Goertzel Algorithm direct form

[curious_mind]

I was referring to this site https://www.codeproject.com/Articles/841228/Second-Order-Goertzel-and-Reinsch-Algorithm to understand the goertzel algorithm. I am not clear as the article explains that input and output sections are interchanged to make the input stage recursive, while output is computed only once. Can anybody explain how this could be possible.

 

[kripacharya]

I think the article explains exactly this in great detail. The purpose is to not repetitively do the complex calculation except once at the end.

 

[curious_mind]

OK, I do have confusion about the final equation after N iterations. Referring to the link, https://stackoverflow.com/questions/11579367/implementation-of-goertzel-algorithm-in-c
Where does real = (q1 - q2 * cosine); imag = (q2 * sine); come from?