On Estimating the Error in Determining the Impulse Function in Experimental Methods

I. E. Naats

(Presented by the scientific seminar of the computing laboratory of TPI)

It is known that by the impulse function of an object (system, chain, etc.) one understands the response of this object to a disturbance in the form of an impulse of infinitely large amplitude and infinitely small duration (the energy of such an impulse is equal to unity). Mathematically, such a disturbance is described by the \(\delta\)-function. In practice, however, it cannot be realized in the experimental determination of the impulse function. For these purposes one has to resort to impulses of finite amplitude and finite duration \(\delta\); therefore the output signal will be described by a time function \(g(t)\), different from the impulse function \(\gamma(t)\).

It is usually assumed that if \(\delta \ll \tau\), where \(\tau\) is the constant of the object, then the output signal \(g(t)\) differs little from the impulse function \(\gamma(t)\), and the change in shape may be neglected. However, in a number of cases—in particular, in the experimental method of determining the response of an object to an external disturbance, for which \(\delta\) is comparable with \(\tau\), and especially in measuring systems using pulse-code modulation—allowance for this fact becomes necessary.

In the present work the change in the shape of the output signal of an object is investigated in comparison with the impulse characteristic when the duration of the input signal \(\delta\) is comparable with the object constant \(\tau\).

Below we shall assume that the impulse function of the object \(\gamma(t)\) is known, and it is required to determine the response of this object to a rectangular pulse of duration \(\delta\) and amplitude \(1/\delta\), i.e., a pulse of unit energy. Analytically, such a signal is written in the following form:

\[ x(t)=\frac{1}{\delta}\,[\sigma(t)-\sigma(t-\delta)], \tag{1} \]

where

\[ \sigma(t)= \begin{cases} 1, & t \ge 0,\\ 0, & t < 0 \end{cases} \]

is the unit step function.

The solution of the stated problem, as is known, is given by Duhamel’s integral:

\[ g(t)=\int_{0}^{t} x(t-\lambda)\gamma(\lambda)\,d\lambda . \tag{2} \]

Taking \(\gamma(t)\) to be an even function and taking (1), (2) into account, we write

\[ g(t)=\frac{1}{\delta}\int_{-\delta/2}^{\delta/2}\gamma(t+\xi)\,d\xi . \tag{3} \]

Relation (3) can be interpreted as follows: the function \(g(t)\) is obtained from \(\gamma(t)\) by local averaging of the latter about the point \(t\) over the interval \([-\delta/2,\delta/2]\). In other words, the function \(g(t)\) at the point \(t\) is equal to the mean value of the function \(\gamma(t)\) on the interval \([-\delta/2,\delta/2]\) centered at the point \(t\). Such averaging gives the function \(g(t)\) a smoother character of variation, and this “smoothing of irregularities” is manifested the more strongly, the larger the magnitude of the interval \(\delta\) over which the averaging extends [1].

Having clarified the nature of the relationship between the functions \(g(t)\) and \(\gamma(t)\), it is not difficult to determine the magnitude of the deviation of one function from the other. To this end, represent the function \(\gamma(t)\) by a Taylor series in a neighborhood of the point \(t\):

\[ \gamma(t+\xi)=\gamma(t)+\xi\gamma'(t)+\frac{\xi^2}{2}\gamma''(t)+\cdots . \tag{4} \]

Substituting (4) into (3) and treating \(\xi\) as the variable, we integrate over the interval \([-\delta/2,\delta/2]\). We obtain

\[ g(t)=\gamma(t)+\frac{\delta^2}{24}\gamma''(t). \tag{5} \]

Hence we determine the deviation of \(g(t)\) from \(\gamma(t)\):

\[ g(t)-\gamma(t)=\Delta(t)=\frac{\delta^2}{24}\gamma''(t). \tag{6} \]

From the obtained relation it is clear that as \(\delta\to 0\), \(g(t)\to\gamma(t)\), i.e., when the duration of the input signal decreases, the response of the object tends to the impulse response at each point \(t\). With the aid of relation (6) one can estimate the maximum deviation of the function \(g(t)\) from \(\gamma(t)\) for a given \(\delta\). Denoting by

\[ M_2=\max|\gamma''(t)|, \]

we have

\[ \Delta=\frac{\delta^2}{24}M_2 . \tag{7} \]

As a concrete example, consider one of the most widespread elements of information systems using the principle of pulse-code modulation, a low-pass filter [2]. The impulse function of the filter is

\[ \gamma(t)=\frac{\sin \frac{\pi}{\tau}t}{\frac{\pi}{\tau}\cdot t}. \tag{8} \]

Substituting (8) into (3), we find

\[ g(t)=\frac{1}{\pi}\frac{\tau}{\delta}\left[ Si\,\frac{\pi}{\tau}\left(t+\frac{\delta}{2}\right) - Si\,\frac{\pi}{\tau}\left(t-\frac{\delta}{2}\right) \right]. \tag{9} \]

For small values of \(\delta\), function (9) differs only slightly from function (8); however, as \(\delta\) increases, this difference becomes

significant. In particular, this is accompanied by a certain decrease in the amplitude of the central maximum and by its broadening around the point \(t = 0\), as well as by a sharp decrease in the amplitudes of the secondary maxima of function (9) and their rapid attenuation. Taking this fact into account is very important in designing systems of discrete sampling [3].

Formula (9) is valid for any values of \(\delta\) and \(\tau\). In particular, for \(\delta = \tau\),

\[ g(t)=\frac{1}{\pi}\left[ Si\,\pi\left(\frac{t}{\tau}+\frac{1}{2}\right)-Si\,\pi\left(\frac{t}{\tau}-\frac{1}{2}\right)\right]. \tag{10} \]

In measuring systems it is often important to know the change of the signal \(g(t)\) as a function of \(\delta\) in the region of the central maximum (at the sampling instant). For function (9)

\[ r''(0)=-\frac{1}{3}\frac{\tau^2}{\tau^2} \tag{11} \]

and, substituting into (6), we find

\[ \Delta(0)=-\frac{\pi^2}{72}\frac{\delta^2}{\tau^2}, \tag{12} \]

from which it is seen that at the point \(t = 0\) a decrease in amplitude occurs depending on the ratio \(\delta/\tau\). For function (10) this decrease is equal to \(\pi^2/72 = 0.137\). The deviation for all other points of interest is determined in a completely analogous manner.

Conclusions

1. The relationship between the impulse response of an object and its response to a rectangular signal of arbitrary duration has been considered.

2. A method is given for calculating the deviation of the indicated quantities (item 1) from one another for arbitrary ratios of the parameters of the object and the input disturbance. Application of the method is shown using the example of the analysis of a low-pass filter.

REFERENCES

1. K. Lanczos. Practical Methods of Applied Analysis, p. 237, Fizmatgiz, 1961.

2. K. Shannon. Works on Information Theory and Cybernetics, p. 415, IL Publishing House, 1963.

3. S. Mason, H. Zimmermann. Electronic Circuits, Signals, and Systems, p. 537, IL Publishing House, 1962.