+-----------------------------------+----------------------------------+
| ##                                | **THALES AEROSPACE**             |
| ###### WORKING PAPER -- JRG-78 {# |                                  |
| working-paper-jrg-78 .unnumbered} | Manor Royal                      |
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+-----------------------------------+----------------------------------+

+--------+-------------------------+----------+----------------------+
| **F    | **A. G. Stove**         | *        | 31 March 2007        |
| rom:** |                         | *Date:** |                      |
|        | andy.s                  |          |                      |
|        | tove@uk.thalesgroup.com |          |                      |
|        |                         |          |                      |
|        | +44 (0) 1293 644291     |          |                      |
+--------+-------------------------+----------+----------------------+
| *      | **ITU WP1A-8B-8C JRG DG | Ref.:    | ####### Sp           |
| *To:** | 02**                    |          | ectra of FMCW Signal |
|        |                         |          | s {#spectra-of-fmcw- |
|        |                         |          | signals .unnumbered} |
+--------+-------------------------+----------+----------------------+
| *      | **UK WP8B**             |          |                      |
| *Cc:** |                         |          |                      |
+--------+-------------------------+----------+----------------------+

# INtroduction

This paper presents an approach to reconciling the previous papers on
the spectra of FMCW radars1-5. The analysis is approximate since the
data in references 2 and 4 suggests that an exact match between theory
and practice is unlikely, therefore there is no need to calculate the
former with high precision.

# theoretical analysis

The key paper appears to be reference 5, which produces an analytic form
for the spectrum of a single FMCW ramp. Unfortunately, this seems to
contain an error, in that the definition of β =2π(*fc* - *f*) is not
compatible with equations (2) and (5). Assuming that equation (5) and
the relations below it are compatible then it seems that the correct
definition of β should be

β = π(*fc* - *f*)*. (Eqn.* 1

If we also assume that the instantaneous frequency at time -*T*/2 is 0,
then *fc* = *B*/2

In that case *x*±*Τ*/2  reduces to √(2π/α)(0,B - *f*)*,* i.e. the two
values at the limits of the integration are the sideband levels as a
function of the frequency deviation, ∆*f,* with respect to the nominal
edge of the sweep.

In view of the need to modify the definition of β to arrive at this
answer, some confirmation is desirable. Reference 6 also quotes the
integral Fourier transform of a chirp, and gives a similar form.
Unfortunately, to make this compatible to the result above, we must
assume that the time-domain waveform of which the transform is quoted is
actually

*s*(*t*) = -(2*A*2*S*/*p*12)sin \[ π*V*2(*t*-*t*0)2/(*p*12)\] (*Eqn.* 2

rather than, as quoted

*s*(*t*) = -(2*A*2*S*/*p*12)sin \[ π*V*2(*t*-*t*0)2/(2*p*12)\]

Note that if the changes given in equations (1) and (2) are made, then
references 5 and 6 become compatible, but if they are not, the two
original versions are not compatible.

If we assume that the modification is correct, then if we consider the
roll off at one end of the nominal excursion, then the contribution from
the \'other\' end may be assumed to take the constant value of 0.5, that
being its limiting value.

Following equation (7) of reference 5, and taking the first two terms in
the approximations in section (C) of that reference, we may assume that

C(*x*2) - C(*x*1) ≈ √(α/2π)/(√2π∆*f*) (*Eqn*. 3

where the factor of √2 in the demoninator in equation 3 represents an
average value of sinusoidal terms, and similarly

S(*x*2) - S(*x*1) ≈ -√(α/2π)/(√2π∆*f*) (*Eqn*. 4

so that

\|*v*(*f*)\|2 ≈ (Τ/2Β).(α/2π)(1/π∆*f*)2 (*Eqn*. 5

The roll-off with respect to the within-beam signal is then

*A* = (α/2π)(1/π∆*f*)2 (*Eqn*. 6

= (*B*/2*T*)(1/π∆*f*)2 (*Eqn*. 7

= 2*BT*/(2π∆*fT*)2 (*Eqn*. 8

Compared with reference 1 this suggests that the unity gain point is
expanded out by a factor of √(2*BT*) compared with the assumed
sin(*x*)/*x* roll-off.

We will use this new factor to modify the roll-off assumed in reference
1 and compare this with the experimental and simulated data. The formula
given in figure 3 of reference 1 can be simplified to

*A =* 20 log10(2π∆*fT*)2 (*Eqn*. 9

If we include the time-bandwidth product effect, this becomes

*A = -*10 log10 (2*BT*) + 20 log10(2π∆*fT*)2 (*Eqn*. 10

## Effect of Flyback

We may approximate the effect of the flyback by considering that the
effective value of *T* for the flyback is the flyback period *Tf*. The
power density of the flyback signal is reduces by a factor (*Tf*/*T*)
due to the lower energy in the flyback and again by another factor of
(*Tf*/*T*) because the power will be spread over a wider bandwidth.
Following equation 9 we would then have for the sidelobes of the flyback

*A =* 20 log10 (*Tf*/*T*) + 20 log10(2π∆*fTf*)2 (*Eqn*. 11

which reduces back to equation 9 itself. This shows that, below the
level determined by the forward sweep/flyback time ratio, the out of
band emissions due to the mainlobe and sidelobe will be the same. The
residual level may then vary between 6dB higher than estimated by
equation (9) to zero, depending on how the signals add up. Reference 3
suggests that for a triangular sweep (*Tf* =*T*) the out-of-band
emissions cancel out to first order. In practice this is a dangerous
assumption when we consider the imperfections of practical signals. When
we consider that the time-bandwidth product of the flyback is less than
that of the signal, the relative out-of-band emission power of the
flyback will be further reduced by another factor of the ratio between
the flyback time and the \'forward\' sweep time so that for sawtooth
sweeps, the effect of the flyback spectrum can actually be ignored.

# COMPARISON WITH OTHER DATA

## Experimental Results in Reference 2.

The results shown for the YIG-tuned source in reference 2 may be summed
up as follows:

[Test]{.underline} [Time-Bandwidth Product]{.underline} [Sweep
Time]{.underline} [Measured Width]{.underline} [Predicted]{.underline}

1 47dB 1ms 0.8/2.3MHz@-45dB ±11MHz

2 33dB 100μs 0.6/1.6MHz@35dB ±4MHz

3 18.5dB 100μs 0.2/1.4MHz@-35dB ±750kHz

4 18.5dB 100μs 0.2/1.1MHz@-45dB ±2.5ΜHz

[Table 1: Predicted and Measured Widths Including √TB
Factor]{.underline}

This formula thus seems to predict wider spectra than are actually seen,
however, it can seen by comparison with table 2, below, that the match
is better than if the time-bandwidth product factor is ignored, i.e. we
revert essentially to equation (9):

[Test]{.underline} [Time-Bandwidth Product]{.underline} [Sweep
Time]{.underline} [Measured Width]{.underline} [Predicted]{.underline}

1 47dB 1ms 0.8/2.3MHz@-45dB ±30kHz

2 33dB 100μs 0.6/1.6MHz@35dB ±90kHz

3 18.5dB 100μs 0.2/1.4MHz@-35dB ±90kHz

4 18.5dB 100μs 0.2/1.1MHz@-45dB ±300kHz

[Table 2: Predicted Widths Excluding √TB Factor]{.underline}

## Experimental Results from Reference 3

If we now compare these formulae with those in table 1 of reference 3,
and include the time-bandwidth factor we have:

[Test]{.underline} [Time-Bandwidth Product]{.underline} [Sweep
Time]{.underline} [Measured Width]{.underline} [Predicted]{.underline}

1 1MHz 100ms 2.2kHz @ -20dB ±5kHz

6.4kHz @ -40dB ±50kHz

2 10MHz 10ms 22kHz @-20dB ±50kHz

64kHz @ -40dB ±500kΗz

3 100MHz 1ms 220kHz @ -20dB ±500kHz

640kHz @ -40dB ±5ΜHz

[Table 3: Predicted Widths Including √TB Factor]{.underline}

This gives predictions are generally over-generous by about a factor of
5.For reference we can also calculate the widths excluding the
time-bandwidth product factor:

[Test]{.underline} [Time-Bandwidth Product]{.underline} [Sweep
Time]{.underline} [Measured Width]{.underline} [Predicted]{.underline}

1 1MHz 100ms 2.2kHz @ -20dB ±16Hz

6.4kHz @ -40dB ±160Ηz

2 10MHz 10ms 22kHz @ -20dB ±160Hz

64kHz @ -40dB ±1.6kΗz

3 100MHz 1ms 220kHz@-20dB ±1.6kHz

640kHz @ -40dB ±16kHz

[Table 4: Predicted Widths Excluding √TB Factor]{.underline}

These predictions are obviously too narrow.

## Comparison Between References 2 and 3

The final cross-check is to compare the formulae in reference 3 with the
results from reference 2. The measured results are at the -35dB or -45dB
points. These will be compared with the predicted values at -40dB. It
was noted in reference (4) that generating data with phase
discontinuities is very difficult, so the equations for φ = 0 seem most
appropriate.. Since the time bandwidth product seems to be in natural,
rather than logarithmic units, the terms which are negative exponentials
of the latter will be effectively zero for any reasonable time-bandwidth
product. Hence we can simplify equation (16) of reference 3 to:

*B*-40 = *B*\[1.1078 -6.8221 (*BT*)-0.25 +14.5749(*BT*)-1/3\] (*Eqn*. 12

The half-width of the roll off is thus:

∆*B*-40 = ±*B*\[0.0539 -3.4111 (*BT*)-0.25 +7.2875(*BT*)-1/3\] (*Eqn*.
13

[Test]{.underline} [Time-Bandwidth Product]{.underline} [Sweep
Time]{.underline} [Measured Width]{.underline} [Predicted @
-40dB]{.underline}

1 47dB 1ms 0.8/2.3MHz@-45dB ±1.2MHz

2 33dB 100μs 0.6/1.6MHz@35dB ±2.4MHz

3 18.5dB 100μs 0.2/1.4MHz@-35dB ±450kHz

4 18.5dB 100μs 0.2/1.1MHz@-45dB ±450kHz

[Table 5: Predicted Widths Using Formula from Reference 3]{.underline}

# CONCLUSIONS

References 5 and 6 agree that the exact formulation of the roll-off rate
shows that it depends on the differences between Fresnel integrals.
Reference 5 shows that this results in an out-of-band roll-off at
-20dB/decade. The analysis above suggests that, compared with the simple
analysis in reference 1 these terms will broadens the out-of-band
spectrum by the square root of the time-bandwidth product of the signal.
This gives a better match to the observed data than the formula in
reference 1, but is over-pessimistic by about a factor of something like
5, although it still underestimates the out-of-band signals on the
\'lower\' side of the experimental data from reference 2.

The formulae from reference 3 probably give the best overall match to
the experimental data, but still over-estimate the out-of-band width on
the upper side by about a factor of 2 and underestimate it on the lower
side by about the same amount. Since this is an ad-hoc fit to the data,
it is unsuitable for a general recommendation until its limits are
known.

Ideally one would calculate a better approximations to the formulae in
reference 5 and derive theoretically-based formulae to predict the
widths of the spectra. However, it is believed that this is not a
priority, since any recommendation would take have to take account of
the practical effects noted in reference 2. Since it had a relatively
long time constant due to the inherent difficulty of driving its
electro-magnetic circuits, the YIG circuits measured therein probably
constitute the worst case of different transmitter/exciter designs.

Since the width of the spurious emissions from FMCW radars is
proportionately much less than for most other types of radars, even if
better approximations to the analytic formulae could be used, the gain
in spectral efficiency would be very modest.

The following formula is therefore proposed:

a\) a roll-off of *A = -*10 log10 (2*BT*) + 20 log10(2π∆*fT*)2 together
with

b\) a broadening of 1MHz + 0.01B on either side to account for
limitations in the accuracy of the sweep generator.

The measurements should be made in as narrow a bandwidth as practical,
in order accurately to measure the shape of the response. The
measurements should also make allowance for the width of the analyser
filters at the response level being measured. Reference 2 gives some
typical examples, but they will not be repeated here as they are, in
principle, specific to one analyser, although further investigation may
show that the filter shape factors used by different manufacturers may
be quite similar.

It may be noted that no data has yet been able to be presented of
radiated measurements of spectra of FMCW radars. Reference 2 argues that
the difference between indirect and radiated measurements is likely to
be less significant for these radars than, for example, for
magnetron-based radars, but it would still be desirable to be able to
have some radiated measurements to support the indirect measurements
which have been made.

# REFERENCES

1: Stove, A. G., \'Bandwidths of FMCW Signals,\' Working Paper dated
December 2005

2: Stove, A. G., \'Measurement of Spectra of Microwave FMCW Signals,\'
Working Paper dated January 2006

3: USA, \'Theory of FMCW Radar Waveforms,\' JRG-70,. September 2006

4: USA, \'Results of FMCW Simulation and Measurements,\' JRG 69,
September 2006

5: Turley, M. D. E., \'FMCW Radar Waveforms in the HF Band,\' JRG 73,
September 2006

6: Cherniakov, M., \'Bistatic Radar Principles and Practice,\'
Chichester, 2007, Wiley, ISBN 978-0-470-02630-4, pp264-8.