Pulse compression
Encyclopedia
Pulse compression is a signal processing
technique mainly used in 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.
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:
, 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
to denote that noise. To detect the incoming signal, matched filter
ing 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 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
, the result of the intercorrelation between the noise and the transmitted signal, remains a white noise of same characteristics as 
since it is not correlated to 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 red), in this case a real truncated sine, of duration 
seconds, of unit amplitude, and frequency 
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 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
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 
(where c is the speed of the wave in the medium), and since this distance corresponds to a round-trip time, we get:
. The energy put into that signal is:
Similarly, the energy in the received pulse is
. If 
is the standard deviation of the noise, the signal-to-noise ratio (SNR) 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.
In radar
or sonar
applications, linear chirp
s 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
, begins at 
and linearly sweeps the frequency band 
centered on carrier 
, it can be written:
N.b., the chirp is written that way so the phase of the chirped signal (that is, the argument of the complex exponential), is:
thus the instantaneous frequency is (by definition):
which is the intended linear ramp going from
at 
to 
at 
.
Since this cross-correlation is equal (save for the
attenuation factor), to the autocorrelation function of 
, this is what we consider:
It can be shown that the autocorrelation function of
is:
The maximum of the autocorrelation function of
is reached at 0. Around 0, this function behaves as the sinc (or cardinal sine) term. 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 (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.
. If 
is the power of the signal before compression, and 
the power of the signal after compression, we have:
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:
is a commonly used technique; in this case, the pulse is divided in
time slots of duration 
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 
(which is equivalent of changing the sign of the signal). The precise way of choosing the sequence of 
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 of the Barker codes are their simplicity (as indicated above, a
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
.
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
technique mainly used in radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
, sonar
Sonar
Sonar is a technique that uses sound propagation to navigate, communicate with or detect other vessels...
and echography
Echography
Echography may refer to;* Sonography or Medical ultrasonography.* An Echograph, more commonly called Ultrasound Display....
to increase the range resolution
Angular resolution
Angular resolution, or spatial resolution, describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object...
as well as the signal to noise
Signal to Noise
Signal to Noise is a graphic novel written by Neil Gaiman and illustrated by Dave McKean. It was originally serialised in the UK style magazine The Face, beginning in 1989, and collected as a graphic novel in 1992, published by Victor Gollancz Ltd in the UK and by Dark Horse Comics in the US.The...
ratio. This is achieved by modulating
Modulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a high-frequency periodic waveform, called the carrier signal, with a modulating signal which typically contains information to be transmitted...
the transmitted pulse and then correlating
Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...
the received signal with the transmitted pulse.
Signal description
The simplest signal a pulse radar can transmit is a sinusoidal pulse of amplitude, and carrier frequencyFrequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the 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 (in reality, Doppler effectDoppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...
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
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
and Gaussian (this generally holds in reality); we write
to denote that noise. To detect the incoming signal, matched filterMatched filter
In telecommunications, a matched filter is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template...
ing 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
Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...
of the received signal with the transmitted signal is computed. This is achieved by convolving
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
the incoming signal with a conjugated
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
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
, the result of the intercorrelation between the noise and the transmitted signal, remains a white noise of same characteristics as since it is not correlated to 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 red), in this case a real truncated sine, of duration seconds, of unit amplitude, and frequency 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 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
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 (where c is the speed of the wave in the medium), 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. |
| Before matched filtering | After matched filtering |
|---|---|
 |
 |
 |
 |
Required energy to transmit that signal
The instantaneous power of the transmitted pulse is. The energy put into that signal is:Similarly, the energy in the received pulse is
. If is the standard deviation of the noise, the signal-to-noise ratio (SNR) 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.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:- 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
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
or sonar
Sonar
Sonar is a technique that uses sound propagation to navigate, communicate with or detect other vessels...
applications, linear chirp
Chirp
A chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
s 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
, begins at and linearly sweeps the frequency band centered on carrier , it can be written:N.b., the chirp is written that way so the phase of the chirped signal (that is, the argument of the complex exponential), is:
thus the instantaneous frequency is (by definition):
which is the intended linear ramp going from
at to at .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
attenuation factor), to the autocorrelation function of , this is what we consider:It can be shown that the autocorrelation function of
is:The maximum of the autocorrelation function of
is reached at 0. Around 0, this function behaves as the sinc (or cardinal sine) term. 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 (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.
| 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 (usually, its value is 20 to 30). |
 |
 |
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. If is the power of the signal before compression, and the power of the signal after compression, we have: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:
| Result 3 |
|---|
After pulse compression, the power of the received signal can be considered as being amplified by  . This additional gain can be injected in the radar equation. |
 |
 |
)Pulse compression by phase coding
There are other means to modulate the signal. Phase modulationPhase modulation
Phase modulation is a form of modulation that represents information as variations in the instantaneous phase of a carrier wave.Unlike its more popular counterpart, frequency modulation , PM is not very widely used for radio transmissions...
is a commonly used technique; in this case, the pulse is divided in
time slots of duration 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 (which is equivalent of changing the sign of the signal). The precise way of choosing the sequence of 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 of the Barker codes are their simplicity (as indicated above, a
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 effectDoppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...
if that change is larger than
.