Efficient Pulse Compression of Stepped-frequency Chirps in Pulse Radar
Timothy Styles (SE), Cambridge University Engineering Department Fourth-year undergraduate project, Group E, 2001/2002
An algorithm is developed to increase the resolution of pulse radar using stepped-frequency linear chirps. 8 carriers are spaced at half the chirp bandwidth. This increases the resolution by a factor of 4. Phase corrections are made to the signals before they are combined. The required phase correction is found to be proportional to sampling time and carrier frequency. The correction can be made efficient if the sampling clock and carrier stepping oscillators are locked together. The algorithm is implemented in Matlab and demonstrated using real data. Targets include a single corner reflector and a series of three corner reflectors.
Introduction
An efficient pulse compression algorithm was required to process pulse radar measurements in real time. The results were then used to check the functionality of the radar and improve the set-up of the hardware. The need for a real time processing tool was identified by the signature management group at Thales defence. The group take measurements of cooperative targets using a versatile pulse radar, which can be used for high-resolution imaging.
The radar is currently able to sample the demodulated (base band) signal at 500MHz, which gives a range resolution of about 1m. A higher bandwidth can be simulated by use of a 'Stepped-frequency' technique where several 500MHz measurements are taken on different carrier frequencies. The separate signals are then combined during processing to increase the range resolution.
Pulse Compression
An ideal pulse radar transmits a short pulse of high magnitude, which is then detected by the receiver after some delay. The length of the pulse defines the smallest range difference (delta-R) that can be resolved. A longer pulse has greater power, allowing measurements to be taken at a greater distance, but the pulse has a lower resolution. One solution is to transmit a long waveform and compress it to a narrow pulse when it is received. This requires the waveform to have an impulsive autocorrelation function (ACF), such as the linear chirp.
The linear chirp can use the maximum available bandwidth and has a flat spectrum. This gives it the desired ACF, and the transmit power is constant at all parts of the waveform. It can be used to modulate the carrier frequency down by –delta-f and then linearly ramp up to a frequency above the carrier, at +delta-f. As the waveform is digitally generated, the value for delta-f is limited by the Nyquist criterion to half the sampling frequency. The Thales radar is capable of transmitting and receiving the chirp with 2ns resolution, so the sampling frequency is 500MHz and the limit on ?f is 250MHz. If the carrier frequency is 10GHz, the transmitted waveform will ramp up linearly from 9.75GHz to 10.25GHz.
The width of the compressed pulse is limited by the bandwidth of the chirp, and so a chirp of 500MHz will theoretically compress to a pulse that is 2ns wide – equal to the sampling period of the received signal. The signal is compressed using a matched filter, which is obtained by connecting the transmitter to the receiver and recording the transmitted waveform, this also cancels out fixed phase shifts. There is little point in windowing the filter as it is not part of a continuous waveform. Windowing only increases the width of the pulse and this has an adverse affect on the stepped-frequency pulse compression.
Stepped-Frequency Pulse Compression
A longer chirp with a higher bandwidth can be simulated by transmitting short chirps on separate carriers. The carriers are spaced by half the chirp bandwidth, giving a 50% overlap in the frequency domain. Transmitting 8 chirps on separate carriers spaced 250MHz apart gives the same bandwidth as a single chirp with a 2.25GHz bandwidth. See figure 1.
The chirps are independent as the sampling bandwidth limit reduces the cross correlation to zero. Neighbouring chirps have some cross correlation where they overlap, but it is distorted by the sampling and can be ignored. This is shown in figure 2.
The received signal is a series of complex samples from the in-phase and quadrature demodulators. The samples are evenly spaced by the period of the sampling oscillator and the sampling frequency defines the bandwidth of the signal. The received signal is correlated with the transmitted waveform to obtain the 'range profile', which is a series of cells, each having the width of a pulse. Combining chirps, as in figure 1, increases the bandwidth and the width of the cell is reduced proportionally giving a higher resolution. To do this the spectrum of each chirp is first padded out, as shown in figure 3.
The spectrum of chirp 6 is shown in figure 3a, and the halves are swapped and padded in figure 3b. This places the DC point at a higher frequency relative to the carrier of chirp 1. When the spectrum is padded, the number of cells in the range profile is increased. This is shown in figure 4. The full spectrum is 4.5 times the size of the chirp bandwidth, so there are 4.5 crosses for every circle in figure 4.
The pulse width is still the same – about 2 samples (4ns = 0.6 meters). The high-resolution range profiles from the different carriers can be added together, and they add constructively in some cells and cancel out in other cells.
Combining Stepped-Frequency Range Profiles
The range profiles have the same magnitude, but as they are on different carriers they change phase at different rates. A single pulse can be made to constructively interfere at its peak by rotating the range profiles so that they have the same phase at the peak. This is called ‘Constant phase correction.’ The range profiles will add together at the peak, but neighbouring cells will cancel out as they have evenly spaced phases and equal magnitudes. The pulse is only two high-resolution range cells wide, which is 1/4.5 the width of the original pulse. See figure 5.
The phases of the range profiles around the peak are shown in figure 6. The phase change is introduced by the delay between the samples. The rate of phase change is therefore different for each carrier frequency. The expected phase difference between carriers is proportional to the frequency difference and the delay. This agrees with the gradients in figure 6.
Phase difference = Carrier frequency difference x Delay between samples
The phase of range cell 6 is the same for all the carriers, so this point will add in phase. On the neighbouring range cells the carriers are evenly spread across one phase rotation, so they will cancel exactly. The next cells out are spread across two phase rotations, and will again cancel out. Unfortunately the constant phase adjustment will only work for the peak selected. The separate range profiles must be phase corrected so that they also have the same rate of phase change with time. The correction is a linear phase spiral, which is tighter when the carrier frequency is further from the chirp 1 reference carrier.
The phase should be corrected for each sample, but the correction should not distort the waveform, so it is applied to the compressed range profiles. This is referred to as 'Linear phase correction'. The result is shown in figure 7 – the range profile of 3 corner reflectors.
In figure 7a the constant phase correction works for the first pulse. Unfortunately the phase correction does not work for the other two pulses as the delay introduces an extra phase shift. This is accounted for in figure 7b so that every group of 4.5 cells interferes correctly, giving a single peak for each pulse. The group of cells to the side of each pulse produce a second peak, called a side-pulse. This is because the pulse in the low-resolution range profile is more than one cell wide, and it is made worse by windowing.
The phase correction was applied to the high-resolution range profiles by phase shifting groups of 4.5 cells. The delay between samples was 2ns, and the frequency difference between carriers was 250MHz, so for the second carrier a phase correction of pi radians was added to the first group of 4.5 cells. 2pi radians was added to the second group and so on. The result was a change of sign for every other group in the range profile. For the third carrier the corrections were multiples of 2pi radians, and so they could be ignored. For the fourth carrier they were multiples of 3?, and so a change of sign was again applied to every other group.
The different carriers were generated by adding combinations of 250MHz, 500MHz and 1GHz to the base carrier. For the phase correction to remain a simple change of sign the oscillators were locked together. The 500MHz sampling oscillator was used as the 500MHz carrier shift. It was also divided and multiplied to generate the 250MHz and 1GHz shifts. The signals could be split into two sets: The first set did not require any phase corrections (odd carriers). The second set (even carriers) required a change of sign to the odd groups of cells. The high-resolution range profiles were added together in the frequency domain to form two sets. They were then converted to the time domain and the change of sign was applied to the sets when they were added together. This reduced the total number of steps involved.
The received signal and matched filter were padded out with zeros to a power of two length, this allowed the use of efficient FFTs. To make the full spectrum a power of two length the ends were ignored so that the resolution was increased by 4 times, as opposed to 4.5 times. The final algorithm requires 8 x 1024 point FFTs, 8192 complex multiplications, 2 x 4096 point FFTs and 4096 complex additions (for a 1024 sample signal). The FFTs are carried out in-place, so the memory required is equivalent to twice the number of samples.
The Algorithm
Conclusions
The stepped-frequency measurements can be combined to form a high-resolution range profile if the correct phase shift is applied. This phase correction is simply a change of sign if the frequency difference between carriers is a factor of the sampling frequency. This requires the following from the radar system:
- The sampling oscillator and carrier frequency shift oscillators are locked together.
- Phase ambiguities and fixed phase shifts due to different path lengths are cancelled out by using a matched filter.
- The oscillators remain locked together between recording the matched filter and the received signal.
The matched filter should ideally be generated from a strong reflection in the received signal.
Acknowledgments
Thanks go to Rob Davis and Andy Brownlow in the Signature Management group at Thales for producing the requirement, some example data and for commenting on the results. Thanks also go to Malcolm Macleod for explaining the methods currently used, as well as supervising and guiding the project.
References
- 1977 – P.S. Brandon: A proposed practical pulse-compression system, which gives effective coherent integration of thousands of pulses in a radar, but is insensitive to target motion. CUED/B-Elect/TR.49
- 1991 – F. McGroary and K. Lindell: A stepped chirp technique for range resolution enhancement. Proc. of the National Telesystems Conference NTC’91, held Mar. 26-27 in Atlanta, GA, IEEE 91CH3010-6, pp. 121-126, 1991