Rectangle Window: Large Fresnel Integral Limits Approximation

We recall from section 2 that we may apply the approximations in equations 3 and 4 provided that the limits of integration are much greater or lesser than 1 or -1 respectively. To make the problem more tractable, we presently restate the condition on the limits as: the limits of integration are greater than or lesser than $1\cdot c$ or $-1\cdot c$ respectively, where $c$ is a constant we deem sufficiently large. Now, given a specified frame length $N$ and chirp signal characterized by $\alpha$, we may conceptualize these limits as functions of $k$. To verify that these limits satisfy the requirement, we must solve the inequalities

$$\sqrt{\ensuremath{\frac{2\alpha}{\pi}}}\left(-\ensuremath{\frac{N... ...(\ensuremath{\frac{N}{2}}-\ensuremath{\frac{\pi k}{\alpha K}}\right) > 1\cdot c$$ (56)

for $k$ to obtain

$$k < \ensuremath{\frac{NK\alpha}{2\pi}}- c\sqrt{\ensuremath{\frac{... ...\ensuremath{\frac{NK\alpha}{2\pi}}+ c\sqrt{\ensuremath{\frac{K^2\alpha}{2\pi}}}$$ (57)

or simply

$$\vert k\vert < \ensuremath{\frac{NK\alpha}{2\pi}}- c\sqrt{\ensuremath{\frac{K^2\alpha}{2\pi}}}.$$ (58)

Concerning our choice of $c$, we note that choosing the ``risky'' $c=1$ yields fractional error bounds (deduced from figure 2) of about 14%. Choosing $c = 2$ drops this bound to 2.3%, and $c=2.5$ leads to 1.3%. A decaying exponential may be roughly fit to the fractional error characteristic for purposes of choosing $c$ based on the fractional error tolerated.⁷

Now, given a desired accuracy, we may view the large limits requirement either as a function of $k$ or as a function of $\alpha$. If the former, we may consider that the model is only valid for certain bins. This is not a complete picture, however, since we see below that in some cases, no choice of $k$ will satisfy equation 60. Thus we must also consider the large limits requirement as a constraint on $\alpha$ values for which we may consider the model valid. The relationship of each of $k$ and $\alpha$ to the large limits requirement is now discussed separately.

Satisfying Requirements for the Large Limits Approximation by Choice of $k$

Despite small errors when $c\geq 2$, equation 60 may raise the concern that $k$ has been restricted significantly. To see if this is true, we compare the above bound to the positive and negative frequency bins representing the maximum frequency present in the signal, $\pm k_{\max} = \ensuremath{\frac{NK\alpha}{4\pi}}$ (equation 15). To do this rigorously, we define $b$ as a constant between 0 and 1 indicating what fraction of $k_{\max}$ contains a valid range for our approximation. We can then solve for $b$ to determine if we have significantly restricted the range of bins in which our approximation is valid. Expressing the constrained $k$ in equation 60 as $b\cdot\vert k_{\max}\vert$, our constraint becomes:

$$b\cdot\vert k_{\max}\vert < \ensuremath{\frac{KN\alpha}{2\pi}}- c\sqrt{\ensuremath{\frac{K^2\alpha}{2\pi}}}.$$ (59)

Substituting from equation 15, this is

$$b\ensuremath{\frac{\alpha K N}{4\pi}}< \ensuremath{\frac{\alpha K N}{2\pi}}- c\sqrt{\ensuremath{\frac{K^2\alpha}{2\pi}}}.$$ (60)

Solving for $b$, we obtain

$$b < 2 - \ensuremath{\frac{2c}{N}}\sqrt{\ensuremath{\frac{2\pi}{\alpha}}},$$ (61)

and solving for $c$, we obtain

$$c < (2 - b)\ensuremath{\frac{N}{2}}\sqrt{\ensuremath{\frac{\alpha}{2\pi}}}.$$ (62)

We take a moment to consider consequences of this result. First, We see a tradeoff between accuracy of the approximation (achieved by requiring a large $c$ value) and the size of the frequency range in which our approximation is valid (represented by $b$). Simply put, the higher quality approximation we wish to make, the smaller the range in which we may consider the approximation valid. The range of validity will become crucially important when we backsolve for $\alpha$.

Second, we note that certain values of $c$ and $\alpha$ can lead to $b$ values outside the range (0,1]. A reasonably high $\alpha$ and low $c$ in equation 63 may actually cause $b>1$, suggesting that the approximation is valid outside the range representing frequencies present in the signal. This case will not give rise to any special analysis, as we do not find that consideration of bins outside $k_{\max}$ enhances our analysis. Furthermore, doing so strains our inverse midpoint integral approximation above.

A more significant problem is that for a very small $\alpha$, we get $b$ values less than or equal to zero, implying that no frequency bins are valid for the approximation. Specifically, solving for $\alpha$ we have that

$$\alpha > 2\pi\left(\ensuremath{\frac{N}{2c}}(2-b)\right)^{-2}.$$ (63)

Though it may not be obvious from inspecting this equation, this is a nontrivial constraint on $\alpha$ for speech and music signals, even when $c\leq 1$. In such practical cases, the large limits approximation is often invalid by this metric, and an alternative model must be used.

To make this more concrete, we offer the example of a musical tone of frequency 440 Hz, sounded with a 14 Hz frequency vibrato at 6 pulses per second. Listening to such a signal reveals that this range and speed represent the largest musically realistic (``Wagnerian opera-singer'') values for such parameters; anything larger or faster sounds machine-like. A frequency range half this size is in fact more typical (and to some, will sound more tasteful). Thus, the $\alpha$ representing the underlying quasi-linear piecewise frequency functions in our Wagnerian singer signal is a representative maximum. Given these parameters, we calculate the maximum $\alpha$ value by considering an FFT analysis frame that happens to catch the ideal point in time where, in one half vibrato pulse, the frequency range will change 14 Hz (from 433 Hz to 447 Hz) over a time period of 83.3 ms. Assuming a sampling rate $F= 10000$ Hz for convenience, we thus have a 14 Hz frequency change over 833 samples, or .0168 Hz per sample, yielding $a=.00840$. This in turn leads to $\alpha = \ensuremath{\frac{2\pi}{10000}}a = 5.28\cdot 10^{-5}$. Comparing this to the minimum value required by equation 65 requires that we choose $b$ and $c$ values, which we modestly require to be 0.75 and 1.5, respectively. Doing so leads to $\alpha > 2\pi\left(\ensuremath{\frac{833}{2\cdot1.5}}(2-0.75)\right)^{-2} = 5.22\cdot10^{-5}$. We thus see the razor-thin margin for validity of this model; requiring greater accuracy in $c$ or the more ``tasteful'' vibrato alluded to above would invalidate the model. Hence, the need for modified models summarized in appendix A and to be rigorously discussed in future writing.

Effect of $\alpha$ on Satisfying the Large Limits Approximation

We take a moment to further explore the relationship between $\alpha$, $k$ and the large limits approximation. As can be seen from inspecting equations 65 and 58, there is a nonlinear relationship between the limits and $\alpha$, and therefore between the validity of the large limits approximation and $\alpha$. In figure 6, we plot the limits as a function of $k$ for each of 9 $\alpha$ values. In each plot, we use a horizontal range of $k \leq k_{\max}$, which necessarily is different for each plot since $k_{\max}$ is a function of $\alpha$. We recall that the lower and upper limits must be outside -1 and 1 respectively in order to use the large limits approximation, and see that these point have been marked by horizontal dotted lines. The key observation here is that as $\alpha$ becomes small, the range over which the approximation holds decreases, until no range remains. Hence, our abovementioned claim that this approximation is not valid for very small $\alpha$. We do, however, observe that the limits in those cases fall in a range where other approximations may be used. This is the modification alluded to just above, and will be covered in future writing. Such modifications are also summarized in appendix A.

Figure 6: The lower (solid) and upper (dashed) limits as given in equation 58 as a function of $k$ for 9 values of $\alpha$ (show above each plot). We see that as $\alpha$ becomes small, a smaller range - and eventually no range - of $k$ is valid. In this example, $F=8000$ and $K=N=201$ for all $\alpha$ values.

Application of the Large Limits Approximation

Having restricted $k$ and $\alpha$ as above, our approximations may be applied to $\Re{Y_a(K)}$ and $\Im{Y_a(K)}$. Defining $l_1 =\sqrt{\frac{2\alpha}{\pi}}\left(-\frac{N}{2}-\frac{\pi k}{\alpha K}\right)$ and $l_2 =\sqrt{\frac{2\alpha}{\pi}}\left(\frac{N}{2}-\frac{\pi k}{\alpha K}\right)$ for notational simplicity, we may now write

$$\int_{\pm l_1}^{\pm l_2} \cos\left(\frac{\pi}{2}u^2\right) du \approx \pm\left(1 + \ensuremath{\frac{1}{-l_1\pi}}\sin\left(\ensuremath{... ...ath{\frac{1}{l_2\pi}}\sin\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right)$$ (64)

$$% \int_{\pm l_1}^{\pm l_2} \sin\left(\frac{\pi}{2}u^2\right) du \approx \pm\left(1 - \ensuremath{\frac{1}{-l_1\pi}}\cos\left(\ensuremath{... ...th{\frac{1}{l_2\pi}}\cos\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right),$$ (65)

which we substitute into equations 56 and 57 to obtain

$$\Re{Y_a(k)} \approx \sqrt{\frac{\pi}{2\alpha}} \left( \cos\left(\frac{\pi^2 k^2}{K^2\... ...c{1}{l_2\pi}}\sin\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right) \right.$$ (66)

$$+ \left. \sin\left(\frac{\pi^2 k^2}{K^2\alpha}\right) \left(1 - \... ...ac{1}{l_2\pi}}\cos\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right)\right)$$

$$x % \Im{Y_a(k)} \approx \sqrt{\frac{\pi}{2\alpha}} \left( \cos\left(\frac{\pi^2 k^2}{K^2\... ...c{1}{l_2\pi}}\cos\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right) \right.$$ (67)

$$- \left. \sin\left(\frac{\pi^2 k^2}{K^2\alpha}\right) \left(1 + \... ...c{1}{l_2\pi}}\sin\left(\ensuremath{\frac{\pi}{2}}(l_2)^2\right) \right)\right).$$

We now apply the trigonometric identities

$$\cos(A)\cos(B) = \ensuremath{\frac{1}{2}}(\cos(A+B) + \cos(A-B))$$ (68)

$$\cos(A)\sin(B) = \ensuremath{\frac{1}{2}}(\sin(B+A) + \sin(B-A))$$ (69)

$$\sin(A)\sin(B) = \ensuremath{\frac{1}{2}}(\cos(A-B) - \cos(A+B))$$ (70)

and, letting $\phi=\frac{\pi^2 k^2}{K^2\alpha}$ to make the notation cleaner, we get:

$$\Re{Y_a(k)} \approx \sqrt{\frac{\pi}{2\alpha}} \left[ \cos(\phi) + \ensuremath{\frac{... ...i\right) + \sin\left(\ensuremath{\frac{\pi}{2}}l_2^2-\phi\right)\right) \right.$$

$$\left. + \sin(\phi) - \ensuremath{\frac{1}{2}}\ensuremath{\frac{1... ...2\right) + \sin\left(\phi-\ensuremath{\frac{\pi}{2}}l_2^2\right)\right) \right]$$ (71)

$$% \Im{Y_a(k)} \approx \sqrt{\frac{\pi}{2\alpha}} \left[ \cos(\phi) - \ensuremath{\frac{... ...2\right) + \cos\left(\phi-\ensuremath{\frac{\pi}{2}}l_2^2\right)\right) \right.$$

$$\left. - \sin(\phi) - \ensuremath{\frac{1}{2}}\ensuremath{\frac{1... ...\right) - \cos\left(\phi+\ensuremath{\frac{\pi}{2}}l_2^2\right)\right) \right].$$ (72)

Regrouping common terms, we see that

$$\Re{Y_a(k)} \approx \sqrt{\frac{\pi}{2\alpha}} \left[ \cos(\phi) + \ensuremath{\frac{... ...h{\frac{1}{l_2\pi}}\sin\left(\phi-\ensuremath{\frac{\pi}{2}}l_2^2\right)\right]$$ (73)

$$% \Im{Y_a(k)} \approx \sqrt{\frac{\pi}{2\alpha}} \left[ \cos(\phi) + \ensuremath{\frac{... ...\frac{1}{l_2\pi}}\cos\left(\phi-\ensuremath{\frac{\pi}{2}}l_2^2\right) \right].$$ (74)

After so much discussion, this is not an entirely satisfying result; though the above equations are highly accurate, they are cumbersome and not readily manipulated into an elegant closed form expression for the phase of $Y_a(k)$. Fortunately, we realize that the approximations can be further simplified if we can eliminate the second and fourth sinusoidal terms in each. We will show that this can be done in either of two ways:

• Assume directly that the limits are well-approximated by infinity (rather than simply being much greater than 1). This is, in actuality, claiming that the ``ringing" seen in Fresnel integrals has died out by the end of the particular integration frame. (Recall figure 3, which shows several such frames.) This allows us to simplify equations 66 and 67 to equal simply 1 (the infinite limits approximation), resulting in the dropping of the aforementioned sinusoidal terms. Making this claim without actually using a longer window, however,is a mathematical disaster. Doing so introduces so much Fresnel approximation error that the model becomes irrelevant. Hence, this approach is relegated to an appendix.
• Lifter out the offending sinusoids. As we will see, this can in fact be done by choosing a different time domain windowing function. The importance of this method is that it allows us to maintain the same error bounds as for the rectangle window with the large limits approximation, while yielding the desired invertible expression.

In the next subsection, we consider the second method. Therein we also show a very elegant fact: two ways of conceptualizing the attenuation of the sinusoids are mathematically equivalent. Those methods are the liftering mentioned in the second bullet above, and consideration of the Hann window as a direct attenuator of Fresnel ringing in the time domain.