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23 for the convolution. 25 the value (n — I) as the index of h is always taken modulo N. If n — I does not lie within the interval 0 , . . , (N — 1), suitable multiples of N are added or subtracted. 27 is known as the cyclic convolution. If the signal s consists of a single impulse with the value so = 1 for / = 0, and all the other sn — 0, then yn = hn. The filter function may be produced in such a way and is therefore also called the impulse response. 23 with the FT of s and h. With larger values Af this computation is then much more economical by application of the FFT compared to a direct convolution of s and h in the time domain.

On the other hand, we have shown above that under the assumed conditions # is the estimate with minimum variance. , ftfc has to be estimated then the variances are determined using Fisher's information matrix [5]. 70 can be given another useful form [5]. 73) The second term on the right-hand side cancels p by forming the expectation. This term is then zero, because Jp Jz = 1 and the derivative (32/dfti ftj) Jpdz = 0. 5 Estimation of a signal In some applications the signal has to be estimated in the presence of a disturbance signal.

2 Transition to continuous variables We assume a variable § to be within the interval (x, x + Ax). 2). 9) p(x) is named the probability density function (abbreviated PDF) of x if Ax becomes infinitesimally small, that is Ax —• dx. 11) The value of p(x) is always positive. 12) By this distribution function we may indicate the probability of a noise voltage being below a threshold JC. 14) This may be extended to the important case of N variables. , *#). 19) Example 2: probability density of the sum of two variables As an illustration let us consider the following example.