# JRG-35 Rev1 {#jrg-35-rev1 .unnumbered}

# United States of America  Derivation of the Necessary Bandwidth (-20 dB Bandwidth) and -40 dB Bandwidth Formulas for Un-Modulated Pulse Waveforms {#united-states-of-america-derivation-of-the-necessary-bandwidth--20-db-bandwidth-and--40-db-bandwidth-formulas-for-un-modulated-pulse-waveforms .unnumbered}

## Background {#background .unnumbered}

At the July 2004 meeting of JRG 1A-1C-8B, a contribution was submitted
that addressed the derivation of the necessary bandwidth (-20 dB
bandwidth)[^1] and -40 dB bandwidth for un-modulated pulse waveforms.
This document supersedes document JRG-09, and provides a more detailed
derivation of the formulas in ITU-R Recommendations SM.853-1 and
SM.1541.

In Recommendation ITU-R SM.1541 un-modulated pulses are also referred to
as non-FM pulsed emissions. In these waveforms there is no intentional
frequency or phase modulation during the pulse. Though the following
derivation assumes the ideal case, real radar output devices introduce
incidental angle modulation and other effects during the pulse. When
establishing bandwidth formulas it is necessary to study the effects of
incidental angle modulation of real systems to see whether the spectrum
differs significantly from the findings of this paper.

## Introduction {#introduction .unnumbered}

The following defines the pulse waveform considered in this derivation:

![](media/image1.wmf){width="4.010416666666667in"
height="1.5104166666666667in"}

Figure 1. Pulse model

In Figure 1 if t~1~ \> 0, the indicated waveform is called a trapezoidal
pulse. If t~1~ = 0, the waveform is called a rectangular pulse. The
period of the pulse sequence is T. 1/T is sometimes called the pulse
repetition rate (PRR).

We begin the introduction by describing the spectrum of a periodic
rectangular pulse waveform. The equation for the magnitude of the line
spectra of a rectangular pulse is derived in Appendix A. The final form,
(A.28), is reproduced as follows:

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

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

where f~n~ = n/T = nPRR, where n is an integer.

The derivations in this paper are not concerned with absolute levels,
but relative. Therefore we consider the following, which has a maximum
amplitude of unity.

  ----------------------------------------------------------------- -----
  ![](media/image3.wmf)                                             ()

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

In the determination of bandwidth formulas we are interested in the
envelope of the spectrum. We note that the maximum value of the sine
function in () is 1, and therefore propose an envelope function as
follows (substituting f~n~ with the continuous f, and t~0~ with the
pulse width τ):

  ----------------------------------------------------------------- -----
  ![](media/image4.wmf)                                             ()

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

() and () are plotted in Figure 2 after converting to decibels.[^2]^,^
[^3]

The equation for the magnitude of the line spectra of a periodic
trapezoidal pulse is derived in Appendix A. The final form, (A.24), is
reproduced as follows:

  ----------------------------------------------------------------- -----
  ![](media/image5.wmf)                                             ()

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

Noting that t~0~ + t~1~ is the 50% amplitude pulse width (τ), and that
t~1~ represents both rise and fall times measured between 0% and 100%
amplitude (δ), we make these substitutions in the following:

  ----------------------------------------------------------------- -----
  ![](media/image6.wmf)                                             ()

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

The line spectra are proportional to the following, and reach a maximum
value of unity when f~n~ = 0.

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

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

Similar to () we can substitute 1 for the sine functions of () to arrive
at an envelope function as follows:

  ----------------------------------------------------------------- -----
  ![](media/image8.wmf)                                             ()

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

Figure 3 is a plot of both () and ():

In both Figure 2 and Figure 3 we note that the spacing between line
spectra is the PRR. We also note that the width of each spectral lobe is
1/τ, which comes from the fact that as f~n~ approaches integer multiples
of 1/τ, the sine argument approaches integer multiples of π, which
yields the nulls of the sinc function.[^4] In Figure 3 there is another
overarching lobing structure of a greater frequency (1/δ) which comes
from the multiplication of the additional sinc function in () as
compared to ().

![](media/image9.wmf){width="9.010416666666666in"
height="3.0104166666666665in"}

Figure 2. One Side of the Spectrum of a Periodic Rectangular Pulse
Plotted with Envelope

![](media/image10.wmf){width="9.010416666666666in"
height="3.0104166666666665in"}

Figure 3. One Side of the Spectrum of a Periodic Trapezoidal Pulse
Plotted with Envelope

There are a couple more things to note about the envelopes in both
figures. The first point is that we can define the frequency values of
both envelopes at which their relative attenuation levels are zero
decibels. For the rectangular pulse we set the attenuation to 1
(corresponding to 0 dB) in () to get the following:

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

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

The frequency corresponding to 0 dB attenuation for the trapezoidal
pulse envelope is determined similarly using ():

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

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

The second point is to note that both envelopes have constant slopes
when plotted on a logarithmic frequency scale, Log(f). The following
shows the definition for the slope of a straight line:

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

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

For the rectangular pulse envelope, applying () using () for y gives the
following:

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

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

For the trapezoidal pulse envelope, applying () using () gives:

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

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

So we have shown that the envelope of the rectangular pulse starts at a
frequency of () and slopes off at ‑20 dB/decade, and the envelope of the
trapezoidal pulse starts at () and slopes downward at ‑40 dB/decade.

A final insight comes from plotting both envelopes against the spectrum
of a trapezoidal pulse.

![](media/image16.wmf){width="6.510416666666667in"
height="3.0104166666666665in"}

Figure 4. Trapezoidal Pulse Power Spectrum Plotted with Trapezoidal and
Rectangular Pulse Envelopes.

Figure 4 shows that if we base X dB bandwidth formulas for a trapezoidal
pulse on () alone, we will overestimate the bandwidth for 0 dB
bandwidths up to a certain magnitude X dB level. The frequency at which
that level occurs (f~i~) corresponds to the intersection of envelopes
and can be calculated by setting the right-hand sides of () and () equal
to each other:

  ----------------------------------------------------------------- -----
  ![](media/image17.wmf)                                            ()

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

Now if we substitute () into () we see that the attenuation level at the
intersection (A~i~) of both envelopes depends on the relationship
between pulse width and rise time:

  ----------------------------------------------------------------- -----
  ![](media/image18.wmf)                                            ()

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

So for attenuation levels larger than A~i~ we obtain a more accurate
bandwidth for the trapezoidal pulse by basing formulas on the
rectangular pulse envelope.

Since the appropriate formula to use for both ‑20 dB and ‑40 dB
bandwidths could be based on either envelope depending on the
relationship between pulse width and rise time, the remainder of the
derivation considers both cases.

## X dB Bandwidth Formulas Based on Rectangular Envelope {#x-db-bandwidth-formulas-based-on-rectangular-envelope .unnumbered}

To arrive at bandwidth formulas based on the rectangular pulse envelope
we rearrange () and note that since f is a one-sided bandwidth, the
desired bandwidth will be twice this:

  ----------------------------------------------------------------- -----
  ![](media/image19.wmf)                                            ()

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

## X dB Bandwidth Formulas Based on Trapezoidal Envelope {#x-db-bandwidth-formulas-based-on-trapezoidal-envelope .unnumbered}

To arrive at bandwidth formulas based on the trapezoidal pulse envelope
we rearrange () similarly:

  ----------------------------------------------------------------- -----
  ![](media/image20.wmf)                                            ()

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

Next we note that rise time is normally given as the 10% to 90% rise
time (t~r~), so we use the following to perform the conversion:

  ----------------------------------------------------------------- -----
  ![](media/image21.wmf)                                            ()

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

Making this substitution into () gives the following:

  ----------------------------------------------------------------- -----
  ![](media/image22.wmf)                                            ()

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

Finally, to assist in determining which envelope to use for the proper
bandwidth of a given trapezoidal pulse, we need to convert () for use
with t~r~. It is also important to note by inspection of Figure 4, that
regardless of what X dB bandwidth is desired, the proper envelope to use
is always the one that will produce the smaller bandwidth:

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

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

## ‑20 dB Bandwidth {#db-bandwidth .unnumbered}

Up to this point we have been dealing with the amplitude spectrum, which
for real signals is the square root of the power spectrum. Therefore ,
the attenuation which is 20 dB below the maximum power is A^2^ = 1/100.
Thus A = 1/10. Substituting this into () gives the ‑20 dB bandwidth for
the rectangular pulse envelope as follows:[^5]

  ----------------------------------------------------------------- -----
  ![](media/image24.wmf)                                            ()

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

The ‑20 dB bandwidth for the trapezoidal pulse envelope is found by
substituting A = 0.1 into ():[^6]

  ----------------------------------------------------------------- -----
  ![](media/image25.wmf)                                            ()

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

Thus, B~-20\ dB~ for un-modulated pulse is the smaller of () and ().[^7]
Through application of () we see that the smaller bandwidth will be ()
when τ/t~r~ ≥ 12.5. Otherwise, () is used.

## ‑40 dB Bandwidth {#db-bandwidth-1 .unnumbered}

The attenuation which is 40 dB below the maximum power is A^2^ =
1/10000. Thus A = 1/100. Substituting this into () gives the ‑40 dB
bandwidth for the rectangular pulse envelope as follows:

  ----------------------------------------------------------------- -----
  ![](media/image26.wmf)                                            ()

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

The ‑40 dB bandwidth for the trapezoidal pulse envelope is found by
substituting A = 0.01 into ():

  ----------------------------------------------------------------- -----
  ![](media/image27.wmf)                                            ()

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

Thus, B~-40\ dB~ for un-modulated pulse is the smaller of () and ().
Through application of () we see that the smaller bandwidth will be ()
when τ/t~r~ ≥ 125. Otherwise, () is used.

The constant of 5.7 in (23) is based on the theoretical emission bounds
of an un-modulated pulse waveform. Since the -40 dB bandwidth is used to
define the emission mask for radar systems, it is important to include
allowances for incidental angle modulation and other effects during the
pulse.

Thus, the formula for the -40 dB bandwidth for an un-modulated pulse
waveform in Recommendation ITU-R SM.1541 replaces the constant 5.7 with
a K factor as shown in (24). The value of *K* is 6.2 for radars with
output power greater than 100 kW, and 7.6 for lower‑power radars and
radars operating in the radionavigation service in the 2900-3100 MHz and
9200-9500 MHz bands.

In Recommendation ITU-R SM.1541 (Equation 39), the -40 dB bandwidth is
given as the lesser of:

  ----------------------------------------------------------------- --------
  ![](media/image28.wmf)                                            \(24\)

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

![](media/image29.wmf){width="6.510416666666667in"
height="3.0104166666666665in"}

Figure 5. Example of Trapezoidal Pulse Spectrum Where Envelopes
Intersect at ‑40 dB Bandwidth

Figure 5 shows a more exaggerated view of the benefit of using the
rectangular pulse envelope for determining X dB bandwidths out to its
intersection with the trapezoidal pulse envelope. It also shows an
interesting case where ideal trapezoidal pulse emissions outside of the
‑40 dB bandwidth attenuate at the rate of ‑40 dB/decade, without
filtering. This will be true as long as: τ/δ ≤ 100, or τ/t~r~ ≤ 125. At
any rate, the ideal spectrum of a trapezoidal pulse always drops off at
‑40 dB/decade, without filtering, outside of the 20Log(δ/τ) dB
bandwidth, which is 2/(πδ).

#####  Periodic Pulse Amplitude Spectrum Derivation

###### Trapezoidal Pulse {#trapezoidal-pulse .unnumbered}

The graphic in Figure 1 depicts the pulse under consideration in this
derivation:

Following is the definition of a single trapezoidal pulse h(t):

  ----------------------------------------------------------------- -------
  ![](media/image30.wmf)                                            (A.1)

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

Following is the definition for the Fourier Series:

  ----------------------------------------------------------------- -------
  ![](media/image31.wmf)                                            (A.2)

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

where f~n~ = n/T, where n is an integer.

Following is the corresponding definition of the Fourier coefficients
C~n~ for this pulse:

  ----------------------------------------------------------------- -------
  ![](media/image32.wmf)                                            (A.3)

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

Since there are three portions of h(t), the following 3 integrals apply,
where I~1~ + I~2~ + I~3~ = I:

  ----------------------------------------------------------------- -------
  ![](media/image33.wmf)                                            (A.4)

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

  ----------------------------------------------------------------- -------
  ![](media/image34.wmf)                                            (A.5)

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

  ----------------------------------------------------------------- -------
  ![](media/image35.wmf)                                            (A.6)

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

In an attempt to make I~1~ more manageable, use the change of variables
x = t + t~1~ + t~0~/2. Then t = x -- t~1~ -- t~0~/2, and dt = dx. When t
= --t~0~/2, x = t~1~. When t = -t~1~ - t~0~/2, x = 0

  ----------------------------------------------------------------- -------
  ![](media/image36.wmf)                                            (A.7)

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

In an attempt to make I~3~ more manageable, use the change of variables
x = t~1~ + t~0~/2 - t. Then t = ‑x + t~1~ + t~0~/2, and dt = -dx. When t
= t~0~/2, x = t~1~. When t = t~0~/2 + t~1~, x = 0

  ----------------------------------------------------------------- -------
  ![](media/image37.wmf)                                            (A.8)

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

The following integral solutions and identity are used in the remainder
of this derivation:

  ----------------------------------------------------------------- -------
  ![](media/image38.wmf)                                            (A.9)

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

  ---------------------------------------------------------------- --------
  ![](media/image39.wmf)                                           (A.10)

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

  ---------------------------------------------------------------- --------
  ![](media/image40.wmf)                                           (A.11)

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

In order to make the derivation more manageable, substitute "a" for
j2πf~n~. I~1~ + I~2~ + I~3~ becomes:

  ---------------------------------------------------------------- --------
  ![](media/image41.wmf)                                           (A.12)

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

Substituting integral solutions gives:

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$I = e^{a(t_{1} + \frac{t_{0}}{2})}\frac{1}{a^{2}t_{1}}\left\lbrack e^{- \text{at}_{1}}\left( - \text{at}_{1} - 1 \right) + 1 \right\rbrack + \frac{1}{a}\left\lbrack e^{\frac{\text{at}_{0}}{2}} - e^{\frac{- \text{at}_{0}}{2}} \right\rbrack + e^{- a(t_{1} + \frac{t_{0}}{2})}\frac{1}{a^{2}t_{1}}\left\lbrack e^{\text{at}_{1}}\left( \text{at}_{1} - 1 \right) + 1 \right\rbrack$$   (A.13)

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

The following equations step through the remainder of the derivation:

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$I = - \frac{e^{\frac{\text{at}_{0}}{2}}}{a} - \frac{e^{\frac{\text{at}_{0}}{2}}}{a^{2}t_{1}} + \frac{e^{a(t_{1} + \frac{t_{0}}{2})}}{a^{2}t_{1}} + \frac{e^{\frac{\text{at}_{0}}{2}}}{a} - \frac{e^{\frac{- \text{at}_{0}}{2}}}{a} + \frac{e^{\frac{- \text{at}_{0}}{2}}}{a} - \frac{e^{\frac{- \text{at}_{0}}{2}}}{a^{2}t_{1}} + \frac{e^{- a(t_{1} + \frac{t_{0}}{2})}}{a^{2}t_{1}}$$   (A.14)

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

  ---------------------------------------------------------------- --------
  ![](media/image42.wmf)                                           (A.15)

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

  ----------------------------------------------------------------- --------
  ![](media/image43.wmf)                                            (A.16)

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

  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$I = \frac{1}{a^{2}t_{1}}\left\lbrack - e^{a(t_{0} + t_{1}\frac{)}{2}}e^{\frac{- \text{at}_{1}}{2}} + e^{a(t_{0} + t_{1}\frac{)}{2}}e^{\frac{\text{at}_{1}}{2}} - e^{- a(t_{0} + t_{1}\frac{)}{2}}e^{\frac{\text{at}_{1}}{2}} + e^{- a(t_{0} + t_{1}\frac{)}{2}}e^{\frac{- \text{at}_{1}}{2}} \right\rbrack$$   (A.17)

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

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------
  $$I = \frac{1}{a^{2}t_{1}}\left\lbrack e^{a(t_{0} + t_{1}\frac{)}{2}}\left( e^{\frac{\text{at}_{1}}{2}} - e^{\frac{- \text{at}_{1}}{2}} \right) - e^{- a(t_{0} + t_{1}\frac{)}{2}}\left( e^{\frac{\text{at}_{1}}{2}} - e^{\frac{- \text{at}_{1}}{2}} \right) \right\rbrack$$   (A.18)

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

  ---------------------------------------------------------------- --------
  ![](media/image44.wmf)                                           (A.19)

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

  ---------------------------------------------------------------- --------
  ![](media/image45.wmf)                                           (A.20)

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

  ---------------------------------------------------------------- --------
  ![](media/image46.wmf)                                           (A.21)

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

  ---------------------------------------------------------------- --------
  ![](media/image47.wmf)                                           (A.22)

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

  ---------------------------------------------------------------- --------
  ![](media/image48.wmf)                                           (A.23)

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

  ---------------------------------------------------------------- --------
  ![](media/image49.wmf)                                           (A.24)

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

###### Rectangular Pulse {#rectangular-pulse .unnumbered}

The rectangular pulse is also depicted by Figure 1 except that t~1~ = 0.
Thus, following is the definition of a single rectangular pulse h(t):

  ---------------------------------------------------------------- --------
  ![](media/image50.wmf)                                           (A.25)

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

  ---------------------------------------------------------------- --------
  ![](media/image51.wmf)                                           (A.26)

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

Using (A.9) this becomes:

  ---------------------------------------------------------------- --------
  ![](media/image52.wmf)                                           (A.27)

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

  ---------------------------------------------------------------- --------
  ![](media/image53.wmf)                                           (A.28)

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

As a test of the validity of the final forms of C~n~ for both
trapezoidal and rectangular pulse, we note the expected result that
substituting t~1~ = 0 into (A.24) yields (A.28).

[^1]: Recommendation ITU-R SM.853-1 defines the necessary bandwidth for
    un-modulated pulsed emissions as 20 dB below the peak envelope
    value.

[^2]: A voltage (V) is converted to decibels (VdB) by applying the
    following: VdB = 20Log(V)

[^3]: All logarithms in this paper are base 10.

[^4]: The sinc function is defined as sinc(x) = sin(πx)/(πx), thus ()
    could be rewritten as sinc(f~n~τ)

[^5]: The actual calculation in () should be 6.37/t, but the published
    formula is 6.36/t. The difference is insignificant, and probably due
    to a rounding error.

[^6]: The actual calculation in () should be 1.80/sqrt(tt~r~), but the
    published formula is 1.79/sqrt(tt~r~). The difference is
    insignificant, and probably due to a rounding error.

[^7]: All practical pulses have non-zero rise times, and can thus be
    approximated as trapezoidal