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We know that the array manifold vector, also called the steering vector, for an M element array is given by:

$$ \mathbf{a}(\theta, \phi) = [e^{-j \omega \tau_{1}(\theta, \phi)}, e^{-j \omega \tau_{2}(\theta, \phi)}, \ldots, e^{-j \omega \tau_{M}(\theta, \phi)}]^{T} $$

Or

$$ \mathbf{a}(\theta, \phi) = [e^{-j \mathbf{\kappa}^{T} \mathbf{p}_1}, e^{-j \mathbf{\kappa}^{T} \mathbf{p}_2}, \ldots, e^{-j \mathbf{\kappa}^{T} \mathbf{p}_M}]^{T} $$

where:

  1. $ \mathbf{\kappa} = - \frac{\omega}{c} \mathbf{\rho} = - \frac{2 \pi}{\lambda} \mathbf{\rho}$,
  2. $\mathbf{\rho} = \begin{bmatrix} \text{sin}(\theta) \cdot \text{cos}(\phi) \\ \text{sin}(\theta) \cdot \text{sin}(\phi) \\ \text{cos}(\theta) \end{bmatrix}$ is the target's approach vector using azimuth-x and inclination coordinates
  3. $\mathbf{p}_{m} = \begin{bmatrix} p_{x_{m}} \\ p_{y_{m}} \\ p_{z_{m}} \end{bmatrix}$ is the m-th receiver's spatial position

when we transmit and receive an LFM waveform, the concept of a time independent $\omega$ or $\lambda$ no longer makes sense because the frequency changes linearly with time.

Question: What do we use instead?

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    $\begingroup$ This is the narrowband vs wideband question. If the ratio of the bandwidth to the carrier is "small enough" (narrowband), then just use the center frequency. If not, then you have to consider all or some subset of the frequencies (perhaps the min, middle, and max frequencies). How small is "small enough"? That depends on your requirements. $\endgroup$ Commented Nov 6, 2023 at 21:33

1 Answer 1

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As with many questions like these...it depends. We won't concentrate on LFM waveforms, the following is true for any waveform transmitted.

Virtually all of the introductory material to steering a phased array uses the narrowband assumption. That is, the transmitted waveform has a small bandwidth. More accurately, it has a small bandwidth ratio when compared to the signal's carrier frequency. Given the high, low, and center frequencies $f_1$, $f_2$, and $f_c$, respectively, the fractional bandwidth (FBW) is small:

$$FBW = \frac{f_2 - f_1}{f_c} << 1$$

Since radar applications are higher in frequency compared to other phenomena, they tend to produce smaller FBW's.

With this assumption, small bandwidth signals can be time-delayed via phase-shifts. This is good enough because a single phase shift will achieve a group delay that's pretty constant across all frequencies. You go to sum them up and get a good response since they're aligned. In this case, the carrier frequency is the one to use.

Now when this assumption breaks down and the $FBW$ is no longer small, the array experiences a relatively drastic change in frequency as the waveforms is transmitted/received. You can attempt to apply a single phase shift, but the gain you achieve will be less since the signals will no longer align and be somewhat distorted. In this case, you must use a wideband beamformer, which is another area altogether.

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  • $\begingroup$ Fair enough -- in my specific application, FBW << 1. I am working at around 60 Ghz f_c with about 2 GHz bandwidth. In this case, is the center frequency, or starting frequency, or ending frequency most appropriate for computing the phase shift? Not sure what 'carrier frequency' refers to in an LFM waveform... $\endgroup$
    – The Dude
    Commented Nov 7, 2023 at 16:45
  • $\begingroup$ Usually the carrier is whatever frequency you mix with the baseband signal to transmit. Unless you have a specific need otherwise, just use the center value. $\endgroup$
    – Envidia
    Commented Nov 7, 2023 at 17:03
  • $\begingroup$ @TheDude Also note that besides a hit to SNR, this issue also manifests as array pointing errors. $\endgroup$
    – Envidia
    Commented Nov 7, 2023 at 17:11
  • $\begingroup$ Yea I am working through the array pointing errors, which is the main motivation for this question :-/ Thanks. I will award you the answer :-) $\endgroup$
    – The Dude
    Commented Nov 7, 2023 at 21:36
  • $\begingroup$ @TheDude Btw, I have another post going into the pointing errors: dsp.stackexchange.com/a/86203/26009 $\endgroup$
    – Envidia
    Commented Jan 6 at 18:22

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