# Can we detect multiple point target within a narrow-band transmit signal?

As I understand matched filter is optimal linear filter for maximizing the Signal to noise ratio (SNR) in presence of additive stochastic noise. Consider a radar/sonar system with narrow-band transmit signal and I guess generally people use matched filter for detection, specifically for range resolution. As per my understanding, under this condition we cannot resolve if there are multiple target within the signal, my first question: is it correct? We can detect multiple target within the signal if we use linear chirp with reasonable bandwidth. Multiple target within the signal refers to point targets within transmission time, (the return comes from more than one target).

My second question (sorry not a technical question) : Is anyone in the community is interested to resolve targets within transmission time with narrow-band transmit signal in the expense of high (O(N)=$N^3$, $N$ is number of samples) computational cost?

• $N^3$ is not a complexity measure unless you define what $N$ is. Also, I probably wouldn't already call that "heavy" unless $N$ for whatever reason is very large. – Marcus Müller Jun 9 '16 at 8:32
• … and that is in the order of hundreds, thousands, millions, billions? – Marcus Müller Jun 9 '16 at 8:39
• seriously, you will have to put down some numbers here to justify this is anything interesting; there's really a lot of radar algorithms out there, and I doubt you're inventing something interesting if the only thing you know about the algorithm is its complexity. – Marcus Müller Jun 9 '16 at 8:42
• no. that is not true. $O(N) = N^3$ (I even assumend the $O$ for you, here) is an asymptotic measure, and other effects will usually out-complex it if $N$ is not large. But anyways, that's not how it should work around here: ask a new question, describe your algorithm in there (instead of just its bad properties) and ask specific questions. – Marcus Müller Jun 9 '16 at 8:46
• Yes, tried to answer the first question as extensively as feasible in my answer :) hope you like it. Your second question does seem relevant to you, so you should probably ask it; I'm just giving feedback on the way you've asked it (which I don't really like, because it's vastly underdefining what you do) :) I really don't have the time to work on radar right now. – Marcus Müller Jun 9 '16 at 8:53

As I understand matched filter is optimal linear filter for maximizing the Signal to noise ratio (SNR) in presence of noise.

some restrictions on the noise apply (mainly: should be additive), but yes!

Consider a radar/sonar system with narrow-band pulse

A pulse that is narrow band is not really much of a pulse, if you'd ask me, but that's more of an opinion but a scientific fact.

However: the smaller the bandwidth, the longer the duration of the pulse. Basically, for pulse radar, bandwidth $b$ is inversely proportional to pulse duration $T$.

specifically for range resolution

low bandwidth is very bad for range resolution. A bit of fiddling with the radar equations will show that your resolution is bounded by the half distance your wave travels during $\frac 1b$ (monostatic rada). Now, as mentioned above, for the pulse radar, that leads to a resolution of $c\frac 1{2T}$.

As per my understanding, under this condition we cannot resolve if there are multiple target within the pulse, my first question: is it correct?

Exactly. See statement above.

However! You can get something that is equivalent to a higher bandwidth by multiple observations under coherent conditions. I'd doubt you'll easily distinguish two targets by that without a secondary source of information, though.

We can detect multiple target within the pulse if we use linear chirp with reasonable bandwidth

That's why people use chirps: The same bandwidth $b$ as a pulse of duration $\frac1b$, but you can spread the energy over a larger time $T\gg \frac 1b$. So, chirps, unlike radar pulses have sub-1/signal duration resolution, because their duration is not inherently linked to the bandwidth.

Is anyone in the community is interested to resolve close target with narrow-band pulse in the expense of heavy computational cost?

So again, you do not want to go narrowband and pulse at the same time. That's just the worst of two worlds.

What you do is apply proper signal processing gain and knowledge on the target as the signal to get higher resolution than a single transmission of given bandwidth would have, and mostly, you combine a lot of observations.

For the "far far away" target, I'd encourage you to learn what a synthetic aperture radar does, in terms of signal-processing. There's pulse compression (like you know it from the chirp) and there's the combination of a lot of observations to mimic a really narrow beamwidth and get a resolution that is far higher than one could expect from a single observation.

For the closer target: OFDM Radar is your topic. You can use the existing OFDM waveforms, for example of vehicular IEEE 802.11p "WiFi", and exploit the fact that you know a lot about these and have very good estimators for channel state that you can apply on the reflections. Martin Braun wrote a dissertation on that (which, by the way, also does a very decent job of explaining typical boundaries in estimator's variance, accuracy and radar estimation possibilities in general).

• You should really define that. Heavy is so unclear. I don't know what you mean with that - modern radars spent a lot of CPU on estimating, tracking, data-aiding, equalizing... – Marcus Müller Jun 8 '16 at 23:48
• The same as for "heavy" goes for "close" – Marcus Müller Jun 8 '16 at 23:49