Outline
This section introduces the theory and application of potential ways to process and display the results from a pulse compression radar system. It shows how processing the chirp signal can lead to improved range resolution, target identification and noise reduction. An example program illustrates the method behind this technique and examples are given for ways to implement it into the system.
Time Domain Correlation
In order to detect the instance of a received pulse it is necessary to correlate all incoming signals with the transmitted pulse. In this way, we are essentially trying to find a match with the shape of the original reference chirp such as to validate it as a pulse that has reflected off a target which may be hidden in background noise.
Fortunately there is a simple method for facilitating such a comparison. It relies on the fact that correlation is similar to convolution which in the time domain is equivalent to multiplication in the frequency domain.
If we first consider the Convolution Theorem, which shows that the Fourier transform of the convolution of 2 signals is equal to the product of the Fourier transforms each of the signals separately-:
It then carries that the inverse Fourier transform of this product will find the convolution-:
By understanding that correlation is simply the same as convolution, except that we use the complex conjugate of one of the signals, we can simply change the equations to accommodate-:
And the inverse of the product is-:
The procedure for performing the correlation in a DSP environment then becomes-:
- Calculate an FFT of 1 instance of the original chirp and store it in memory. Successive chirps, for all intensive purposes, are identical and therefore do not need to be calculated again.
- Reverse the polarity of the imaginary component of the FFT data to produce the complex conjugate of the chirp. This will formulate the reference chirp signal used to find reflected pulses from the receiver.
- Take regular windows from the received signal and calculate the FFT. The sample size of these windows should be equivalent to that of the reference chirp and should ideally overlap and add as often as processing power allows. The more the windows overlap, the more accurate the results will be.
- Multiply the received FFT windows with the complex conjugate of the reference chirp.
- Perform an inverse FFT on the product. This will provide the correlated result as required.
- Display the results
An example of the signal flow in the system.
Example
A short program was written to illustrate how this works. As the purpose is to illustrate the theory, time is represented by a number of samples. The actual frequencies are not defined as no sample rate is taken into consideration.
Chirp (Linear Sweep)
First of all a transmission chirp is generated, an example of typical behaviour seen at the receiver is then simulated. This includes some possible reflections from targets which are then mixed with noise. By applying the aforementioned correlation we can see how well the technique works.
From Fig1(a) we can see the initial chirp, (b) shows 3 reflected pulses; the first is on its own and has a gain of 0.5 compared with the original, the second and third pulses overlap and are of 0.4 and 0.3 gain respectively. Of course in reality, due to the radiation pattern of the antenna, as well as that of the target itself, dissipating the signal strength, the received pulses are likely to be significantly smaller than the original high gain transmission pulse. Part (c) shows the received signal hidden in Gaussian distributed random noise. This holds as a good approximation of the type of noise the system may experience from a typical outdoor environment (reflections from small insignificant objects, existing RF signals from TV, radio, etc.).
Fig1(d) shows the result after the correlation procedure. Here, the FFT window is set to an 80% overlap and the outcome is normalised by first considering the perfect reflection by correlating the initial pulse with itself and defining its peak as 1. We can clearly see that the pulses have been compressed around a centre residing at the first instance of the reception and, for all intensive purposes, now resemble a positive spike at this point. The distance of this spike in time from the start of the initial transmission can be used to determine the distance and the height can be used to determine the size of the object (larger objects will reflect more signal and hence more will be received).
Figure 1: Time Correlation with Chirp (top to bottom) a)Transmitter b)Receiver c) Receiver + noise d)Correlation output
Chirp Weighting
There are some small artefacts that can be found at when the received pulse stops. This is due to the partial correlation found when the analysis window leaves the received pulse compared with relatively zero correlation immediately after. While this would not be much of a problem for most applications it could potentially limit the object size resolution. For example, a pulse from a small object slightly after and overlapping a larger object may be lost in such an artefact. The effect of this could be reduced by smoothing out the sudden change from pulse to no pulse by using some pre-processing such as an analogue low-pass filter. It could also be reduced by introducing weighting to the transmission chirp.
Fig2 shows the same method but with the chirp multiplied by a Hanning window function. We can see that this effect is reduced but the output pulse now have time domain side-lobes. Essentially this widens the spikes which could cause detriment to the minimum distance measurable between targets.
Figure 2: Time Correlation with Hanning-weighted Chirp (top to bottom) a)Transmitter b)Receiver c) Receiver + noise d)Correlation output
Of course there are many other window functions that would bear slightly different results and it all depends upon the requirements of the system. It would also be relatively easy to change this on the fly and perhaps process results from a windowed pulse and a normal one, to gain more detail about the target.
Justification of Chirp Method
There are many benefits to using a chirp method such as the reduced power requirement. However, as a further demonstration of the effectiveness of the chirping let us examine how a simple fixed-frequency pulse behaves under the same conditions.
Fig3 shows that when we use a fixed-frequency pulse the correlation result is a symmetrical diamond shape centred around the first instance of the received pulse. While this may be acceptable for determining the position of a target that is isolated, it is difficult to distinguish overlapping received pulses. The signal is not compressed as in the chirp method and actually occupies twice as much time space at the output.
Figure 3: Time Correlation with Fixed-Frequency (top to bottom) a)Transmitter b)Receiver c) Receiver + noise d)Correlation output
Processing Issues
Ideally a radar system should be able to provide results in as close to real-time as possible. This is especially important in systems such as air traffic control where lives are at stake. Unfortunately, the correlation process is very demanding of processing power to achieve this. The numerous overlapping FFT operations on high frequency resolution data require powerful digital signal processing and fast analogue-to-digital conversion.
While it is possible to meet such demand with conventional DSPs it can be expensive and may require a complex multi processing environment which is costly and inconvenient. An emerging solution to this problem using FPGAs is discussed here.
