# Seven term Blackman Harris window

I am currently trying to evaluate a high resolution ADC for which I need to window my output data for the FFT. Up until now I have been using the default 4 term Blackman harris window in MATLAB. However, I came across a few app notes that recommend using a seven term Blackman Harris for the SNR ranges that I am interested in. I have tried looking quite a bit but I cannot seem to find an equation for implementing the seven term BH window, neither is it available in MATLAB. Can someone point me to a source where I can find this? As far as I am aware Harris' classic paper on windows doesn't provide the equation either.

• In addition to the complete answer that was provided, I have found this and this paper useful when thinking about windows. – A_A Aug 8 '18 at 10:11
• I wrote this function for a few different windows including the blackmanharris 7 term: github.com/Matlab-Toolbox/create_window – Morgan Apr 17 '20 at 11:13

In this document you can find the coefficients of a seven-term Blackman Harris window. Ignoring the bizarre notation, it seems like the window is defined by

$$w[n]=\sum_{k=0}^6a_k\cos\left(\frac{2\pi kn}{N}\right),\quad n=0,1,\ldots,N-1\tag{1}$$

where $N$ is the window length and the coefficients $a_k$ are given by

\begin{align} a_0&=0.27105140069342\\ a_1&= - 0.43329793923448\\a_2&=0.21812299954311\\a_3&= - 0.06592544638803\\a_4&= 0.01081174209837\\a_5&= - 0.00077658482522\\a_6&= 0.00001388721735\end{align}

Unfortunately, they do not mention where those coefficients come from. Somebody must have run an optimization routine to minimize the sidelobe level for a given mainlobe width.

EDIT:

I think I found the relevant paper here. This is a corresponding Matlab function.