17 On the Stability of a Discrete Convolution with Measured Impulse. . . 175 where the equality ωi =2πfi is used. If an odd number Nis assumed and the symmetry of the vector of Fourier coefficients is taken into account the time signal can be reconstructed as well with y tj =c0 +2 N−1 2 i=1 |ci| cos ωitj +arg(|ci|) (17.4) If Nis a straight number, Eq. (17.4) would have a slightly different form. 17.2.2 Error Due to the Approximation of an Integral with the Trapezoidal Rule Let us consider a function f (x) in the interval [a,b], see Fig. 17.1. For the trapezoidal rule, the integral of f (x) between a and b is approximated by the integral of a straight line with the endpoints f (a) and f (b). In numerous mathematics books it can be verified that an upper bound of the error can be given as |eab| ≤ Δx 3 12 # # f (xab)# # (17.5) where f holds the second derivative of f with respect tox andxab leads to the maximum value of f in the interval [a,b]. The symbol x is an abbreviation for the length of the interval b −a. If it is assumed, that an interval [a,b] is subdivided into M equidistant subintervals, each with length x, an upper bound of the error can be given as |eAB| ≤M Δx 3 12 # # f (xAB)# # (17.6) where xAB leads to the maximum value of f in the interval [a,b]. 17.2.3 Discrete Convolution: Error Due to Trapezoidal Rule For a linear time invariant system the output y(t) can be computed as a function of the input x(t) by means of the convolution y(t) = t $ 0 h(t −τ) x (τ) dτ = t $ 0 h(τ) x (t −τ) dτ (17.7) whereh(t) is the IRF between the considered degrees of freedom. For the following considerations a time invariant test signal x(t) =1 is assumed so that the convolution simplifies to y(t) = t $ 0 h(τ) dτ (17.8) Fig. 17.1 Trapezoidal rule in the interval [a,b]
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