clean derivation with summary table

content/posts/2024/generator-functions.md

@@ -47,7 +47,19 @@ to transmit amplitude modulated symbols. As we will see the approach of using
 generator functions will allow us to synthesize high-precision waveforms with
 exact frequency characteristics with just a few predetermined coefficients.
 
-## Initial Conditions
+At first glance, we can reason that many of the desirable properties that one would 
+like to see here are similar to that of window functions (e.g. Hanning or
+Kaiser Windows)[^1]. This is because we are interested in both the time and 
+frequency properties of the generated sequence simultaneously. The key 
+difference here however is that we are constrained to polynomial dynamics.
+As a result the text-book approach for approximating a sum of weighted cosines
+may not be the best approach. Although taking a padé-approximant, using a 
+rational polynomial basis, may be a good choice in some cases. More generally
+nesting or convolving our polynomial basis will result in higher oder 
+polynomal. In order to realize a transcendental response we would need to
+revisit the feedback coefficient for each integrator.
+
+## Determining Initial Conditions
 
 There are a few ways to go about defining a polynomial \\(P(x)\\). Either in terms of
 the roots or in terms of the characteristic equation. Both are useful, 
@@ -104,6 +116,7 @@ def mapping_coefficients(order: int) -> np.array:
         # for each element calculate new coefficient
         # Based on expanding d/dx * P(x) * (x+1)
         coef[elem + 1] = (order - elem - 1) * (base[elem] + base[elem + 1])
+    # m_n will always be n!
     coef[0] = base[0] * order
     return coef
 
@@ -124,11 +137,55 @@ def initial_condition(self, poly_coef: np.array) -> np.array:
 
 ```
 
+It is worthwhile to point out that not all polynomial functions can be realized
+with this method. While not all zeros in \\( P(x) \\) have to be real, we do 
+require the characteristic coefficients \\( a_n \\) and thereby \\( c_n \\) to
+to be real numbers. 
+
 ## Frequency Response
 
-Here we will consider polynomials of even orders with real roots such
+Here we will consider a simplified scenario to exemplify the frequency \
+characteristics for generated polynomials.
+Specifically polynomials of even orders with real roots such
 that we can decompose the polynomial \\(P(x)\\) as a product of several 
 elements in the form of \\( (x+p_1)(x+p_2) \\). We can show that the 
-fourier-transform of of this element is in the form of \\( 16d^2 sinc(d ω)^4 \\)
-where \\( d = (p_1 - p_2) \\).
+fourier-transform of of this element is in the form of \\( sinc(d ω)^2 \\)
+where \\( d = (p_1 - p_2)/2 \\) such that we can derive relations for the 
+polynomial \\( P(x) = d^2-x^2 \\) and scale them accordingly.
 
+$$
+\hat{P}(x) = \int^{d/2}_{d/2} (d^2-x^2) cos(k x) dx 
+\quad = \quad 
+\frac{8 \sin{ (d k) }}{k^3} -  \frac{8 d \cos{ (d k ) }}{k^2}
+$$
+
+We can numerically solve for some of the filter properties of interest and 
+compare to other simple windows. There is little suprise in the table below
+as the roll-off and rejection is closely related to the 3dB bandwidth.
+Here we see that the frequency response of P(x) is somewhere inbetween a 
+rectangular window and that of the raise-cosine or Hann window.
+
+| Property      | 2nd Order Poly len(2d) | Rectangle len(2d)     | Hann len(2d)          |
+|---------------|------------------------|-----------------------|-----------------------|
+| DC Value      | \\(\frac{2d^3}{3}\\)   | \\(2d^2\\)            | \\(2d^2\\)            |
+| 3db Bandwidth | \\(\sim 2.498/d\\)     | \\(\sim 1.895/d\\)    | \\(\sim 3.168/d\\)    |
+| 1st Null      | \\(\sim 4.5/d\\)       | \\(\frac{\pi}{d}\\)   | \\(\frac{2\pi}{d}\\)  |
+| Roll Off      | 40 dB / decade         | 20 dB / decade        | 60 dB / decade        |
+
+For completeness we also include the analytical expression for the Hann window
+frourier transform.
+
+$$
+\hat{H}(x) = \int^{d/2}_{d/2} (1+cos(\frac{2\pi x}{d})) cos(k x) dx 
+\quad = \quad 
+\frac{2 \pi^2 sin(d k)}{k(d^2 k^2-\pi^2)}
+$$
+
+
+
+
+## References:
+
+[^1]: A. Nuttall, ''Some windows with very good sidelobe behavior,'' IEEE 
+Trans. Acoust., Speech, Signal Process. , vol. 29, no. 1, pp. 84-91, February 
+1981 [Online]:  http://dx.doi.org/10.1109/TASSP.1981.1163506.