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| title | date | draft | toc | images | math | tags | ||||
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| Polynomial Generator Functions | 2024-12-28T12:33:01+01:00 | false | false | true |
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Previously I went over some interesting techniques for [synthesizing sinusoids]({{< relref "../2022/synthesizing-sinusoids" >}}). There were some interesting points to take away from that discussion. First the CIC topology essentially calculated the Nᵗʰ moments or derivatives in a signal and essentially reconstructed a corresponding response using a polynomial with those moments or piece-wise derivatives.
This leads to a interesting question when synthesizing a time-limited waveform: What are the properties for using polynomial generator functions for waveform synthesis if we follow a simple cascade of integrators? The general structure for a cascade of integrators is shown below.
.-. .-. .-.
c₃ -->| Σ +-----*----->| Σ +-----*----->| Σ +-----*----> N sequence Waveform
'-' | '-' | '-' |
^ v ^ v ^ v
| .-----. | .-----. | .-----.
| | z⁻¹ |<-- | | z⁻¹ |<-- | | z⁻¹ |<-- Initial Conditions
| '--+--' | '--+--' | '--+--' 3ʳᵈ Order Polynomial
| | | | | | c₂, c₁, c₀
'-----' '-----' '-----'
The concept here is that the registers are initialized with coefficients derived from the desired polynomial response and sequenced for a fixed set of cycles before we trigger a reset. This is some what a simplified scenario but one could imagine the sequence can be used to modulate a RF carrier in order 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.
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, especially when we consider the derivative components at the extents of our synthesized waveform.
P(x) = (x-p_1) (x-p_2) (x-p_3) ... (x-p_n)
P(x) = a_n x^n + ... + a_2 x^2 + a_1 x + a_0
We can back calculate the corresponding initial conditions for our cascade
of integrators by considering the super-position of each component a_n
seperately. Lets denote our initial conditions as \( c_n + ... + c_2 + c_1 + c_0 \)
for our Nᵗʰ order polynomial. As shown in the diagram above the coefficient
\(c_n\) is directly accumulated on the left most integrator. It should be obvious
that the first two coefficients: \(a_1\) & \(a_0\) from our characteristic equation
directly related correspond to \(c_1\) & \(c_0\) respectively. Now we can infer the
mapping by recursion and equating the derivative components. For example \(P(x) = a_2 x^2\): the
contribution from \(a_2\) on the 2nd order derivative is calculated as taking
the 2nd derivative of \(P(x)\) and evaluating its value with x=0 which gives
us \(c_2 = 2 * a_2\). Now equating the 1st derivative from \(a_2\)
similarly gives \(c_1 = a_2\) and finally \(c_0 = 0\). If there were lower
order terms, the contribution from a_1 for example would be calculated
independently and added together.
This was a simpler example but one can reason that if the mapping for a particular \( a_n \) is \( m_n, ... , m_1, m_0 \) such that for all \( n \) the initial conditions are \(c_n = m_n a_n\). Then for some given mapping of a Nᵗʰ order polynomial we can add one more integration stage to the far right integrating the output to realize a N+1ᵗʰ order polynomial. This is equivalent to multiplying the response with \( x + 1 \). Now it should be clear that when we equate the derivative terms the N+1ᵗʰ order terms can be derived from the Nᵗʰ order terms simply by adding the appropriate contributions after the aforementioned multiplication. That is \( k_n = m_n + m_{n-1} \) where \( k_n \) are the mapping terms for the N+1ᵗʰ order polynomial. Interestingly the mapping here generates a set of basis coefficients related to the sterling-numbers of the second kind. More specifically the A019538 sequence. Using python and numpy as np we can write the following recursive function:
def mapping_coefficients(order: int) -> np.array:
"""Determine nth coefficient scaling factor for initial condition."""
assert order >= 0
# Start with coefficient from n-1
if order == 0:
return np.array([1])
elif order == 1:
return np.array([1, 0])
else:
base = mapping_coefficients(order - 1)
coef = np.zeros((order + 1,))
for elem in range(order - 1):
# 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
This function will derive the \( m_n \) mapping values based on our recursive derivation above. In order to then determine the initial conditions we similarly iterate over the characteristic coefficients \( a_n \) and accumulate all contributions to resolve the initial conditions \( c_n \).
def initial_condition(self, poly_coef: np.array) -> np.array:
"""Set register values based on polynomial coefficients."""
init_coef = np.zeros((poly_coef.size,))
for index, elem in enumerate(poly_coef):
init_coef[index:] += elem * mapping_coefficients(poly_coef.size - index - 1)
return init_coef
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 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 \( sinc(d ω)^4 \)
where \( d = (p_1 - p_2)/2 \) such that we can resolve the transform for
\( P(x) = (x+d)(x-d) \)
First consider a simple box function of width \( d/4 \) and its corresponding fourier transform:
S(x) = \begin{cases}
1 &\text{if |x| <= d/4 } \\\\
0 &\text{otherwise }
\end{cases} \quad \xrightarrow{\mathfrak{F}} \quad \frac{d}{2} sinc( \frac{d\omega}{4} )
We can auto-convolve \(S(x)\) twice in order to realize a parabola with roots at +/- d. First formulate the associated triangle function \(T(x)\):
T(x) = S(x) \ast S(x) = \int_{-d/2}^{d/2} S(\tau) S(x-\tau) d\tau \quad = \begin{cases}
0 &\text{if x < -d/2 } \\\\
(x+d/2) &\text{if x < 0 } \\\\
(d/2-x) &\text{if x < d/2 } \\\\
0 &\text{otherwise}
\end{cases}
The second auto-convolution of \(T(x)\) gives back the parabola \(P(x)\):
P(x) = T(x) \ast T(x) = \int_{-d}^{d} T(\tau) T(x-\tau) d\tau = \begin{cases}
d^2/4 - x^2/4 &\text{if |x| <= d } \\\\
0 &\text{otherwise }
\end{cases}
P(x) \quad \xrightarrow{\mathfrak{F}} \quad \frac{d^4}{16} sinc(\frac{d\omega}{4} )^4
References:
-
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. ↩︎