Transforms impulse properties

EXPLOSIVES ate solid, liquid and gaseous substances possessing the property, when subjected C3 either heat, impact, friction or other initial impulse, of undergoing a very rapid exothermic self propagating transformation or decomposition with the formation of mote stable materials (usually gases), accompanied by the production of a very loud noise (report) and the development of very great pressures and very high temperatures. This action is called explosion or detonation... 

Many of its properties follow readily from the properties of <5(x). Of greatest interest, however, is the fact that its Fourier transform is a similar function that has reciprocal spacing between its impulses ... 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

Experimental NMR data are typically measured in response to one or more excitation pulses as a function of the time following the last pulse. From a general point of view, spectroscopy can be treated as a particular application of nonlinear system analysis [Bogl, Deul, Marl, Schl]. One-, two-, and multi-dimensional impulse-response functions are defined within this framework. They characterize the linear and nonlinear properties of the sample (and the measurement apparatus), which is simply referred to as the system. The impulse-response functions determine how the excitation signal is transformed into the response signal. A nonlinear system executes a nonlinear transformation of the input function to produce the output function. Here the parameter of the function, for instance the time, is preserved. In comparison to this, the Fourier transformation is a linear transformation of a function, where the parameter itself is changed. For instance, time is converted to frequency. The Fourier transforms of the impulse-response functions are known to the spectroscopist as spectra, to the system analyst as transfer functions, and to the physicist as dynamic susceptibilities. 

As expected from the scaling property, the Fourier transform of an impulse is a function that is infinitely stretched , that is, the Fourier Transform is 1 at all frequencies. Using the duality principle, a signal x(t) = 1 for all t will have a Fourier transform of 6( ), that is, an impulse at time 00 = 0. This is to be expected - a constant signal (a d.c. signal in electrical terms) has no variation and hence no information at frequencies other than 0. 

The STFT thus results in a spectrum that depends on the time instant to which the window is shifted. The choice of Gaussian functions for the short-duration window gives excellent localization properties despite the fact that the functions are not limited in time. Alternatively, STFT can also be viewed as filtering the signal at all times using a bandpass filter centered around a given frequency/whose impulse response is the Fourier transform of the short-duration window modulated to that frequency. However, the duration and bandwidth of the window remain the same for all frequencies. 

Many DSP concepts can be demonstrated by examples which involve a great deal of computation. A list of some of the concepts is as follows convolution, filtering, quantization effects, etc. The curriculum begins with discrete Fourier transform (DFT). DFT is derived from discrete-time Fourier transform expression. The continuous and discrete Fourier transform are covered in Signals and Systans. The flow of the topics is as follows DFT, properties of DFT, Fast Fourier Transform, Infinite Impulse Response filter and Finite Impulse Response fillers and filter structures. If the topics are linked to a project with each block of the project demonstrating the various topics of the curriculum, it is easier for the student to comprehend what is being taught. 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

Figure 2A shows a portion of the train of impulses Vs(t) that has a spectrum Vs(cf>) for -00 < co < 00, where 0) is the radian frequency 2nf. This spectrum can be obtained using property 6 of the Fourier transform in Table 1, which shows that the transform of the product of two functions is the convolution of the transforms. The convolution of Vdco) with the periodic train of impulses Up co) of Figure 1 results in periodically repeated copies of Vdco), separated by l/Tg, a portion of which is shown in Figure 2A. 

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