# Pulse compression

Pulse compression is a signal processing technique commonly used by radar, sonar and echography to increase the range resolution 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.

The simplest signal a pulse radar can transmit is a sinusoidal-amplitude pulse, ${\displaystyle A}$ and carrier frequency, ${\displaystyle f_{0}}$, truncated by a rectangular function of width, ${\displaystyle \scriptstyle T}$. The pulse is transmitted periodically, but that is not the main topic of this article; we will consider only a single pulse, ${\displaystyle s}$. If we assume the pulse to start at time ${\displaystyle t\,=\,0}$, the signal can be written the following way, using the complex notation:

Let us determine the range resolution which can be obtained with such a signal. The return signal, written ${\displaystyle \scriptstyle r(t)}$, 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 and Gaussian (this generally holds in reality); we write ${\displaystyle \scriptstyle B(t)}$ 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 ${\displaystyle \scriptstyle (t)}$ for this cross-correlation. We have:

If the reflected signal comes back to the receiver at time ${\displaystyle \scriptstyle t_{r}}$ and is attenuated by factor ${\displaystyle \scriptstyle K}$, this yields:

Since we know the transmitted signal, we obtain:

## Related Topics

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag msub invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mn invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mspace invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mspace invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mn invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mspace invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mo invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag msub invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag math invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag semantics invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mrow invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mstyle invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag mi invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389

Warning: DOMDocument::loadHTML(): Tag annotation invalid in Entity, line: 1 in /home/ashver/webapps/infospaze/index.php on line 389