Pulse compression is a
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
technique commonly used by
radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
,
sonar
Sonar (sound navigation and ranging or sonic navigation and ranging) is a technique that uses sound propagation (usually underwater, as in submarine navigation) to navigate, measure distances (ranging), communicate with or detect objects on o ...
and
echography to increase the range
resolution
Resolution(s) may refer to:
Common meanings
* Resolution (debate), the statement which is debated in policy debate
* Resolution (law), a written motion adopted by a deliberative body
* New Year's resolution, a commitment that an individual mak ...
as well as the
signal to noise ratio. This is achieved by
modulating the transmitted pulse and then
correlating the received signal with the transmitted pulse.
Simple pulse
Signal description
The simplest signal a pulse radar can transmit is a sinusoidal-amplitude pulse,
and
carrier frequency
In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has a ...
,
, truncated by a
rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
\operatorname(t) = \Pi(t) =
\left\{\begin{array}{rl ...
of width,
. The pulse is transmitted periodically, but that is not the main topic of this article; we will consider only a single pulse,
. If we assume the pulse to start at time
, the signal can be written the following way, using the
complex notation:
:
Range resolution
Let us determine the range resolution which can be obtained with such a signal. The return signal, written
, is an attenuated and time-shifted copy of the original transmitted signal (in reality,
Doppler effect can play a role too, but this is not important here.) There is also noise in the incoming signal, both on the imaginary and the real channel, which we will assume to be
white
White is the lightest color and is achromatic (having no hue). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully reflect and scatter all the visible wavelengths of light. White o ...
and
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
(this generally holds in reality); we write
to denote that noise. To detect the incoming signal,
matched filtering is commonly used. This method is optimal when a known signal is to be detected among
additive white Gaussian noise.
In other words, the
cross-correlation of the received signal with the transmitted signal is computed. This is achieved by
convolving the incoming signal with a
conjugated and time-reversed version of the transmitted signal. This operation can be done either in software or with hardware. We write
for this cross-correlation. We have:
:
If the reflected signal comes back to the receiver at time
and is attenuated by factor
, this yields:
:
Since we know the transmitted signal, we obtain:
:
where
, is the result of the intercorrelation between the noise and the transmitted signal. Function
is the triangle function, its value is 0 on