The characteristics, of a special function generator which generates output by convoluting input signals with the internally stored impulse response are calculated and the method of its basic design is given. The function generator of this type may be widely used in future, especially as a speech synthesizer and a model of industrial system for the adaptive control, for it has many advantages:(1) Impulse responses of various types can be easily realized only rewriting the memory, (2) Impulse respone can be changed almost instantaneously, and (3) By changing the time base, the frequency characteristics can be easily varied proportionally. On the other hand the following conditions cannot be avoided practically: (1) The range of integration in the following equation is limited by the finite time duration L, g(t)=∫^^∞__0 f(t-τ)K(τ)dτ (2-1) where g(t), f(t), K(τ) denote the output, input and impulse response respecitvely. (2) The output g(t) is obtained only as a discrete time series. The second condition gives us no problem, if the Lagrange interpolation method is utilized. So the first condition is most essential. The anthors calculated the effect of L and formulas for the integration error, the ratio of peak to dip in frequency characteristics, and of the growth factor at the resonant frequency (see Eqs. (2-8)(2-9), Fig. 2 and Table 1). These equations can be derived by the vector conception shown in Fig. 1, which is a graphical representation of Eqs. (2-7). Furthermore, the authors culcuatred the effects of sampling on the error, and obtained Eq. (2-11) for the ratio of output when f(t) and K(τ) are both discrete to that when f(t), K(τ) are both continuous, Eq. (2-16) for the ratio of output when f(t) is continuous and K(τ) is discrete to that when f(t) and K(τ) are both continuous, and others. When the damping constant α is sufficiently small compared to the angular frequencies ω and β, we have Eq. (2-13) instead of Eq. (2-11) and we may use Fig. 3 and determine the least sampling rate using Eq. (2-15) or (2-17). Therefore the total number (n) of sampling can be determined as the product of L and sampling rate. Concerning the time rate (Δ) of g(t), the authors gave the following formula, <nt(C)>/N≦Δ (5-1) N<<nt(C)>/<t(AD)> (5-2) in which N is the number of parallel operations, t(C) the cycle time of memory, and t(AD) the time necessary for analogue-digital conversion. It is shown from Eqs. (5-1) and (5-2) that the possible high speed operation is limited only by t(AD). All of the formulas mentioned above are summarized in Table 2, which will be conveniently used in the design of the impulse of basic design of speech synthesizer based on the transfer functions of Japanese vowels, given in Table 4. It is seen in Table 3 that the almost satisfactory performances can be obtained from the memory of 0. 5 μsec. cycle time. An electrical circuit which realizes the above mentioned idea, was constructed. Its block diagram and time chart of operation are shown in Fig. 7 and Fig. 8 respectively. Photographs of waveforms of Japanese vowels synthesized by use of this circuit are also shown in Figs. 12〜15.