An update to the sliding DFT

Those properties allow us to create Table 1, listing the appropriate arctan approximation based on the octant location of complex x. So we have to check the signs of Q and I, and see if |Q | > |I |, to determine the octant location and then use the appropriate approximation in Table 1. The maximum angle approximation error is 0.26◦ for all octants. When θ is in the fifth octant, the above algorithm will yield a θ ′ that’s more positive than +π radians. If we need to keep the θ ′ estimate in the range of −π to +π , we can rotate any θ residing in the fifth quadrant +π/4 rad (45◦) by multiplying (I + jQ ) by (1 + j ), placing it in the sixth octant. That multiplication yields new real and imaginary parts defined as