Pulse compression on arvau

Wikipedia / Pulse compression

Pulse compression is a signal processing technique commonly used by radar, sonar and echography to either increase the range resolution when pulse length is constrained or increase the signal to noise ratio when the peak power and the bandwidth (or equivalently range resolution) of the transmitted signal are constrained. This is achieved by modulating the transmitted pulse and then correlating the received signal with the transmitted pulse.

Simple pulse

Signal description

The ideal model for the simplest, and historically first type of signals a pulse radar or sonar can transmit is a truncated sinusoidal pulse (also called a CWcarrier wave and carrier frequency, truncated by a rectangular function, the signal can be written the following way, using the complex notation:

Range resolution

, is an attenuated and time-shifted copy of the original transmitted signal (in reality, Doppler effect (this generally holds in reality, where a bandpass to denote that noise. To detect the incoming signal, a matched filter is commonly used. This method is optimal when a known signal is to be detected among additive noise having a normal distribution.

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 for this cross-correlation. We have:

, this yields:

Since we know the transmitted signal, we obtain:

hertz. Two echoes (in blue) come back with delays of 3 and 5 seconds and amplitudes equal to 0.5 and 0.3 times the amplitude of the transmitted pulse, respectively; these are just random values for the sake of the example. Since the signal is real, the intercorrelation is weighted by an additional factor.

so that the maxima of both pulses can be separated. If this condition is not met, both triangles will be mixed together and impossible to separate.

(where c is the speed of the wave in the medium), and since this distance corresponds to a round-trip time, we get:

Energy and signal-to-noise ratio of the received signal

. The energy put into that signal is:

is the standard deviation of the noise which is assumed to have the same bandwidth as the signal, the signal-to-noise ratio (SNR) at the receiver is:

improves the SNR, but reduces the resolution, and vice versa.

Pulse compression by linear frequency modulation (or chirping)

Basic principles

In order to have a large enough pulse, and still have a good SNR at the receiver, without poor resolution, pulse compression may be used. The basic principle is the following:

a signal is transmitted, with a long enough length so that the energy budget is correct
this signal is designed so that after matched filtering, the width of the intercorrelated signals is smaller than the width obtained by the standard sinusoidal pulse, as explained above (hence the name of the technique: pulse compression).

In radar or sonar applications, linear chirps are the most typically used signals to achieve pulse compression. The pulse being of finite length, the amplitude is a rectangle function, it can be written:

The chirp definition above means that the phase of the chirped signal (that is, the argument of the complex exponential), is the quadratic:

thus the instantaneous frequency is (by definition):

and writing the chirp phase via the integration of frequency:

This transmitted signal is typically reflected by the target and undergoes attenuation due to various causes, so the received signal is a time-delayed, attenuated version of the transmitted signal plus an additive noise of constant power spectral density, and zero everywhere else:

Cross-correlation between the transmitted and the received signal

To compute the correlation of the received signal with the transmitted signals, two actions must be taken:

- The first action is a simplification. Instead of computing the cross-correlation an auto-correlation may be computed, which assumes that the autocorrelation peak is centered at zero. This will not change the resolution and the amplitudes:

is to be determined so that energy is conserved through correlation.

Now, it can be shown is:

is the correlation of the reference signal with the received noise.

Width of the signal after correlation

is reached at 0. Around 0, this function behaves as the sinc. The −3, hence the pulse compression name.

Since the cardinal sine can have sidelobes, which cause issues, a common practice is to filter the result by a window (Hamming, Hann, etc.). In practice, this can be done at the same time as the adapted filtering by multiplying the reference chirp with the filter. The result will be a signal with a slightly lower maximum amplitude, but the sidelobes will be filtered out.

Energy and peak power after correlation

, then it is possible to conserve the energy before and after correlation. The peak (and average) power before correlation is:

Since, before compression, the pulse is box-shaped, the energy before correlation is:

, so the energy after correlation is:

If energy is conserved:

so that the peak power after correlation is:

is correctly scaled to conserve energy through correlation).

Signal-to-noise gain after correlation

The energy of the signal does not vary during pulse compression. However, it is now located in the main lobe is conserved:

which yields an increase in power after pulse compression:

. This spectral density remains the same after matched filtering.

and identical input power, this equivalent sinusoidal pulse has an energy:

. Through conservation of energy, we have:

, the same filtering effect is obtained on the noise in both cases after correlation. This means that the net effect of pulse compression is that, compared to the equivalent CW pulse, the signal-to-noise ratio because the signal is amplified but not the noise.

As a consequence: For technical reasons, correlation is not necessarily done for actual received CW pulses as for chirped pulses. However during baseband shifting the signal undergoes a bandpass which has the same net effect on the noise as the correlation, so the overall reasoning remains the same (that is, the SNR makes only sense for noise defined on a given bandwidth, here being that of the signal).

, and squeezing the pulse by correlation. This works best only for a limited number of signal types which, after correlation, have a narrower peak than the original signal, and low sidelobes.

Stretch processing

can be hundreds of megahertz or even exceed 1GHz.) Stretch Processing is a technique for matched filtering of wideband chirping waveform and is suitable for applications seeking very fine range resolution over relatively short range intervals. .

If the transmitted waveform is the chirp waveform:

can be expressed as:

and the echo will become:

is the wavelength of electromagnetic wave in air.

After conducting sampling and discrete Fourier transform can be solved:

can be obtained:

, respectively.

As a consequence: , which is equal to the period of the transmitted impulse train. As a result, the Fourier transform of y(t) is actually a sinc function with Rayleigh resolution apart.

Consequently,

and,

which is the same as the resolution of the original linear frequency modulation waveform.

Stepped-frequency waveform

. In addition, the effective wavelength of the electromagnetic wave changes during the frequency sweep of a chirp signal, and therefore the antenna look direction will be inevitably changed in a Phased array system.

from the preceding pulse. The baseband signal can be expressed as:

, but the analog components can be reset to support the frequency of the following pulse during the time between pulses. As a result, the problem mentioned above can be avoided.

, individual pulses are processed through the simple pulse matched filter:

and the output of the matched filter is:

where

, we can get:

where l means the range bin l. Conduct DTFT (m is served as time here) and we can get:

and the differential range can be obtained:

where c is the speed of light.

. As a result:

and therefore the differential range resolution is :

which is the same of the resolution of the original linear-frequency-modulation waveform.

Pulse compression by phase coding

There are other means to modulate the signal. Phase modulation (which is equivalent of changing the sign of the signal); this is known as binary phase-shift keying phases can be done according to a technique known as Barker codes.

The advantages 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.

Other pseudorandom binary sequences have nearly optimal pulse compression properties, such as Gold codes, JPL codes or Kasami codes, because their autocorrelation peak is very narrow. These sequences have other interesting properties making them suitable for GNSS positioning, for instance.

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.