Abstract --- This paper describes a processing technique for combining stepped-frequency waveforms efficiently to obtain higher range resolution. Essentially this method involves the reconstruction of a wider portion of the target's reflectivity spectrum by combining the individual spectra of the transmitted narrow-bandwidth pulses in the frequency domain. This paper describes the signal processing steps involved, and shows simulation results which validate and illustrate the method.
Keywords --- Stepped-frequency processing, synthetic range profile,
SRP, target reflectivity spectrum.
Synthetic range profile (SRP) processing is a very effective method to obtain high-resolution downrange profiles of targets [10]. This method, however, has the unfortunate drawback that target energy spills over into consecutive coarse range bins due to the matched filter operation, causing ``ghost images'' in the resulting range profile [3]. This is the main reason why it is not regarded as a suitable method for processing SAR images. A time-domain method which does not have this drawback has already been demonstrated [5, 6]. However this method has been found to be inefficient due to the upsampling requirement of the narrow-bandwidth signals.
Instead of recombining the stepped-frequency waveforms in the time-domain,
they can also be recombined in the frequency domain. Essentially the aim
is to reconstruct a larger portion of the target's reflectivity spectrum
by combining the individual spectra of the transmitted narrow-bandwidth
pulses in the frequency domain. The target reflectivity function 
represents the target's reflection properties at range r mapped
into the time domain, where 
. The principle of enhancing the range resolution by recovering a larger
band of the reflectivity's spectrum was introduced by Prati and Rocca [7],
but in the context of coherent combination of SAR images taken from different
observation angles. In this paper we show how this concept may be applied
to the concept of stepped-frequency processing, and describe the signal
processing steps involved.
where A is the amplitude, 
the pulse length and 
the chirp rate. The spectrum is approximately rectangular with bandwidth 
. This is a good approximation for linear FM chirp pulses with a high time-bandwidth
product.
The transmitted RF signal is modelled by
where 
is the centre frequency. The received signal is given by the convolution
of the scene reflectivity function
and the transmitted pulse:
After coherent demodulation, the signal at baseband is given by
In accordance with the Nyquist criteria, the I and Q
channels of this complex signal are each sampled at a sampling rate of 
. Fourier transforming 
we obtain
Thus, the spectral representation of the received signal can be viewed
as a windowed version of the target's reflectivity spectrum 
, where the position of the window is determined by the centre frequency,
and the shape is determined by the spectrum of the transmitted pulse.
Range compression is achieved by convolving the received signal with
a linear compression filter H(f), designed as a trade-off
between compressing the received signal into a narrow pulse (with desirable
low sidelobe properties) and improving the signal to noise ratio. In the
frequency domain the range-compression operation is given by
where 
. The time-domain signal is
The phase of H(f) is always chosen to cancel the phase
of P(f), thus 
. The amplitude of H(f) is chosen according to the desired
time-domain impulse response. For example, if we assume that P(f)
is strictly bandlimited to 
, and if 
across that bandwidth, then
This is simply a bandlimited, shifted version of the target's reflectivity spectrum. Inverse transforming, the time-domain waveform is
where 
. Thus the time-domain impulse response is a sinc function with
a 
dB resolution of 
. The sidelobe response can be reduced by introducing an appropriate frequency-domain
window function, at the expense of widening the mainlobe.
and bandwidth
, but stepped appropriately in frequency by appropriate choice of the carrier
frequency. If the frequency step 
, the desired reconstruction is possible. Assuming a sequence of n
adjacent windows (indexed by 
),
a broad region of the frequency spectrum can be reconstructed, symmetrical
about zero, by shifting each spectrum at baseband by an amount
in the positive direction, and adding together the shifted versions.
If we assume a linear superposition of the shifted subspectra, the
reconstructed spectrum as shown in Figure 1
is
where
is the carrier frequency of pulse i and 
is the frequency spectrum at baseband of pulse i. The shift 
can, by substitution and simplification, be seen to be the centre frequency 
of the entire reconstructed spectrum:
The reconstructed spectrum can thus be expressed as
where
The time-domain signal can then be obtained by inverse transforming to yield
If, for example, we consider the case where the frequency step 
, then the total bandwidth of the reconstructed spectrum is 
, as is shown in Figure 1. If the combined compression
filter
is chosen such that 
, then the time-domain signal is
and the 3dB resolution is 
, which is a factor of n better than would have been achieved without
reconstruction.
spaced at 
equal to, or marginally less than
. If 
, the reconstructed spectrum will contain gaps and the resulting time-domain
range profile will possess undesirable properties in the form of repeated
artifacts spaced at multiples of 
. However even if there are gaps, an improvement in mainlobe resolution
is still possible.
, starting at 
, with
being the complex sample rate.
was sampled starting at
, each subspectrum has to be multiplied by the delay compensation factor 
.
.
(see Figure 3) to obtain the discrete equivalent
of 
(see Figure 4).
. This re-shifts the time domain back to a sampled set beginning at 
.
.
after applying compression filter,
is the most critical and important step in this method. It is important
to consider the following points when constructing this filter:
. Ideally, for a real system, this should be achieved by measuring the
return of a real, physical point target, for example a corner reflector,
and recombining the individual spectra as described in the previous section.
For such a target 
, where 
is a constant, and therefore the combined baseband spectrum would be 
. This method automatically includes receiver distortions.
must also be range-compressed first.
, the filter
is piecewise defined as follows:
where the region 
includes the major part of the reconstructed spectrum, containing all the
subspectra boundaries. This will effectively smoothen any ripples at the
subspectra boundaries, yielding the desired time-domain impulse response.
The first section and the last section of
, where the target spectrum decays to zero, is equated to the conjugate
of
. This prevents the amplification of noise for a real system, while still
cancelling the phase correctly. Note that
has to be scaled appropriately so that no discontinuities arise at 
and 
.
| first centre frequency |  |
5.2625 GHz |
| frequency step size | |
25 MHz |
| number of steps | n | 4 |
| total radar bandwidth |  |
105 MHz |
| pulse length | |
5 
s |
| chirp bandwidth | B | 30 MHz |
| ADC (complex) sample rate |  |
32 MHz |
Figure 2 shows the magnitude of the reconstructed
target reflectivity spectrum. The ``ripples'' at the three subspectra boundaries
are clearly visible. After applying the compression filter shown in Figure
3 one obtains the spectrum shown in Figure 4,
which closely approximates the desired 
function. A window function has been applied to this spectrum to reduce
sidelobe levels. Inverse Fourier transforming this spectrum yields the
time-domain high-resolution profile. A dB-plot of a section of this profile
is shown in Figure 5, with sidelobe levels at approximately
-35dB.
is such that the reconstructed target spectrum contains ripples at the
subspectra boundaries. It is therefore important that great care is invested
in the design of the compression filter.
