# Pulse compression

Pulse compression

Pulse compression is a signal processing technique mainly used in radar, sonar and echography to augment 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.

Simple pulse

Signal description

The simplest signal a pulse radar can transmit is a sinusoidal pulse of amplitude $A$ and carrier frequency $f_0$, truncated by a rectangular function of width $T$. The pulse is transmitted periodically, but that is not the main topic of this article; we will consider only a single pulse $s$. If we assume the pulse to start at time $t=0$, 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 $r\left(t\right)$, 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 $B\left(t\right)$ 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 an additive white Gaussian noise.

In other words, the cross-correlation of the received signal with the transmitted signal is computed. This comes down to convolving the incoming signal with a conjugated and mirrored 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 $t_r$ and is attenuated by factor $K$, this yields:

Since we know the transmitted signal, we obtain:

where $B\text{'}\left(t\right)$, the result of the intercorrelation between the noise and the transmitted signal, remains a white noise of same characteristics as $B\left(t\right)$ since it is not correlated to the transmitted signal. Function $Lambda$ is the triangle function, its value is 0 on $\left[-infty,-1/2\right] cup \left[1/2,+infty\right]$, it augments linearly on $\left[-1/2, 0\right]$ where it reaches its maximum 1, and it decreases linearly on $\left[0,1/2\right]$ until it reaches 0 again. Figures at the end of this paragraph show the shape of the intercorrelation for a sample signal (in red), in this case a real truncated sine, of duration $T=1$ seconds, of unit amplitude, and frequency $f_0=10$ hertz. Two echoes (in blue) come back with a delay of 3 and 5 seconds, respectively, and have an amplitude equal to 0,5 and 0,3; those are just random values for the sake of the example. Since the signal is real, the intercorrelation is weighted by an additional 1/2 factor.

If two pulses come back (nearly) at the same time, the intercorrelation is equal to the sum of the intercorrelations of the two elementary signals. To distinguish one "triangular" envelope from that of the other pulse, it is clearly visible that the times of arrival of the two pulses must be separated by at least $T$ so that the maxima of both pulses can be separated. If this condition is false, both triangles will be mixed together and impossible to separate.

Since the distance travelled by a wave during $T$ is $c.T$ (where "c" is the celerity of the wave in the medium), and since this distance corresponds to a round-trip time, we get:

! Result 1
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The range resolution with a sinusoidal pulse is $c.frac\left\{T\right\}\left\{2\right\}$ where $T$ is the pulse length and $c$ the celerity of the wave.

Conclusion: to augment the resolution, the pulse length must be reduced.

Pulse compression by phase coding

There are other means to modulate the signal. Phase modulation is a commonly used technique; in this case, the pulse is divided in "N" time slots of duration "T/N" for which the phase at the origin is chosen according to a pre-established convention. For instance, it is possible not to change the phase for some time slots (which comes down to just leave the signal as it is, in those slots) and de-phase the signal in the other slots by $pi$ (which is equivalent of changing the sign of the signal). The precise way of choosing the sequence of $\left\{0, pi \right\}$ phases is done according to a technique known as Barker codes. It is possible to code the sequence on more than two phases (polyphase coding). As with a linear chirp, pulse compression is achieved through intercorrelation.

The advantages [J.-P. Hardange, P. Lacomme, J.-C. Marchais, "Radars aéroportés et spatiaux", Masson, Paris, 1995, ISBN 2-225-84802-5, page 104. Available in English: "Air and Spaceborne Radar Systems: an introduction", Institute of Electrical Engineers, 2001, ISBN 0852969813] of the Barker codes are their simplicity (as indicated above, a $pi$ de-phasing is a simple sign change), but the pulse compression ratio is lower than in the chirp case and the compression is very sensitive to frequency changes due to the Doppler effect if that change is larger than "1/T".

Notes

ee also

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