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I am studying a 1D beamforming for wide-band signals, so that applying only phase weights to each individual antenna would steer the beam for the central frequency. For other frequencies the steering would not be the desired one. I understand the optimum way to apply the beamforming would be to apply delay weights instead, meaning a different phase is correctly applied to each frequency.

Imagine I apply first a phase shift to each channel. This means this phase shift is only valid for the central frequency. If I want then to apply a correction for all the frequencies, could I apply a 'delta delay' to each channel? The corresponding correction in the frequency domain would be a phase ramp, but the central frequency requires no delta-phase correction. However, I am uncertain because I'm not sure if this linear phase ramp corresponds to a delay in fact. As I said, no extra phase shift is needed for the central frequency. The point here is that I am doing a 'correction' of the phase, instead of using delay steering vectors only.

What I found is that this correction does not seem to be able to be applied through a delay, rather just a different phase per frequency. But it's striking because theoretically I could relate a phase ramp to a time domain delay.

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  • $\begingroup$ What bandwidth are we talking about ? Something very narrowband (like wireless communication) or wideband (multiple octaves like a speaker or microphone array) ? $\endgroup$
    – Hilmar
    Mar 21 at 11:22
  • $\begingroup$ It's about 12% relative bandwidth $\endgroup$
    – Albert
    Mar 21 at 12:04

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I am studying a 1D beamforming for wide-band signals, so that applying only phase weights to each individual antenna would steer the beam for the central frequency. For other frequencies the steering would not be the desired one. I understand the optimum way to apply the beamforming would be to apply delay weights instead, meaning a different phase is correctly applied to each frequency.

Exactly. That's why as far as I'm aware, all wideband systems that do MIMO are OFDM systems: you do the MIMO / beamforming per subcarrier than, and that's narrowband enough so that one value suffices.

Imagine I apply first a phase shift to each channel. This means this phase shift is only valid for the central frequency. If I want then to apply a correction for all the frequencies, could I apply a 'delta delay' to each channel?

You need to be as granular (actually, probably twice) as your channel is frequency selective. So, if you could build an $N$-tap equalizer to flatten your SISO channel frequency response (to convert your channel impulse response to a singular impulse), you'll need at least $N$ taps per antenna to support your beamforming (again, probably more).

The math for this exists, it's just not very commonly done, because it's a) computationally intense, and as far as I understand (never done the math myself) b) not much better than simply dividing your wideband channels into $N$ narrowband channels and then going at them with independent classical, single-coefficient-per-antenna beamforming, and commercially pretty importantly, c) everyone uses OFDM already (even if that brings tears to the eyes of people who wrote excellent synchronizers and equalizers for the pre-OFDM cellular era).

You'll find wideband beam former in the audio domain, and specific radar application. It's not trivial to define what the right metric is: Is it narrowness of the beam at the "worst" frequency? Or is it quadratic deviation from some desired beam shape, integrated again over all frequencies? Is it mean squared average narrowness over all frequencies?

You probably would want to start looking into MVDR beamforming; that's (as far as I'm aware) the methodology best-explained in literature. To get there, get a good textbook: I flicked through Optimum Array Processing once, it seemed fined. I think we used to have "Array Signal Processing" by Johnson, and Dudgeon in the institute's library, but can't quite remember. If you have access to a university library, take the question "I want to learn about wideband beamforming, and been recommended these two books. Do we happen to have them or related books available?" to a librarian. They are probably a great help telling good from bad literature, entry-level from super-specific, simplified from overly detailed.

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    $\begingroup$ This is a solid answer. I would also add that spectral estimation books give a good foundation for narrowband beamforming techniques as they are closely related. I like the Stoica and Moses book, but the Marple or Kay books will also work. $\endgroup$
    – Baddioes
    Mar 21 at 16:06
  • $\begingroup$ Could you please give some references for "The math for this exists, it's just not very commonly done, because it's a) computationally intense, and as far as I understand (never done the math myself)"? $\endgroup$
    – AlexTP
    Mar 22 at 16:17
  • $\begingroup$ @Baddioes yeah, I "grew up" with Kay, but I later found Stoica & Moses' Spectral Estimation to be a bit more on point in the right places and a bit more explanatory in the places where that is needed. I don't know whether that's just me growing or the book actually being nicer to read. I guess it's mostly down to preferences. $\endgroup$ Mar 22 at 16:38
  • $\begingroup$ @AlexTP let me look up my literature list for one Master Thesis I advised, if I still have that (not with the institute any more) $\endgroup$ Mar 22 at 16:39
  • $\begingroup$ @AlexTP sorry, managed to find the thesis, but he didn't discuss effort. In that, case, we can only broadly speak: For a sufficiently narrow channel, finding weights boils down to finding an optimum in a system of equations under boundary conditions; in general, that's $\mathcal(N^3)$ time-complex, $N$ being the number of weights. In the channelization case, we need to divide our $K$-taps channel in $\mathcal O(K)$ subchannels first, the complexity of applying the base change matrix of $K^2$ be ignored, in the face of having complexity do $\mathcal O( KN^3)$ . $\endgroup$ Mar 22 at 19:40
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The final boss of beamsteering is the true-time delay beamformer. It is what is approximated for narrowband signals when applying a single phase-shift to the entire signal.

If I want then to apply a correction for all the frequencies, could I apply a 'delta delay' to each channel? The corresponding correction in the frequency domain would be a phase ramp, but the central frequency requires no delta-phase correction.

You could do this, but it will only be an approximation. You can use the Fourier transform property

$$x(t - t_0) \xrightarrow{\mathscr{F}} X(f)e^{-j2\pi ft_0}$$

Apply the "phase ramp" in the frequency domain to approximate the time delay, but since you're usually in the digital domain, you can only act on discrete frequency bands. Your bin size (and therefore sample rate and number of samples) affects how close you can get to an ideal delay. I however would not consider this a "true" time delay beamformer, but it is a decent wideband beamformer.

The "true" part comes when you physically (either in analog or digitally) delay the signal. Here, there is no such things as weights as we're usually used to. This is classically achieved in the analog domain using inductor-capacitor networks. In modern systems, the signals are sampled and delayed using a combination of whole-sample (integer) delays and fractional delay filters. This type is more complex and expensive to achieve, but it is a "true" time delay beamformer.

Also note the conservation of complexity. The true time delay beamformer is very simple, you just delay the signals. However physically doing it is the hard part. The inverse are the other wideband beamformers, where the math is more complex but applying them is easy.

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A paper on designing broadband sensor arrays is:

The key parts are:

  • The sensors are logarithmically spaced, apart from the center of the array where they are linearly spaced.
  • The filters on each sensor are chosen to keep the "aperture" of the array constant with wavelength.

The beam pattern is chosen, as you suggest, by changing the filter phase with frequency appropriately.

Darren, Robert, and I used Darren's work to do broadband nulling.

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