Chirp compression

Furthermore, the process offers good range resolution because the half-power beam width of the compressed pulse is consistent with the system bandwidth.

The basics of the method for radar applications were developed in the late 1940s and early 1950s,[1][2][3] but it was not until 1960, following declassification of the subject matter, that a detailed article on the topic appeared the public domain.

Such a delay characteristic ensures that all frequency components of the chirp pass through the device, to arrive at the detector at the same time instant and so augment one another, to produce a narrow high amplitude pulse, as shown in the figure: An expression describing the required delay characteristic is This has a phase component ψ(f), where which has a linear slope with frequency, as required.

The chirp signals reflected from targets are amplified in the receiver and then processed by the compression filter to give narrow pulses of high amplitude, as previously described.

By modifying the spectrum to have a bell-shaped profile, by means of a weighting (or windowing, or apodization) function, lower level sidelobes are obtained.

[4][18] When windowing is implemented, some signal attenuation occurs and there is a broadening of the main pulse, so both signal-to-noise ratio and range resolution are impaired by the process.

[19][20] Windowing techniques can still be applied to the compressed pulse spectra, to lower sidelobe levels, in a similar manner to linear chirps.

These sidelobes tend to reach a maximum at locations ±T/2 on each side of the main lobe of the compressed pulse[21] and they are a consequence of the Fresnel ripples present on the frequency spectrum.

An alternative method of obtaining a bell-shaped spectral shape in order to achieve lower sidelobes, is to sweep the frequency band in a non-linear manner.

The inherent temperature sensitivity of the quartz substrates was overcome by mounting both the transmit and receive filters in a common package, so providing thermal compensation.

The increased precision offered by SAW technology, enabled time sidelobe levels approaching −30 dB to become achievable by radar systems.

In the closed form solution, just presented, the compressed waveform has the standard sinc function response, because a rectangular shape was assumed for the amplitude of the pulse spectrum.

To investigate fully the consequences of these ripples it is advisable to consider each situation individually, either by evaluating convolution integrals, or more conveniently, by means of FFTs.

In an example of the process, the spectrum of a chirp pulse, with a time-bandwidth product of 250, is shown on the left, and is approximately rectangular in profile.

As an example, consider first the figure shows the compressed spectrum of a linear chirp, which has fast rise and fall times, with T×B = 100 and where Blackman-Harris weighting has been applied.

[11][12][22][33] More recently, digital techniques with mathematically derived look-up tables have provided a convenient way of introducing reciprocal ripple correction.

In order to achieve a practical solution Judd[22] proposed that the total length of the compression pulse be truncated to 2T, whereas Butler[11] suggested 1.6T and 1.3T.

After compression, the resulting pulses will show some loss in amplitude, a time (range) shift and degradation in sidelobe performance.

In addition, linear chirps which use phase pre-distortion to lower sidelobe levels, as described in an earlier section, are found to be tolerant of Doppler.

It is not possible to determine a precise Doppler frequency at which r-r fails because the Fresnell ripples on chirp spectra do not have a single dominant component.

However, as a rough guide, r-r correction ceases to be of benefit when To ensure that a compressed pulse has low time sidelobes, its spectrum should be approximately bell-shaped.

Non-linear chirps, however, have the advantage that by achieving the spectral shaping directly, close-in sidelobe levels can be made low with negligible mismatch loss (typically less than 0.1 dB).

The left hand figure shows the spectrum of a non-linear chirp, with a time bandwidth product of 40, aiming to have a Blackman-Harris profile.

Cook,[23] using paired-echo distortion methods,[41] estimated that in order to keep sidelobe levels below −30 dB, the maximum allowed Doppler frequency is given by

In the two examples below, the chirps have a non-linear sweep characteristic which gives a spectrum with Taylor weighting which, used alone, will achieve a sidelobe level of −20 dB on its compressed pulses.

If the compression process is carried out in the analogue domain before digitization (by a SAW filter, for example), the resulting high-amplitude pulses will place excessive demands on the dynamic range of the A/D converters.

In addition, filters employed in the frequency conversion processes of the transmitter and receiver all contribute to gain and phase variations across the system passband, especially near to band edges.

There has been a growth in interest in adaptive filters for pulse compression, made possible by the availability of small fast computers, and some relevant articles are mentioned in the next section.

These techniques will also compensate for hardware deficiencies, as part of their optimization procedure[59] The growth in digital processing and methods had a significant influence in the field of chirp pulse compression.

The availability of computers has led to a growth in numerical processing and much interest in adaptive networks and optimization methods, to achieve these aims.