Chirp compression

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

${\displaystyle Y(f)=\exp[j\pi (f-f_{0})^{2}/k]}$

This has a phase component ψ(f), where

${\displaystyle \psi (f)=\pi .(f-f_{0})^{2}/k}$, and the instantaneous delay t_d is given by

${\displaystyle t_{d}=-{\frac {1}{2\pi }}.{\frac {d\psi }{df}}={\frac {1}{k}}.(f_{0}-f)}$

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,[6][7][8][9] a SAW device,[10][11][12][13][14] or by means of digital signal processing [15][16][17]

The characteristic of a single linear chirp pulse, of unit amplitude, can be described by

${\displaystyle s_{1}(t)=\operatorname {rect} \left({\frac {t}{T}}\right).e^{2\pi j(f_{0}t+kt^{2}/2)}}$

where rect(z) is defined by rect(z) = 1 if |z| < 1/2 and rect(z) = 0 if |z| > 1/2

The phase response φ(t) is given by

${\displaystyle \phi (t)=2\pi (f_{0}t+kt^{2}/2)}$

and the instantaneous frequency f_I is

${\displaystyle f_{i}={\frac {1}{2\pi }}.{\frac {d\phi }{dt}}=f_{0}+k.t}$

So, during the T second duration of the pulse, the frequency changes in a linear fashion from f₀ – kT/2 to f₀ + kT/2. With the net frequency sweep defined as B, where B = (F1- F2), then k = B/T, as stated previously.

The spectrum of this waveform may be found from its transform

${\displaystyle S_{1}(f)=\int _{-\infty }^{\infty }s_{1}(t).e^{-j2\pi .ft}.dt=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{j2\pi .[(f_{0}-f)t+{\frac {kt^{2}}{2}}]}.dt}$

which is an integral which has been evaluated in chirp spectrum.

The spectrum of the compressed pulse can be found from

${\displaystyle S_{\text{out}}(f)=Y(f).S_{1}(f)}$

Where Y(f) is the spectrum of the compression filter.

The time domain waveform ${\displaystyle s_{\text{out}}(t)}$ of the compressed pulse can be found as the inverse transform of ${\displaystyle S_{\text{out}}(f)}$. (This procedure has been described in a paper by Chin and Cook.[9][43])

Here it is more convenient to find ${\displaystyle s_{\text{out}}(t)}$ from the convolution of the two time domain responses, i.e.

${\displaystyle s_{\text{out}}(t)=s_{1}(t)^{*}y(t)}$

where the convolution of two arbitrary functions is defined by

${\displaystyle f(t)^{*}g(t)=\int _{-\infty }^{\infty }f(\tau ).g(t-\tau ).d\tau }$

However, in order to use this method, the impulse response of Y(f) is first needed. This is y(t), which is obtained from

${\displaystyle y(t)=\int _{-\infty }^{\infty }Y(f).e^{j2\pi .ft}.df=\int _{-\infty }^{\infty }e^{j\pi .(f-f_{0})^{2}/k}.e^{j2\pi .ft}.df}$

${\displaystyle {{\text{Put}}\quad f-f_{0}=u\quad {\text{so}}\quad f=u-f_{0}\quad {\text{and}}\quad df=du.\qquad {\text{Also when}}\quad f=\pm \infty \quad {\text{then}}\quad u=\pm \infty }}$

${\displaystyle y(t)=\int _{-\infty }^{\infty }e^{j\pi .u^{2}/k}.e^{j2\pi .(u+f_{0})t}.du=e^{j2\pi .f_{0}t}\int _{-\infty }^{\infty }e^{j\pi .u^{2}/k}.e^{j2\pi .ut}.du}$

A table of standard integrals[44] gives the following transform ${\displaystyle \int _{-\infty }^{\infty }\exp(-\pi \beta .u^{2}).\exp(j2\pi .ut).du={\frac {1}{\sqrt {\beta }}}.\exp \left(-{\frac {\pi .t^{2}}{\beta }}\right)}$

Comparing the equations, they are equivalent if β = -j/k, so y(t) becomes

${\displaystyle y(t)={\sqrt {jk}}.e^{j2\pi .f_{0}t}.e^{-j\pi .t^{2}k}={\sqrt {\frac {jB}{T}}}.e^{j2\pi .f_{0}t}.e^{-j\pi .t^{2}.k}={\sqrt {\frac {jB}{T}}}.e^{j2\pi (f_{0}t-t^{2}k/2)}}$

[Note: the same transform can also be found in Fourier transforms, no. 206, but with α replacing πβ]

With y(t) determined, the output s_out(t) can be obtained from the convolution of s₁(t) and y(t), i.e.

${\displaystyle s_{\text{out}}(t)={\sqrt {\frac {jB}{T}}}.\int _{-T/2}^{T/2}e^{j2\pi (f_{0}\tau +k\tau ^{2}/2)}\cdot e^{j2\pi (f_{0}(t-\tau )-k(t-\tau )^{2}/2)}.d\tau }$

which can be simplified to

${\displaystyle s_{\text{out}}(t)={\sqrt {\frac {jB}{T}}}.e^{j2\pi (f_{0}t-kt^{2}/2)}.\int _{-T/2}^{T/2}e^{j2\pi kt\tau }.d\tau }$

now as ${\displaystyle \int _{}^{}e^{ax}.dx={\frac {e^{ax}}{a}}}$ then

${\displaystyle \int _{-T/2}^{T/2}e^{j2\pi .kt\tau }.d\tau ={\frac {1}{j2\pi kt}}.{\Bigl [}e^{j2\pi .kt\tau }{\Bigr ]}_{\text{-T/2}}^{\text{T/2}}={\frac {1}{j2\pi kt}}[e^{j2\pi kT/2}-e^{-j2\pi kT/2}]}$

${\displaystyle ={\frac {1}{\pi kt}}.{\frac {e^{j\pi Bt}-e^{-j\pi Bt}}{2j}}={\frac {1}{\pi kt}}.\sin(\pi Bt)}$

and finally

${\displaystyle s_{\text{out}}(t)={\frac {1}{\pi kt}}\cdot {\sqrt {\frac {jB}{T}}}\cdot \sin(\pi Bt).\exp[j2\pi (f_{0}t-kt^{2}/2)]}$

${\displaystyle ={\sqrt {T.B}}\cdot {\frac {\sin(\pi Bt)}{\pi Bt}}\cdot {\sqrt {j}}\cdot \exp[j2\pi (f_{0}t-kt^{2}/2)]}$

So, for a unit amplitude linear chirp, with a pulse duration T seconds and a frequency sweep B Hz (i.e. with a ‘time-bandwidth product’ T.B), pulse compression gives a waveform with a magnitude given by

${\displaystyle \left|{\text{Compressed Output}}\right|\approx {\sqrt {T.B}}\times {\frac {\sin(\pi Bt)}{(\pi Bt)}}}$

which has the form of the familiar sinc function. The collapsed pulse width τ, is of the order 1/B (with τ measured at the −4 dB points). Consequently, a pulse width reduction has occurred given by the ratio T/τ where ${\displaystyle {\frac {T}{\tau }}\approx T\times B}$

Also there is a signal amplification of ${\displaystyle {\sqrt {T\times B}}}$

The main parameters are displayed in the figures below. The product T.B gives the compression ratio of the system and it equates, approximately, to the improvement in signal to noise ratio of the main lobe of the compressed pulse relative to that of the original chirp.

To ensure that a compressed pulse has low time sidelobes, its spectrum should be approximately bell-shaped. With linear chirp pulses this can be achieved by applying a window function either in the time domain or in the frequency domain, i.e. by amplitude modulating the chirp waveforms or by applying weighting to the compressed pulse spectra. In either case there is a mismatch loss of 1½dB, or more.

An alternative way to obtain the required spectral shape is to use a non-linear frequency sweep in the chirp. In this case, to achieve the required spectral shape, the frequency sweep changes very rapidly at band edges and more slowly around band centre. Consider, as an example, the frequency versus time plot that achieves the Blackman-Harris windowing profile. When T×B =100, the spectrum of the compressed pulse and the compressed waveform are as shown.

The required non-linear characteristic can be derived using the method of stationary phase.[24][57] As this technique does not take account of the Fresnel ripples, these have to be dealt with in additional ways, as was the case with linear chirps.

In order to achieve the required spectral shape for low time sidelobes, linear chirps require amplitude weighting and consequently incur a mismatch loss. 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). Another benefit is that the far out sidelobes, due to Fresnel ripples on the spectrum, tend to be lower than for a linear chirp with the same T×B product (4 to 5 dB lower with large T×B).

However, for chirps where the T×B product is low, the far-out sidelobe levels of the compressed pulse can still be disappointingly high, because of high amplitude Fresnel ripples on the spectrum. As with linear chirps, results can be improved by means of reciprocal ripple correction but, as previously, truncation of the compression waveform results in the appearance of gating sidelobes.

An example of reciprocal ripple and truncation is shown below. 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. The right-hand figure shows the compressed pulse for this spectrum,

The next figures shows the spectrum after r-r compensation, but with truncation of the compression waveform to 1.1T, and the final compressed waveform.

Doppler tolerance of non-linear chirps

A major disadvantage of non-linear chirps is their sensitivity to Doppler frequency shifts. Even modest values of Doppler will result in broadening of the main pulse, raising of the sidelobe levels, increase in mismatch loss and the appearance of new spurious sidelobes.

An example of a non-linear chirp pulse and the effects of Doppler are shown. The non-linear characteristic is chosen to achieve −50 dB sidelobes using Taylor weighting. The first figure shows the compressed pulse for a non-linear chirp, with bandwidth 10 MHz, pulse duration 10usec, so T×B = 100, and with no Doppler shift. The next two figures shows the pulse degradation cause by 10 kHz and 100 kHz Doppler, respectively. In addition to the waveform degradation, the mismatch loss increases to 0.5 dB. The final figure shows the effect of 100 kHz Doppler on a linear chirp which has had amplitude weighting applied to give the same spectral shape as that of the non-linear chirp. The greater tolerance to Doppler is clearly seen.

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

${\displaystyle f_{d}\leq {\frac {0.06}{T}}}$

so, for a 10μs pulse, the maximum Doppler frequency that can be tolerated is 6 kHz. However, more recent work suggests that this is unduly pessimistic.[33] In addition, as the new sidelobes, when at a low level, are very narrow. Consequently, it may be possible to ignore them initially, as they may not be resolvable by the receiver's D to A.

Using a combination of non-linear and linear characteristics to improve Doppler tolerance

A way of reducing the susceptibility of non-linear chirps to Doppler is to use a ‘hybrid’ scheme, where part of the spectral shaping is achieved by a non-linear sweep, but with additional spectral shaping achieved by amplitude weighting.[11][12] Such a scheme will have greater mismatch loss than a true non-linear scheme, so the advantage of greater Doppler tolerance has to be weighed against the disadvantage of the increased mismatch loss.

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. To achieve lower level sidelobes, this spectral shape is augmented by amplitude weighting so that the final target sidelobe level for the compressed pulses is −50 dB. Comparing the results for Doppler shifts of 10 kHz and 100 kHz with those shown earlier it is seen that the new spurious sidelobes, caused by the Doppler, are seen to be 6 dB lower than before. However, the mismatch loss has increased from 0.1 dB to 0.6 dB, but this is still better than the 1.6 dB figure for linear chirps.

The amplitude of random noise is not changed by the compression process, so the signal to noise ratios of received chirp signals are increased in the process. In the case of a high power search radars, this extends the range performance of the system, while for stealth systems the property will permit lower transmitter powers to be used.

As an illustration, a possible received noise sequence is shown, which contains a low amplitude chirp signal obscured within it. After processing by the compressor, the compressed pulse is clearly visible above the noise floor.

When pulse compression is carried out in digital signal processing, after the incoming signals are digitised by A/D converters, it is important that level of the noise floor is correctly set. The noise floor at the A/D must be high enough to ensure that the noise is adequately characterised. If the noise level is too low, Nyquist will not be satisfied, and any embedded chirp will not be recovered correctly. On the other hand, setting the noise level unnecessarily high will reduce the dynamic range capability of the system.

For systems using digital processing, it is important to carry out the chirp compression in the digital domain, after the A/D converters. 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.[17]

The transmitter and receiver subsystems of a radar are not distortion free. In consequence system performance is often less than optimum. In particular, the time sidelobe levels of the compressed pulses are found to be disappointingly high.

Some of the characteristics which degrade performance are:

• Gain slope, or non-linear phase slope, across the system passband.
• Amplitude and phase ripple across the passband (which may be caused by mismatches on interconnecting cables[58] as well as by imperfections in amplifiers).

Delay modulation by the transmitter (if power supply regulation is poor).

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. In particular, major contributors to overall non-linear phase characteristics are the low-pass filters preceding the A/D converters, which are usually sharp-cut filters chosen to ensure maximum bandwidth while minimizing aliased noise. The transient response characteristics of these filters contribute another (unwanted) source of time sidelobes.

Fortunately it is possible to compensate for several system properties, provided they are stable and can be characterized adequately when a system is first assembled. This is not difficult to implement in radars using digital look-up tables, since these tables can be easily amended to include compensation data. Phase pre-corrections can be included in the expander tables and phase and amplitude corrections can be included in the compressor tables, as required.

So, for example, the earlier equation, defining the compressor characteristic to minimize spectral ripple, could be expanded to include additional terms to correct for known amplitude and phase impairments, thus:

${\displaystyle K(\omega )={\frac {C'(\omega )}{\Phi (\omega ).A(\omega ).H(\omega )}}}$

where, as before, H(ω) is the initial chirp spectrum and C'(ω) is the target spectrum, such as a Taylor window, but now additional terms have been included, namely, Φ((ω) ) and A(ω) which are the phase and amplitude characteristics that require compensation.

A compressor chirp waveform that includes phase correction data will have additional ripple components present at each end of the waveform (pre-shoots and after-shoots). Any truncation procedure should not remove these new features.

In addition, it is easy to time shift the compressed pulses by ±t₀, by multiplying the compressor spectrum by the unity amplitude vector, i.e.

${\displaystyle \cos(\omega .t_{0})\pm j.\sin(\omega .t_{0})=\exp(\pm j\omega .t_{0})}$.

Time shift can be useful to position the main lobes of compressed pulses at a standard location, regardless of chirp pulse length. However, care has to be taken with the overlap and save or overlap and discard algorithm, should time shift be used, to ensure only valid waveform sequences are retained.

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. An introduction to these techniques is provided in a chapter of the Radar Handbook (3rd ed.), edited by Skolnik.[17]

The main aims of most investigations into pulse compression has been to obtain narrow main lobes, with low sidelobe levels, a tolerance to Doppler frequency shifts and to incur low system losses. 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. For example, see the comparison of the various techniques made by Damtie and Lehtinen[60] and, also, various articles by Blunt and Gerlach on these topics.[61][62][63][64] A number of other contributors in the field has included Zrnic et al.[65] Li et al.[49] and Scholnik.[59]

A number of other works, with a variety of approaches to pulse compression, are listed below:

• New methods of generating non linear chirp waveforms and of improving their Doppler tolerance has been investigated by Doerry[66][67]
• Further studies of Hyperbolic chirps have been carried out by Kiss,[68] Readhead,[69] Nagajyothi and Rajarajeswari[70] and Yang and Sarkar.[71]
• Convolution windows have been investigated by Sahoo and Panda who show that they can result in very low sidelobes yet be Doppler tolerant, but may suffer from some pulse broadening.[72] Wen and his co-workers have also discussed convolution windows.[73][74]
• Some new window functions have been proposed by Samad[75] and Sinha and Ferreira,[76] which claim improved performance over the familiar functions.
• Several techniques to lower the sidelobe levels of the compressed pulses for non-linear FM chirps are compared by Varshney and Thomas.[77]
• In a paper by Vizitui,[78] sidelobe reduction is considered where phase pre-distortion is applied to non-linear FM chirps, rather than to linear chirps. Lower sidelobes and some improvement in Doppler tolerance is claimed.

There have been extensive investigations of phase modulation for pulse compression schemes, such as biphase (binary phase-shift keying) and polyphase coding methods, but this work is not considered here.

• Chirp
• Chirp spectrum
• Pulse compression
• Spread spectrum