# Frequency response of antennas

Electromagnetics and antennas are not my main area of expertise. However, I have a specific question about antennas that I want to get right in order to make sure my signal processing is right.

What is the frequency response of an antenna?

I am OK with Fraunhofer answers. Clearly, the frequency response, as a function of $$\\omega\$$, can generally differ with angle, commonly denoted $$\\phi\$$ and $$\\theta\$$ in polar coordinates. And clearly this is reflected in the antenna pattern as it varies with frequency, but this does not tell the whole story.

I have carefully studied basic diffraction theory in such books as Antenna Theory and Design by Stutzman and Thiele, Theory of Remote Image Formation by Blahut, Fourier Optics by Goodman, Engineering Optics by Iizuka, and I even spent some time with RadLab 12.

I have also been familiar with various radar books and papers for quite some time. A couple of examples are Principles of Aperture and Array System Design by Steinberg and Introduction to Radar Systems by Skolnik.

All of these references include, as necessary, frequency dependence in the antenna pattern. What is disturbing is that the works that carefully develop the field include a $$\j\omega\$$ or $$\1/j\lambda\$$ term that is missing in the other works; in some cases it gets normalized out when defining the antenna pattern as "a constant" and in the less rigorous works it does not even appear in the casually-presented development of aperture or array theory. For example, in Stutzman and Thiele, the $$\E\$$ field is calculated as $$\E=-j\omega \mu A\$$ where $$\A\$$ is the vector potential calculated from a spatial integral. On the other hand, Skolnik does an informal presentation that omits the leading frequency dependence.

So this is my problem: It seems that antennas have a rising frequency response that is independent of angle, and that many non-antenna-specialist works and authors are unaware of this or are interested in only the antenna pattern. Is it possible that the field of radar signal processing has gotten this wrong? Or have I completely misunderstood diffraction and antenna theory?

For radar, of course, the energy passes through the antenna twice so the net effect is proportional to $$\\omega^2\$$.

For anyone interested, my application is imaging radars and I need to know if this angle-independent rising response needs to be compensated in the signal processing, especially for wide-band systems.

In case anyone would like to comment further, does the reflected radar signal also undergo a $$\j\omega\$$ effect as it leaves the ground patch, since the reflecting ground patch is just another antenna?

• What do you mean by "rising frequency response"? Commented May 12, 2022 at 12:15
• "Rising frequency response" - are you referring to aperture gain, which for a given size aperture, increases with frequency? Commented May 12, 2022 at 13:22
• @SteveSh By rising frequency response I mean the $j\omega$ term in the E-field expression, a compressed example of which I gave from Stutzman. The field is generally $j\omega$ * a spatial integral which in the far field factors as $j\omega$ * 1/r * a unit-amplitude, spatially varying complex exponential, * a Fourier transform of the aperture field or current density. The $j\omega$ term indicates a field amplitude which increases linearly with frequency. "Gain" might come from this but I believe gain is a single gross number that describes the antenna, not specifically the field. Commented May 12, 2022 at 22:16
• I'm sorry, but I just don't see it. I've looked through some of my EM references, and the only places I've found where something like jw, or f shows up is in equations describing the time-varying nature of an EM wave, as in cos(wt). Commented May 13, 2022 at 10:58
• @SteveSh It's everywhere in field equations. I'll give you a specific example from Antenna Theory and Design by Stutzman and Thiele 9-36 or 8-34 depending on edition. The same expression is in e.g. Iizuka, and Goodman, with minor notation changes. $$E_{\theta}=j\beta\frac{\exp\left(-j\beta r\right)}{2\pi r}E_{0}L_{x}L_{y}\sin\left(\phi\right)\frac{\sin\left[\left(\beta L_{x}/2\right)u\right]}{\left(\beta L_{x}/2\right)u}\frac{\sin\left[\left(\beta L_{y}/2\right)v\right]}{\left(\beta L_{y}/2\right)v}$$ Here, $\beta=\omega/c$. In other places you will find $\lambda=2\pi/\omega$. Commented May 13, 2022 at 22:53

The calculations of antenna pattern usually begin with a reasonably assumed current distribution on a wire or on an aperture (magnetic current in this case). Of course, the current distribution is the main factor that determines the pattern. A change in the frequency changes the current distribution and the pattern in a way that cannot be expressed in an explicit relation. Not only the $$\omega ,\beta ,\lambda$$ appearing or not-appearing in the final pattern formula is a judge.