Phase-comparison monopulse

We can define the beam pattern (array factor) of a uniform linear array (ULA) with N elements, as:[5]

${\displaystyle B_{\theta }\left(\theta \right)={\vec {w}}^{H}{\vec {v}}_{\theta }\left(\theta \right)=\sum _{n=0}^{N-1}w_{n}^{*}\left[{\vec {v}}_{\theta }\left(\theta \right)\right]_{n}=\sum _{n=0}^{N-1}w_{n}^{*}e^{j\left(n-{\frac {N-1}{2}}\right){\frac {2\pi }{\lambda }}dcos\theta }}$, where ${\displaystyle {\vec {v}}_{\theta }}$ is the array manifold vector and ${\displaystyle {\vec {w}}}$ is a vector of complex weights representing amplitude and phase adjustments applied to each antenna element. The manifold vector, ${\displaystyle {\vec {v}}_{\theta }}$, fully encapsulates all of the spatial properties of the array. ${\displaystyle d}$ is the distance between elements of the array, and ${\displaystyle \theta }$ is the angle of arrival of an incident plane wave, defined from end-fire, i.e., ${\displaystyle \theta =90^{\circ }}$ is a signal from array broadside.

It is common to perform a variable substitution to ${\displaystyle \psi }$-space, where ${\displaystyle \psi ={\frac {2\pi }{\lambda }}dcos\theta }$, and therefore we have:

${\displaystyle B_{\psi }\left(\psi \right)=\sum _{n=0}^{N-1}w_{n}^{*}e^{j\left(n-{\frac {N-1}{2}}\right)\psi }}$

and we can more easily see that ${\displaystyle \psi }$ is simply the phase shift between adjacent elements. The ${\displaystyle {\frac {N-1}{2}}}$ term simply references the absolute phase to the physical center of the array.

Notice that this result is the same if we instead first sum each half of the array, then add those results together.

${\displaystyle B_{\psi }\left(\psi \right)=\sum _{n=0}^{{\frac {N}{2}}-1}w_{n}^{*}e^{j\left(n-{\frac {N-1}{2}}\right)\psi }+\sum _{n={\frac {N}{2}}}^{N-1}w_{n}^{*}e^{j\left(n-{\frac {N-1}{2}}\right)\psi }}$

The weight vector is a combination of a steering vector that steers the beam in a steered direction, ${\displaystyle \psi _{S}}$, using phase adjustments and an amplitude taper that is often applied to reduce sidelobes. Thus, ${\displaystyle \left[{\vec {w}}\right]_{n}=a_{n}e^{j\left(n-{\frac {N-1}{2}}\right)\psi _{S}}}$, and

${\displaystyle B_{\psi }\left(\psi _{\Delta }\right)=e^{j\left({\frac {N-1}{2}}\right)\psi _{\Delta }}\sum _{n=0}^{N-1}a_{n}e^{-jn\psi _{\Delta }}}$, where ${\displaystyle \psi _{\Delta }=\psi _{S}-\psi }$.

We can clearly see now that the beam pattern, in ${\displaystyle \psi }$-space, is the spatial equivalent of the discrete time Fourier transform (DTFT) of the array amplitude tapering vector times a linear phase term. The advantage of ${\displaystyle \psi }$-space is that the beam shape is identical no matter where it is steered, and is only a function of the deviation of the desired target phase from the actual target phase.

Let us now assume an un-tapered, normalized array with ${\displaystyle a_{n}={\frac {1}{N}}}$. The beam pattern can be easily shown to be the familiar aliased sinc (asinc) function:

${\displaystyle B_{\psi }\left(\psi _{\Delta }\right)={\frac {1}{N}}{\frac {sin\left(N{\frac {\psi _{\Delta }}{2}}\right)}{sin{\frac {\psi _{\Delta }}{2}}}}}$

This pattern is also known, for monopulse purposes, as the "sum" pattern, as it was obtained by summing all of the elements together. Going forward we will suppress the ${\displaystyle \Delta }$ subscript and instead use only ${\displaystyle \psi }$ with the understanding that it represents the deviation of the steered target phase and the actual target phase.

Let us now develop the monopulse "difference" or "del" pattern by dividing the array into two equal halves called subarrays. We could have just as easily derived the sum pattern by first determining the pattern of each subarray individually and adding these two results together. In monopulse practice, this is what is actually done. The reader is left to show that ${\displaystyle {\vec {v}}_{\psi }\left(\psi \right)}$ is conjugate symmetric, so it can be re-written in terms of only its first half, ${\displaystyle {\vec {v}}_{\psi _{1}}\left(\psi \right)}$ using an exchange matrix, ${\displaystyle {\textbf {J}}}$, that "flips" this vector.

${\displaystyle {\textbf {J}}={\begin{bmatrix}0&\cdots &0&1\\\vdots &\ddots &1&0\\0&\cdot ^{\cdot ^{\cdot }}&\ddots &\vdots \\1&0&\cdots &0\end{bmatrix}}}$

Note that ${\displaystyle {\textbf {J}}\cdot {\textbf {J}}={\textbf {I}}}$. Assuming that N is even (we could just as easily develop this using an odd N),[6]

${\displaystyle {\vec {v}}_{\psi }\left(\psi \right)={\begin{bmatrix}{\vec {v}}_{\psi _{1}}\left(\psi \right)\\\cdots \\{\textbf {J}}{\vec {v}}_{\psi _{1}}^{*}\left(\psi \right)\end{bmatrix}}}$

If we assume that the weight matrix is also conjugate symmetric (a good assumption), then

${\displaystyle {\vec {w}}={\begin{bmatrix}{\vec {w}}_{1}\\\cdots \\{\textbf {J}}{\vec {w}}_{1}^{*}\end{bmatrix}}}$

and the sum beam pattern can be rewritten as:[7]

${\displaystyle B_{\psi }\left(\psi \right)=\Sigma _{\psi }\left(\psi \right)={\vec {w}}^{H}{\vec {v}}_{\psi }\left(\psi \right)={\begin{bmatrix}{\vec {w}}_{1}^{H}&\vdots &{\vec {w}}_{1}^{T}{\textbf {J}}\end{bmatrix}}{\begin{bmatrix}{\vec {v}}_{\psi _{1}}\left(\psi \right)\\\cdots \\{\textbf {J}}{\vec {v}}_{\psi _{1}}^{*}\left(\psi \right)\end{bmatrix}}={\vec {w}}_{1}^{H}{\vec {v}}_{\psi _{1}}\left(\psi \right)+{\vec {w}}_{1}^{T}{\vec {v}}_{\psi _{1}}^{*}\left(\psi \right)=2Re\left[{\vec {w}}_{1}^{H}{\vec {v}}_{\psi _{1}}\left(\psi \right)\right]}$

The difference or "del" pattern can easily be inferred from the sum pattern simply by flipping the sign of the weights for the second half of the array:

${\displaystyle \Delta _{\psi }\left(\psi \right)={\begin{bmatrix}{\vec {w}}_{1}^{H}&\vdots &-{\vec {w}}_{1}^{T}{\textbf {J}}\end{bmatrix}}{\begin{bmatrix}{\vec {v}}_{\psi _{1}}\left(\psi \right)\\\cdots \\{\textbf {J}}{\vec {v}}_{\psi _{1}}^{*}\left(\psi \right)\end{bmatrix}}={\vec {w}}_{1}^{H}{\vec {v}}_{\psi _{1}}\left(\psi \right)-{\vec {w}}_{1}^{T}{\vec {v}}_{\psi _{1}}^{*}\left(\psi \right)=2Im\left[{\vec {w}}_{1}^{H}{\vec {v}}_{\psi _{1}}\left(\psi \right)\right]}$

Again assuming that ${\displaystyle a_{n}={\frac {1}{N}}}$, the del pattern can be shown to reduce to:

${\displaystyle \Delta _{\psi }\left(\psi \right)={\frac {2}{N}}Im\left[\sum _{n=0}^{{\frac {N}{2}}-1}e^{-j\left(n-{\frac {N-1}{2}}\right)\psi }\right]={\frac {2}{N}}{\frac {sin^{2}\left(N{\frac {\psi }{4}}\right)}{sin{\frac {\psi }{2}}}}}$

Mono-pulse sum and difference (del) patterns

The monopulse ratio is formed as:

${\displaystyle {\frac {\Delta _{\psi }}{\Sigma _{\psi }}}={\frac {{\frac {2}{N}}{\frac {sin^{2}\left(N{\frac {\psi }{4}}\right)}{sin{\frac {\psi }{2}}}}}{{\frac {1}{N}}{\frac {sin\left(N{\frac {\psi }{2}}\right)}{sin{\frac {\psi }{2}}}}}}={\frac {2sin^{2}\left(N{\frac {\psi }{4}}\right)}{sin\left(N{\frac {\psi }{2}}\right)}}={\frac {1-cos\left(N{\frac {\psi }{2}}\right)}{sin\left(N{\frac {\psi }{2}}\right)}}=tan\left(N{\frac {\psi }{4}}\right)}$

One can see that, within the 3dB beam width of the system, the monopulse ratio is almost linear. In fact, for many systems a linear approximation is good enough. One can also note that the monopulse ratio is continuous within the null-to-null beam width, but has asymptotes that occur at the beam nulls. Therefore, the monopulse ratio is only accurate to measure the deviation angle of a target within the main lobe of the system. However, targets detected in the sidelines of a system, if not mitigated, will produce erroneous results regardless.

Mono-pulse ratio within 1 beamwidth of the main response axis