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.
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Signal description
The simplest signal a pulse radar can transmit is a sinusoidal-amplitude pulse,
Range resolution
Let us determine the range resolution which can be obtained with such a signal. The return signal, written
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
If the reflected signal comes back to the receiver at time
Since we know the transmitted signal, we obtain:
where
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
Since the distance travelled by a wave during
Required energy to transmit that signal
The instantaneous power of the transmitted pulse is
Similarly, the energy in the received pulse is
The SNR is proportional to pulse duration
Basic principles
How can one have a large enough pulse (to still have a good SNR at the receiver) without poor resolution? This is where pulse compression enters the picture. The basic principle is the following:
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. If the transmitted signal has a duration
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):
which is the intended linear ramp going from
The relation of phase to frequency is often used in the other direction, starting with the desired
Cross-correlation between the transmitted and the received signal
As for the "simple" pulse, let us compute the cross-correlation between the transmitted and the received signal. To simplify things, we shall consider that the chirp is not written as it is given above, but in this alternate form (the final result will be the same):
Since this cross-correlation is equal (save for the
It can be shown that the autocorrelation function of
The maximum of the autocorrelation function of
Since the cardinal sine can have annoying sidelobes, 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, which is more important.
Improving the SNR through pulse compression
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
which yields:
Besides, the power of the noise does not change through intercorrelation since it is not correlated to the transmitted pulse (it is totally random). As a consequence:
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
The advantages of the Barker codes are their simplicity (as indicated above, a