# Sign of manifold vectors in monostatic phased-array radar

I am reading some notes where the impulse response of phased-array to is given by;

$$\beta S(\theta, \phi) \bar{S}^H(\theta, \phi) \cdot \delta(t-\tau)$$

where

\begin{align} \bar{S}(\theta, \phi) = \exp(+j\bar{r}k(\theta, \phi)) \newline S(\theta, \phi) = \exp(-jrk(\theta, \phi)) \end{align}

$$\bar{r}$$ is a vector with the coordinates of the antenna elements of the transmitter array while $$r$$ is the vector of the coordinates of the receiver elements. In this case (monostatic radar) the two are the same. $$k$$ is the wavenumber of the wave that propagates in the direction given by $$\theta$$ and $$\phi$$. My question is why are $$S$$ and $$\bar{S}$$ defined the way they are. It would make more sense to me if they were the conjugate of that. That way the antennas that are closer to the target would receive signals with a more advanced phase which would make more sense to me.

• If you start with the signals at the carrier frequency and include its removal, does that explain the sign of the phase? Mar 19, 2020 at 11:21