Magnitude Approximation

Inspection of equations 87 through 91 reveals that the magnitude of $Y_a(k)$ is approximated by

$$\fbox{$ \displaystyle \vert Y_a(k)\vert \approx \sqrt{\f... ...left(\ensuremath{\frac{2\pi^2 k}{KN\alpha}}\right)\right) $}$$ (90)

under the same constraints on $k$ as for the phase approximation.

Figure 11 illustrates the the magnitude of an example $Y(k)$ as well as the estimate obtained using equation 92. We see the great accuracy of the approximation. Again, the error bound for the large limits approximation is shown (dot-dash). We see that the error propagates in a more well-modeled way than for the phase approximation.

Figure 11: The magnitude of an example $Y(k)$(solid) (with $\alpha = 0.004280$, $F=8000$, $K=N=201$, and a Hann windowing function) and the approximation obtained using equation 93 (dashed). The error bound for the large limits approximation (dot-dash) is also shown.