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From what I understand, PS beamforming applied a phase by multiplying the complex expontential ($e^{i\theta}$), while TTD applies a delay ($x(t-d)$)

It makes sense that multiplying a complex number to a signal is simple/inexpensive (PS beamforming) but why is applying a delay complex/expensive?

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Here I am assuming all data is complex baseband (IQ) with digital beamforming. There is an analogous approach for analog beamforming.

In phase only beamforming there is a single complex multiply per-sample per-channel with all of the complex data being summed across channels to generate a sample of output.

Time delay beamforming is more complex. The delay operation is in general not an integer number of sample delays. This requires the use of fractional samples delay filers in each of the channels. As a result, the single complex multiply from phase only beamforming is replaced with a filtering operation. The exact degree of computational growth will depend on things such as allowable error and the ratio of the signal bandwidth to the sample rate.

For an example implementation see Time Delay Digital Beamforming for Wideband Pulsed Radar Implementation (PDF).

Phase only beamforming is typically sufficient in narrowband applications. Time delay beamforming is required for wideband applications. (Some may even use this as the criteria for calling a beamforming system narrow or wideband.)

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    $\begingroup$ @DanBoschen Usually phase only is sufficient for "narrow band" applications and time delay is required for wider bandwidths. Equalization is insufficient as the response is direction dependent, unless you're going to equalize per channel and then it is the same as time delay beamforming. $\endgroup$
    – GrapefruitIsAwesome
    Commented May 4 at 19:26
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    $\begingroup$ @DanBoschen You can also perform the TTD approximation in the frequency domain by applying the time shift property across the samples, which is a kind of equalization and can be calculated on the fly knowing the desired pointing direction. $\endgroup$
    – Envidia
    Commented May 4 at 19:41
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    $\begingroup$ In addition to "wideband" versus "narrowband" consideration for time versus phase-only shift is the beam width. For a large number of antennas (e.g. Starlink with 1280 antennas), the beams are quite narrow and more susceptible to beam squint than a device with a handful of antennas, even for a reasonably "narrow" bandwidth relative to carrier frequency. Another consideration is whether each antenna element is fed by a separate digital stream that can have fractional time shift applied versus an architecture with analog fanout and phase shifting where time shift becomes more difficult. $\endgroup$
    – vml
    Commented May 4 at 19:53
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    $\begingroup$ Excellent points @vml. Also, combo systems are sometimes used where you've got a whole bunch of elements combined into subarrays, and the subarrays are steered with true time delays, while the elements within a subarray are steered with phase shifts. $\endgroup$
    – Gillespie
    Commented May 4 at 22:58
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    $\begingroup$ Also @DanBoschen, there are ways to pre-process wideband signals s.t. they can then be treated as narrowband. E.g., first take the FFT of the signal on each element. Then in each frequency bin, you can apply a transformation across elements that will compensate for the bandwidth effect. After this, you can proceed as if the signals are all at the carrier. $\endgroup$
    – Gillespie
    Commented May 4 at 23:06

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