Time-delay

A signal x t) that is delayed by time is expressed as x(r — If we denote the Fourier 

To simplify the integration, we use a dummy variable v and make the substitution v=t- td. 

So delaying a signal by td seconds is equivalent to multiplying its Fourier transform by jf apply this to a unit impulse, x t - - td), we get 

At first sight, this may seem strange since the Fourier transform for a shifted impulse seems very different from that for a normal impulse, which simply had a Fourier transform of 1. Recall, however, that the magnitude of will be 1, so the magnitude spectrum will be the same as the delta function. The phase of is simply a linear function of td - tiiis is as we should expect the longer the delay td the more the phase spectrum will be shifted. It should be noted that the above result is tiie Fourier-transform result of the z-transform delay derived in Equation (10.35). 

Where instantaneous reaction is not imperative, susceptibility to false alarms can be reduced by requiring the fire signal to be present for a predetermined period of time. However, the time delay reduces the advantages of high speed early detection. In most applications, the tradeoffs between false alarms and the damage incurred in the first few seconds of a fire have been inconsequential. 

So delaying a signal by td seconds is equivalent to multiplying its Fourier transform by e If 

At first sight, this may seem strange as the Fourier Transform for a shifted impulse seems very different for a normal impulse which simply had a Fourier Transform of 1. Recall however, that the magnitude of will be 1, and so the magnitude spectrum will be the same as the delta 

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The geometrical consistency of all possible triplets of time delays (AT31, AT21 and AT32) is validated and invalid combinations are rejected. 

Where Ui denotes input number i and there is an implied summation over all the inputs in the expression above A, Bj, C, D, and F are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic models trom u to y, as well as of the noise model from e to y). Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system close to the input). Likewise Fj(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise N(t). 

We first supposed that the field radiated into the piece by the transducer is known, thanks to the Champ-Sons model. Then, the main approximation used consists in making far field assumptions in the beam defect interaction area. In the case of a focused transducer we assume that the incident wavefronts on the defect are plane. This is equivalent to say that the defect is located in or near the transducer focal area and that a defect located outside this zone does not cause a significant echo. In the case of planar contact transducer, the incident wavefronts on the defect are assumed to be spherical The incident field on the defect is therefore approximated by the product of a spatial function gfp,0,z)describing the amplitude distribution in the beam and a time-delayed waveform < ) ft) representing the plane or spherical propagation in the beam. The incident field on the defect can therefore be approximated for ... 

We recall that, for flat array, the steered angle beam, 9, depends only on ultrasound celerity in the propagation medium, ci, the time delay between each element, 5t, and the pitch element, p, through the relation ... 

Similarly, the focusing capability of an array is the strongest focused beam which can be steered. The simplest way to evaluate it is to test a theoretical focusing time delay law, in the near-field and in the natural direction of propagation of the array. The beam pattern characteristics depth, lateral size and length of the focal spot must be found consistent with modelling and no lobe must appear above a predetermined level. 

The required acoustic verifications depend on what the probe is made for. If the probe is used as an angular scamiing system with a fix set of elements, then we think it is only needed to characterize the array behavior with a few selected time delay laws to isolate the angular steering capability and the foeusing capability as explained before. 

To determine time dependent behaviours of the specimen up to 25 measurements in series with different time delays are possible. To prevent mistakes in application many help comments appear when inputs are necessary or differences between the calibration and the measurement are detected. All calibration conditions, a description for the specimen and results can be printed or saved by the hard disk. To reduce the input expenditure, the last configuration is made to current values when the program is stopped ore leave. 

We will explore the effect of three parameters 2 -and < )> that is, the time delay between the pulses, the tuning or detuning of the carrier frequency from resonance with an excited-state vibrational transition and the relative phase of the two pulses. We follow closely the development of [22]. Using equation (Al.6.73). 

Figure Al.6.7. Schematic diagram illustrating the different possibilities of interference between a pair of wavepackets, as described in the text. The diagram illustrates the role of phase ((a) and (c)), as well as the role of time delay (b). These cases provide the interpretation for the experimental results shown in figure Al.6.8. Reprinted from [22],...

Figure Al.6,8 shows the experimental results of Scherer et al of excitation of I2 using pairs of phase locked pulses. By the use of heterodyne detection, those authors were able to measure just the mterference contribution to the total excited-state fluorescence (i.e. the difference in excited-state population from the two units of population which would be prepared if there were no interference). The basic qualitative dependence on time delay and phase is the same as that predicted by the hannonic model significant interference is observed only at multiples of the excited-state vibrational frequency, and the relative phase of the two pulses detennines whether that interference is constructive or destructive.

Figure Al.6.8. Wavepacket interferometry. The interference contribution to the exeited-state fluoreseenee of I2 as a fiinotion of the time delay between a pair of ultrashort pulses. The interferenee eontribution is isolated by heterodyne deteetion. Note that the stnieture in the interferogram oeeurs only at multiples of 300 fs, the exeited-state vibrational period of f. it is only at these times that the wavepaeket promoted by the first pulse is baek in the Franek-Condon region. For a phase shift of 0 between the pulses the returning wavepaeket and the newly promoted wavepaeket are in phase, leading to eonstnietive interferenee (upper traee), while for a phase shift of n the two wavepaekets are out of phase, and interfere destnietively (lower traee). Reprinted from Seherer N F et 0/1991 J. Chem. Phys. 95 1487.

Second-order effects include experiments designed to clock chemical reactions, pioneered by Zewail and coworkers [25]. The experiments are shown schematically in figure Al.6.10. An initial 100-150 fs pulse moves population from the bound ground state to the dissociative first excited state in ICN. A second pulse, time delayed from the first then moves population from the first excited state to the second excited state, which is also dissociative. By noting the frequency of light absorbed from tlie second pulse, Zewail can estimate the distance between the two excited-state surfaces and thus infer the motion of the initially prepared wavepacket on the first excited state (figure Al.6.10 ). 

CN] —> I + CN. Wavepacket moves and spreads in time, with its centre evolving about 5 A in 200 fs. Wavepacket dynamics refers to motion on the intennediate potential energy surface B. Reprinted from Williams S O and lime D G 1988 J. Phys. Chem.. 92 6648. (c) Calculated FTS signal (total fluorescence from state C) as a fiinction of the time delay between the first excitation pulse (A B) and the second excitation pulse (B -> C). Reprinted from Williams S O and Imre D G, as above. 

Figure Al.6.20. (Left) Level scheme and nomenclature used in (a) single time-delay CARS, (b) Two-time delay CARS ((TD) CARS). The wavepacket is excited by cOp, then transferred back to the ground state by with Raman shift oij. Its evolution is then monitored by tOp (after [44])- (Right) Relevant potential energy surfaces for the iodine molecule. The creation of the wavepacket in the excited state is done by oip. The transfer to the final state is shown by the dashed arrows according to the state one wants to populate (after [44]).

Figure Al.6.22 (a) Sequence of pulses in the canonical echo experiment, (b) Polarization versus time for the pulse sequence in (a), showing an echo at a time delay equal to the delay between the excitation pulses.

Figure Al.6.25. Modulus squared of tire rephasing, (a), and non-rephasing, R., (b), response fiinetions versus final time ifor a near-eritieally overdamped Brownian oseillator model M(i). The time delay between the seeond and third pulse, T, is varied as follows (a) from top to bottom, J= 0, 20, 40, 60, 80, 100,...

Figure Al.6.28. Magnitude of the excited-state wavefimction for a pulse sequence of two Gaussians with time delay of 610 a.u. = 15 fs. (a) (= 200 a.u., (b) ( = 400 a.u., (c) (= 600 a.u. Note the close correspondence with the results obtained for the classical trajectory (figure Al. 6.27(a) and (b)). Magnitude of the ground-state wavefimction for the same pulse sequence, at (a) (= 0, (b) (= 800 a.u., (c) (= 1000 a.u. Note the close correspondence with the classical trajectory of figure Al.6.27(c)). Although some of the amplitude remains in the bound region, that which does exit does so exclusively from chaimel 1 (reprinted from [52]).

Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

Figure Al.6.31. Multiple pathway interference interpretation of pump-dump control. Since each of the pair of pulses contains many frequency components, there are an infinite number of combination frequencies which lead to the same fmal energy state, which generally interfere. The time delay between the pump and...

Knopp G, Pinkas I and Prior Y 2000 Two-dimensional time-delayed coherent anti-Stokes Raman spectroscopy and wavepacket dynamics of high ground-state vibrations J. Raman Spectrosc. 31 51... 

For fluorescent compounds and for times in die range of a tenth of a nanosecond to a hundred microseconds, two very successftd teclmiques have been used. One is die phase-shift teclmique. In this method the fluorescence is excited by light whose intensity is modulated sinusoidally at a frequency / chosen so its period is not too different from die expected lifetime. The fluorescent light is then also modulated at the same frequency but with a time delay. If the fluorescence decays exponentially, its phase is shifted by an angle A([) which is related to the mean life, i, of the excited state. The relationship is... 

FigureBl.5.16 Rotational relaxation of Coumarin 314 molecules at the air/water interface. The change in the SFI signal is recorded as a fimction of the time delay between the pump and probe pulses. Anisotropy in the orientational distribution is created by linearly polarized pump radiation in two orthogonal directions in the surface. (After [90].)...

Time delay ber / een laser puJse arid RF-pulse. ms... 

Time delay betzcecn laser poise a.nd RF-pulse, us... 

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