Range Resolution

FMCW radar systems are capable of very-range resolution that enables not just high-range measurement accuracy but allows production of high-high-range resolution profiles (HRRP). This also permits the FMCW technique to be used in a synthetic aperture radar (SAR) mode to generate high-resolution two-dimensional imagery. The range resolution of a linear FMCW homodyne radar is fundamentally limited by the transmit bandwidth. In addition, it will also have limits imposed by transmit-and-receive waveform overlap, receiver frequency resolution, and frequency sweep nonlinearities. For any radar wave-form, the range resolution is linearly proportional to the time resolution (i.e., the pulse length for an unmodulated pulsed radar) or inversely proportional to the transmit wave-form bandwidth (or the modulation bandwidth for a pulsed radar or an FMCW radar):

DR0¼cDT

2 ¼ c

2DF (2.3-24)

where

DR0¼ ideal range resolution, DT ¼ time resolution, and

DF ¼ bandwidth of the transmit waveform.

For PILOT, the bandwidths of the transmitted FMCW waveforms are 70 MHz, 28 MHz, and 7 MHz with corresponding ideal range resolution of 2.1 m, 5.4 m, and 21.4 m, respectively.

0.0 –15 –10 –5 0

Clutter RCS (dBsm)

5 10 15

4.4 8.9 13.3 17.8 22.2

Range (km)

26.7 31.1 35.6 40.0 44.4 FIGURE 2.3-14 ^¢

Sea Clutter RCS as a Function of Range.

As seen earlier range measurement is also a function of the beat frequency which must therefore be estimated as accurately as possible. The PILOT radar system uses a sawtooth linear FMCW waveform with 1-kHz modulation frequency and a corre-sponding 1-ms modulation period. The three range settings of PILOT (4.4 km, 11 km, and 44 km) have corresponding total frequency deviations of 70 MHz, 28 MHz, and 7 MHz, respectively. For ranges out to 4.4 km, with corresponding transit times up to 30 ms, the frequency slew rate slope is 70 MHz/ms that yields a maximum beat fre-quency of 2.07 MHz. For the 11-km and 44-km maximum range modes, the slope is decreased proportionally so that the maximum beat frequency remains 2.07 MHz. The parameters for PILOT are shown in Table 2.3-7.

Figure 2.3-15 shows the beat frequency, f_b, spectrum from 0 MHz to 2.048 MHz for the 70-MHz bandwidth and a modulation period of 1 ms. A single stationary point target echo results in a beat frequency at 1.4 MHz, corresponding to its range of 3 km. For example,

TABLE 2.3-7 ¢ PILOT Waveform and Receiver Parameter Summary

Range Setting 4.4 11 44 km Given

FMCW waveform Sawtooth Sawtooth Sawtooth Given

Frequency deviation, peak to peak 70 28 7 MHz Given

Ideal time resolution 14 36 143 ns Calculated

Ideal range resolution 2.1 5.4 21.4 m Calculated

Range resolution claimed 2.4 6.0 24.0 m Given

Modulation frequency 1 1 1 kHz Given

Modulation index 70,000 28,000 7,000 Calculated

Modulation period 1 1 1 ms Calculated

Frequency slew rate 70 28 7 MHz/ms Calculated

Beat frequency/range ratio 467 187 47 Hz/m Calculated

Range/beat frequency ratio 0.002 0.005 0.021 m/Hz Calculated

Maximum transit time 30 74 296 ms Calculated

Overlap 97.0% 92.6% 70.4% Calculated

Maximum beat frequency 2.07 2.07 2.07 MHz Calculated

Minimum beat frequency interval 970 926 704 ms Calculated Minimum beat frequency spectral width 1,031 1,080 1,421 Hz Calculated

Range resolution limit 2.2 5.8 30.5 m Calculated

Analog-to-digital converter sample rate 4.096 4.096 4.096 MHz Assumed

FFT length 4,096 4,096 4,096 Points Given

FFT length 1,000 1,000 1,000 ms Calculated

FFT frequency sample spacing 1,000 1,000 1,000 Hz Calculated

FFT range sample spacing 2.1 5.4 21.4 M Calculated

Window Hamming Hamming Hamming Assumed

Window frequency resolution (6 dB) 1.81 1.81 1.81 Sample Calculated Window frequency resolution (6 dB) 1,810 1,810 1,810 Hz Calculated Window range resolution (6 dB) 3.9 9.7 38.8 M Calculated Doppler frequency shift 62.0 62.0 62.0 Hz/(m/s) Calculated Range error due to Doppler 0.13 0.33 1.33 m/(m/s) Calculated

Figure 2.3-16 is a zoom of the 1.4-MHz single target return showing the beat fre-quency spectrum from 1.38 MHz to 1.42 MHz (i.e., corresponding to ranges from 2,957 m to 3,043 m). The plot shows sidelobes at approximately13 dBs that are consistent with a sin x/x or sinc function response for a single-point target.

Figure 2.3-17 shows transmit-and-receive frequency as a function of time for two targets. The plot shows the beat frequency as a function of time. For the far-range target,

–500

Beat Frequency Spectrum for Target at 3-km Range

Magnitude Squared (dB)

200 400 600 800 1,000

Frequency (kHz)

1,200 1,400 1,600 1,800 2,000 Frequency Deviation 70 MHz

1,380 1,385 1,390 1,395 1,400 1,405 1,410 1,415 1,420 –50

Expanded Beat Frequency Spectrum for Target at 3-km Range Frequency Deviation 70 MHz

the beat frequency is larger, as is the transit-time gap after the beginning of the quency sweep. In the beat frequency spectrum, the nearer-range target has a lower fre-quency than the far-range target. The spectral width is greater for the longer-range target because the overlap in time is less.

The beat frequency bandwidth, B_b, of a single-point target is given by

B_b¼ 1

T_m td tsweep recovery

(2.3-26) where

B_b¼ beat frequency bandwidth of target, T_m¼ modulation period,

t_d¼ round-trip propagation time delay, and t_s¼ sweep recovery time.

Thus, again, for the PILOT radar with 1-ms modulation period, for a target at a range of 44-km with a 296-ms transit time, and assuming negligible sweep-recovery time, the spectral width of difference frequency is 1,421 kHz, or 42 percent wider than a short-range target, as shown here:

Bb¼ 1

T_m td ¼ 1

1;000 ms  296 ms¼ 1

704ms¼ 1;421 kHz (2.3-27) Partial overlap between the transmit and receive waveforms causes this broadening of the spectral width of the beat frequency. Figure 2.3-18 shows the beat frequency spectrum for a single point at 30-km range with 7-MHz frequency deviation. The beat frequency is the same as in Figure 2.3-15 because of the ten-time increase in range and transit time is offset by the ten time reduction in reduction in frequency deviation and sweep slope. However, the target return broadens by over 20 percent due to the change in overlap from 98 percent to 80 percent.

Transmit and Receive Frequency

Beat Frequency

Beat Freq. Spectra

1/(T_m–t_d1)

f_b1 f_b2 Frequency

f_b2 f_b1

t_d1

∆F

T_m

Time

Time

FIGURE 2.3-17 ^¢ Homodyne Linear FMCW Overlap.

A further source of broadening of the beat frequency is due to target motion causing ambiguity between range and Doppler known as range-Doppler coupling. Figure 2.3-19 shows the beat frequency spectrum for a single-point target at 3-km range with 32-m/s relative radial velocity. The target velocity is sufficient to shift the target return beat frequency by 2 kHz to 1.402 MHz as shown here:

fb¼DF

1,380 1,385 1,390 1,395 1,400 1,405 1,410 1,415 1,420 Frequency (kHz)

Beat Frequency Spectrum for Target at 30-km Range Frequency Deviation 7 MHz

1,380 1,385 1,390 1,395 1,400 1,405 1,410 1,415 1,420 Frequency (kHz)

Beat Frequency Spectrum for Target at 3-km Range

Modulation Period 1 ms

This is also equivalent to shifting the return by two 2-m range bins. This 1.402-MHz beat frequency thus ambiguously also corresponds to a stationary target at 3,004-m range.

When the target is moving, there is also a beat frequency chirp during the mod-ulation period due to the relative velocity between the radar and target in addition to the beat frequency components due to range and velocity. If the relative velocity is less than the range resolution divided by the modulation period, then the error will be less than a frequency or range bin. For the preceding moving target example, the beat frequency chirp is only 34 Hz as shown by the following computation:

f_bchirpðtÞ ¼4DF For the 7-MHz frequency deviation used by the PILOT in the far-range mode, the FM modulation would decrease by 3 Hz. Figure 2.3-20 shows how this results in an increase in the beat frequency sidelobe levels for a case in which the velocity is very high, taking a value of 1.6 km/s. A velocity of this magnitude is so high that the beat frequency is approximately 1.493 kHz higher at the end of the modulation period than at the beginning. Even with a velocity this high, the error only corresponds to approxi-mately 1.5 range bins (note also that sidelobes are only marginally higher).

Having examined the spectrum of the beat frequency under various different target conditions, we now consider how well the different beat frequencies can be resolved.

This enables the limits on range resolution to be evaluated. The FMCW radar receiver will include frequency filters to form the range bins. The spectral output of the range bin filter will be the convolution of the mixer output beat frequency spectrum and the filter

1,480 1,485 1,490 1,495 1,500 1,505 1,510 1,515 1,520 Frequency (kHz)

Beat Frequency Spectrum for Target at 3-km Range

Modulation Period 1 ms

spectral characteristic. The bandwidth of the matched filter will be approximately equal to the inverse of the modulation period less the transit time and any sweep recovery time. Wider bandwidth filters may be used when the frequency sweep nonlinearities broaden the mixer output spectrum beyond the inverse of the modulation period. While analog range bin filters can be used to form the range bins, digital signal processing is attractive for many modern FMCW radar applications. Coarser beat frequency resolu-tion and consequently degraded range resoluresolu-tion will result from using sequences shorter than modulation period. Weighting to reduce frequency and range sidelobes will also broaden the filter bandwidth. The zero-padding technique can be used for narrower bandwidth filters, thus improving both frequency and range resolution.

For the ADC at the mixer output to comply with the sampling theorem, the sampling rate f_smust be at least two times the maximum beat frequency, or

2fbmax fs (2.3-29)

where

f_bmax¼ maximum beat frequency and f_s¼ sampling frequency.

Substituting for f_bmax

2 DF Tm

2R_max

c þ2V_max l

 

fs (2.3-30)

Thus, for a modulation bandwidth of 500-MHz frequency and a modulation period of 1 ms, a stationary target at a range of 1 km results in a beat frequency, f_bmax, of 3.3 MHz.

Thus, f_smust be sampled at a rate at least greater than twice this value.

2DF Tm

2R_max

c ¼ 2ð500 MHz=msÞð6:7 msÞ ¼ 2ð3:3 MHzÞ ¼ 6:7 MHz fs (2.3-31) Matching the samples processed in the FFT to the modulation period, and requiring that the number of samples be a power of 2, yields

Tm¼N f_s ¼2ⁿ

f_s (2.3-32)

where N¼ 2ⁿis the number of samples, so that fs ¼ 2ⁿ

T_m (2.3-33)

Substituting,

2DF T_m

2R_max c 2ⁿ

T_m (2.3-34)

so that

4DFRmax

c 2ⁿ (2.3-35)

So a 2¹³or 8,192-point FFT is needed for the 500-MHz frequency deviation and 1-km range as shown here:

4DFRmax

c ¼4ð500 MHzÞð1 kmÞ

310⁸m=s ¼ 6;667 2ⁿ¼ 2¹³¼ 8,192 (2.3-36) This can be accomplished by sampling the 1-ms modulation period at 6.7 MHz and appending 1,525 zeroes to pad the sequence length to 8,192, or increasing the ADC sample rate to 8.192 MHz to get 8,192 samples in 1 ms.

Some linear FMCW homodyne FMCW radar systems use coherent-processing intervals that are shorter than the modulation period. This reduces the signal-processing burden since FFTs with fewer points require less processing. Examples include Lear Astronics and TSC [19, 20]. It may also be desirable to only process the interval from the maximum transit time to the end of the modulation period to ensure that any window function applied to the data includes returns from all ranges, avoiding degradation of filter sidelobe response due to misalignment.

Figure 2.3-21 shows the beat frequency spectrum for two stationary-point targets with equal RCS located at a nominal range of 3 km with a separation of 4 m. To be able to resolve these as two targets, they have to be resolvable in beat frequency. The null depth between the two target peaks is very sensitive to changes in the range separation that are comparable to fractions of a wavelength because of interference between the range sidelobes of the returns from the two targets. The range sidelobes are approxi-mately 15 dB below the peaks in this figure. Weighting can be used to reduce the sidelobes at the expense of mainlobe broadening; Figure 2.3-22 shows the reduction in range sidelobes to approximately 40 dB below the peak with a Hamming weighting for

–50

1,380 1,385 1,390 1,395 1,400 1,405 1,410 1,415 1,420 Frequency (kHz)

Beat Frequency Spectrum for Target at 3-km Range

Modulation Period 1 ms

the same two targets. Note that the depth of the null between the two targets is now much less, and any further reduction could render them unresolved.

The achieved range resolution may be estimated as the square root of the sum of the squares of the beat frequency spectral width and the receiver frequency resolution; it is given by

T_m¼ modulation period, c¼ propagation velocity, DF ¼ total frequency deviation, R¼ range,

t_sr¼ sweep recovery (or other unprocessed) time, and Dfreceiver¼ receiver frequency resolution.

In document Principles of Modern Radar - Volume 3 (Page 69-77)