spectrum.

The phase weighting is applied by using a non-linear FM modulation.

# **Figure 2 -40 dB Taylor Weighting** 

![](media/image2.wmf)

The amount of distortion applied depends on the range side lobes that
are required several weightings are possible. A good standard weighting
to use is a -40 dB Taylor this is typical for modern radars. An example
with a 5 MHz NLFM chirp is shown in Figure 2 above. Further information
can be found in Reference 1.

The ITU formula (1) for the --40 dB bandwidth contains a factor *B~c~*
which is equal to the total frequency excursion of the chirp (the
formula is based on a liner change of frequency with time). The use of a
non-linear chirp gives rise to a different spectral shape.The objective
of this investigation was to compare the spectrum of linear and
non-linear chirped radar pulses with the same *B~c~* factor. These
calculated results are then compared to the specifications limits
defined by the ITU.

Pulse lengths of 6, 25, 50 & 100 s are considered with rise times of
50,100 & 150 ns.

The ITU-R 1541 has a design objective for the roll-off in the OOB domain
of --40 dB/dec. Currently they have been required too roll-off at
-20dB/dec from the specified bandwidth at the -40dB point to the -60dB
level.

This report deals only with the proposed design objective for -40dB/dec
roll-off it has been established that if the calculated bandwidth at the
-40 dB point is met then the current requirement for a -20 dB/dec
roll-off is not problematic for these classes of waveforms.

This report studies how a trapezoidal pulse with a chirp of 3.25MHz
conforms to the current ITU calculated 40 dB bandwidth and the OOB
design objective, for combinations of various pulse lengths and rise
times.

The equation (1) for the --40 dB bandwidth of a chirped trapezoidal
pulse is:

$B_{- \text{40}}\  = \ \frac{K}{\sqrt{t\  \cdot \ t_{r}}} + 2\ \left( B_{c} + \frac{A}{t_{r}} \right)$
(1)

Where *A*[^1] is 0.105 when *K* = 6.2, and 0.065 when *K* = 7.6.

The results are compared for pulses with the same *B~c~* this is so a
comparison to the ITU mask can be made directly. These pulses are not
necessary equivalent in radar terms. Non-linear FM compressed pulses do
not have an equivalent compressed pulse length to linear FM pulses.
Non-linear pulses have a slightly wider compressed pulse length. An
alternative comparison may be compare pulses with an equivalent
compression ratio this however would require to compare pulses of
different *B~c~* values.

**2) Methods of Calculation**

To investigate the emitted spectrum of pulsed radar a theoretical model
was developed, which represented the pulse in time this was then Fourier
transformed into its frequency spectrum. This raises the point that the
spectrum from a practical radar will not be identical and these
theoretical wave forms and the results obtained should only be used as a
guide.

The model was developed in Matlab and consists of several functions. One
is a 40dB Taylor, frequency vs. time function ^\[1\]^. The model is
capable of producing the spectrum for linear and non-linear frequency
modulations. The following figures are the results for pulse lengths of
6,25,50 and 100 s with rise times of 50,100 and 150 ns. All of the
pulses were made symmetric so the fall time was equal to the rise time.
The red line is that of the ITU specification with a -40dB roll off
^\[2\]^ it is based on the K = 7.6 and A = 0.065 values for calculating
the allowed bandwidth at the -40dB point

**3) Results**

**Figure 3**

![](media/image3.wmf){width="6.013888888888889in"
height="4.519444444444445in"}

**Figure 4**

![](media/image4.wmf){width="5.993055555555555in"
height="4.499305555555556in"}

**Figure 5**

![](media/image5.wmf){width="6.017361111111111in"
height="4.518055555555556in"}

**Figure 6**

**Figure 7**

![](media/image7.wmf){width="6.013888888888889in"
height="4.514583333333333in"}

**Figure 8**

**Figure 9**

![](media/image9.wmf){width="6.020833333333333in"
height="4.520833333333333in"}

**Figure 10**

![](media/image10.wmf){width="6.020833333333333in"
height="4.520833333333333in"}

**Figure 11**

![](media/image11.wmf){width="6.013888888888889in"
height="4.514583333333333in"}

**Figure 12**

![](media/image12.wmf){width="6.020833333333333in"
height="4.520833333333333in"}

**Figure 13**

![](media/image13.wmf){width="6.020833333333333in"
height="4.520833333333333in"}

**Figure 14**

![](media/image14.wmf){width="6.020833333333333in"
height="4.520833333333333in"}

**4) Analysis of Results**

It can be seen that the critical part of the spectrum is that between 40
and 60dB the summary of the results is shown in Tables 1 & 2 .

These tables show the bandwidths calculated for linear and non-linear FM
compared to the ITU calculated mask.

The values are shown in MHz for the 40,44,48,52,56 and 60dB points. It
can be seen from the table and the figures that in general a longer rise
time give more leeway in meeting the ITU specification.

Another point, which is obvious from all the data available, is that for
the set of waveforms used the non-linearly modulated pulses use less
bandwidth than the linear cases with the same B factor.

**Table 1 Comparison of Bandwidths in MHz**

  -------- -------- -------- -------- ------- -------- -------- ------- -------- ---------- -------
  6s       T/t~r~            **40     ** **   ** **    **44     ** **   ** **    **48dB**    
                             dB**                      dB**                                 

  rise              Linear   Non      ITU     Linear   Non      ITU     Linear   Non Linear ITU
  time                       Linear                    Linear                               

  50 ns    120      10.13    9.60     11.49   11.95    11.29    14.46   13.14    12.77      18.21

  100 ns   60       6.67     6.31     8.81    7.71     7.19     11.09   8.69     8.20       13.96

  150 ns   40       5.70     5.21     7.69    6.21     5.86     9.68    6.70     6.38       12.19

  25s                                                                                        

  rise              Linear   Non      ITU     Linear   Non      ITU     Linear   Non Linear ITU
  time                       Linear                    Linear                               

  50 ns    500      8.03     6.99     7.95    9.74     8.95     10.01   11.55    10.71      12.60

  100 ns   250      5.71     5.27     6.30    6.50     6.11     7.94    7.49     6.89       9.99

  150 ns   167      4.92     4.49     5.65    5.65     5.25     7.11    6.21     5.89       8.95

  50s                                                                                        

  rise              Linear   Non      ITU     Linear   Non      ITU     Linear   Non Linear ITU
  time                       Linear                    Linear                               

  50 ns    1000     6.39     5.53     6.95    8.35     7.26     8.75    10.04    9.15       11.02

  100 ns   500      4.98     4.49     5.60    5.87     5.38     7.05    6.62     6.19       8.87

  150 ns   333      4.30     3.86     5.07    5.05     4.60     6.38    5.76     5.34       8.04

  100s                                                                                       

  rise              Linear   Non      ITU     Linear   Non      ITU     Linear   Non Linear ITU
  time                       Linear                    Linear                               

  50 ns    2000     5.36     4.32     6.25    5.70     5.79     7.87    8.95     7.60       9.90

  100 ns   1000     4.40     3.79     5.10    5.70     4.64     6.42    6.11     5.53       8.09

  150 ns   667      3.81     3.39     4.66    5.70     3.97     5.87    5.27     4.74       7.39
  -------- -------- -------- -------- ------- -------- -------- ------- -------- ---------- -------

**Table 2 Comparison of Bandwidths in MHz**

  ------- -------- -------- -------- ------- -------- -------- ------- -------- -------- -------
  6s      T/t~r~            **52     ** **   ** **    **56     ** **   ** **    **60      
                            dB**                      dB**                      dB**     

  rise             Linear   Non      ITU     Linear   Non      ITU     Linear   Non      ITU
  time                      Linear                    Linear                    Linear   

  50 ns   120      14.29    13.96    22.92   25.61    14.82    28.86   28.61    27.79    36.33

  100 ns  60       13.30    12.45    17.57   14.80    14.28    22.12   21.11    15.85    27.85

  150 ns  40       10.87    9.29     15.34   11.88    11.36    19.31   16.86    14.27    24.31

  25s                                                                                     

  rise             Linear   Non      ITU     Linear   Non      ITU     Linear   Non      ITU
  time                      Linear                    Linear                    Linear   

  50 ns   500      12.91    12.31    15.86   14.07    13.51    19.97   14.99    14.59    25.14

  100 ns  250      8.57     8.01     12.58   12.87    8.93     15.83   14.55    13.91    19.93

  150 ns  167      6.69     6.41     11.26   10.44    8.51     14.18   11.81    11.29    17.85

  50s                                                                                     

  rise             Linear   Non      ITU     Linear   Non      ITU     Linear   Non      ITU
  time                      Linear                    Linear                    Linear   

  50 ns   1000     11.81    10.91    13.87   13.11    12.43    17.47   14.27    13.68    21.99

  100 ns  500      7.69     7.06     11.17   8.70     8.16     14.07   13.19    9.02     17.71

  150 ns  333      6.31     5.96     10.12   6.76     6.50     12.74   10.92    8.73     16.04

  100s                                                                                    

  rise             Linear   Non      ITU     Linear   Non      ITU     Linear   Non      ITU
  time                      Linear                    Linear                    Linear   

  50 ns   2000     10.70    9.42     12.47   12.29    11.21    15.70   13.52    12.66    19.76

  100 ns  1000     6.93     6.33     10.18   8.04     7.24     12.81   8.94     8.33     16.13

  150 ns  667      5.92     5.48     9.31    6.46     6.05     11.72   6.87     6.59     14.75
  ------- -------- -------- -------- ------- -------- -------- ------- -------- -------- -------

The averages of the percentage bandwidth used are shown below in Table
3.

**Table 3 Average % of ITU Bandwidth**

  ----------------------- ----------------------- -----------------------
  50 ns                   82.30%                  linear

  100 ns                  74.10%                  linear

  150 ns                  68.50%                  linear

  50 ns                   73.90%                  non-linear

  100 ns                  66.20%                  non-linear

  150 ns                  62.40%                  non-linear
  ----------------------- ----------------------- -----------------------

The above data in Table 3 clearly supports the conclusions made thus far
it is also clear that the difference between the two modulations is less
pronounced for longer rise-time pulses. An alternative way is to look at
the maximum bandwidth that is achieved. This is shown in Table 4 below.
Looking at the maximum percentage used for different rise times can also
give some insight into, which is the best rise time to use. These are as
follows:

**Table 4 Maximum % of ITU Bandwidth**

  ----------------------- ----------------------- -----------------------
  50 ns                   101.00%                 linear

  100 ns                  90.60%                  linear

  150 ns                  87.20%                  linear

  50 ns                   89.40%                  non-linear

  100 ns                  83.60%                  non-linear

  150 ns                  79.50%                  non-linear
  ----------------------- ----------------------- -----------------------

From this it is clear that a 50 ns rise time with linear chirp falls
outside the allowed limits.

A point to note is that the bandwidth used at a certain dB level can be
misleading as to how well a pulse is going to meet the specification as
a peak that sits just below a certain level could be just above in a
real system.

A good example of this can be seen on the figure for Figure 6 a 25 micro
second pulse with a 50 ns rise time. There is a peak that is very close
to the spec line but as it lies just under the 60-dB level the bandwidth
used at 60-dB is given as 59%.

This may not be an accurate reflection of how well the pulse would meet
the specification in a real system. Looking at the figures this envelope
effect is not such a problem for larger rise times.

5)  **Conclusions**

*5.1 Theoretical Results*

The results of this initial investigation show that:

-   In the majority of cases the theoretical --40 dB bandwidths
    calculated are less than those allowed in the ITU specification for
    a equal *B~c~* factor,

-   Theoretical spectrums also meet the requirements of the --40 dB/dec
    roll-off design objective given in ITU-R 1541 Reference 3.

-   Compliance with the design aim is eased by the --40 dB roll-off
    starting at the calculated ITU --40 dB bandwidth point not the
    actual -- 40 dB bandwidth point.

-   Non-linear FM using typical weighting gives narrower bandwidths than
    the linear case in all the cases tested[^2].

In this initial paper only one chirp has been considered further work is
necessary to confirm the findings above on other practical radar
waveforms.

The ITU formula is based on the pulse length, rise-time and chirp and
assumes linear FM. As non-linear FM gives a reduced bandwidth then there
is a possible argument that this needs to be taken into account in the
formula. The problem with adding a factor for the FM modulation is how
one can specify the shape of the chirp curve in a relatively simple
manner. A possible way may be just to use a multiplying parameter\< 1
based on a "typical" non-linear waveform. This parameter my have to be
dependent on pulse length and rise time. Further study would be needed
to develop such a parameter.

*5.2 Practical Issues*

These calculations are based on theoretical waveforms they do not take
into account any of the practical distortions that can occur in the
transmitter chain. They should not be viewed as what can be achieved
from radars using such waveforms; they represent the best that could be
achieved from a theoretical pulse. Due allowance needs to be made for
these distortions when producing a practical specification mask.

Driven systems using current digital signal generation techniques can
produce low-power waveforms that approach the theoretical performances
calculated in this paper that are compliant with the design objective
for the ITU mask.

However for radar to operate these low-level waveforms have to be
converted into high power pulses with adequate energy to detect a target
reliably. The main problem comes from distortions created in the
up-conversion and amplification chain of the radar transmitter.

In order to generate high power pulses with usable efficiency class C
(or equivalent) amplification techniques must be used. Such amplifiers
exhibit non-linearity and other phase effects such as AM to PM
conversion.

It is particularly difficult to model and control the high power
transmitter/amplifier side, and such it is very difficult to model and
hence control the final transmitter/amplifier spectra; in particular
controlling the fall-time giving rise to pulses with unsymmetrical rise
and fall-times. In some amplifier systems controlling the effect of the
rise time and in particular the fall-time is problematic.

It is possible fixed frequency high power transmitter chains can
maintain compliance with the design objective however in many cases to
achieve this, filtering is required to "shape" the final spectrum.
Typical modern Solid State ATC radars have been shown to have a spectrum
that meets the ITU objective and is very "clean" these systems generally
use high power filtering after high power amplification.

Whilst some of practical systems can meet the proposed design objective
compliance may be compromised if the roll-off starts at the true --40 dB
bandwidth points not as is current the calculated ITU --40 dB bandwidth
points.

Systems that cannot employ such efficient filters, such as frequency
agile or active array systems have to make use of waveform control only
and thus it is more difficult to achieve the fast roll-off.

Even the best-designed frequency agile transmitters distort cleanly
generated signals and result in probable non-compliances against the
design objective very often in the OOB domain, or even in the \"close\"
(i.e. within the agility bandwidth) spurious domain.

**6) References**

\[1\] Merill Skolnik, Radar Handbook

\[2\] John Holloway, Preliminary Parametric Study of the Bandwidth
Requirements for FM modulated Pulse Radars. ITU RCG Meeting London June
2003.

\[3\] RECOMMENDATION ITU-R SM.1541 Unwanted emissions in the out-of-band
domain.

[^1]: ^4^ The term *A*/*tr* adjusts the value of *B*--40 to account for
    the influence of the rise time, which is substantial when the
    time-bandwidth product *Bct*, is small or moderate and the rise time
    is short.

[^2]: These two pulses may not be equivalent in operational radar terms,