# Theory of FMCW Radar Waveforms

## Introduction

The current -20 dB bandwidth formula for FMCW within the ITU is the
following:[^1]

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  ![](media/image1.wmf)                                             ()

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The current -40 dB bandwidth for FMCW is the following:

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  ![](media/image2.wmf)                                             ()

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Where F~o~ is the carrier frequency, and B~c~ is the total chirp
deviation. The F~o~ term in () seems to provide for carrier drift within
the -40 dB formula, so the formulas indicate that the FMCW waveform is
"brickwall", at least down to the -40 dB bandwidth.

There is concern internationally that the bandwidth formulas are too
narrow especially for applications whose center frequency is in the HF
range.[^2] In addition, it is irregular to include frequency tolerance
in a bandwidth formula.[^3] Unfortunately the origin of these formulas
is unknown.

This paper develops the theory of the FMCW waveform, evaluates current
bandwidth formulas, and proposes a methodology to find appropriate
bandwidth formulas.

## FMCW Signal

There may be different forms of the FMCW waveform. The one we consider
in this section is linear FM where the flyback portion is also chirped.
This waveform is conceptually a normalized linear periodic signal m(t)
which is frequency modulated onto a carrier.

![](media/image3.wmf){width="6.395833333333333in" height="2.09375in"}

Figure 1. Sawtooth waveform showing non-zero flyback. Period is τ +
t~fb~.

Figure 1 depicts the frequency deviation of the waveform under
consideration. If t~fb~ = τ the resulting signal may be called
"triangular" FMCW. If the ratio of t~fb~/τ is small, or zero, the
waveform might be called "sawtooth".

m(t) is frequency modulated onto a carrier such that the maximum
excursions about the carrier will be ±0.5B~c~, as shown in the
following. Thus the FMCW signal's frequency varies linearly over a range
of B~c~ centered on the carrier, chirping up in frequency in a time of
τ, and chirping back down again in a time of t~fb~. A critical
consideration in this analysis is that the signal is constant envelope.
Any amplitude weighting would affect the spectrum, and must be
considered in a separate analysis.

![](media/image4.wmf){width="3.4583333333333335in" height="2.09375in"}

Figure 2. FMCW is a carrier whose frequency varies linearly between F~o~
± 0.5B~c~ in time τ

To facilitate the analysis we break this signal down into the
convolution of two waveforms as shown in the following Figures 3a and
3b:

![](media/image5.wmf){width="4.9375in" height="1.6395833333333334in"}

Figure 3a

![](media/image6.wmf){width="4.9375in" height="1.6395833333333334in"}

Figure3b. A forward and return chirp convolved with a sequence of delta
functions

Using the properties of the Fourier analysis we know that the spectrum
of Figure 1 will be the spectrum of Figure3ba multiplied by the spectrum
of Figure3bb. Since the spectrum of a series of delta functions in time
separated by period T is a series of delta functions in frequency
separated by 1/T, the resultant will be the Fourier Transform of
Figure3ba as a function of line spectra spaced by 1/(τ+t~fb~). Thus we
only need to analyze Figure3ba.

Figure3ba consists of a forward and return chirp both spanning the same
frequency range, but at different rates (Hz/s). We condense these into
one figure (see Figure ) where the forward chirp occurs between --τ and
0, and the return chirp occurs from 0 to t~fb~.

The following is a general expression of the voltage versus time
waveform x(t) for FM modulation, where m(λ) varies between ±1:

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  ![](media/image7.wmf)                                             ()

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![](media/image8.wmf){width="4.53125in" height="2.8333333333333335in"}

Figure 4. m(λ) truncated to an interval --τ ≤ λ ≤ t~fb~

Performing the integration of () for a positive chirp (using Figure )
gives the following:[^4]

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  ![](media/image9.wmf)                                             ()

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Differentiating the argument of () with respect to time shows the
instantaneous frequency

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  ![](media/image10.wmf)                                            ()

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Applying -τ ≤ t ≤ 0 to () shows that the frequency varies between F~o~
-- 0.5Δω/π and F~o~ + 0.5Δω/π. Since we want it to vary between
±0.5B~c~, this means that Δω = πB~c~. The final form for the positive
chirp is the following:

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  ![](media/image11.wmf)                                            ()

  ----------------------------------------------------------------- -----

where --τ ≤ t \< 0

For the negative chirp it is:

  ----------------------------------------------------------------- -----
  ![](media/image12.wmf)                                            ()

  ----------------------------------------------------------------- -----

where 0 ≤ t \< t~fb~

The sampled output of () can be appended to that of () to give one
period of the voltage sequence x(n).

## Implementing Continuous Phase (CP) FMCW

The FMCW waveform is continuous frequency because of the chirped
flyback. Also, referring to Figure , the waveform is continuous-phase
(CP) across the high frequency chirp transition because of the way the
equations are written; but there is not necessarily phase continuity
across the lower end. Because it has been suggested that phase
discontinuities will broaden the spectrum, this section derives
adjustments to FMCW that will force phase continuity across the periodic
waveform. The power spectrum of the equations of section 1.2, after
adjustment for differing amounts of phase discontinuity according to the
equations of this section, can be examined to see if, and by how much,
spectrum is improved through CP.[^5]

The derivation is straightforward and begins by setting the arguments of
() and () equal to each other.

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  ![](media/image13.wmf)                                            ()

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The 2πn term is added because adding any integer multiple of 2π does not
change the value of a sine wave. The term is added for analyzing the
non-continuous-phase (NCP) case to ensure that phase differences are the
same across different waveform periods.

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  ![](media/image14.wmf)                                            ()

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  ![](media/image15.wmf)                                            ()

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  ![](media/image16.wmf)                                            ()

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  ![](media/image17.wmf)                                            ()[^6]

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Figure () shows that CP (corresponding to = 0) is achieved simply by
ensuring that the product of center frequency and waveform period is an
integer. For computer analyses, setting F~o~ to 0 is the simplest way to
achieve CP. For NCP, () allows one to fix the amount of phase
discontinuity so we can compare resulting X dB bandwidths across a
number of different waveform periods. This also allows one to research
whether bandwidths change depending on the amount of phase
discontinuity. One might guess that smaller phase discontinuities would
lead to smaller bandwidths.

Figure 4 and Figure 5 show a single period of an FMCW waveform as
implemented by () and (), where the center frequency is adjusted
according to () to achieve NCP and CP, respectively.

![](media/image18.wmf){width="6.46875in" height="2.46875in"}

Figure 4. Example of phase discontinuity across periods

![](media/image19.wmf){width="6.46875in" height="2.46875in"}

Figure 5. F~o~ adjusted according to () to achieve phase continuity

## Rectangular FM chirp

It may be helpful to note that the FMCW waveform is the same as a
rectangular FM pulse whose duty cycle is set to 100%. Figure 6 shows a
rectangular pulse centered at 10 MHz, which chirps up a total of 10 MHz
in 1 μsec, and down again in 1 μsec, with a duty cycle of less than
100%. Figure 7 shows how FMCW is formed simply by raising the same
signal to a 100% duty cycle. In the frequency domain the continuous
function of the Fourier Transform of a single pulse is sampled with
frequency elements spaced apart by the PRF, ensuring that one of them
coincides with the fundamental frequency. When this is done different
patterns emerge in the spectrum depending on the PRF (see Figure 8).
Although the spectrum appearance varies with duty cycle, the envelope of
rectangular FM pulse spectrum is independent of duty cycle unless the
duty cycle is exactly equal to 100%.[^7] This is due to the effect of
pulsing on the FM chirp, which convolves the spectrum with that of the
pulse shape. This effect abruptly goes away when the signal is no longer
pulsed and indicates that the FM pulse bandwidth formulas cannot
converge (with increasing duty cycle) to those we choose for FMCW.

![](media/image20.png){width="5.472222222222222in"
height="3.2006944444444443in"}

Figure 6. A forward and return chirp convolved with a sequence of delta
functions

![](media/image21.png){width="5.491666666666666in"
height="3.2041666666666666in"}

Figure 7. Delta function spacing matches width of forward and return
chirp

![](media/image22.png){width="6.499305555555556in"
height="8.498611111111112in"}

Figure 8. Spectrum of triangular FM rectangular pulse with varying duty
cycles

## Sample Spectrums

Following are spectrum plots using the equations in this paper. Section
1.5.1 shows the effects of using differing amounts of phase
discontinuity with reference to (). The graphs show the discontinuity as
a fraction of 2π radians, as shown in (). For a given set of FMCW
parameters, the broadest spectrum occurs when approaches π. As expected,
by the time reaches 2π the spectrum returns to the CP case. Section
1.5.2 shows the effect of varying the ratio of flyback to forward chirp
time. As expected the broadest spectrum occurs when the flyback is zero,
with the narrowest spectrum occurring when flyback and forward chirp
times are equal. The curves are distinguished from each other by color,
but they are also distinguished with the order of the curves in the
legend being the same as those in the plot.

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  ![](media/image23.wmf)                                            ()

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### Samples using different amounts of phase discontinuity

![](media/image24.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 9

![](media/image25.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 10

![](media/image26.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 11

![](media/image27.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 12

![](media/image28.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 13

### Samples using different flyback ratios

![](media/image29.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 14

![](media/image30.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 15

![](media/image31.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 16

![](media/image32.png){width="6.499305555555556in"
height="3.9993055555555554in"}

Figure 17

The figures in section 1.5.2 lead to a few observations. The spectrum of
FMCW is a non-linear function of the phase discontinuity () between
periods with the spectrum being narrowest when is an even integer
multiple of π, and broadest at odd integer multiples of π. For all cases
of NCP the spectrum eventually rolls off at 20 dB/decade. Thus
out-of-band (OOB) spectral advantages of unfiltered FMCW are not
realized unless the signal is CP. For the CP cases the spectrum roll-off
is 60 dB/decade for all t~fb~/τ ratios except for 0, in which case it is
40 dB/decade. Absolute bandwidths decrease with increasing t~fb~/τ up to
a value of unity. The spectrum drop from the chirp edge is steeper with
higher B~c~τ products, particularly with higher τ. Since current
bandwidth formulas predict that FMCW is very steep they appear to be
assuming that B~c~τ and τ are large.

## Evaluation of Current Bandwidth Formulas

This section looks at theoretical FMCW based on the FFT in comparison
with the current bandwidth formulas. The plots of this section are based
on FFTs generated from 120 different combinations of FMCW parameters.
The chirps ranged from 10 kHz to 100 MHz over a span of 100 nanoseconds
to 1 second, and t~fb~ ranged from 0 to 100% of τ.

### -20 dB Bandwidth

This subsection shows plots of the -20 dB bandwidth versus τ, and
several plots of different chirp widths. Each plot contains the ITU
formula (called "RSEC20") along with 4 curves of flyback time as a
percentage of τ (e.g. "10%tfb/tau" indicates the curve where t~fb~ is
fixed at 10% of τ). The plots also specify either "Phi = 0Pi" or "Phi =
1Pi" indicating either CP or 180 degree phase discontinuity,
respectively.

![](media/image33.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 18a

![](media/image34.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 19b**

![](media/image35.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 19a

![](media/image36.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 20b**

![](media/image37.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 20a

![](media/image38.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 21b**

![](media/image39.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 21a

![](media/image40.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 22b**

![](media/image41.wmf){width="6.489583333333333in" height="3.4375in"}

Figure 22a

![](media/image42.wmf){width="6.489583333333333in" height="3.4375in"}

**Figure 23b**

### -40 dB Bandwidths

This subsection shows the plots of the -40 dB bandwidth versus τ, and
plots of different chirp widths. Each plot contains the ITU formula
(called "RSEC40") along with 4 curves of flyback time as a percentage of
τ (e.g. "100%tfb/tau" indicates the curve where t~fb~ is as long as τ).
The plots also specify either "Phi = 0Pi" or "Phi = 1Pi" indicating
either CP or 180 degree phase discontinuity, respectively. In the
application of () F~o~ was picked to be as close to 1 GHz as possible,
and the RSEC40 curve was based on 1 GHz. F~s~ could vary from curve to
curve but was limited so that M never exceeded 6x10^6^ due to computing
limitations.

![](media/image43.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 23a

![](media/image44.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 24b**

![](media/image45.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 24a

![](media/image46.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 25b**

![](media/image47.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 25a

![](media/image48.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 26b**

![](media/image49.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

Figure 26a

![](media/image50.wmf){width="6.489583333333333in"
height="3.4479166666666665in"}

**Figure 27b**

![](media/image51.wmf){width="6.489583333333333in" height="3.4375in"}

Figure 27a

![](media/image52.wmf){width="6.489583333333333in" height="3.4375in"}

**Figure 28b**

## Summary Observations 

1.  The OOB roll-off for NCP ( ≠ 0) chirped flyback FMCW is 20
    dB/decade.

2.  The OOB roll-off for CP ( = 0) chirped flyback FMCW with zero
    flyback (t~fb~/τ = 0) is 40 dB/decade.

3.  The OOB roll-off for CP chirped flyback FMCW with non-zero flyback
    (t~fb~/τ \> 0) is 60 dB/decade.

4.  As t~fb~/τ approaches unity the bandwidths decrease for both CP and
    NCP.

5.  The FMCW waveform is Time-Bandwidth product (B~c~τ) dependant. The
    higher B~c~τ is, regardless of the center frequency, the steeper the
    spectrum becomes, meaning that the ratio of -20 and -40 dB
    bandwidths approaches unity. This is valid for both CP and NCP
    waveforms.

6.  Given the same B~c~τ, the spectrum is narrower with larger τ than
    larger B~c~.

7.  The current FMCW -20 dB bandwidth formula underestimates the
    theoretical spectrum, although the fit improves with increasing
    B~c~τ.

8.  As long as B~c~τ ≥ 1000, the ITU FMCW -20 dB bandwidth
    underestimates the FFT by less than 10% regardless of phase
    discontinuity. For cases where t~fb~ = τ and the discontinuity is
    small ( ≤ 0.1π), B~c~τ can be as small as 100 and still keep the
    error below 10%.

9.  The current FMCW -40 dB bandwidth formula usually underestimates the
    theoretical spectrum, but it may be adequate for a high B~c~τ and
    high center frequency.

10. For CP FMCW with large F~o~, the current -40 dB bandwidth formula
    overestimates the bandwidth (see Figure 24), even for small B~c~τ
    (see Figure 23).

11. For small F~o~, the current FMCW -40 dB bandwidth formula can
    underestimate the bandwidth, even for large B~c~τ CP FMCW.

12. Since spectrum is not dependent on center frequency it is
    inappropriate for the bandwidth formula to be a function of center
    frequency.

# Bandwidth Proposals

## Introduction

The previous portion of this paper concluded that the current bandwidth
formulas for FMCW are inadequate, and provided a background for
developing a methodology to derive more suitable formulas. This section
analyzes a large data set of FFT bandwidths of various FMCW waveforms,
organizes the data, makes simplifying observations, and curve fits the
data with various proposed bandwidth formulas. Appendices A and B
provide the code segment and sample spreadsheet used to generate the
data set analyzed in this section, so that anyone with Matlab® or Excel®
can duplicate and follow the process for generating and evaluating the
proposals set forth herein.

The previous analysis showed that phase discontinuities broaden the
spectrum. As the phase discontinuity is increased from 0 to π, the
bandwidths monotonically increase, though not linearly. It is unknown
whether or not phase discontinuities exist in real FMCW system
implementations, or whether they can be controlled if they do. Due to
this uncertainty it may be prudent to analyze different cases of NCP in
addition to CP. Therefore CP ( = 0) and two cases of NCP ( = 0.5π and =
π) are addressed in the remainder of this paper.

## Data Generation

Using the code segment in APPENDIX A, and the sample input spreadsheet
in Appendix B, bandwidths were generated for 480 different combinations
of FMCW parameters. There were 120 simulations each for phase
discontinuities () of 0, 0.1π, 0.5π, and π. The simulations for each
used 5 values for B~c~ {0.01, 0.1, 1, 10, 100}, 6 consecutive elements
of {0.01, 0.1 1, 10, 100, 1000, 10000, 100000, 1000000} for τ, and 4
values for t~fb~/τ {0, 0.01, 0.1, 1}.[^8] The simulations show that the
bandwidths increased in magnitude with phase discontinuity. In order to
compare FFT bandwidths with RSEC20 and RSEC40, their ratio was
calculated. When all the data were sorted by B20/RSEC20, a pattern
emerged as shown in Table 1.[^9]

Table 1. Sample of data generated

  -------------------------------------------------------------------------------------------------------------------------------
  F~o~          B~c~   τ        B~c~\*τ   t~fb~    t~fb~/τ   /π    F~s~   B20           B40           B20/RSEC20    B40/RSEC40
  ------------- ------ -------- --------- -------- --------- ----- ------ ------------- ------------- ------------- -------------
  99.99999975   1      100000   100000    100000   1         0.1   30     1.002222895   1.006471772   1.002222895   0.97715706

  99.9999975    10     10000    100000    10000    1         0.1   300    10.02222895   10.06471772   1.002222895   1.003461387

  99.999975     100    1000     100000    1000     1         0.1   3000   100.2222895   100.6471772   1.002222895   1.006169921

  100           100    1000     100000    1000     1         0     3000   100.2632928   100.8307749   1.002632928   1.008005348

  100           10     10000    100000    10000    1         0     300    10.02632928   10.08307749   1.002632928   1.005291874

  100           1      100000   100000    100000   1         0     30     1.002632928   1.008307749   1.002632928   0.978939563

  99.999875     100    1000     100000    1000     1         0.5   3000   100.3343887   103.609783    1.003343887   1.035787094

  99.9999875    10     10000    100000    10000    1         0.5   300    10.03343887   10.3609783    1.003343887   1.032998833

  99.99999875   1      100000   100000    100000   1         0.5   30     1.003343887   1.03609783    1.003343887   1.005920224

  90.90909045   1      100000   100000    10000    0.1       0.1   60     1.004207114   1.013668586   1.004207114   0.984144258

  90.90908636   10     10000    100000    1000     0.1       0.1   600    10.04207114   10.13668586   1.004207114   1.010636676

  90.90904545   100    1000     100000    100      0.1       0.1   6000   100.4207114   101.3668586   1.004207114   1.013364577

  100           1      100000   100000    10000    0.1       0     60     1.004702024   1.016131416   1.004702024   0.986535355

  100           100    1000     100000    100      0.1       0     6000   100.4702024   101.6131416   1.004702024   1.015826668

  100           10     10000    100000    1000     0.1       0     600    10.04702024   10.16131416   1.004702024   1.013092139
  -------------------------------------------------------------------------------------------------------------------------------

Table 1 shows the pattern that every three consecutive values of in
column B20/RSEC20 are identical. After sorting the table by , and noting
that RSEC20 for FMCW is equal to the chirp B~c~, the following
observation was made. For each , the BW formula is some factor
multiplied by the chirp, where the factor is a function of B~c~τ and
t~fb~/τ.

  ----------------------------------------------------------------- -----
  ![](media/image53.wmf)                                            ()

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When B40 was divided by the chirp a pattern of similar values emerged,
showing that () also applies to the -40 dB bandwidth.[^10]

To find the factors, hereafter called "B20factor" and "B40factor" (which
are B20/B~c~ and B40/B~c~, respectively), additional sorting and
filtering was done to produce the following figures.

![](media/image54.wmf){width="6.510416666666667in"
height="2.5104166666666665in"}

Figure 28. B20/B~c~ vs B~c~τ curves for different t~fb~/τ, for the CP
case ( = 0)

![](media/image55.wmf){width="6.520833333333333in" height="2.53125in"}

Figure 29. B40/B~c~ vs B~c~τ curves for different t~fb~/τ, for the CP
case ( = 0)

![](media/image56.wmf){width="6.510416666666667in"
height="2.5104166666666665in"}

Figure 30. B20/B~c~ vs. B~c~τ curves for different t~fb~/τ, for the NCP
case ( = 0.5π)

![](media/image57.wmf){width="6.510416666666667in"
height="2.4791666666666665in"}

Figure 31. B40/B~c~ vs. B~c~τ curves for different t~fb~/τ, for the NCP
case ( = 0.5π)

![](media/image58.wmf){width="6.520833333333333in" height="2.53125in"}

Figure 32. B20/B~c~ vs. B~c~τ curves for different t~fb~/τ, for the NCP
case ( = π)

![](media/image59.wmf){width="6.510416666666667in"
height="2.4791666666666665in"}

Figure 33. B40/B~c~ vs. B~c~τ curves for different t~fb~/τ, for the NCP
case ( = π)

Each of these figures contain four curves for different values of
t~fb~/τ along with an additional curve which is an equation fitted to
the t~fb~/τ = 0.01 curve.[^11]^,^ [^12] The form of B20factor and
B40factor equations is arbitrary, and it may be possible to improve
them, if necessary. At B~c~τ = 0.1 the curves rose sharply making it
difficult to curve fit with the larger values of B~c~τ. Therefore, all
equations that follow are valid only for B~c~τ ≥ 1.

All data points are spaced one decade apart, and are connected by
straight lines. The B20factor and B40factor curves were all drawn with
100 evenly spaced points per decade, to show how the curve is shaped
between decades.

The figures show that bandwidths decrease with increasing B~c~τ and
t~fb~/τ. The bandwidth for t~fb~ = 0 does not drop to within 110% of the
bandwidth for t~fb~ = τ until B~c~τ exceeds 100 for B20, and 10000 for
B40.

The following equations are the final forms for -20 and -40 dB bandwidth
formulas, based on the cases presented in the above sections:

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  ![](media/image60.wmf)                                            ()

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  ![](media/image61.wmf)                                            ()

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  ![](media/image62.wmf)                                            ()

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  ![](media/image63.wmf)                                            ()

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  ![](media/image64.wmf)                                            ()

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  ![](media/image65.wmf)                                            ()

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where B~c~τ ≥ 1

Formulas () and () were compared to the output of real waveforms
generated by an Agilent Vector Signal Generator (VSG) programmed to
generate various FMCW waveforms. Summary Tables were made that compared
the -20 and -40 dB bandwidth values to validate that the two equations
are a good approximation of FFT theory and also to begin to investigate
implementation factors. The results are given in Ref. [^13]. The
implementation factors for these equations will require additional
research and measurements.

**APPENDIX A**

This section lists a Matlab® script file used to generate the data of
this paper.

\% FMCWchirpedFlyback.m

\% Calculate the spectrum of a complex CW chirp. Plot the power spectrum

clear all; format long g;

Plotit=0; % 1 if you want to plot, 0 otherwise

PlotTime=0; % 1 if you want to plot voltage vs time, 0 otherwise

semi=1; % 1 if you want to plot spectrum on logx scale, 0 otherwise

BWs=1; % 1 if you want to calculate X dB bandwidths, 0 otherwise

Intplt=1; % 1 if you want to interpolate these BWs, 0 otherwise

Writeout=1; % 1 if you want to write an output matrix to Excel, 0
otherwise

b1=3;b2=20;b3=30;b4=40; %bX are the various calculated X dB bandwidths

\[indata,colnames\]=xlsread(\'FMCWinputs.xls\'); %read input parameters
from Excel file

Fo_v=indata(:,1); %center frequency vector

Bc_v=indata(:,2); %chirp vector

tau_v=indata(:,3); %tau vector

tfb_v=indata(:,4); %flyback time vector

Fs_v=indata(:,5); %sampling frequency vector

outdata={\'Fo\',\'Bc\',\'tau\',\'tfb\',\'Fs\',\'M\',\'N\',\[\'B\'
num2str(b1)\],\[\'B\' num2str(b2)\],\...

\[\'B\' num2str(b3)\],\[\'B\' num2str(b4)\]};

for Cnt = 1:length(Bc_v),

Fo = Fo_v(Cnt); % center frequency of chirp, MHz

Bc = Bc_v(Cnt); % chirp bandwidth

tau=tau_v(Cnt); %length of up chirp, usec

tfb=tfb_v(Cnt); %flyback time

Fs=Fs_v(Cnt); %sampling frequency

dt=1/Fs;

ChtRng=2000; %range of frequencies to be plotted, MHz

if ChtRng\>Fs,

ChtRng=Fs;

end

DR=140; %total dynamic range to be displayed, dB

t=-tau:dt:0-dt; %time for first part of LFM

x = cos(2\*pi\*Fo\*t+pi\*Bc\*(t.\*t/tau+t)) +
j\*sin(2\*pi\*Fo\*t+pi\*Bc\*(t.\*t/tau+t));

%Complex time domain is necessary to depict the spectrum properly

if tfb\>0,

t2=0:dt:tfb-dt; %time for second part of LFM

x2 = cos(2\*pi\*Fo\*t2-pi\*Bc\*(t2.\*t2/tfb-t2)) +
j\*sin(2\*pi\*Fo\*t2-pi\*Bc\*(t2.\*t2/\...

tfb-t2)); % complex time domain signal

x=\[x,x2\]; clear x2; %free up x2 memory

t = \[t,t2\]; clear t2; %time for pulse

end

M=length(t);

N=M; %# FFT points. Must equal M for FMCW

DF=Fs/N; %frequency resolution

if \~(Plotit & PlotTime), clear t; end %free up memory

X = 20\*log10(abs(fftshift(fft(x,N)))); %generate the power spectrum

X=X-max(X); %normalize the power spectrum

if \~(Plotit & PlotTime), clear x; end %free up memory

f = -Fs/2:DF:Fs/2-DF; % frequency scale, MHz

if BWs, % Calculate the 4 XdB bandwidths specified

if Intplt, %if DF is large you can improve accuracy through
interpolation

u=max(find(X\>-b1));v=min(find(X\>-b1));

B1=interp1(X(u:u+1),f(u:u+1),-b1)-interp1(X(v-1:v),f(v-1:v),-b1);

u=max(find(X\>-b2));v=min(find(X\>-b2));

B2=interp1(X(u:u+1),f(u:u+1),-b2)-interp1(X(v-1:v),f(v-1:v),-b2);

u=max(find(X\>-b3));v=min(find(X\>-b3));

B3=interp1(X(u:u+1),f(u:u+1),-b3)-interp1(X(v-1:v),f(v-1:v),-b3);

u=max(find(X\>-b4));v=min(find(X\>-b4));

B4=interp1(X(u:u+1),f(u:u+1),-b4)-interp1(X(v-1:v),f(v-1:v),-b4);

else

B1 = f(max(find(X\>-b1))+1) - f(min(find(X\>-b1))-1);

B2 = f(max(find(X\>-b2))+1) - f(min(find(X\>-b2))-1);

B3 = f(max(find(X\>-b3))+1) - f(min(find(X\>-b3))-1);

B4 = f(max(find(X\>-b4))+1) - f(min(find(X\>-b4))-1);

end

elseif semi,

u=max(find(X\>-b1));v=min(find(X\>-b1));

B1=interp1(X(u:u+1),f(u:u+1),-b1)-interp1(X(v-1:v),f(v-1:v),-b1);

end

if Plotit,

figure(1); % plot

tstring=\[\'F_o=\',num2str(Fo),\'MHz; B_c=\',num2str(Bc),\';
\\tau=\',num2str(tau),\...

\'\\musec; t\_{fb}=\',num2str(tfb),\'; F_s=\',num2str(Fs),\';
m=\',num2str(M)\];

if PlotTime,

subplot(2,1,1);

plot(t,real(x),\'b\')

xlabel(\'time\');

ylabel(\'amplitude\')

title(tstring);

subplot(2,1,2); % plot the spectrum

tstring=\[\'n=\',num2str(N)\];

else

tstring=\[tstring,\'; n=\',num2str(N)\];

end

if BWs & (B4\>ChtRng),ChtRng=B4;end

if semi,

h=semilogx(f-Fo,X,\'b-\');

axis(\[10\^floor(log10(B1/2)),10\^ceil(log10(ChtRng/2)),-DR,0\]);

xlabel(\'frequency offset (MHz)\');

else

plot(f,X,\'b-\');

axis(\[Fo-ChtRng/2,Fo+ChtRng/2,-DR,0\]);

xlabel(\'frequency (MHz)\');

end

set(gca,\'gridlinestyle\',\'-\')

set(gca,\'minorgridlinestyle\',\'-\')

ylabel(\'Power (dB)\')

if BWs,

tstring=\[tstring,\'; B\',num2str(b1),\'=\',num2str(B1),\';
B\',num2str(b2),\'=\',\...

num2str(B2),\'; B\',num2str(b3),\'=\',num2str(B3),\';
B\',num2str(b4),\'=\',num2str(B4)\];

end

title(tstring);

grid on;

end

outdata=cat(1,outdata,{Fo,Bc,tau,tfb,Fs,M,N,B1,B2,B3,B4}); %create
output data

end

if Writeout,

xlswrite(outdata) %open output data in Excel

end

APPENDIX B

This section lists a M.S. Excel® spreadsheet called \'FMCWinputs.xls\'
which contains the inputs for the Matlab script for the = 0 case. This
method of input was used for ease because input parameters may need to
be tweaked due to the FFT sensitivities of sampling rates on data
accuracy, and the memory and speed limitations of different computers.

  ------------------------------------------------------------------------
  Fo     Bc          tau               tfb               Fs
  ------ ----------- ----------------- ----------------- -----------------
  0      0.01        10                0                 1000

  0      0.01        10                0.1               1000

  0      0.01        10                1                 1000

  0      0.01        10                10                1000

  0      0.01        100               0                 100

  0      0.01        100               1                 100

  0      0.01        100               10                100

  0      0.01        100               100               100

  0      0.01        1000              0                 100

  0      0.01        1000              10                100

  0      0.01        1000              100               100

  0      0.01        1000              1000              100

  0      0.01        10000             0                 100

  0      0.01        10000             100               100

  0      0.01        10000             1000              100

  0      0.01        10000             10000             100

  0      0.01        100000            0                 60

  0      0.01        100000            1000              60

  0      0.01        100000            10000             55

  0      0.01        100000            100000            30

  0      0.01        1000000           0                 6

  0      0.01        1000000           10000             6

  0      0.01        1000000           100000            6

  0      0.01        1000000           1000000           3

  0      0.1         1                 0                 10000

  0      0.1         1                 0.01              10000

  0      0.1         1                 0.1               10000

  0      0.1         1                 1                 10000

  0      0.1         10                0                 1000

  0      0.1         10                0.1               1000

  0      0.1         10                1                 1000

  0      0.1         10                10                1000

  0      0.1         100               0                 1000

  0      0.1         100               1                 1000

  0      0.1         100               10                1000

  0      0.1         100               100               1000

  0      0.1         1000              0                 1000

  0      0.1         1000              10                1000

  0      0.1         1000              100               1000

  0      0.1         1000              1000              1000

  0      0.1         10000             0                 600

  0      0.1         10000             100               600

  0      0.1         10000             1000              600

  0      0.1         10000             10000             300

  0      0.1         100000            0                 60

  0      0.1         100000            1000              60

  0      0.1         100000            10000             60

  0      0.1         100000            100000            30

  0      1           1                 0                 10000

  0      1           1                 0.01              10000

  0      1           1                 0.1               10000

  0      1           1                 1                 10000

  0      1           10                0                 1000

  0      1           10                0.1               1000

  0      1           10                1                 1000

  0      1           10                10                1000

  0      1           100               0                 1000

  0      1           100               1                 1000

  0      1           100               10                1000

  0      1           100               100               1000

  0      1           1000              0                 1000

  0      1           1000              10                1000

  0      1           1000              100               1000

  0      1           1000              1000              1000

  0      1           10000             0                 600

  0      1           10000             100               600

  0      1           10000             1000              600

  0      1           10000             10000             300

  0      1           100000            0                 60

  0      1           100000            1000              60

  0      1           100000            10000             60

  0      1           100000            100000            30

  0      10          0.1               0                 100000

  0      10          0.1               0.001             100000

  0      10          0.1               0.01              100000

  0      10          0.1               0.1               100000

  0      10          1                 0                 10000

  0      10          1                 0.01              10000

  0      10          1                 0.1               10000

  0      10          1                 1                 10000

  0      10          10                0                 10000

  0      10          10                0.1               10000

  0      10          10                1                 10000

  0      10          10                10                10000

  0      10          100               0                 10000

  0      10          100               1                 10000

  0      10          100               10                10000

  0      10          100               100               10000

  0      10          1000              0                 6000

  0      10          1000              10                6000

  0      10          1000              100               6000

  0      10          1000              1000              3000

  0      10          10000             0                 600

  0      10          10000             100               600

  0      10          10000             1000              600

  0      10          10000             10000             300

  0      100         0.01              0                 1000000

  0      100         0.01              0.0001            1000000

  0      100         0.01              0.001             1000000

  0      100         0.01              0.01              1000000

  0      100         0.1               0                 100000

  0      100         0.1               0.001             100000

  0      100         0.1               0.01              100000

  0      100         0.1               0.1               100000

  0      100         1                 0                 100000

  0      100         1                 0.01              100000

  0      100         1                 0.1               100000

  0      100         1                 1                 100000

  0      100         10                0                 100000

  0      100         10                0.1               100000

  0      100         10                1                 100000

  0      100         10                10                100000

  0      100         100               0                 60000

  0      100         100               1                 60000

  0      100         100               10                60000

  0      100         100               100               30000

  0      100         1000              0                 6000

  0      100         1000              10                6000

  0      100         1000              100               6000

  0      100         1000              1000              3000
  ------------------------------------------------------------------------

[^1]: See ITU-R SM.1541 equations (38) and (44).

[^2]: The dependence of () on center frequency may cause the most severe
    underestimation of bandwidth for systems in the HF range.

[^3]: The Radio Regulations definition of "assigned frequency band"
    specifically distinguishes the necessary bandwidth from the
    frequency tolerance.

[^4]: Positive chirps are defined to be those that sweep from low to
    high frequency.

[^5]: It is assumed that the power spectrum will be calculated using the
    FFT of the x(n) sequence described at the end of Section 1.2

[^6]: Since n is an arbitrary integer, the liberty has been taken to
    reverse the sign of n in ()

[^7]: The fact that the envelope doesn't change allows FM pulse
    bandwidth formulas to be independent of duty cycle.

[^8]: 5 x 6 x 4 = 120 simulations

[^9]: B20 is the FFT -20 dB bandwidth. RSEC20 is the Annex J 20 dB
    bandwidth. RSEC40 is the RSEC 40 dB bandwidth

[^10]: This is seen by dividing the B40 column by the B~c~ column (note
    these values *are not* shown in Table 1).

[^11]: t~fb~/τ = 1% is assumed to be a worst-case value in this
    derivation, because t~fb~ = 0 seemed unrealistic. In addition, for
    most of the plots there was not much difference between the two
    values.

[^12]: The curve fitting used the least mean square method

[^13]: "Results of FMCW Simulation and Measurements"