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+--------+-------------------------+----------+----------------------+
| **F    | **A. G. Stove**         | *        | 30 January 2006      |
| rom:** |                         | *Date:** |                      |
|        | andy.s                  |          |                      |
|        | tove@uk.thalesgroup.com |          |                      |
|        |                         |          |                      |
|        | +44 (0) 1293 644291     |          |                      |
+--------+-------------------------+----------+----------------------+
| *      | **WP1A-8B-8C JRG DG     | Ref.:    | Measurements of      |
| *To:** | 02**                    |          | Spectra of Microwave |
|        |                         |          | FMCW radars          |
+--------+-------------------------+----------+----------------------+

.

# INtroduction

This paper complements the earlier paper^1^, which looked at the
theoretical analysis of the spectrum of a linear chirp. That paper
predicted that the spectrum would be substantially contained within the
width of the nominal excursion, but with a roll-off at each end with
sidelobes falling in a sin(*x*)/*x* pattern, with a lobe width
proportional to the time duration of the sweep.

The measurements given in this paper confirm the first statement, but
show that, whilst the fall-off beyond this is indeed very steep, it does
not take the form predicted. It does show, however, that for the
YIG-tuned oscillator used, the broadening is substantially independent
of the sweep bandwidth (the necessary bandwidth) and that a fixed
allowance is probably a better first-order match to the actual behaviour
of the spectrum than one which us either proportional to the necessary
bandwidth, as is done with pulsed radars, or to the carrier frequency,
as is the case with the current recommendation.

# THE RADAR TRANSMitter

The radar is an experimental X-band FMCW radar. The master oscillator is
a YIG-tuned oscillator, and by controlling the bias and the amplitude
and duration of the ramp applied to the magnetic circuit, the centre
frequency and the duration and frequency excursion of the modulation may
be adjusted.

Apart from the limited stability of the centre frequency, it is
representative of the transmitter of the PILOT FMCW navigation radar as
it was when the radar was first developed in 1988, but it is not known
how representative it is of its current form.

# measurement technique

The output of the YIG is sent directly to the rotary joint and
transmitter. For the purpose of this experiment the signal was connected
directly to an HP8592B spectrum analyser. Since no carrier filter was
used, the dynamic range of the measurements was limited to about 50dB by
the need to use the analyser's attenuators to protect it.

Although this is an 'indirect' measurement, it is probably very
representative of what would be obtained by a direct measurement. Since
the resolution on an FMCW radar depends on the fidelity of the
transmitted chirp, any distortion caused by the rotary joint and antenna
will degrade its performance, and since the out of band emissions being
measured are very close to the necessary bandwidth, they are likewise
very unlikely to be significantly altered by the rotary joint or
antenna. The further-from-carrier spurii were not measured, but other
observations have indicated that the only other signals which are likely
to be measurable are the harmonics of the YIG, which may to be only 20dB
below the carrier at the YIG output.

The data points were read off the spectrum analyser using its marker
facility, recorded and then re-plotted.

Contrary to what is often recommended for the measurement of out-of-band
signals for pulse radars, where IF bandwidths comparable with the 3dB
bandwidth of spectrum are proposed, these measurements were performed in
the narrowest practical IF bandwidth, in order to measure the spectrum
to the highest possible resolution. The video bandwidth used was that
chosen automatically by the analyser to match the IF bandwidth. The data
was captured in 'peak hold' mode, to 'fill in' any notches caused by
interaction between the analyser scan period and the radar's sweep
repetition period. The display was then frozen and the data values
recorded at leisure.

The spectrum analyser's filters^2^ have a shape factor of about 15 at
--60dB, so at the --50dB level the shape factor would have been about
12.5, so the measured spectra would have been about 12MHz wider if
measured in 1MHz bandwidth than in 100kHz, and the filter response would
have dominated the observed roll-off.

# results

## Wideband Chirp, about 50MHz in about 1ms

The first experiment measured the spectrum of a 'typical' FMCW
navigation radar waveform. Figure 1 shows the spectrum of a relatively
wideband chirp.

![](media/image2.png){width="5.279166666666667in"
height="3.948611111111111in"}

Figure 1: Spectrum of a Wideband Chirp

The measurements were made with an IF bandwidth of 100kHz and a video
bandwidth of 30kHz.

The 'necessary bandwidth,' between the 'peaks' at the end of the
spectrum is 54.5MHz. The ramp length was 0.92ms, with a flyback period
of about 10μs and there was a 'CW' section about 30μs long, giving an
overall sweep repetition interval of 0.96ms. The time-bandwidth product
of the signal is thus about 50,000 (47dB).

Although the dynamic range is only 50dB, the fact that the signals are
still falling rapidly when they reach the noise floor at about --45dBm
suggests that the spectrum will be similarly contained down to at least
the --60dB level. The 'floor' at the ends of the spectrum represents the
analyser noise floor.

The rise from the noise floor to the lower peak takes place over about
2.3MHz and the fall at the higher end over about 800kHz, the accuracy of
this measurement being limited by the resolution of the marker.

There is no evidence of ripple within this spectrum, although some was
observed for a much narrower bandwidth. The 'peaks' at either end are
probably partly due to the Gibbs' phenomenon and partly to the
slowing-down on the ramp at its ends. The fact that the spike at the
upper end is higher than that at the lower is certainly due to the short
CW section between the end of the flyback and the start of the sweep.

## Wideband Chirp, about 20MHz in about 100μs

The second experiment investigated the effect of speeding-up the chirp.
Figure two therefore shows the results obtained with a high
time-bandwidth product but with a much faster chirp:

![](media/image3.png){width="5.279166666666667in"
height="3.948611111111111in"}

Figure 2: Spectrum of a Fast Wideband Chirp

This measurements was made with both the IF and video bandwidths being
30kHz.

The ramp length was 100μs, with a flyback period of about 5μs and there
a 'CW' section about 5μs long, giving an overall sweep repetition
interval of 0.11ms. The frequency excursion was less than for the
earlier measurement because of the limitations in the sweep generator.

The 'necessary bandwidth,' between the 'peaks' at the end of the
spectrum is 20.75MHz. The time-bandwidth product of the signal is thus
about 2075 (33dB).

The rise from the noise floor to the lower peak takes place over about
1.62MHz and the fall at the higher end over about 630kHz and there is
again no evidence of ripple within this spectrum.

## Narrowband Chirp, about 700kHz in about 100μs

The third experiment investigated the spectrum of signals with a much
lower time-bandwidth product.

![](media/image4.png){width="5.279166666666667in"
height="3.948611111111111in"}

Figure 3: Spectrum of a Narrowband Chirp

This measurement was again made with both the IF and video bandwidths
being 30kHz. Note that for this and the next measurement, the rise at
the lower end of the sweep has been drawn using a parabolic
interpretation between the measured end points, since this gives a good
representation of what was observed.

The sweep timing was the same as for the previous experiment, i.e. the
ramp length was 100μs, with a flyback period of about 5μs and a 'CW'
section about 5μs long, giving an overall sweep repetition interval of
0.11ms. The frequency excursion was reduced to 675kHz, so the
time-bandwidth product of the signal is thus about 67.5 (18.5dB).

The rise from the noise floor to the lower peak takes place over about
1.36MHz and the fall to --44dBm takes place over 188kHz, with the
remaining 3dB of fall taking place over about a further 300kHz. There is
again no evidence of ripple within this spectrum.

## Narrowband Chirp, measured in a narrower bandwidth

Figure four shows the same signal as figure three, but measured in IF
and video bandwidths each of 10kHz.

![](media/image5.png){width="5.279166666666667in"
height="3.948611111111111in"}

Figure 4: Narrower Measurement Bandwidth

The spectrum is almost identical to that measured in 30kHz bandwidth,
within the limitation of the number of points recorded. The change in
centre frequency was due to a re-adjustment of the sweep bias. This
result shows that the measurement bandwidth is having only a very minor
effect on the measurement of the spectrum.

The rise from the noise floor to the lower peak takes place over about
1.06MHz and the fall at the higher end was again over about 188kHz and
there is again no evidence of ripple within this spectrum.

## In-Band Ripple Measured in 1kHz Bandwidth

Figure 5 shows ripple in the passband of the same signal as was shown in
the last two figures, but measured with IF and video bandwidths both
equal to 1kHz. It should be noted that even in this narrower bandwidth
there was no evidence of ripple in the out-of-band region.

![](media/image6.png){width="5.279166666666667in"
height="3.948611111111111in"}

Figure 5: In-Band Ripple

The measurements were made between the peak at the lower end of the
spectrum and that at the top. The effect is similar to the Gibbs'
phenomenon, but is probably also influenced by the spectrum of the
flyback.

## Narrowband Chirp, Lower Duty Cycle

In another experiment, the sweep period and excursion were left at 100μs
and 675kHz, but the CW section was increased to about 900μs, reducing
the active duty cycle to 10%. Apart from the corresponding increase in
the height of the CW 'spike' at the upper end of the spectrum, the
spectrum was identical to that at high duty cycle, so the duty cycle
seems to be having a negligible effect.

# ANALYSIS

If the expected sidelobe structure in the out-of-band signals is in fact
present, it would have to be hidden under the observed roll-off. The
ripple in the passband is not believed to be significant, but the fact
that ripple can be seen in the passband when the resolution is
sufficiently high is also further evidence that there is no ripple in
the out-of-band sections.

The observed width of the roll-off is believed to be due to three
factors:

a\) imperfections in the waveform generator which generates the voltage
ramp which drives the YIG.

b\) time lags within the YIG circuit itself and

c\) the effects of the shape factor of the spectrum analyser filters.

Note that for a 30kHz wide filter the width at the --50dB point will be
about ±200kHz which is significant in comparison to the observed widths.

A possible explanation of the results is that the width of the
out-of-band response is composed of the following factors:

> a\) A filter width at the --50dB point of ±200kHz for the 30kHz
> filter, ±60kHz for the 10kHz filter and ±600kHz for the 100kHz filter,
>
> b\) A factor of about 200kHz width on the upper side and 1MHz on the
> lower side, which is independent of the sweep excursion and

c\) A factor of 10kHz/MHz which is dependent on the excursion.

If the 'fixed' and 'excursion dependent' components add up and the
effect of the spectrum width is added as a root-of-sums-of-squares then
the comparison between the predictions of this explanation and the
observed widths of the roll-off are as shown below:

[Sweep]{.underline} [Measurement]{.underline} [Predicted
\_\_\_\_]{.underline} \_\_\_\_\_\_[Measured \_\_\_\_]{.underline}

[Bandwidth]{.underline} [Bandwidth]{.underline} [Lower
Width]{.underline} [Upper Width]{.underline} [Lower Width]{.underline}
[Upper Width]{.underline}

54.50 0.1 1.67 0.97 2.30 0.80

20.75 0.03 1.22 0.45 1.62 0.63

0.68 0.03 1.02 0.28 1.36 0.20

0.68 0.01 1.01 0.22 1.06 0.20

[Table 1: Predicted and Measured Out-Of-Band Responses
(MHz)]{.underline}

The values are all in Megahertz. The correspondence between the
predicted and measured widths are reasonable. Better predictions would
undoubtedly be possible, but these would probably become more
specifically related to a particular radar type and so are not worth
exploring in this document, since the implicit aim of these measurements
has been to obtain data which can be used to derive generic
recommendations.

#  conclusions

It is believed that a YIG-tuned oscillator is the best representative of
a typical microwave FMCW radar source. It is highly linear, but probably
has more significant out-of-band signals than other types of oscillator
since its inductive tuning circuit is harder to drive at high speed.
Direct-digital synthesis and varactor tuning, the two other principle
means of generating FMCW sweeps, will probably produce even cleaner
spectra.

These imperfections mask the expected sidelobe structure in the
out-of-band region.

The spectrum of an FMCW radar may thus be described as being
trapedzoidal having a rectangular top with a width equal to the
'necessary bandwidth' of the signal and step sides. It may also have a
peak at one end due to any CW section before the sweep starts. The width
of each edge of the spectrum at the --50dB point may be taken as
±(1MHz + 10kHz per MHz of necessary bandwidth) wider than the necessary
bandwidth. A practical specification may double this allowance to
provide the difference between the 'typical' values and the 'must not
exceed' limits.

A decision must also be made as to the appropriate measurement
bandwidth: if the conventional process is followed, of using a bandwidth
comparable to the necessary bandwidth for the measurements, then a
significant adjustment must be made to the spectral masks to account for
this.

Since FMCW radars are typically simple, low-cost systems then it is
appropriate to include an allowance for drifts in the centre frequency,
due to the limited stability of the oscillators which arises from the
requirement to be able to modulate this. A suitable allowance would be a
few megahertz, say a limit of ±5MHz.

If the allocated band is wider than the sum of these factors, i.e. of
the necessary bandwidth, the allowance for the out-of-band roll off and
the allowance for frequency drift, then it would be practical to omit
the two latter factors and constrain the centre frequency of the radar
to be sufficiently far from the band edge so that the whole of the
necessary bandwidth and the out-of-band roll-off will always be within
the allowed bandwidth.

The allowance for drift is more related to the sweep bandwidth than it
is to the carrier frequency and so is probably best not made a function
of the centre frequency, at least over the range 1GHz to 10GHz.

# REFERENCES

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

2: Hewlett Packard: Manual for HP8592B Spectrum Analyser.