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 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, of amplitude and carrier frequency,, truncated by a rectangular function 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. There is also noise in the incoming signal, both on the imaginary and the real channel. The noise is assumed to be band-limited, that is to have frequencies only in ; we write 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 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, it increases linearly on where it reaches its maximum 1, and it decreases linearly on until it reaches 0 again. Figures at the end of this paragraph show the shape of the intercorrelation for a sample signal, in this case a real truncated sine, of duration seconds, of unit amplitude, and frequency hertz. Two echoes 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.
If two pulses come back 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 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.
Since the distance travelled by a wave during is , and since this distance corresponds to a round-trip time, we get:
| Result 1 |
| The range resolution with a sinusoidal pulse is where is the pulse Duration and,, the speed of the wave. Conclusion: to increase the resolution, the pulse length must be reduced. |
Energy and signal-to-noise ratio of the received signal
The instantaneous power of the received pulse is. The energy put into that signal is:If is the standard deviation of the noise which is assumed to have the same bandwidth as the signal, the signal-to-noise ratio at the receiver is:
The SNR is proportional to pulse duration, if other parameters are held constant. This introduces a tradeoff: increasing improves the SNR, but reduces the resolution, and vice versa.
Pulse compression by linear frequency modulation (or ''chirping'')
Basic principles
How can one have a large enough pulse without poor resolution? This is where pulse compression enters the picture. 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.
The chirp definition above means that the phase of the chirped signal, is the quadratic:
thus the instantaneous frequency is :
which is the intended linear ramp going from at to at.
The relation of phase to frequency is often used in the other direction, starting with the desired 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 on, and zero everywhere else:
Cross-correlation between the transmitted and the received signal
We now endeavor to compute the correlation of the received signal with the transmitted signals. Two actions are going to be taken to do this:- The first action is a simplification. Instead of computing the cross-correlation we are going to compute an auto-correlation which amounts to assuming that the autocorrelation peak is centered at zero. This will not change the resolution and the amplitudes but will simplify the math:
- The second action is, as shown below, is to set an amplitude for the reference signal which is not one, but. Constant is to be determined so that energy is conserved through correlation.
Now, it can be shown that the correlation function of with is:
where is the correlation of the reference signal with the received noise.
Width of the signal after correlation
Assuming noise is zero, the maximum of the autocorrelation function of is reached at 0. Around 0, this function behaves as the sinc term, defined here as. The −3 dB temporal width of that cardinal sine is more or less equal to. Everything happens as if, after matched filtering, we had the resolution that would have been reached with a simple pulse of duration. For the common values of, is smaller than, hence the pulse compression name.Since the cardinal sine can have annoying sidelobes, a common practice is to filter the result by a window. 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, which is more important.
| Result 2 |
| The distance resolution reachable with a linear frequency modulation of a pulse on a bandwidth is: where is the speed of the wave. |
| Definition |
| Ratio is the pulse compression ratio. It is generally greater than 1. |
Energy and peak power after correlation
When the reference signal is correctly scaled using term, then it is possible to conserve the energy before and after correlation. The peak power before correlation is:Since, before compression, the pulse is box-shaped, the energy before correlation is:
The peak power after correlation is reached at :
Note that if this peak power is the energy of the received signal before correlation, which is as expected.
After compression, the pulse is approximal by a box having a width equal to the typical width of the function, that is, a width, so the energy after correlation is:
If energy is conserved:
... it comes that: so that the peak power after correlation is:
As a conclusion, the peak power of the pulse-compressed signal is that of the raw received signal.
Signal-to-noise gain after correlation
As we have seen above, things are written so that the energy of the signal does not vary during pulse compression. However, it is now located in the main lobe of the cardinal sine, whose width is approximately. If is the power of the signal before compression, and the power of the signal after compression, energy is conserved and we have:which yields an increase in power after pulse compression:
In the spectral domain, the power spectrum of the chirp has a nearly constant spectral density in interval and zero elsewhere, so that energy is equivalently expressed as. This spectral density remains the same after matched filtering.
Imagining now an equivalent sinusoidal pulse of duration and identical input power, this equivalent sinusoidal pulse has an energy:
After matched filtering, the equivalent sinusoidal pulse turns into a triangular-shaped signal of twice its original width but the same peak power. Energy is conserved. The spectral domain is approximated by a nearly constant spectral density in interval where. Through conservation of energy, we have:
Since by definition we also have: it comes that: meaning that the spectral densities of the chirped pulse, and the equivalent CW pulse are very nearly identical, and are equivalent to that of a bandpass filter on. The filtering effect of correlation also acts on the noise, meaning that the reference band for the noise is and since, 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 has improved by a factor because the signal is amplified but not the noise.
As a consequence:
| Result 3 |
| After pulse compression, the signal-to-noise ratio can be considered as being amplified by as compared to the baseline situation of a continuous-wave pulse of duration and the same amplitude as the chirp-modulated signal before compression, where the received signal and noise have undergone a bandpass filtering on . This additional gain can be injected into the radar equation. |
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 filtering on which has the same net effect on the noise as the correlation, so the overall reasoning remains the same.
This gain in the SNR seems magical, but remember that the power spectral density does not represent the phase of the signal. In reality the phases are different for the equivalent CW pulse, the CW pulse after correlation, the original chirped pulse and the correlated chirped pulse, which explains the different shapes of the signals despite having the same power spectrum in all cases. If the peak transmitting power and the bandwidth are constrained, pulse compression thus achieves a better peak power by transmitting a longer pulse, compared to an equivalent CW pulse of same peak power and bandwidth, 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.