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I believe your thinking is correct. For bandpass filters, for each z-plane pole in the positive-frequency range there's a conjugate pole in the z-plane's negative-frequency range. So for bandpass filters there will all be an even number of total z-plane poles (two poles, four poles, six poles, etc.). When using MATLAB's ellipord command for bandpass filters that command returns the number of poles in the z-plane's positive-frequency range which is half the actual number of z-plane poles.

This fourth edition of a well-established textbook takes students from fundamental ideas to the most modern developments in optics. Illustrated with 400 figures, it contains numerous practical examples, many from student laboratory experiments and lecture demonstrations. Aimed at undergraduate and advanced courses on modern optics, it is ideal for scientists and engineers. The book covers the principles of geometrical and physical optics, leading into quantum optics, using mainly Fourier transforms and linear algebra. Chapters are supplemented with advanced topics and…

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Practical bandpass filters will be even-ordered. Below Matt L points out that modifying an even-ordered filter’s transfer function by placing a z-plane zero at DC can make a bandpass filter odd-ordered, which is true. But placing a z-plane zero at DC so badly distorts the original even-ordered filter’s frequency response that the new odd-ordered filter becomes unusable.

Then he also mentions that the MATLAB code ellipord returns half the actual order, so I figured the two statements must be connected somehow. I can't ask him since it's currently 3 AM and I couldn't find anything alone as to why these two statements are true.

My professor mentioned that the order of a band-pass and a notch filter must always be even, when showing an example of designing a digital filter using the bilinear transformation.

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Of course, there are specific designs for which an odd filter order doesn't make much sense. E.g., for a notch filter (with a notch frequency greater than zero) you want exactly one zero at the notch frequency plus its mirror image at the negative notch frequency. So for each notch frequency you get two zeros, and, consequently, two poles.

Henry Lipson was Professor of Physics at the University of Manchester Institute of Science and Technology, UK. He was a pioneer in the development of optical Fourier transform methods for solving problems in X-ray crystallography.

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Ariel Lipson is Senior Physicist at BrightView Systems Ltd, Israel. He has contributed to three successful start-up companies in optics, which have influenced several of the topics discussed in this book.

Stephen G. Lipson is Professor of Physics and Electro-optics in the Physics Department of the Technion - Israel Institute of Technology, Israel. He holds the El-Op Chair of Electro-Optics at Technion where he has taught courses in optics both at elementary and advanced levels.

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Maybe that's a just a matter of semantics. You can certainly cascade an even order high pass with an odd order lowpass and you get something that's an odd order filter that sure looks like a bandpass.

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In principle there is no reason why the filter order of a general bandpass or bandstop filter must be even. Such a restriction is a consequence of a specific design procedure. In classic IIR filter design (Butterworth, Chebyshev, Cauer) you start with an analog prototype lowpass filter. Bandpass or bandstop filters are then obtained by a frequency transformation. And it is this frequency transformation that doubles the order of the prototype lowpass filter, hence the even filter order for that specific design method. Note that the bilinear transform has nothing to do with that restriction on the filter order.

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