# Why is there a tradeoff between simplicity of a window function and its ability to be adjustable and controllable?

While studying and applying different window functions in a project I am working on, I found that known windows (here is an example paper which cataloged numerous windows) suffer from 2 kinds of trade-offs:

1. Simple (from processing speed/complexity point of view) windows (predominantly variations of $$cos$$ and $$exp$$) whose DFT cannot be controlled*.
2. Complex windows whose DFT can be controlled* (Dolph-Chebyshev, Kaiser, Ultraspherical).

There seems to be no simple window whose DFT characteristics can be controlled (Dolph's window can control the side-lobe level for example but is costly to implement in a real-time application). I realize there are alternatives to overcome this, such as using a LUT, but that is outside the scope of this question.

Question: Is this true? If so, is there any discussion on it (textbook/article/paper)? I am unable to find a reason why a simple window whose DFT can be controlled cannot exist.

*By control, I don't mean just having a variable(s) in window equation (adjustable), but also predicting how a characteristic (main lobe width or side-lobe levels) would change when the variable(s) changes. The chapter "Arrays: Linear, Planar and Circular" of Balanis's "Antenna Theory" shows a way to design the Dolph-Chebyshev window to match user requirements. Similar methods can be applied to Kaiser and Ultraspherical windows.

Edit for clarification: I am talking about adjustable time-domain windows whose frequency response can be controlled. Example of adjustable but not controllable window: $$w(n) = sin(\frac{\pi n}{N})^{\alpha}$$. It's frequency response does vary with $$\alpha$$ but we don't know how much the BW or SLL would vary. Example of adjustable and controllable window: the Chebychev window. We can adjust it by varying $$\beta = \cosh\!\big(\tfrac{1}{N} \cosh^{-1}(10^\alpha)\big)$$ and control the SLL of it's frequency response (the SLLs are guaranteed to be -20$$\alpha$$dB). There are not many windows of the later category and the few which are present are complex to implement (resource-wise, they take more mathematical operations, memory, speed)

• Balanis' book is 40+ years old. The definition of "costly" has wildly changed since then. Typically you design the window once and then store the coefficients and the actual application of the window is a simple vector multiply. Today's computers and embedded hardware have no problems designing any window you like. Jan 31 at 13:25
• @Hilmar, I certainly agree that today's computers and hardware have no problem. But it would be a problem in an environment like FPGA with limited additions/multiplications and limited power/speed (I am an amateur, there are probably much better explained examples). Feb 1 at 3:07