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Phased array pulse design

null1null1 wrote 11/12/2022 at 15:13 • 5 min read • Like

Continuous wave or CW

First a couple of links to some background explanations:

Link: Radar tutorial on CW

Link: Radar tutorial on range resolution


CW pulse

CW pulse consist of a constant frequency and a duration (pulse length).

Range resolution is:

where c is the speed of sound (in air 330m/s).

So to get a range resolution of 2cm, we need a pulse length of 121us.

If sampling frequency is 48000Hz, the CW pulse is approx. 6 samples long.

Pulse length vs. bandwidth

There is an inverse relationship between pulse length and bandwidth of the transmitted signal. 

The short pulse, have a wide bandwidth.

Likewise, if pulse is 10 times longer we get

Narrow bandwidth on a longer pulse.

With a narrow bandwidth, will be able to process the signal to only listen to those specific frequencies and filter out noise. Further, sending long pulses require less energy to transmit to reach the same distance. Drawback is that range resolution worse (10x to be exact).

Spectrum has a main lope, the signal we want to process, but it also has a lot of side lopes, which is unwanted signals.

We can manage the side lopes using window functions. There are a lot of different windows function designs, with different properties. Boxcar (rectangular), Hamming, Kaiser, Chebychev to name a few. The latter are pretty good at suppressing side lopes, but the cost is a wider main lope.

Some perspective, a wide main lope will make our objects in the beam formed image wider. Objects will look wider than they are, so a wide main lope is not what we want.

Again, we want narrow beams, good range resolution so we can identify the objects in the image.

An example with a Kaiser window function applied to the transmitted (long) pulse

Side lopes are almost gone, but the main lope is wider.

CW pulses are easy to understand, require less post-processing to generate great looking images. But total range might suffer, as short pulses requires more energy.

Intrapulse modulation (or FM)

If we, instead of transmitting short pulses, transmit longer pulses with a frequency change, a sweep from frequencies F1 to F2, built in, AND post process the signal with range compression (using a match filter), the properties of range resolution is defined as:


where BW is the bandwidth of the transmitted signal. Notice that the range resolution is no longer dependant on pulse length (in theory at least). The larger the BW is, the better range resolution we get.

Example: range resolution of an FM transmitted signal where f1=12000Hz and f2=14000Hz is approx 8.25cm. 

Example of FM pulse, linear chirp.

Link: Radar tutorial on Intrapulse Modulation

Pulse compression

Pulse compression is a filter that matches the transmitted pulse with the received data. It filters out data that is not correlated with the pulse, hence increase S/N significantly.

(source wikipedia: https://en.wikipedia.org/wiki/Pulse_compression)

Red pulse is transmitted and when pulse hits objects, the are reflected, the blue echos are returned. When applying the match filter, the impulse response is shortened to the short "blips", seen to the right (hence compression of the pulse).

I real life sonars, data is echos from the objects and surroundings and the compression filter aka match filter, removed every thing else but the matching pulse.

A sonar ping under water, will contain background noise from the surroundings, maybe engine noise from a ship or similar. This is effectively removed so only data echoed from the transmitted pulse remains.

Match filter can be processed in frequency-domain

where MD is the resulting data after match filtering, RD is the raw received data and pulse is the transmitted pulse.

Or in time-domain

In my project, I use the frequency-domain implementation, since its running in CUDA 

Above is basically just a FIR filter. You could use the same implementation to remove low frequency noise. Make sure that pulses are symmetric to keep the zero-phase. 

Back to phased array project...





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