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1
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A
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;

(
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for \frac{1 - β}{2 T} < ❘ f ❘ \leq \frac{1 + β}{2 T}

\begin{matrix} H_{n} (f) = 0; for ❘ f ❘ > \frac{1 + β}{2 T} & (3) \end{matrix}

In equation 2, the generalization of the raised cosine filter that has a calculated frequency response value for frequencies that fall within a band that is within a range of

$\frac{1 - β}{2 T} < ❘ f ❘ \leq \frac{1 + β}{2 T} .$

In equation 3, when the absolute value of the frequency is greater than

$\frac{1 + β}{2 T},$

the frequency response is zero.

It should be understood that the frequency behavior H_n(f) is even-symmetric around f=0 and odd-symmetric in the vicinity of the frequency being equal to plus or minus one divided by two times the roll-off frequency. Additionally, smoothness in the frequency space translates into better decay behavior in time by applying the inverse Fourier transform. In some implementations, the raised-cosine Nyquist equation may be generalized by factorizations of the frequency functions.

Furthermore, the generalization of the raised cosine filter of equation 2 has a closed-form expression for the time-domain response as shown in the following equation 4, which is the inverse Fourier transform of equation 1.

$\begin{matrix} h_{n} (t) = \frac{\sin (π t (1 - β))}{π t} + \frac{\cos (π t) \sin (πβ t)}{π t} + \sum_{k = 0}^{k = n} A_{2 k + 1} \frac{4 β^{2} t \sin (π t) \cos (π β t)}{{(2 k + 1)}^{2} π - 4 π β^{2} t^{2}} & (4) \end{matrix}$

The time-domain filter may be rewritten into matrix form as shown in equation 5 below, where the matrix determines the higher order number of continuous derivatives.

$\begin{matrix} [\begin{matrix} 1 & 1 & \dots & 1 \\ 1^{2} & 3^{2} & ⋮ & {(2 n + 1)}^{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 1^{2 n} & 3^{2 n} & \dots & {(2 n + 1)}^{2 n} \end{matrix}] [\begin{matrix} A_{1} \\ A_{3} \\ ⋮ \\ A_{2 n + 1} \end{matrix}] = [\begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}] & [5] \end{matrix}$

The above time-domain equations are valid except at points where the denominator terms are greater than zero, in which case the continuation of the function may be applied at that point. The matrix in equation 5 is a square matrix and both the determinant and the inverse have closed form solutions, which means that the solution vector also has a closed-form solution. The closed-form solutions are piecewise continuous in the frequency domain. The matrix may be used to determine higher order numbers of continuous derivatives. Additionally, it is possible to linearly combine such filters and retain the Nyquist properties according to equation 6 below.

$\begin{matrix} h (t) = \sum_{k = 0}^{n} a_{k} h_{k} (t); subject to \sum a_{k} = 1 & (6) \end{matrix}$

The Nyquist filter can be down-sampled such that the down-sampling factor is equal to the number of filters (i.e., the number of antennas within the phased-array antenna 206) so that the fractional delay filter 120 is critically sampled (once per sample period).

In general, radio frequency signals are received asynchronously, so it is assumed that the communications links are not synchronized. Accordingly, the filter equations may be written to include a prescribed time error as shown in equations 7 and 8 below.

$\begin{matrix} h_{n} (t + τ) where t = i ϵ Z, & (7) \end{matrix}$

- where the time t is an integer value i within the sample space Z. The variable t represents the prescribed time error or fractional offset. In equation 8 below, the closed-form expression for the time-domain response of the generalization of the raised cosine filter in equation 4 is rewritten to include the prescribed time error.

h
n

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i
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=

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1
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