# What is the advantage of using a Stepped Frequency waveform over FMCW?

What is the advantage of using a Stepped Frequency Waveform instead of a traditional linear FMCW ?

• perhaps for people who don't wanna do the math, with a stepped frequency waveform you can wait for the transients die off and measure the gain and phase shift accurately at that constant frequency, then move on to the next frequency. but a linear chirp will work if you do the math right. Mar 30 '21 at 18:10
• None that I'm aware off. Mostly a little easier to program Mar 30 '21 at 20:30

If you're defining the stepped-frequency waveform as a pulse train where each pulse has a different frequency, then it indeed does have advantages. One of the main ones being bandwidth requirements, which we'll go over.

Both FMCW and stepped-frequency techniques use frequency-domain range finding via the DFT. They map beat frequencies resulting from the mixing process to a range. For a given LFM chirp of bandwidth $$\beta$$ and pulse width $$\tau$$ the mapping is given by

$$f_b = \frac{-2R\beta}{c\tau}$$

The range resolution capability of any waveform with captured bandwidth $$\beta$$ is

$$\Delta R = \frac{c}{2\beta}$$

In order to get fine resolution, $$\beta$$ must be large. In a FMCW system, the entire instantaneous bandwidth $$\beta$$ must be supported for a single chirp. Once you begin exceeding 100 MHz or so, this can become a big problem.

Stepped-frequency waveforms go around this issue by achieving the desired bandwidth by collecting multiple pulses, with each pulse's carrier frequency increasing by some $$\Delta F$$ (usually linearly). By doing this, the receiver has to only support the bandwidth of one individual pulse of length $$\tau$$, which is approximately $$1/\tau$$.

The desired range resolution can be achieved since the bandwidth necessary is now captured over $$M$$ pulses and is given by

$$\Delta R = \frac{c}{2\beta} = \frac{c}{2M(\Delta F)}$$

As you may already be thinking, there are disadvantages to this. The timing complexity increases and you must spend more time collecting pulses in order to achieve the desired range resolution.

## Edit: Follow-up Questions

Does a step-frequency waveform ease the load on transmitter because its easy to generate?

Traditionally yes, it is more difficult to generate high chirp-rate LFM pulses when compared to simple single-frequency modulated pulses since analog devices were used. Back in the day, having oscillators that could provide the required chirp-rate were expensive to design and build. Today were we usually use a DAC, that limitation has been eliminated. You simply have to meet the sampling criteria, handle the sample throughput, some filtering and you're off to the races.

I would like to understand what kind of problems arise in LFM when the receiver has to handle a large BW?

An example of one of the many issues involves the antenna. Antenna's have defined beam patterns for a particular frequency and are designed with this in mind. When using LFM, the frequency changes intra-pulse and so does the wavelength. This means that the antenna pattern is changing as the pulse is being transmitted. For small bandwidths this can be tolerated. Once you start having hundreds of MHz of bandwidth, the antenna pattern can change drastically, which is usually undesirable.

In a stepped-frequency waveform, is the frequency constant within a chirp? If yes, then you would not get the beat frequency by mixing the transmitted and reflected signal?

In the waveform I described, there are no chirps. It is a train of simple modulated pulses, where each pulse has a higher (or lower if downchirping) frequency than the previous pulse. An example looks as follows:

We collect each individual pulse after it is reflected from a target and "stack" them to construct a pseudo-LFM sequence. Due to the mixing process, this will yield a beat frequency just like in FMCW, and we can take the DFT to process range.

There is a trade-off in the $$\Delta F$$ chosen and the number of pulses. You ideally want $$\Delta F$$ to be small so that you approach an ideal LFM pulse, but then that requires more pulses to capture the bandwidth $$\beta$$.

• Thank you for the explanation ! I still have 2 questions : Apr 8 '21 at 12:31
• 1. Does a step-frequency waveform ease the load on transmitter because its easy to generate? 2. I would like to understand what kind of problems arise in LFW when the receiver has to handle a large BW ? 3. In a stepped-frequency waveform , is the frequency constant within a chirp ? If yes, then you would not get the beat frequency by mixing the transmitted and reflected signal Apr 8 '21 at 12:46
• @Hp179 I've updated my answer to address some of your questions. Apr 14 '21 at 21:52
• Brilliantly explained and very well understood. Thank you @Envidia Apr 15 '21 at 15:35