Rectangular Window

When $\alpha$ is very small, $l_1$ and $l_2$ will take on the same sign for almost all frequency bins $k$, rendering equations 66 and 67 invalid. Fortunately, the large limits approximation may still be used, but while bearing in mind that the limits of integration in those expressions will have the same sign and be close in value to each other. Specifically, we may now state that

$$\int_{\pm l_1}^{\pm l_2} \cos\left(\frac{\pi}{2}u^2\right) du \approx \pm\left(\ensuremath{\frac{1}{-l_1\pi}}\sin\left(\ensuremath{\fra... ...ath{\frac{1}{l_2\pi}}\sin\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right)$$ (116)

$$% \int_{\pm l_1}^{\pm l_2} \sin\left(\frac{\pi}{2}u^2\right) du \approx \pm\left(- \ensuremath{\frac{1}{-l_1\pi}}\cos\left(\ensuremath{\f... ...h{\frac{1}{l_2\pi}}\cos\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right) .$$ (117)

Continuing algebraically in a similar fashion as before, we apply trigonometric identities and obtain

$$\Re{Y_a(k)} \approx \ensuremath{\frac{-1}{l_2\pi}}\cos\left(i\left(\ensuremath{\frac{... ...t(\ensuremath{\frac{\pi}{2}}-\phi+\ensuremath{\frac{\pi}{2}}l_1^2\right)\right)$$ (118)

$$\Im{Y_a(k)} \approx \ensuremath{\frac{-1}{l_2\pi}}\sin\left(i\left(\ensuremath{\frac{... ...(\ensuremath{\frac{\pi}{2}}-\phi+\ensuremath{\frac{\pi}{2}}l_1^2\right)\right).$$ (119)

We make note that this only applies when both $l_1$ and $l_2$ have magnitude much greater than one, and have the same sign. This will occur only when $\alpha$ is very small and $k$ is at least slightly greater than zero. At $k=0$, the large limits approximation is simply not valid.¹¹

In order to extract the parameter $\alpha$ from a model based on equations 122 and 123, we must make a critical analysis of the situation: a closed form expression for the phase cannot be obtained unless the arguments of the sinusoids in these equations are identical. It can be shown that this in fact occurs when we choose $k=\ensuremath{\frac{K}{N}}p$ where $p$ are nonzero integers less than $\ensuremath{\frac{N}{2}}$. Doing so shows that, for odd $p$,

$$\angle Y_a(k) = \ensuremath{\frac{\alpha N^2}{4}}-\ensuremath{\frac{\pi}{2}},$$ (120)

and that for even $p$,

$$\angle Y_a(k) = \ensuremath{\frac{\alpha N^2}{4}}+\ensuremath{\frac{\pi}{2}},$$ (121)

which we may solve for $\alpha$ to obtain

$$\fbox{$\displaystyle \alpha \approx \ensuremath{\frac{4}{N... ...th{\frac{K}{N}}p\right) +\ensuremath{\frac{\pi}{2}}\right) $}$$ (122)

for odd $p$ and

$$\fbox{$\displaystyle \alpha \approx \ensuremath{\frac{4}{N... ...th{\frac{K}{N}}p\right) -\ensuremath{\frac{\pi}{2}}\right) $}$$ (123)

for even $p$.

Thus, to estimate $\alpha$, we simply substitute values of the angle of the FFT at $k=\ensuremath{\frac{K}{N}}p$ into the appropriate expression above. We note that all exponential and sinusoidal arguments take on values between 0 and $2\pi$ in this analysis.