The impulse response

While the above clearly showed how our filter created an output signal y [n ] from our given input signal x[n], it isn t immediately clear how the filter will modify other, different input signals. Rather than having to examine the output with respect to a wide range of different input signals, it would be useful to have a characterisation of the effect of 

A unit impulse, 5, is a special function, which has value 1 at n = 0 and 0 elsewhere  

Consider now an example first-order HR filter, with a = —0.8 and h[0] = 1. Its difference equation is therefore 

Rgure 10.14 Example impulse responses for FIR and HR filters (a) FIR filter and (b) HR filter. Note that, while the number of coefficients in the HR filter is actually less than for the FIR filter, the response will continue indefinitely (only values up to 30 samples are shown). 

The output of the example FIR filter for an impulse is given in figure 10.14a, which shows that the impulse response is simply the coefficients in the filter. Because of this, the duration of the impulse will always be the order of the filter. As the order is finite, the impulse response is finite which gives rise to the name of the filter. 

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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. 

Often an unit impulse is not available as a signal to get the impulse response function g(x). Therefore an other characteristic signal, the unit step, is be used. 

The step response function h(x) is also determined by the integral equation (1). The relationship between step response h(x) and the impulse response g(x) is represented by... 

The superposition integral (1) corresponds to a division of the input signal u(x) into a lot of Dirac impulses 5 x). which are scaled to the belonging value of the input. The output of each impulse 5fx) is known as the impulse response g(x). That means, the output y(x) is got by addition of a lot of local shifted and scaled impulse responses. 

Chapter 4.3. discusses the explained theory for choosed examples. For several cracks the output is pre-calculated by using the impulse response and compared with measurement data. 

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. 

The first example presents the importance of the impulse response function for the comparison of several sensors with the same arrangement from chapter 3.1.. 

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

The following examples represent the importance of the impulse response for the comparison of different magnetic field sensors. For presentation in this paper only one data curve per method is selected and compared. The determined signals and the path x are related in the same way like in the previous chapter. 

This is visible in the behaviour of the impulse responses as well (fig, 7), There the amplitude of curve (1) (gWng, )) is the highest, of curve (2) ig(L,)) the lowest, but the maxima are not located at the same place. 

The difference in widths of the impulse responses are small. Especially visible the pulse response of the inductive sensors, curves (1) and... 

For calculation the known data are the. .input signal", cracks of different widths, and the impulse response. The material of the crack model is assigned to the value 0, the air to 1. 

The determined eddy-eurrent parameter is the inductance of the eomplex impedance measured by impedance analyzer at j=100 kHz. Therefore the impulse response function from chapter 4.2.1. is used for calculation. The depth of the cracks is big in comparison to coil size. For presentation the measured and pre-calculated data are related to their maxima (in air). The path X is related to the winding diameter dy of the coil. 

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 function h(t) to be restored is the impulse response of the medium x(t) is the transmitted pulse measured by reflection on a perfect plane reflector, for example the interface between air and water and y(t) is the observed signal. 

The impulse response funetion, equation (3.26) is shown in Figure 3.11. 

The response-factor approach is based on a method in which the response factors represent the transfer functions of the wall due to unit impulse excitations. The real excitation is approximated by a superposition of such impulses (mostly of triangular shape), and the real response is determined by the superposition of the impulse responses (see Figs. 11.33 and 11.34). ... 

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... 

As an example of the use of Eq. (3-321), we shall calculate the output power density spectrum of an RC filter whose input consists of white noise. The RC filter in question is shown in Fig. 3-14. It is a simple matter to verify that the impulse response of this filter is given by... 

Impulse Response and the Differential Distribution. Suppose a small amount of tracer is instantaneously injected at time 1 = 0 into the inlet of a reactor. All the tracer molecules enter together but leave at varying times. The tracer concentration at the outlet is measured and integrated with respect to time. The integral will be finite and proportional to the total quantity of tracer that was injected. The concentration measurement at the reactor outlet is normalized by this integral to obtain the impulse response function. ... 

For a mean filter with a window width of T, the impulse response is given by... 

What is the pulse response in the effluent If we do not have the patience of 10 s and dump all the extra tracer in at one shot, what is the impulse response ... 

With the impulse input, we use the impulse response in Eq. (2-41) instead, and Eq. (2-44) becomes... 

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) ... 

The decay ratio for the impulse response from equation (iv) =... 

The impulse response data of a pilot hydrofiner shown with the first figure are fitted by... 

The filtering operation may be implemented by ascertaining the filter impulse response required and then performing convolution or a FFT/IFFT combination to correlate p(t) against the impulse response. 

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