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+--------+-------------------------+----------+----------------------+
| **F    | **A. G. Stove**         | *        | 14 December 2005     |
| rom:** |                         | *Date:** |                      |
|        | andy.s                  |          |                      |
|        | tove@uk.thalesgroup.com |          |                      |
|        |                         |          |                      |
|        | +44 (0) 1293 644291     |          |                      |
+--------+-------------------------+----------+----------------------+
| *      | **ITU WP1A-1B-8B JRG    | Ref.:    | **BANDWIDTHS OF FMCW |
| *To:** | DG1**                   |          | SIGNALS**            |
+--------+-------------------------+----------+----------------------+

One way to visualize the spectrum of an FMCW waveform is to begin with
the recognition that the complex representation of an infinitely long
chirp:

s = exp(jt2), (*Eqn*. 1

where is the chirp rate, is its own Fourier transform1.

S = exp\[j/2) (*Eqn*. 2

A qualitative explanation of this begins with the realization that the
Fourier transform of an impulse is a continuous signal, and then
considering that what the chirp has done is to spread the frequency
components out in time, so that the component is delayed by a time,
which is proportional to This delay adds an extra phase shift to the
component at frequency alpha, which leads to a net phase shift whilst
the \'real\' signal in the time domain is a chirp and the corresponding
representation of the complex signal is a spiral. Figure 1 shows the
chirp and figure 2 a representation of the complex signal as a spiral
which gets tighter and tighter as the frequency (or time) moves away
from zero.

![](media/image1.png){width="3.0305555555555554in"
height="2.2666666666666666in"}

Figure 1: Representation of a Chirp Signal

![](media/image2.png){width="3.63125in" height="2.7159722222222222in"}

Figure 2: Complex Representation of a Chirp Signal as a Spiral

Clearly, the actual signal is not of infinite extent, but will be
limited to a length of ±*T*/2. This will mean that the signal in the
frequency domain will be convolved with a sinc(*T*/2), where sinc (*x*)
= sin(*x*)/*x*.

A key point is that, to first order, we can connect a frequency
component with the time at which that frequency is the instantaneous
frequency of the chirp, therefore the gross effect of the finite
duration of the signal will be that the truncation of the sequence will
truncate the spectrum at ±*T*/2. This \'ideal\' rectangular spectrum
will be convolved with the sinc(*T*/2) function. The net effect will be
to produce a rectangular spectrum with width ±*T*/2. with, at the ends a
sinc-shaped envelope with a lobe spacing of 1/*T*. This may be modelled
by a rectangle of width ±*T*/2 augmented by a role off at a rate of
about 1/\|*T* where is the distance from the end of the flat section.
This last term, however, cannot be quite right, since this it not exceed
unity at the corner point, so a better approximation to the roll-off is
1/(1+\|*T).*

In terms of cyclical frequency (Hertz) rather than radians per second
this is a width of *T*/2 and a role off as 1/(1+2*f*\|). This is shown
in figure 3

![](media/image3.png){width="5.083333333333333in"
height="2.8958333333333335in"}

[Figure 3: Representation of the Spectrum of a Chirp
Signal.]{.underline}

Experimental results indicate that this is a plausible representation of
the spectrum. The illustration in figure 4 shows a typical simulated
result, but this may not represent all the actual details.

![](media/image4.png){width="4.230555555555555in"
height="3.165277777777778in"}

[Figure 4: Simulation of the Spectrum of a Chirp Signal.]{.underline}

The effect of repeating the waveform will be to split the spectrum into
lines spaced at the sweep repetition frequency. If the signal has a
finite flyback time, the total spectrum will be the complex sum of the
main ramp and of the flyback, which will have a roll-off related to the
width of the flyback, but a relative level proportional to the flyback
time as a proportion of the total repetition time.

It is the spectrum of the repeated ramp, with no flyback, which should
be the same as that of a pulsed chirp when the duty cycle of the latter
reaches 100%.

The \'necessary width\' is the sweep bandwidth, but the spectrum must
also include an allowance for the roll-off. If the current allowance is
a rectangular spectrum of twice the sweep bandwidth will not necessarily
give enough width, if the roll-off at the edges is not below the
spurious level by the time the wider rectangular width is reached, but
in general it is over-generous.

It may be noted that the approximations given above may not be
sufficiently accurate for low-time -bandwidth waveforms.

It is hoped that we will be able to produce some photographs of actual
spectra before Xmas, but unfortunately this is unlikely to be achieved
before the \'initial\' report is due on 15th December.

**[REFERENCE]{.underline}**

Campbell, G. A. and Foster, R. M., \'Fourier Integrals for Practical
Applications,\' Princeton, 1948, van Nostrand, Equation 705.1