Synthesis of a Linear Filter for Time Functions
Abstract Generated abstract
The paper addresses pulse overlap in high-density digital magnetic recording, where the reproduced readout pulse can exceed the bit period and thereby distort amplitude and timing. It proposes synthesizing a linear time-function filter that compresses readout pulses by using spectral regions for ensembles of possible input and output pulse shapes rather than requiring exact Gaussian pulse models. The transfer function is approximated through a Padé representation of an exponential form, realized as a passive lattice or active equivalent filter, and tested on a magnetic storage device. Laboratory results show approximately twofold pulse-width compression and suggest that, under otherwise equal conditions, recording density can be increased by about 1.8 to 2.0 times.
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SYNTHESIS OF A LINEAR FILTER FOR TIME FUNCTIONS
Yu. P. Zabashta
(Presented by the Department of Computer Engineering)
The development of digital magnetic recording is proceeding along the path of achieving a high density of information recording, both longitudinal and transverse, in order to obtain a more perfect combination of large capacity, high data output rate, and short information sampling time.
The capabilities of existing devices are inevitably limited by the parameters of the head–medium systems. With the development of technology, heads have appeared with a narrow gap for contact recording, or recording close to contact, on a thin, high-quality magnetic medium. Further increase in the density of information recording by improving these components is associated with major technological difficulties.
Attention should be given to the prospect of achieving high information-recording densities by other, simpler means. The duration of the reproduced pulse is 4–6 times greater than the width of the recorded magnetic transition and, at high recording density, becomes greater than the period of one bit [1].
Fig. 1. Process of signal overlap
The resulting overlap of these pulses leads to the generation of a complex readout signal, as a result of which the amplitudes and positions of the peaks of the pulses change significantly because of the influence of neighboring pulses (Fig. 1). The process of pulse overlap, within certain limits, may be considered linear [1]. This makes it possible to apply the principle of superposition and to base further exposition on the analysis of a single readout pulse.
Appropriate processing of the signal by means of linear filtering effectively narrows the readout pulses, reducing interference and thereby increasing the possible bit density of recording.
This processing process, on the other hand, may be represented as an expansion of the effective resolving power of the system by compensating for the effect of high frequencies by the head–medium system.
Thus, on the basis of analysis of the real readout signal, the problem is posed of synthesizing a linear filter that compresses readout pulses. Since this system will process time pulses, this filter may be classified as a filter of time functions.
Linear filters of time functions differ conceptually from the more widespread filters of frequency functions in that they are intended to change the shape of a pulse and, consequently, are defined from the standpoint of a transient mode of operation.
Analysis of the Signal and the Transfer Function
The classical method of synthesizing time functions, as a rule, includes approximation of the input \(r(t)\) and output (corrected) pulse \(q(t)\) by means of time functions that are mathematically easy to handle.
From the Fourier or Laplace transforms of these expressions one can obtain the transfer function of the filter \(K(p)\)
\[ K(p)=\frac{G(p)}{R(p)} , \tag{1} \]
where
\[ G(p)=\int_{-\infty}^{\infty} q(t)e^{-pt}\,dt, \]
\[ R(p)=\int_{-\infty}^{\infty} r(t)e^{-pt}\,dt. \]
In order to make \(K(p)\) physically realizable, a further approximation of \(K(p)\) must be carried out; this will then constitute the real transfer function of the filter being synthesized. The entire process described is tedious and complicated.
Usually [2], in wishing to simplify this procedure, the real readout signal \(r(t)\) and the corrected signal \(q(t)\) are represented by pulses of “bell-shaped” form:
\[ r(t)=e^{-\xi^2 t^2}, \]
\[ q(t)=e^{-\mu^2 t^2}. \]
The compression coefficient at a prescribed level is determined by
\[ \kappa_c=\frac{\xi}{\mu}, \]
and the transfer function \(K(p)\) is
\[ K(p)=\frac{G(p)}{R(p)}, \]
where
\[ G(p)=L[q(t)]=\int_{-\infty}^{\infty} e^{-\mu^2 t^2}e^{-pt}\,dt =\frac{\sqrt{\pi}}{\mu}e^{\frac{p^2}{4\mu^2}}, \]
\[ R(p)=L[r(t)]=\int_{-\infty}^{\infty} e^{-\xi^2 t^2}e^{-pt}\,dt =\frac{\sqrt{\pi}}{\xi}e^{\frac{p^2}{4\xi^2}} \]
and
\[ K(p)=\frac{\xi}{\mu}e^{-\frac{p^2}{4}\left(\frac{1}{\mu^2}-\frac{1}{\xi^2}\right)} . \]
In the case of an actual readout signal, whose shape may differ significantly from a Gaussian one, the classical approach causes a number of difficulties.
Owing to the convenient properties of the readout pulse, it is possible to use a synthesis procedure that deviates from the one described. First, since the readout pulse is, as a rule, symmetric with respect to a certain instant of time, its phase spectrum will be equal to zero. Moreover, the zero-phase regime is also advantageous in the spectrum of the corrected pulse, since it helps to avoid undesirable overshoots. This simplifies the transfer function \(K(p)\), as containing only amplitude functions.
Second, an exact determination of the shape of the input and output pulses of the filter creates unnecessary restrictions and additional difficulties in determining \(K(p)\); moreover, the shape of the readout pulses is not constant and is a function of the head parameters, the magnetic properties of the medium at the point of recording, its linear velocity, etc. On this basis, the following approach is proposed for determining the transfer function of the filter.
Fig. 2. Plot of the input and output ensemble of signals
Depending on the shape, the input signals \(r(t)\) form an ensemble \(W_r\), lying in a certain region \(F_r\) (Fig. 2). From the standpoint of spectral analysis, this is equivalent to specifying the region of variation of the spectral density of the input signals \(S_r(j\omega)\) (Fig. 3).
It is clear that the limits of compression of the input signal are restricted both by its width and by the noise level of the carrier \(\gamma\) at the filter input. Hence it makes sense to consider the spectrum of the input signal, and consequently also the transfer function \(K(p)\), up to the frequency \(\omega_d\) (Fig. 3). Starting from \(\omega_d\), the signal compression coefficient \(K_c\), the desired shape of the output signal \(q(t)\), and the region of its variations \(F_q\) are specified (Fig. 2). It should be noted at once that no rigid requirements are imposed on the shape of the output signal. The main condition is the absence, at the filter output, of parasitic overshoots \(\beta\) (Fig. 2), whose magnitude depends both on the shape of the input signal \(r(t)\) and on the parameters of the synthesized filter. In practice, overshoots \(\beta\) of the order of 10–15% of the signal amplitude are allowed in systems.
Fig. 3. Spectral density of the input and output signals
Thus the region of variation of the spectral density of the output signal \(S_q(j\omega)\) is determined (Fig. 3). Such an approach makes it possible to estimate the degree of influence of a particular frequency band of the spectrum on the signal shape.
The region of existence of the transfer function \(K(j\omega)\) (Fig. 4) is determined by the relation
\[ K(j\omega)_{\min}^{\max}= \frac{S_{q\,\min}^{\max}(j\omega)} {S_{r\,\min}^{\max}(j\omega)}, \tag{2} \]
where \(K(j\omega)_{\min}^{\max}\) are the limiting values of the transfer function.
The fact that a certain region of existence of the transfer function has been specified substantially facilitates the problem of approximating it and makes it possible to choose an approximation function convenient from the standpoint of mathematical processing.
Fig. 4. Transfer function of the filter
3. Synthesis of the Filter
To approximate \(K(p)\), a function of the form \(e^{z}\) was chosen,
\[ K(p)=M_{1}e^{z}, \tag{3} \]
which is transcendental and can be represented by a ratio of polynomials in \(p\).
Nevertheless, such a representation does not yield Hurwitz polynomials in the denominator, since equation (3) will have quadratic symmetry. Therefore, instead of equation (3), the square of the absolute value of \(K(p)\) is considered, equal to
\[ |K(p)|^{2}=Me^{2z}=K(p)\cdot K(-p). \tag{4} \]
The approximation of \(|K(p)|^{2}\) by a ratio of polynomials in \(p\) was carried out with the aid of Padé’s table [2], since it provides very rapid convergence for functions of the form \(e^{z}\). We take
\[ z=\frac{1}{2}\cdot \psi^{2}p^{2}; \tag{5} \]
we obtain
\[ |K(p)|^{2}=Me^{\psi^{2}p^{2}}, \tag{6} \]
which is equivalent to
\[ \begin{aligned} |K(p)|^{2}&=K(p)\cdot K(-p)=\\ &=M\, \frac{\psi^{8}p^{8}-10\psi^{6}p^{6}+45\psi^{4}p^{4}+105\psi^{2}p^{2}+105} {\psi^{8}p^{8}-10\psi^{6}p^{6}+45\psi^{4}p^{4}+105\psi^{2}p^{2}+105}. \end{aligned} \tag{7} \]
For the case under consideration, on the basis of the region of existence of \(K(p)\), \(\psi=0.34\) was adopted. Solving equation (7), we obtain the values of the zeros
\[ \begin{aligned} &\pm 4.874 \pm j\,2.358,\\ &\pm 5.060 \pm j\,0.741 . \end{aligned} \tag{8} \]
and poles
\[ \pm 2.358 \pm j\,4.874,\qquad \pm 0.741 \pm j\,5.060. \]
If the poles and zeros are considered in the complex plane, then it is necessary to take the poles in the left half-plane and the zeros in the right half-plane, since this guarantees a more linear phase characteristic than in other variants [3]. Then the expression for the transfer function takes the form
\[ K(p)=M\, \frac{p^{4}-19.868p^{3}+154.119p^{2}-551.617p+776.700} {p^{4}+6.198p^{3}+62.162p^{2}+166.783p+766.700}. \tag{10} \]
For \(p=j\omega\),
\[ K(j\omega)=M\left[ \frac{ \left(\omega^{4}-154.119\omega^{2}+766.700\right)^{2} +\left(19.868\omega^{3}-551.617\omega\right)^{2} }{ \left(\omega^{4}-62.162\omega^{2}+766.700\right)^{2} +\left(6.198\omega^{3}-166.783\omega\right)^{2} } \right]^{1/2}. \tag{11} \]
There are many ways of realizing this transfer function. We shall illustrate only one of the possible solutions, namely: a symmetrical lattice with constant resistance. The following requirements are imposed on this structure:
- \(K(p)\) must not have poles in the right half-plane or on the imaginary axis.
- \(K(j\omega)\leq 1\) for \(0\leq\omega\leq\infty\).
The transfer function under study, \(K(p)\), given in expression (10), satisfies the first requirement. To satisfy the second, it is necessary to multiply \(K(p)\) by
\[ M\leq \frac{1}{|K(p)|_{\max}}, \tag{12} \]
Fig. 5. Filter circuit
Fig. 6. Active-filter circuit
where \(|K(p)|_{\max}\) is the maximum value of the modulus \(K(p)\); in the case under consideration \(|K(p)|_{\max}=14.30\).
Taking \(M=0.07\), we obtain
\[ K(p)= \frac{0.07p^{4}-1.39p^{3}+10.78p^{2}-38.55p+53.6} {p^{4}+6.198p^{3}+62.16p^{2}+166.80p+766}. \tag{13} \]
In terms of the transfer function \(K(p)\), the branch resistances of the symmetrical lattice are determined as
\[ Z_a(p)=R_0\,\frac{1-K(p)}{1+K(p)}, \tag{14} \]
\[ Z_b(p)=R_0\,\frac{1+K(p)}{1-K(p)}. \]
The filter is loaded by a linear amplifier with an input impedance taken to be equal to \(2\ \text{k}\Omega\). The filter circuit is shown in Fig. 5.
4. Test Results
Practically good results (compression coefficient \(K_c=1.8 \div 2\)) are given by an active filter [4] using one arm \(z_a\) (Fig. 6).
Its transfer function
\[ \frac{e_2}{e_1} = \frac{ \dfrac{1}{R_1+Z_a}-\dfrac{1}{R_0} }{ \dfrac{1}{R_1+Z_a}+\dfrac{1}{R_0}+\dfrac{1}{R_L} } \tag{15} \]
for \(R_L \gg R_0 \gg R_1\) takes the form
\[ \frac{e_2}{e_1}=\frac{R_0-Z_a}{R_0+Z_a}=K(p). \tag{16} \]
The filter was tested on a storage device with floating magnetic heads. Fig. 7 shows a single readout pulse before \((a)\) and after \((b)\) passing through the filter.
Fig. 7. Readout signal 10001000 before \((a)\) and after \((b)\) the filter
The duration of a single input pulse at the 0.5 level is \(5\ \mu\text{s}\), and that of the output pulse is \(2.5\ \mu\text{s}\). The filter compression coefficient is \(K_c=2\). Fig. 8 illustrates the action of the filter on the cyclic code 100111001.
Laboratory tests showed that an active frequency-time filter can be used practically for pulse compression—
reading in information-recording systems on a magnetic medium. In this case, the information-recording density, all other conditions being equal, can be increased by a factor of 1.8–2.0.
Fig. 8. Readout signal 10001100 before (a) and after (b) the filter
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