System impulse responses

L(t) is the convolution of the system impulse response and the excitation waveform. Notice, however, that the lower limit of integration starts at zero not-°° due to reasons of causality the sensor luminescence response begins after excitation t" = 0. In our case... 

Applying the KFH phase to dispersion requires the estimation of the spectral magnitude M(C0 m ) of the system impulse response and the pitch period of the excitation P(m). The duration of the synthetic impulse response is set close to the pitch period P(m) so that the resulting waveform is as dense as possible. The sine-wave analysis produces estimates of the spectral and pitch characteristics. The synthetic system phase derived using the KFH solution, denoted byip fh (CO m), is given by... 

From equation 2 it is plain that G(s) may be considered as a power series in s, the coefficients of which are moments of the system impulse response. Note that the system mean M is equal to i/ao and that if G(s) is expressed in the form of equation 4 then expressions for ao and ai are as specified in 5. 

If the impulse response function g(x) of a system is known, the output signal y(x) of the system is given for any input signal u(x). The integral equation, which is called superposition integral. 

The function g(x) is named impulse response of the system, because it is the response to an unit pulse 5(x) applied at =0 [2]. This unit impulse 5(x), also called Dirac impulse or delta-function, is defined as... 

Eq.(2) describes an impulse with the area of 1 [1-3]. Fig. 1 (left) shows such an unit impulse S(x) and an example for an impulse response g(x) at the output of the system. 

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

All described sensor probes scan an edge of the same material to get the characteristic step response of each system. The derivation of this curve (see eq.(4) ) causes the impulse responses. The measurement frequency is 100 kHz, the distance between sensor and structure 0. Chapter 4.2.1. and 4.2.2. compare several sensors and measurement methods and show the importance of the impulse response for the comparison. 

It should be a symmetrical form in the impulse response of a linear system. 

Figure 7 Impulse responses of different sensor systems...

Methods from the theory of LTI-systems are practicable for eddy-current material testing problems. The special role of the impulse response as a characteristic function of the system sensor-material is presented in the theory and for several examples. 

So, a comparison of different types of magnetic field sensors is possible by using the impulse response function. High amplitude and small width of this bell-formed function represent a high local resolution and a high signal-to-noise-characteristic of a sensor system. On the other hand the impulse response can be used for calculation of an unknown output. In a next step it will be shown a solution of an inverse eddy-current testing problem. 

The Fourier transform H(f) of the impulse response h(t) is called the system function. The system function relates the Fourier transforms of the input and output time functions by means of the extremely simple Eq. (3-298), which states that the action of the filter is to modify that part of the input consisting of a complex exponential at frequency / by multiplying its amplitude (magnitude) by i7(/)j and adding arg [ (/)] to its phase angle (argument). 

This equation assumes a much simpler form if we express it in terms of the system function H(f) instead of the impulse response h(t) namely, as reference to Eq. (3-300) shows... 

Here, h(t) characterize the behaviour of the system, and is called the response function, or the impulse response, because it is identical to the response to a unit impulse excitation. 

The combination of Eqs. (28) and (22) gives the Laplace transform of the impulse response H(p) which allows us to solve Eq. (21). By the inverse transformation, the relation which gives the output of the linear system g(t) (the thermogram) to any input/(0 (the thermal phenomenon under investigation) is obtained. This general equation for the heat transfer in a heat-flow calorimeter may be written (40, 46) ... 

We could go through the Laplace domain by approximating and then inverting. However, there is a direct conversion V. V. Solodovnilcov, Introduction to Statistical Dynamics of Autoinatic Control, Dover, 1960). Suppose we want to find the impulse response of a stable system (defined as g,), given the system s frequency response. Since the Laplace transformation of the impulse input is unity,... 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

Fortunately, it is not always necessary to recover the system RTD curve from the impulse response, so the complications alluded to above are often of theoretical rather than practical concern. In addition, the dispersion model is most appropriately used to describe small extents of dispersion, i.e. minor deviations from plug flow. In this case, particularly if the inlet pipe is of small diameter compared with the reactor itself, the vessel can be satisfactorily assumed to possess closed boundaries [62]. An impulse of tracer will enter the system and broaden as it passes along the reactor so that the observed response at the outlet will be an RTD and will be a symmetrical pulse, the width of which is a function of DjuL alone. 

Urea in kidney dialysate can be determined by immobilizing urease (via silylation or with glutaraldehyde as binder) on commercially available acid-base cellulose pads the process has to be modified slightly in order not to alter the dye contained in the pads [57]. The stopped-flow technique assures the required sensitivity for the enzymatic reaction, which takes 30-60 s. Synchronization of the peristaltic pumps PI and P2 in the valveless impulse-response flow injection manifold depicted in Fig. 5.19.B by means of a timer enables kinetic measurements [62]. Following a comprehensive study of the effect of hydrodynamic and (bio)chemical variables, the sensor was optimized for monitoring urea in real biological samples. A similar system was used for the determination of penicillin by penicillinase-catalysed hydrolysis. The enzyme was immobilized on acid-base cellulose strips via bovine serum albumin similarly as in enzyme electrodes [63], even though the above-described procedure would have been equally effective. 

---