Impulse response functions

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

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. 

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. 

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. ... 

The deviations due to some of these destructive influences are reversible. These are usually described as systematic errors. Many of the degradation processes that affect images and most recorded data are classified as systematic errors. For many of these cases the error may be expressed as a function known as the impulse response function. Much mathematical theory has been devoted to its description and correction of the degradation due to its influence. This has been discussed in some detail by Jansson in Chapter 1 of this volume. In that correction of this type of error usually involves increasing the higher frequencies of the Fourier spectrum relative to the lower frequencies, this operation (deconvolution) may also be classified as an example of form alteration. ... 

Addressing first the limitations of a periodic representation, such as with the DFT or Fourier series, we see that it is evident that these forms are adequate only to represent either periodic functions or data over a finite interval. Because data can be taken only over a finite interval, this is not in itself a serious drawback. However, under convolution, because the function represented over the interval repeats indefinitely, serious overlapping with the adjacent periods could occur. This is generally true for deconvolution also, because it is simply convolution with the inverse filter 1 1/t(w). If the data go to zero at the end points, one way of minimizing this type of error is simply to pad more zeros beyond one or both end points to minimize overlapping. Making the separation across the end points between the respective functions equal to the effective width of the impulse response function is usually sufficient for most practical purposes. See Stockham (1966) for further discussion of endpoint extension of the data in cyclic convolution. 

With convolution and deconvolution, one must be careful to avoid end-point error with this type of function. Convolution with the function beyond the end point of the data will extend inside the interval containing the data about half the length of the impulse response function, so the error will extend about half the length of the impulse response function also (assuming the impulse response function is approximately symmetrical). To minimize this error, the function extending beyond the end points should... 

In data-point units, the original infrared peaks were about 34 units wide (full width at half maximum). This corresponds to an actual width of approximately 0.024 cm-1. The impulse response function was about 25 units wide. After inverse filtering and restoration of the Fourier spectrum, the resolved peaks were 11 and 14 units wide, respectively. This is close to the Doppler width of these lines. 

We shall end this chapter with a few practical remarks concerning the calculation of the inverse-filtered spectrum. In this research the Fourier transform of the data is divided by the Fourier transform of the impulse response function for the low frequencies. Letting 6 denote the inverse-filtered estimate and n the discrete integral spectral variable, we would have for the inverse-filtered Fourier spectrum... 

Data are often normalized so that the area under the curve is preserved. This area is given by the dc spectral term, that is, for /t = 0. To preserve the area in the discrete inverse-filtered result, every term should be multiplied by the dc spectral components of the impulse response function (if the impulse response function has not been normalized earlier). We would then have ford... 

Figures 5-11 illustrate the restoration process in the presence of a drifting base line. These data are methane absorption lines taken with a four-pass Littrow-type diffraction grating spectrometer. For these data 2048 data points were taken. The impulse response function was approximated by a gaussian. The true width of these lines is approximately 0.02 cm-1.

Fig. 7 Result of inverse-filtering the corrected data of Fig. 6 with a Gaussian impulse response function having a FWHM of 39 units. The Fourier spectrum was truncated after the 35th (complex) coefficient.

Figure 9 shows the result of inverse filtering with a Gaussian impulse response function having a FWHM of 46 units. The Fourier spectrum was truncated after the 30th coefficient. Note that the broader impulse response function should result in narrower restored peaks. Restoring 62 (31 complex) coefficients to the Fourier spectrum of the inverse-filtered result of Fig. 9 by minimizing the sum of the squares of the negative deviations produces the result shown in Fig. 10. Note that these peaks are narrower than those...

The researcher may want to combine the computer program used for inverse filtering with that used for spectral continuation so as to perform the complete restoration in one step. The truncation frequency of the inverse-filtered spectrum could be automatically determined from the rms of the noise and the signal, and the amplitude of the spectrum of the impulse response function. 

Time-resolved emission spectra are reconstructed from the normalized impulse response functions (26) ... 

FIGURE 10.7 Impulse response function of the seven-point rectangular smoother window function used in Figure 10.6. Note that the Fourier transform of a step function has the form sin(x)/a . 

FIGURE 10.12 Time-domain smoothing of the noisy data in Figure 10.1 with the impulse response function of Figure 10.7, processed from left to right in this spectrum. The true signal is shown as a dotted line. Note the significant filter lag in this example. 

The change in the population distribution ATj(t) of the i th level due to the pump excitation is ANi(t). Let us assume for simplicity that a single AN(t) population is induced. There is a new absorption associated with this population, given by Aa(ui) = a(uj)AN(t). [Usually many excited states i are involved, and more than one transition starts from each state with rates proportional to aij(u>)ANi, so that the real equation is a sum over i and j] AN(t) depends on the material and is represented by the impulsive response function A(t) (i.e., the response to a delta-like pulse). Given the finite duration of the pulse, shorter but not negligible compared with A(t), the real change in a is described by a convolution ... 

The dielectric is often assumed to be isotropic in order to simplify Eq. (8) by assuming transverse phonon-polaritons the extension to anisotropic media is straightforward (31). In the limit of very short pulse duration compared to the phonon-polariton oscillation period, the time-dependence of the excitation field can be treated as a delta function, and the phonon-polariton response is given by the impulse response function for the spatial excitation pattern used. If crossed excitation pulses are used, then it is simplest to describe the excitation and response in terms of the excitation wavevector or wavevector range. 

---