What is the difference between Goertzel algorithm and cross correlation in time Im reading https://en.wikipedia.org/wiki/Goertzel_algorithm,
when we want to discover the amplitude of a some frequency in a given signal what is the difference between applying Goertzel algorithm and doing cross-correlation in time between our signal and a complex exponential with the desired frequency.
It seems like in the button line Goertzel algorithm does exactly that,
then why couldn't we apply the correlation in the first place and instead used the IIR filter ?

Thank you.
The Goertzel algorithm gets away without calculating a complex product with $e^{j\pi lk}$ n times; thus drastically reducing the computational complexity.