Numerical Solution of the Filter Analysis Problem
Abstract Generated abstract
This paper addresses numerical evaluation of Fourier transform integrals arising in the analysis of low pass radio engineering filters, especially the response of a Butterworth filter to a rectangular pulse of finite duration. It proposes replacing more cumbersome mechanical quadrature by graphically forming the product of the input spectrum and filter transfer modulus, approximating selected frequency intervals with a Lagrange polynomial, and reducing the resulting inverse transform to tabulated integrals. For a Butterworth low pass filter, the method separates the ideal filter response, expressed through sine integrals, from a finite duration correction term, and discusses how filter order and pulse duration affect high frequency contributions and the impulse response. The example indicates that for order five with the cutoff constant comparable to the pulse duration, the high frequency energy contribution is small.
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NUMERICAL SOLUTION OF THE FILTER ANALYSIS PROBLEM
A. N. LAFEROV, I. E. NATS
(Presented by the scientific seminar of the Computing Laboratory of TPI)
In solving many engineering problems one has to use the Fourier transform. In particular, determining the response \(r(t)\) of a low-pass filter (LPF) when a signal whose spectrum is known is applied to the input of the device. The solution of the problem is given by the expression
\[ r(t)=\int_{-\infty}^{\infty} S(\omega)\cdot K(\omega)e^{i\omega t}\,d\omega, \tag{1} \]
where
\(S(\omega)\) — the spectrum of the input signal,
\(K(\omega)=|F(j\omega)|\) — the modulus of the filter transfer function.
Finding such an integral reduces to computing sine or cosine transforms, which for even and odd functions are written as:
\[ r(t)=\int_{0}^{\infty}\psi(\omega) \begin{matrix} \cos\\ \sin \end{matrix} \omega t\,d\omega. \tag{2} \]
In the problem under consideration
\[ S(\omega)=A\delta\,\frac{\sin \frac{\delta}{2}\omega}{\frac{\delta}{2}} \tag{3} \]
— the spectrum of a rectangular pulse of duration \(\delta\);
\[ K(\omega)=\frac{1}{\sqrt{1+\left(\frac{\omega}{\omega_c}\right)^{2m}}} \tag{4} \]
— the spectral characteristic of a Butterworth LPF of order \(m\) with cutoff frequency \(\omega_c\), therefore
\[ \Psi(\omega)=S(\omega)\cdot K(\omega)= \frac{A\delta\sin\frac{\delta}{2}\omega} {\frac{\delta}{2}\omega\sqrt{1+\left(\frac{\omega}{\omega_c}\right)^{2m}}}. \tag{5} \]
There exist special tables for computing the Fourier transform, for cases in which the special function \(\Psi'(\omega)\) is expressed in terms of elementary or special functions.
To compute integral (2), one may construct a mechanical quadrature:
\[ \gamma(t)=\int_{0}^{\infty}\Psi'(\omega)\begin{matrix}\cos\\ \sin\end{matrix}\omega\,d\omega \approx \sum_{\kappa=1}^{n} A_{\kappa}\cdot \gamma(x_{\kappa}). \tag{6} \]
The coefficients \(A_{\kappa}\) and \(x_{\kappa}\) for the nodes can be found in [1]; in this case it is necessary to represent the spectral function (5) in the form
\[ \Psi'(\omega)=\frac{\Phi(x)}{(1+x)^{s+1}}, \tag{7} \]
where \(i=1,2,3\ldots\) and \(s\) is the order of growth.
The representation (7), in turn, entails additional computations. The method of solution described below makes it possible to reduce integral (1) to a tabular one. For this purpose it is proposed, in the integrand expression (1), to multiply \(S(\omega)\) and \(K(\omega)\) geometrically, and then to approximate the separate portions of the resulting graph of the function \(\Psi'(\omega)\).
Fig. 1. Spectral characteristics of an active low-frequency filter in the Butterworth approximation for \(m=3\) and \(m=5\)
In Fig. 1 are shown the graphs of the functions \(S(\omega)\), \(K(\omega)\) for \(m=5\), \(m=3\), at the cutoff frequency \(\omega_c=1/\delta\). The obtained functions \(\Psi'_5(\omega)\) and \(\Psi'_3(\omega)\) are normalized to 1. It is evident from the graph that on the interval \(0 \div 1/\delta\), \(\Psi'_5(\omega)\) coincides with high accuracy with the function \(S(\omega)\), and on the interval \(1/\delta \div 1.6/\delta\) we perform an approximation by the Lagrange polynomial [2] at 4 points: \(1/\delta\), \(1.2/\delta\), \(1.4/\delta\), \(1.6/\delta\).
We neglect the values of the spectral function in the interval \(1.6/\delta \div \infty\) because of their smallness. As a result we have
\[ \Psi_{z}(\omega)= \left. \frac{\sin \omega \frac{\delta}{2}}{\omega \frac{\delta}{2}} \right|_{0}^{1/\delta} - \left. \frac{1}{\delta} \left[ 1.72\left(\frac{\omega}{2\pi}\right)^{3} - 7.09\left(\frac{\omega}{2\pi}\right)^{2} + 9.57\frac{\omega}{2\pi} - 4.2 \right] \right|_{1/\delta}^{1.6/\delta}. \tag{8} \]
The cosine transform of the first term of (8) gives
\[ \zeta(t)= \int_{0}^{1/\delta} \frac{\sin \omega \frac{\delta}{2}\cos \omega t} {\omega \frac{\delta}{2}} \,d\omega = \frac{1}{\delta} \left[ Si(t+0.5)-Si(t-0.5) \right]. \tag{9} \]
The complete expression for the LFF response to a pulse of finite duration \(\delta\) is written as follows:
\[ \begin{aligned} r(t)&=\zeta(t)+h(t)=\frac{1}{\delta}\Biggl\{ \left[Si(t+0.5)-Si(t-0.5)\right] \\ &\quad - \int_{1/\delta}^{1.6/\delta} \left[ 1.72\left(\frac{\omega}{2\pi}\right)^{3} - 7.09\left(\frac{\omega}{2\pi}\right)^{2} + 9.57\left(\frac{\omega}{2\pi}\right) - 4.2 \right]\cos \omega t\,d\omega \Biggr\}. \tag{10} \end{aligned} \]
In this expression all integrals are tabulated. The first term in (10) is the response of the system to a pulse of duration \(\delta\) of an ideal filter, for which
\[ K(\omega)= \begin{cases} 1, & \text{for } \omega \leqslant \omega_c,\\ 0, & \text{for } \omega > \omega_c, \end{cases} \]
where, as follows from (4), \(\zeta(t)\) corresponds to the filter response more accurately the larger \(m\) is. The second term \(h(t)\) introduces a correction obtained due to finite \(\delta\), and with increasing \(\delta\), \(|h(t)|\to 0\). Under the condition \(\delta \ll \tau_c\) (\(\tau_c\) is the filter constant), \(S(\omega)\equiv \text{const}\); the parameter determining the response at the filter output is the order \(m\) (curves \(K_5(\omega)\), \(K_3(\omega)\), Fig. 1). Calculations show that already for \(m=5\) (\(\tau_c=\delta\)) the energy contribution of the high frequencies of the spectral function \(\Psi_z(\omega)\) is no more than \(3.6\%\). For \(\Psi_z(\omega)\) this contribution is much larger.
Fig. 2. LFF responses to a pulse of finite duration for \(m=5\) (upper) and \(m=3\) (lower)
Taking into account that the high-frequency components of the spectrum affect mainly the value of the function in the region \(t=0\), one may assume that \(i'_3(t) > i'_5(t)\). This proposition is illustrated by the oscillograms in Fig. 2: for the upper oscillogram, \(m=5\); for the lower, \(m=3\), and by Fig. 1 \(\bigl(\Psi'_5(\omega)\) in the interval \(1/\delta \div 2/\delta\bigr)\).
Conclusions
The article presents a method for the approximate computation of the inverse Fourier transform for the analysis of radio-engineering devices and, using the example of an active Butterworth low-frequency filter, investigates the influence of the device parameters on the impulse function.
The method presented remains valid for the Fourier transform as well.
REFERENCES
- L. Kruglikova. Tables for the Numerical Fourier Transform. “Nauka i tekhnika,” 1964.
- A. Ango. Mathematics for Electrical and Radio Engineers. “Nauka,” 1964.
- A. Laferov. Diploma thesis. TPI, 1966.