Sine-wave conversion

A third time-conversion technique uses sine-wave signals for time measurement. Two orthogonal sine-wave signals are sampled with the start and the stop pulses. The phase difference between start and stop is used as time information [313]. Currently the sine-wave technique is inferior to the TAC-ADC principle and the TDC principle in terms of count rate. It is not used in single-board TCSPC devices. However, with the fast progress in ADC and signal processor speed the sine wave technique may become competitive with the other techniques. The principle is shown in Fig. 4.16. 

The phase of the oscillator is independent of the phase of the stop pulses. Therefore the start and stop times are converted at random locations on the circle described by the rotating pointer. In a histogram of the time differences of the 

The most severe problem of the circuit shown in Fig. 4.16 is that the start and stop pulses share a common line. The time between subsequent pulses cannot be shorter than the conversion time of the ADCs. The conversion time can be below 20 ns if fast ADCs are used, but then crosstalk between the pulses must be expected. For TCSPC application it is probably better to use separate ADCs for start and stop, driven from the same oscillator. 

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Now the first term on the right is in phase with the applied strain, i.e. it has the form of a sine wave. This can be equated with the storage modulus. Conversely the phase difference between the second term on the right and the applied signal is the difference between sine and cosine waves which can be equated with the loss modulus ... 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

But in Figure 3-12, we have finally modified Dowell s original sine-wave curves. Fourier analysis has been carried out while constructing these curves, and so the designer can apply them directly to the typical (unidirectional) current waveforms of power conversion. We will now use these curves to do the calculations for the primary winding of our ongoing numerical example. 

There are a number of different time measurement teehniques applicable to TCSPC. The TAC-ADC principle of the elassic TCSPC teehnique has been upgraded with a fast, error-cancelling ADC technique. Integrated eireuits for direct time-to digital conversion (TDCs) have been developed, and a sine-wave time-eonversion teehnique has been introdueed. 

Fig. 4.16 Sine-wave principle of time-to-digital conversion...

The TAC parameters determine the time scale and the part of the signal that is reeorded. The available parameters may differ for different TCSPC deviees. Espe-eially deviees based on direct time-to-digital conversion (TDC) or sine-wave eon-version may differ eonsiderably from devices using the TAC/ADC prineiple and reversed start-stop, which will be considered below. 

The frequency was controlled by an on-line computer (Eq. 7), which was also used for the recording of the pressure data from the Baratron transducer. The conversion rate of the analogue-to-digital converter in the interface imit must be fast enough to cope with the 1 to 4 ms response time of the pressure transducer. The pressure response to the voliune change over the whole frequency range was measured in the absence (blank experiment) and presence of sorbent samples to ehminate time constants associated with the apparatus. The FR spectra were derived from the equivalent fimdamental sine-wave perturbation by a Fourier transformation of the volume and pressure square-wave forms. 

Binary chirps (BC) are two-level counterparts of the sine wave chirp. Zero crossings (signum conversions) of the sine wave chirp constitute the BC signal (Fig. 25) ... 

Both harmonic and electrochemical frequency modulation (EFM) methods take advantage of nonlinearity in the E-I response of electrochemiced interfaces to determine corrosion rate [47-50]. A special application of harmonic methods involves harmonic impedance spectroscopy [5i]. The EFM method uses one or more a-c voltage perturbations in order to extract corrosion rate. The electrochemical frequency modulation method has been described in the literature [47-50] and has recently been reviewed [52]. In the most often used EFM method, a potential perturbation by two sine waves of different frequencies is applied across a corroding metal interface. The E-I behavior of corroding interfaces is typically nonlinear, so that such a potential perturbation in the form of a sine wave at one or more frequencies can result in a current response at the same and at other frequencies. The result of such a potential perturbation is various AC current responses at various frequencies such as zero, harmonic, and intermodulation. The magnitude of these current responses can be used to extract information on the corrosion rate of the electrochemical interface or conversely the reduction-oxidation rate of an interface dominated by redox reactions as well as the Tafel parameters. This is an advantage over LPR and EIS methods, which can provide the Z( ) and, at = 0, the polarization resistance of the corroding interface, but do not uniquely determine Tafel parameters in the same set of data. Separate erqreriments must be used to define Tafel parameters. A special extension of the method involves... 

Analysis — Any arbitrary time-dependent function may be synthesized by adding together sine and cosine functions of different frequencies and amplitudes, a process known as synthesis [i]. Conversely, the determination of the amplitudes and frequencies of the sine and cosine waves that make up a time-dependent signal (or noise), for example v(f) or I(t), is known as analysis (or decomposition). Thus, for a signal defined over some time period T, analysis results in the determination of the amplitudes a and b , as a function of frequency in the expression... 

The electrical output signal from a conventional scanning spectrometer usually takes the form of an amplitude-time response, e.g. absorbance vs. wavelength. All such signals, no matter how complex, may be represented as a sum of sine and cosine waves. The continuous fimction of composite frequencies is called a Fourier integral. The conversion of amplitude-time, t, information into amplitude-frequency, w, information is known as a Fourier transformation. The relation between the two forms is given by... 

Each diffraction spot is caused by reflection of X-rays by a particular set of planes in the crystal. If the crystal contains layers of atoms with the same spacing and orientation as a particular set of planes which would satisfy Bragg s law (if the set of planes is physically present), the corresponding diffraction spot will be strong. On the other hand, if only few atoms in a crystal correspond to a particular set of planes, the corresponding reflection will be weak. The complicated structure present in the crystal is transformed by the diffraction process into a set of diffraction spots which correspond to sets of planes (more precisely, sinusoidal density waves), just as our ear converts a complicated sound signal into a series of (sinusoidal) tones when we listen to music. This conversion of a complicated function into a series of simple sine and cosine functions is called a Fourier transformation. 

Fourier analysis permits any continuous curve, such as a complex spectmm of intensity peaks and valleys as a function of wavelength or frequency, to be expressed as a sum of sine or cosine waves varying with time. Conversely, if the data can be acquired as the equivalent sum of these sine and cosine waves, it can be Fourier transformed into the spectrum curve. This requires data acquisition in digital form, substantial computing power, and efficient software algorithms, all now readily available at the level of current generation personal computers. The computerized instmments employing this approach are called FT spectrometers—FTIR, FTNMR, and FTMS instruments, for example. 

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