ABSTRACT
Linear lumped reactance parameter networks loaded by resistors can be used to form pulses of different shapes [1,2]. The most known applications [2] require the networks that form pulses of the so-called quasi-rectangular shape. The network excitation is usually a step voltage, and it is required that the network output response (in this case it is the step response) is a pulse of finite duration. In a realizable network, the fronts of this pulse should have ‘‘rounded’’ corners, and the slopes of the fronts should also be finite. Below this is done using a semi-period of sin2 t function. These pulse-forming networks will be considered first, and it will be shown that the required result
may be obtained using different input excitations and using, of course, different networks. The pulse of the same shape can be obtained when the excitation is a step, an impulse or even a sinusoidal function. The last result looks surprising for linear networks, yet the reader should understand that the output pulse is shaped during the transient period, when the sinusoidal voltage is turning on. When the transient is over, the steady-state response of the network is zero. This reconsideration of the old problem establishes a certain difference between synthesis in time
domain and frequency domain. The output signal time-domain approximation, in case of pulse-forming
networks, should also result in a realizable transfer function. But this synthesis results in many solutions not only because of many possible realizations of the same transfer function. In addition, one may change input excitation (even though in practice the choice of input excitations is limited), then find modified transfer function, and then synthesize new networks. This modification of input signal is not used in the frequency-domain synthesis where it is assumed that the input excitation is the sinusoidal signal. The method of approximation proposed here gives new and useful results even for the case of the step
voltage input excitation and quasi-rectangular shape of the output response. It is shown that if the output pulse is delayed with respect to the input step, then it is possible to shape the output pulse so that its amplitude will be higher than the amplitude of the applied input step voltage. The increase of amplitude is usually connected with the idea of using a transformer, yet the pulse-shaping circuit may provide transformerless change of amplitude. The approximation developed here for pulse forming may also be applied to the synthesis of wideband
amplifier transfer functions. Unfortunately, an initial attempt of using sin2 t for this purpose [3] did not bring any general results, and the required transfer functions were found numerically [4]. But even this attempt was forgotten, and the transfer functions of the wideband amplifiers were found indirectly. They were obtained starting from the frequency domain and investigating the time-domain response of the filter transfer functions. It was found that the step responses of the Bessel filters do not have overshoot. It happened that exactly this transient response is required in realization of the wideband amplifiers. Yet, as shown here, using sin2 t in the approximation of the impulse response allows one to find the realizable transfer functions directly, on the basis of requirements formulated in the time domain to the step response. Finally, the considered approximation method allows one to find realizable transfer functions for the
networks with the step or impulse response representing the sinusoidal pulse of finite duration, and with the pulse envelope that can be of an arbitrary shape. As an example, we consider the case when this envelope is also of sinusoidal shape. The pulses of these shapes find applications in ultra-wideband transmission networks and wavelet signal analysis. These pulse-shaping networks are usually realized by active networks. Yet, the fact that at least one physically realizable network is available is important for using of algorithms, which are able to improve an approximation and find a transfer function that is better suitable for the chosen realization method. Many of the results given here can be found in [5]. Considering that this source of information is not
easily accessible, a sincere effort is done to summarize all available material.
