**Comparison of Radar Emission Spectra Measured in the Frequency Domain
Versus FFT Results from Time Domain Sampling**

**1. Introduction.**

Waveforms in the time domain can be represented in the frequency domain
via the Fourier transform. For purposes of computational practicality
and convenience, discretely time-sampled waveforms are ordinarily
converted to frequency domain representations via the fast Fourier
transform (FFT). In principle, an emission spectrum of a transmitter
should be measurable either directly in the frequency domain or else
indirectly via FFTs of time domain samples, and both techniques should
yield the same emission spectrum. Depending upon operational
contingencies, it could be argued on this basis that each ITU
administration might choose to pursue either approach to emission
spectrum measurements of radars for verification of compliance with such
emission mask criteria as ITU-R Recommendation SM.1541. It has been
suggested that it may therefore be desirable to modify ITU-R
Recommendation M.1177 to include a new provision for the measurement of
radar spectra using FFTs generated from discrete time domain data
samples.

SM.1541 specifies that the emission mask should be applied to the peak
envelopes of radar emission spectra.[^1] This requirement for compliance
with peak emission levels effectively means that any measurement
techniques that are to be used to verify compliance with SM.1541 or
other, similar emission spectrum mask criteria should yield peak
spectrum emission levels. Alternatively, it might be possible that a
technique yielding average spectrum emission levels might be adapted to
provide emission mask compliance data if it were found to be possible to
either convert the average data into peak data, or if there existed a
constant, frequency-independent amplitude offset between the peak and
average emission spectra of a radar transmitter throughout both the
in-band and unwanted emission regions. If such a frequency-independent
offset could be shown to exist, then average emission spectrum envelopes
would show the same ratio between in-band and unwanted emission levels
as peak emission envelopes.

The problem addressed in this contribution is whether time-sampled FFT
radar emission spectra will necessarily be the same as peak-detected
radar emission spectra measured directly in the frequency domain with
the stepped-frequency technique described in M.1177 (or at least might
have the same ratio of in-band to unwanted levels as is observed in
frequency-domain peak-detected measurements).

In order to examine this issue, the United Sates administration has
performed a series of emission spectrum measurements on high frequency
(HF) radars that employ FMCW and gated FMCW waveforms. The measurements
have been performed independently with two different techniques. The
first technique is that given in M.1177, in which peak-sampled emission
spectra are measured in the frequency domain with a spectrum
analyzer[^2] using a stepped-frequency algorithm. The second technique
generates radar emission spectra from FFTs of time-domain data collected
with a type of high-speed digitizer known as a vector signal analyzer
(VSA).[^3] The details of how these measurements were performed and the
resulting emission spectra that were measured with these two techniques
are presented and compared in this contribution.

It is shown that for the radar emission waveforms that have been
examined the spectra are different. This is because the M.1177
stepped-frequency technique produces the *peak* envelope of the measured
radar emission spectra, whereas the FFT results from time-domain
sampling produce *average* emission spectra. Furthermore, the *amplitude
offset* between the frequency-domain spectra and the FFT spectra are not
constant across the in-band and unwanted emission regions. The
time-sampled FFT technique yields unwanted emission levels that are
*lower* relative to the FFT-measured in-band power than the peak levels
of the unwanted emissions relative to the peak in-band emission level.
The FFT technique is demonstrated to not produce a radar emission
spectrum envelope that can be used to verify compliance with existing
emission mask criteria such as SM.1541.

**2. Stepped-Frequency Radar Spectrum Measurement Technique.**

As described in M.1177, measurements of radar emission spectra may be
performed by sequentially tuning a spectrum analyzer to a series of
individually selected frequencies and measuring the maximum,
peak-detected power at each frequency. As a practical matter the
analyzer is computer controlled. The technique relies upon a
*time-based* measurement of power within a bandwidth at each measured
frequency. The spectrum analyzer used for the measurement is effectively
operated as a slow-motion oscilloscope with an RF detector at each
measured frequency. The analyzer dwells on each measurement frequency in
zero-span mode long enough to measure the peak power response at that
frequency. The time required to complete a measurement at each frequency
is slightly longer than the beam-scanning interval of the radar being
measured. (E.g., if a radar scans a spatial beam across the measurement
location every 5 seconds, then the time required for a power measurement
at each frequency might be 5.2 seconds.)

The measurement bandwidth needs to be equal to or less than the radar
emission bandwidth as computed from its pulse characteristics. (E.g., if
a radar transmits 1-µs-long pulses, then the measurement bandwidth
should be 1 MHz or less.) M.1177 provides further guidance on the
technical details of such measurements. To measure the in-band,
out-of-band (OOB), and spurious domain components of a radar's emission
spectrum, the analyzer is stepped in frequency across the band of
interest in increments that are equal to or less than the 3-dB
measurement bandwidth. (E.g., if the measurement bandwidth is 1 MHz at
its 3-dB points, then the interval between measured frequencies should
be 1 MHz or less.)

An M.1177-style measurement system ordinarily uses a low noise amplifier
(LNA) in front of the spectrum analyzer to reduce the system noise
figure. A tunable, narrow-band filter, typically based on
yttrium-iron-garnet (YIG) technology, is used ahead of the LNA to
prevent it from overloading on in-band radar power when the measurement
frequencies are within the OOB and spurious frequency regions. A
variable-level attenuator is used ahead of the filter and the amplifier.
It is used to increase the dynamic range of the measurement system by
being set to zero decibels when measurements are being performed in the
spurious region, set to intermediate values (perhaps 20 to 40 dB) when
the measurement is being performed in the OOB region, and set to maximum
levels (60 to 70 dB) when the measurement is being performed within the
fundamental (in-band) region.

Radar emission spectra measured with this RF front-end design and the
stepped-frequency technique can routinely achieve dynamic ranges of
110 dB to 120 dB. Spectra that are measured with swept-frequency
techniques (which do not allow dynamic adjustment of front-end
attenuation across the spectrum) exhibit much more limited dynamic
range, usually not exceeding 60 dB to 70 dB. Typically, this is not
enough dynamic range to verify compliance with emission masks, as the
measurement dynamic range needs to be at least 10 dB greater than the
ultimate suppression level specified by a mask.[^4]

An attractive feature of the stepped-frequency approach to measuring
radar emission spectra is that particular frequencies where problems
occur in the course of the measurement can be selectively re-measured on
a one-by-one basis. For example, if either the transmitter or the
measurement system experience some sort of failure on any particular
frequency or frequencies, the problematic portion of the spectrum is
easily and immediately re-measured. Furthermore, measurement personnel
can (and ordinarily do) observe the time-domain behavior of the radar
transmitter on a frequency-by-frequency basis as the measurement
proceeds. This allows unusual, unexpected, or anomalous behaviors of the
transmitter to be observed. This feature turned out to be critical in
the analysis of the measurements that are described in this
contribution.

**3. Time-Sampled FFT Radar Spectrum Measurement Technique.**

An alternative spectrum measurement technique is to collect discretely
spaced time-domain samples of radar energy using a high-speed digitizer
and then transform those measurements to the frequency domain using a
FFT. The radar time waveform needs to be directly sampled at a rate that
meets or exceeds the Nyquist requirement for the highest frequency
component in the spectrum that is to be measured. In practice, this rate
must be somewhat higher than twice the highest frequency that is to be
measured.[^5]

The sampling process[^6] records, in effect, the total complex-number
representation of the waveform (i.e., the amplitude, frequency, and
phase information in the radar emission). After the data have been
collected for a number of radar pulse cycles, the stored waveform
samples are processed through an FFT computational engine to produce an
emission spectrum. The FFT spectrum can represent a convolution with any
effective bandwidth that is equal to or narrower than the half of the
rate at which the data were sampled.[^7]As described in the accompanying
JRG contribution, measured data may be zero-padded to force the FFT to
sample at more tightly spaced frequency points, although no new
information is added by this process. As also described in an
accompanying JRG contribution, it may be desirable and sometimes
necessary to sample the emitted radar waveform over multiple pulse
cycles to obtain an accurate FFT spectrum from the collected time
samples.

VSA-based time-sampling systems work as follows. Incoming radar
waveforms are downconverted, digitized, and stored into high-speed time
memory. The captured data are stored as blocks from which time,
frequency, and modulation domain data are derived.[^8] The data may be
replayed through the acquisition device (a VSA) at full or reduced
speed, saved to mass storage, or transferred to other software
applications such as Matlab©.

VSA data bandwidth is limited by the sampling rate (e.g., 95
Msamples/sec) and whatever RF and IF filtering might be installed in the
VSA. VSA sampling allows a wide range of data acquisition options in
which the frequency span of the measurement, the number of calculated
frequency points, and the effective resolution bandwidth of output FFTs
are all interrelated.

Depending upon circumstances, it may be desirable to install an LNA at
the RF input of a VSA measurement system to reduce its noise figure. But
the overall measurement system dynamic range will be restricted to the
smallest instantaneous dynamic range of either the front-end LNA or the
analog-to-digital converter (ADC). It is not possible with a
time-sampling system to use the M.1177 RF front-end combination of LNA,
tunable RF filter, and variable attenuator to extend the measurement
system dynamic range. Conversely, the time-sampling system can
potentially collect a given portion of the radar spectrum much faster
than the stepped-frequency algorithm.[^9]

**4. Stepped-Frequency Versus Time-Sampled FFT Measurements on FMCW and
Gated FMCW HF radars.**

Radars that have been measured for this study are used for scientific
ocean-surface observations. Deployed on beaches and buoys, they are used
to observe wave and current patterns at coastal locations. Emission
characteristics of the radars are given in JRG-\[XX\]. For the purposes
of this contribution, the most important features or these radars are
that they operate in the HF (2-30 MHz) portion of the spectrum and
utilize FMCW and gated FMCW waveforms.

Each of the radar emission spectra were measured first with an
M.1177-type spectrum analyzer-based system and then with a time-sampling
VSA-based system. The two systems are shown in block-diagram schematic
form in Figures 1 and 2. The VSA diagram of Figure 2 expands the
internal processing stages to show its signal processing technique more
clearly. Optional notch filters for the M.1177-type system of Figure 1
were available to measurement personnel but were ultimately deemed to
not be necessary and were not used. A single antenna, a slant-polarized
HF dipole, was used for both measurement systems. Other than the
inherent differences between the use of a spectrum analyzer for the
stepped-frequency algorithm versus the use of a VSA to collect time
samples for FFT processing, the most significant difference between the
two systems was the use of a specialized RF front-end with the spectrum
analyzer to extend the dynamic range of the M.1177 system. As discussed
above, this sort of RF front end cannot extend the dynamic range of a
time-sampling system. Thus the measurement system antenna was directly
connected to the input of the VSA without an LNA or any additional
external filtering or attenuation. There was a built-in 37 MHz-wide IF
bandpass filter inside the VSA, as shown in Figure 2.

For all but one of the radars, the measurement system antenna was
located within 60 feet (about 18 m) of the radar transmitter and a cable
run of fifty feet (about 15 m) of RG-58 coaxial cable connected the
antenna to each of the measurement systems in turn.[^10] Because the
same antenna was used for both measurement systems at a fixed location
for each of the radars, no differences in the emission spectra generated
by the two measurement systems could be attributed to variations in the
measurement antenna or its placement.

![](media/image1.png){width="6.5in" height="3.410416666666667in"}

Figure 1. Block diagram of HF radar spectrum measurement system in which
the ITU-R Recommendation M.1177 technique was implemented.

![](media/image2.png){width="6.5in" height="5.963888888888889in"}

Figure 2. Block diagram of HF radar spectrum measurement system in which
the discretely time sampled FFT technique was implemented.

The measurement bandwidth for the stepped-frequency, spectrum
analyzer-based measurement system met the criterion of M.1177. For the
gated FMCW waveforms this meant the measurement bandwidth was less than
(*B~c~/T*)*^1/2^*, where *B~c~* was the FM bandwidth of the radar[^11]
(e.g.: 49.4 kHz for the 13.46 MHz system) and *T* was the length of time
of the gating structure of the frequency sweep (e.g., 669 us for the
13.46 MHz system). For the FMCW radar, for example, the measurement
bandwidth was 1 kHz. The measurement IF (or resolution) bandwidth of the
spectrum analyzer was variously set to values between 1 kHz to 10 kHz
for the other radars. The frequency-step interval, or dwell time, for
the M.1177 measurements was set at 0.6 sec per frequency step, which was
slightly longer than the 0.5-sec frequency-sweep interval of each of the
radar transmitters.

The VSA samples for each of the radars were collected for a minimum of
10 seconds. This ensured that data for each of the radars were collected
over at least 20 complete frequency-sweep periods. An FFT was performed
on each set of collected data to generate emission spectra after the
measurements were completed and the data were post-processed. A detailed
description of this computational process, including the effect of
zero-padding the time-sampled data and transforming multiple time
sampled pulses to obtain a frequency domain measurement output is
contained in JRG document \[XX\].

Data acquired with the M.1177 technique are peak-detected in a
resolution bandwidth, and therefore the resulting spectra are *peak*
emission envelopes. They are obtained real-time and do not need to be
post-processed. In contrast, FFT spectra generated from VSA data are
*average* emission envelopes. If a given radar spectrum has a *constant*
decibel amplitude offset between its peak and average emission
envelopes, then both spectra will have the same *relative* decibel
amplitude offset between the levels of their in-band emissions and their
unwanted emissions. For purpose of verifying compliance with emission
mask criteria such as SM.1541, it is conceivable that under such a
circumstance an average emission spectrum envelope might be adequate to
show that the emission mask criteria are met, even though SM.1541
specifies that the mask criteria are to be applied to peak radar
spectrum emission levels.

However, if an emission spectrum does *not* have a *constant*
(frequency-independent) decibel amplitude offset between its peak and
average emission envelopes, then the peak and average emission spectra
will have *different* ratios between the levels of their in-band
emissions and their unwanted emissions. For the purpose of verifying
compliance with emission mask criteria such as SM.1541, this sort of
average emission spectrum will *not* be adequate to verify compliance
with emission mask criteria. As shown in the next section, such spectra
do exist.

**5. Comparison of Stepped-Frequency Versus VSA Spectrum Envelopes for
Radar Emissions.**

Figures 3 through 6 show the peak-detected emission spectra for a pair
of HF radars that operate with FMCW (Figure 3) and gated FMCW
(Figures 4-6) waveforms, as measured with the M.1177 system shown in
Figure 1. These spectra could be used to verify compliance of these
radars with the emission criteria of SM.1541.

**5.1 Comparison of Stepped-Frequency Versus VSA Spectrum Envelopes for
FMCW Radar Emissions.**

In order to see the differences between stepped-frequency, peak-detected
spectra and Fourier transformed time-sampled spectra for selected
radars, figures have been generated in which the respective spectra have
been overlaid. These comparison figures have been created for both FMCW
and gated FMCW radar signals.

Figures 7 through 9 show the in-band and OOB measurements of the FMCW
radar using the M.1177 technique (from the spectrum curve of Figure 3)
superimposed on spectra generated from FFTs of time-sampled data that
were acquired on the same radar with the VSA system of Figure 2. The FFT
spectra were generated with and without zero padding for 1 to 8 radar
pulses, respectively.

![](media/image3.png){width="6.096527777777778in"
height="4.4222222222222225in"}

Figure 3. Peak-detected emission spectrum of a FMCW HF radar.

![](media/image4.png){width="6.070833333333334in"
height="4.423611111111111in"}

Figure 4. Peak-detected emission spectrum of a gated FMCW HF radar at
4.5 MHz.

![](media/image5.png){width="6.201388888888889in"
height="4.473611111111111in"}

Figure 5. Peak-detected emission spectrum of a gated FMCW HF radar at
13.5 MHz.

![](media/image6.png){width="6.5in" height="4.653472222222222in"}

Figure 6. Peak-detected emission spectrum of a gated FMCW HF radar at
24.6 MHz.

![](media/image7.png){width="5.759027777777778in" height="4.33125in"}

Figure 7. FMCW waveform with M.1177 data superimposed on an FFT spectrum
of 1-pulse transformed time-sampled data.

![](media/image8.png){width="5.719444444444444in"
height="4.295138888888889in"}

Figure 8. FMCW waveform with M.1177 data superimposed on an FFT spectrum
of 4-pulse transformed time-sampled data.

![](media/image9.png){width="5.679166666666666in"
height="4.270833333333333in"}

Figure 9. FMCW waveform with M.1177 data superimposed on an FFT spectrum
of 8-pulse transformed time-sampled data.

Figures 7 through 9 show that FFT transformed time sampled data are
8-12 dB below the peak detected emission spectra in the OOB and spurious
domains of the FMCW waveform, relative to the measured power in the
in-band part of the spectrum. This effect is due to the time-sampled
waveform in effect being measured by the VSA at an average level, as if
an analog root mean square (RMS) detector had been used in an analog
measurement receiver instead of a peak detector.

Figures 10 and 11 illustrate the source of this effect. In Figure 10,
the radar emission is shown in the time domain at an in-band emission
frequency. In Figure 11, the radar emission is shown in the time domain
as it typically appeared at any given frequency in the unwanted emission
region. In the in-band region, both peak and average detection register
the same output level. But in the unwanted emission region a
short-duration 'tic' in the time domain occurs at the moment that the
radar transmitter undergoes a phase change during its swept-frequency
flyback. This tic (which, as Figure 10 shows, also occurs at the in-band
frequencies but which is much lower in amplitude than the desired
emission in the in-band region) registers on a peak detector but will
not register on an average detector or the equivalent FFT spectrum
output of a set of VSA data.

Comparison of the frequency domain and time domain data from the M.1177
measurement and the time-sampled VSA measurement show that when the
measured in-band spectrum power levels are normalized for the M.1177 and
VSA techniques, *the two techniques yield OOB and spurious emission
levels that differ by 10-12 dB*. The source of the difference is the
existence of a 10-12 dB difference between the peak and average emission
levels in the unwanted emissions, a difference that does not exist for
the in-band emission levels. The FFT spectra generated from

![](media/image10.png){width="5.465972222222222in"
height="3.8756944444444446in"}

Figure 10. Time domain behavior of the FMCW radar of Figure 3 in its
in-band region, as documented during the M.1177 stepped-frequency
measurement. Peak and average detectors register the same output levels
for this emission.

![](media/image11.png){width="5.53125in" height="3.920138888888889in"}

Figure 11. Time domain behavior of the FMCW radar in its unwanted
emission region.

The phase-switching flyback transient at 0.35 sec dominates; peak
detection or a very high sampling rate is required to measure it. The
VSA sampling rate is not high enough to capture the flyback transient.
In contrast, it registers the power level of the emissions at 0.1 sec
and 0.6 sec, which are about 12 dB lower than the transient power level.
The VSA measurement fails to detect the peak and indicates that the
unwanted emission levels are lower than they really are. The VSA
time-sampled data do not capture the radar waveform's transient
phase-switch component because the VSA FFT is, in effect, using RMS
detection on the spectrum.

The figures also illustrate a feature in the spectrum plots that arises
when the time-sampled data are transformed using multiple pulses of the
radar waveform. Transforming multiple time-sampled pulses to generate
the frequency domain plots generates better resolution and gives the
*appearance* of reducing the emissions in the OOB, as compared to the
M.1177 peak detected measurements. Figures 7-9 show this effect when 1,
4, and 8 pulses are processed, respectively.

**5.2 Comparison of Stepped-Frequency Versus VSA Spectrum Envelopes for
Gated FMCW Radar Emissions.**

Figures 12 through 14 show the measurements of the gated FMCW radar
waveforms using the M.1177 technique along with the VSA FFT time-sampled
data with 2x zero padding and multiple pulses[^12].

![](media/image12.png){width="6.022916666666666in"
height="4.270833333333333in"}

Figure 12. Gated FMCW (4.53 MHz) waveform with M.1177 data superimposed
on an FFT spectrum of 520-pulse transformed time-sampled data.

![](media/image13.png){width="6.0152777777777775in"
height="4.235416666666667in"}

Figure 13. Gated FMCW (13.46 MHz) waveform with M.1177 data superimposed
on an FFT spectrum of 750-pulse transformed time-sampled data.

![](media/image14.png){width="5.775in" height="4.33125in"}

Figure 14. Gated FMCW (24.65 MHz) waveform with M.1177 data superimposed
on an FFT spectrum of 520-pulse transformed time-sampled data.

In Figures 12 through 14, the emission spectra measured using the M.1177
stepped-frequency, peak detection technique (from Figures 4-6) are drawn
as red traces. Figures 12 and 14 generally show a good agreement between
the transformed time-sampled waveforms and the frequency domain
measurements. Figure 13 does not show such good agreement between the
results of the two different techniques. In Figure 13 the difference
between the curves in the OOB region is on the order of 15 dB. However,
as explained above, the VSA system did not include an LNA. Its inherent
noise was greater than the received measured signal, and this noise
thereby dominated the data in the OOB region. The OOB measurement of
Figure 13 shows VSA noise rather than radar emission energy. This
happened because the radar transmitter was over 300 feet away from the
measurement system receiver antenna and was partially hidden behind sand
dunes. The M.1177 measurement included an LNA that provided a low enough
noise figure to overcome this problem.

Referring to the data in Figures 12 and 14, the M.1177 spectra and the
FFT spectra made from the VSA data are in good agreement. This is
because the peak and average emission levels of these radar spectra were
either the same across the span of the measurement, or else because
there existed a *constant* decibel-level offset between the peak and
average emission levels of the radar spectra.

**6. Summary.**

The data presented in this contribution demonstrate that radar emission
spectra produced by FFTs of discrete time-sampled data will not
necessarily agree with peak-detection data collected with M.1177-type
measurement systems, such as for example a FMCW waveform with time
transient artifacts. When in-band emission data taken with the same
measurement antenna on the same radar for this type of waveform are
normalized, the FFT spectra can be at least 10-12 dB lower in the
unwanted emission region than the peak-envelope spectra. The difference
arises when the power ratio between the in-band and unwanted emissions
has a different value for peak-detected emission spectra than it does
for average emission spectra.

FFT processing of an increasing number of radar pulses *lowers* the
apparent relative level of the unwanted emissions compared to the level
of the in-band emissions and makes the difference between the results
for the unwanted emission power levels for the two techniques *larger*.
For shorter time sequences with fewer pulses this smaller difference
arises from the effect of the sampling window. The longer time sequences
with more pulses more closely estimate the average power spectrum and
deviate more from the peak-detected measurement. The apparent
improvement in the unwanted radar emission spectrum power levels that is
derived from processing increased numbers of radar pulses is illusory,
if the goal of the measurement is to obtain the peak emission envelope
of the radar spectrum.

For pulsed or gated FMCW waveforms, the two measurement techniques
produced similar results. For these waveforms, a constant
(frequency-independent) decibel amplitude offset existed between the
peak and average emission envelopes of the radar spectra. In this case,
when the peak and average spectra were normalized (to zero decibels in
the in-band emission region), they overlaid each other almost perfectly.

**7. Conclusions.**

While it may be possible to use FFT spectra for radar waveforms that
generate a constant (frequency-independent) decibel amplitude offset
between the peak and average emission levels throughout the in-band and
unwanted emission spectrum regions, the only way to verify that this
situation exists is to use an M.1177 compliant measurement system based
on peak detection. But performing verification with M.1177
stepped-frequency measurements would make an additional set of FFT/VSA
emission spectrum measurements redundant.

Another concern is that the windowing function used for the FFT and the
number of pulses that are measured for the FFT can significantly affect
the shape of the emission spectra and can have the effect of reducing
the levels of the unwanted emissions relative to the measured in-band
power level. This effect is contrary to the goal of obtaining peak
emission spectrum envelopes for verification of compliance with existing
emission spectrum mask criteria such as SM.1541.

Since some radar waveforms generate different offsets between the peak
and average emission levels in the in-band versus unwanted emission
regions, only an M.1177-compliant, peak-detected, stepped-frequency
measurement should be used at present to verify a radar's compliance
with SM.1541 emission mask criteria or other criteria that are based on
peak emission spectrum envelopes. Until or unless the failure of the
FFT/VSA measurement approach to measure peak emission levels is
resolved, it would seem to be premature at this time to consider
modifications to M.1177 to include a new, FFT-based spectrum measurement
technique.

[^1]: ITU-R SM.1541, Section 1.5: "Both the in-band and the unwanted
    emissions should be evaluated in terms of peak values."

[^2]: The spectrum analyzer was an Agilent 4407.

[^3]: The VSA was an Agilent 89600S series model number 89641A, that
    accepts RF inputs up to 6 GHz.

[^4]: For example, if the ultimate suppression level for unwanted radar
    emissions is -60 dBc relative to the measured peak in-band power
    level, the measurement system's dynamic range should be at least
    70 dB. This characteristic will ensure that the measurement system's
    internal noise does not add measurably to the power of the unwanted
    emissions. Similarly, a suppression requirement of -80 dBc would
    require a dynamic range capability of at least 90 dB for the
    measurement system.

[^5]:
    > If a sampling device cannot achieve a sufficiently high sampling
    > rate for a radar RF frequency, then the radar RF energy is
    > downconverted to a lower frequency before it is channelled into
    > the digitizer. In practice this IF frequency is typically 19 MHz.

[^6]: "Sampling process" here includes the ADC sampling followed by
    quadrature detection and decimation.

[^7]: Under some circumstances the maximum effective convolution
    bandwidth can be equal to the sampling rate, but it is safest to
    assume as a general rule that the maximum bandwidth will be less
    than half that wide.

[^8]: The time length of a VSA data record is (*N*-1)/*∆f*, where *N* =
    number of computed frequency points and *∆f* is the frequency span
    of the measurement. The time sample resolution is 1/(*k*\**∆f*),
    where *k* = 2.56 for a VSA baseband data mode.

[^9]: A spectrum frequency range that may require an hour to measure
    with the stepped-frequency M.1177 technique might be collected in
    about 1 minute with a VSA-based system, albeit with potentially less
    dynamic range for the VSA data.

[^10]: This distance placed the measurement and radar antennas within
    their respective near fields. This was acceptable because the goal
    of the measurements was to establish the differences, if any,
    between the outputs of the two measurement systems for identical RF
    inputs. Also, some emission spectrum compliance measurements (as
    opposed to antenna pattern and gain measurements) can be adequately
    performed within near-field distances.

[^11]: See JRG-01 Section 3, along with Table 1 (waveform parameters)
    and Figure 1 (waveform structure) for a discussion on the specifics
    of each CODAR waveform parameter.

[^12]: To see a discussion of why multiple (500 or more) pulses were
    taken in the FFT, refer to Section 5.2 of JRG-01.