The chirp pulse compression process transforms a long duration frequency-coded pulse into a narrow pulse of greatly increased amplitude. It is a technique used in radar and sonar systems because it is a method whereby a narrow pulse with high peak power can be derived from a long duration pulse with low peak power. 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,Dicke R. H.,"Object Detection System", U.S. Patent 2,624,876, submitted Sept. 1945Darlington S.,"Pulse Transmission", U.S. Patent 2,678,997, submitted Sept. 1945Sproule D. O. and Hughes A. J., "Improvements in and Relating to System Operation by Means of Wave Trains", U.K. Patent 604,429, submitted June 1945 but it was not until 1960, following declassification of the subject matter, that a detailed article on the topic appeared the public domain.Klauder J. R., Price A. C., Darlington S. and Albersheim W. J., "The Theory and Design of Chirp Radars", BSTJ Vol. 39, July 1960, pp. 745–808 Thereafter, the number of published articles grew quickly, as demonstrated by the comprehensive selection of papers to be found in a compilation by Barton.Barton D. K. (ed), "Radars, Volume 3, Pulse Compression", Artech House 1975, 1978
Briefly, the basic pulse compression properties can be related as follows. For a chirp waveform that sweeps over a frequency range F1 to F2 in a time period T, the nominal bandwidth of the pulse is B, where B = F2 – F1, and the pulse has a time-bandwidth product of T×B. Following pulse compression, a narrow pulse of duration τ is obtained, where τ ≈ 1/B, together with a peak voltage amplification of .
The chirp compression process – outline In order to compress a chirp pulse of duration T seconds, which sweeps linearly in frequency from F1 Hz to F2 Hz, a device with the characteristics of a dispersive delay line is required. This provides most delay for the frequency F1, the first to be generated, but with a delay which reduces linearly with frequency, to be T second less at the end frequency F2. 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 (f), where
, and the instantaneous delay is given by
which has a linear slope with frequency, as required. In this expression, the delay characteristic has been normalized (for convenience), so as to give zero delay when the frequency f equals the carrier frequency f0. Consequently, when the instantaneous frequency is (f0 − B/2) or (f0 + B/2), the required delay is +T/2 or −T/2, respectively, so k = B/T.
The required dispersive characteristic may be obtained from a lumped element delay network,Bernfeld M., Cook C. E., Paolillo J., and Palmieri C. A., "Matched Filtering, Pulse Compression and Waveform Design", Microwave Journal, Oct 1964 – Jan 1965, (34 pp.)Farnett E. C. & Stevens G. H., "Pulse Compression Radar", Chapter 10 of "Radar Handbook, 2nd Ed.", ed Skolnik M., McGraw Hill 1990Millett R. E., "A Matched-Filter Pulse-Compression System Using a Non-linear F.M. Waveform", IEEE Trans. Aerospace and Electronic Systems, Vol. AES-6, No. 1, Jan 1970, pp. 73–78Cook C. E., "Pulse Compression – Key to More Efficient Radar Transmission", Proc. IRE, Vol.48, March 1960, pp. 310–316 a SAW device,Jones W. S., Kempf R. A. and Hartman C. S., "Practical Surface Wave Chirp Filters for Modern Radar Systems", Microwave Journal, May 1972, pp. 43–50Butler M. B. "Radar applications of s.a.w. dispersive filters", Proc IEE, Vol.27, Pt. F, April 1980, pp.118–124Arthur J. W., Modern SAW-based pulse compression systems for radar applications. Part 1: SAW matched filters, Part 2: Practical systems", Electronics & Communication Engineering Journal, Dec 1995, pp. 236–246, and April 1996, pp. 57–79Andersen Laboratories, "Handbook of Acoustic Signal Processing, Vols. 2 & 3, SAW Filters and Pulse Expansion/Compression IF Subsystems for Radar"MESL Microwave, "SAW Pulse Compression" (technical brochure), http://www.meslmicrowave/saw-pulse-compression/technical-notes/ or by means of digital signal processing Halpern H. M. and Perry R. P., "Digital Matched Filters Using Fast Fourier Transforms", IEEE EASTCON '71 Record, pp. 222–230Arthur J. W., "Digital Waveform Generation for SAW Compression Systems", Tech. Note, Racal MESL, Newbridge, MidlothianAlter J. J. and Coleman J. O., "Digital Signal Processing", Chapter 25 of "Radar Handbook, 3rd edition", Skolnik M. I. (ed.), McGraw Hill 2008
An overview of pulse compression concepts
Compression by matched filter A chirp pulse, however generated, can be considered as the output of one of a pair of filters, which have dispersive characteristics. So, if the transmit filter has a group delay response which increases with frequency, then the receive filter will have one which decreases with frequency and vice versa.
In principle, the transmitted pulses can be generated by applying impulses to the input of the dispersive transmit filter, with the resultant output chirp amplified as necessary for transmission. Alternatively, a voltage controlled oscillator may be used to generate the chirp signal. To achieve maximum transmitted power (and so achieve maximum range) it is normal for a radar system to transmit chirp pulses at constant amplitude from a transmitter run in a near-limiting condition. 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.
In general, the compression process is a practical implementation of a matched filter system. For the compression filter to be matched to the radiated chirp signal, its response is the complex conjugate of the time inverse of the transmit filter's impulse response. So, the output of this matched filter is given by the convolution of the signal h(t) with the conjugate impulse response h*(-t):
Alternately, if the frequency response of the coding filter is H(), then that of the matched filter is H*(), the spectrum of the compressed pulse is |H()|2. The waveform of this spectrum is obtained from the inverse Fourier transform, i.e.
For the case of a linear chirp, with constant amplitude and time duration T, compression by the matched filter gives a waveform with the sinc characteristic, with duration 2T, as shown later. So, in addition to the main pulse, there are a large number of time sidelobes (or, more precisely, range sidelobes) present, the biggest of which are only 13.5 dB below the peak signal level.
In order to achieve a more desirable pulse characteristic (with lower sidelobes, for example), an alternative to the matched filter is often preferred. In this more general case, the compression filter has, say, impulse response g(t) and spectral response G(), so the equations for y(t) becomes:
and
When compared to the performance of the true matched filter, there will be some loss of processing gain, the main pulse lobe will be broader, and the total time duration of the compressed waveform will exceed 2T (usually).
The application of windowing to linear chirps The sinc characteristic of a compressed pulse is a direct consequence of the spectrum of the linear chirp pulse having a rectangular profile. 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.Harris F. J., "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform", Proc. IEEE, Vol.66, Jan 1978, pp.174–204 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. Preferably, the transmitted and received pulses should be modified in equal measure, but when this is impractical, windowing in the compression filter alone is still beneficial.
Doppler tolerance of linear chirps When the frequency sweep of a chirp is linear, the compression process is found to be very tolerant of Doppler frequency shifts on target returns, for a wide range of time-bandwidth products. It is only when T×B is very large (>2000, say) that loss of performance due to Doppler becomes an issue (with main-pulse broadening and increased sidelobe levels). In these situations a chirp with a hyperbolic frequency law can be used, as it has been shown to be fully tolerant to Doppler shifts.Thor R. C., "A large Time-Bandwidth Product Pulse Compression Technique", Trans IRE MIL-6, No. 2, April 1962, pp. 169–173Kroszczynski J. J., "Pulse Compression by Means of Linear-Period Modulation", Proc. IEEE, Vol. 57, No. 7, July 1969, pp. 1260–1266 Windowing techniques can still be applied to the compressed pulse spectra, to lower sidelobe levels, in a similar manner to linear chirps.
Far-out sidelobes There are different concerns when the time-bandwidth product is small. When T×B is less than about 75, the windowing process is not altogether successful, especially when it is applied only within the compressor. In such a situation, even though the close-in sidelobes are lowered by the anticipated amount, further away from the main lobe the sidelobes are found to increase in amplitude once more. These sidelobes tend to reach a maximum at locations ±T/2 on each side of the main lobe of the compressed pulseKowatsch M. and Stocker H. R., "Effect of Fresnel ripples on sidelobe suppression in low time-bandwidth product linear FM", IEE Proc. Vol, 129, Pf.F, No. 1, Feb 1982, pp. 41–44 and they are a consequence of the Fresnel ripples present on the frequency spectrum. This topic is discussed in more detail later.
There are techniques available that will reduce the amplitude of the spectral ripple (see chirp spectrum) and so reduce the amplitude of these far-out sidelobes, but they are not very effective when T×B. is small. In practice, the technique of "reciprocal ripple correction"Judd G. W., "Technique for Realising Low Time Sidelobe Levels in Small Compression Ratio Chirp Waveforms", Proc. IEEE Ultrasonics Symposium, 1973, pp.478–483Cook C. E. and Bernfeld M., "Radar Signals, An Introduction to Theory and Application"; Academic Press 1967, 1987; Artech House 1993 gives good results (where the spectrum of the compression filter is designed to have a ripple characteristic which is the inverse of that of the signal), but the method is less successful when signal returns contain large Doppler frequency shifts.
Non-linear chirps 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 required characteristic is obtained by having rapid changes in frequency near to the band edges, with a slower rate of change around band centre. This is a more efficient way of achieving the required spectral shape than by applying amplitude weighting to the spectrum of the linear chirp because no attenuation of signal power is necessary in order to achieve it.Key E. L., Fowle E. N., Haggarty R. D., "A Method of Designing Signals of Large Time-Bandwidth Product", Proc. IRE Int. Conf. Rec. Pt.4, Mar 1961, pp. 146–154 In addition, the procedure gives far-out sidelobes that tend to be lower than those for the comparable linear sweep version. As the mathematics of non-linear chirps is more complicated than that of linear chirps, many early workers resorted to stationary phase methods to design them.Fowle E. N., "The design of FM pulse-compression signals", IEEE Trans. IT-10, 1964, pp. 61–67