Single sine analysis

The use of FFT (fast Fourier transform) spectral analysis can be the perfect complement to the single sine analysis technique. The FFT can perform very fast measurements since it is not necessary to sweep the stimulus frequency in order to measure multiple frequencies. With the introduction of DSP technology, very fast... 

The techniques commonly utilized for ac-impedance measurements in modern equipment can be subdivided into two main groups - single-sine and multiple-sine techniques. The lock-in technique and frequency-response analysis will be described as representatives of the singlesine techniques and FFTs will be introduced as an example for multiple-sine techniques. 

In the present state-of-the-art equipment, it is possible to measure and plot the electrochemical impedance automatically. Electronic circuitry is designed to generate the frequency sweep of a desired resolution over the range of interest. The generator can be programmed to sweep from a maximum to a minimum frequency in a number of required frequency steps. The commonly used modem equipment AC impedance measurement techniques can be subdivided into two main groups - single sine (lock-in amplification and frequency response analysis) and multiple sine techniques such as fast Fourier transforms (FFT). 

A similar analysis can be made for quasi-periodic signals which consist of a sum of sine waves with slowly-varying amplitude and instantaneous frequency each of which is assumed to pass through a single filter. 

In Fig. 7.4-5 the cosine wave fits exactly eight times, and this shows in its transform, which exhibits a single point at f = 8 X (1/128) = 0.0625. On the other hand, the cosine wave in Fig. 7.4-6 does not quite form a repeating sequence, and its frequency, 77/(0.4 X 128) —0.06136, likewise does not fit any of the frequencies used in the transform. Consequently the Fourier transform cannot represent this cosine as a single frequency (because it does not have the proper frequency to do so) but instead finds a combination of sine and cosine waves at adjacent frequencies to describe it. This is what is called leakage the signal at an in-between (but unavailable) frequency as it were leaks into the adjacent (available) analysis frequencies. 

Hoagland and Prud homme (1985) presented a method of moments analysis of dispersion in a single sinusoidal capillary tube with solid walls (Fig. 3-6C). Their geometric model is defined by the wave length (k), the amplitude (a) and mean radius () of the sine wave (z) that describes the aperture wall ... 

The detailed analysis of the model for the product-activated allosteric enzyme allowed a precise quantification of the role played by enzyme cooperativity in glycolytic oscillations (see chapter 2). However, the analysis of a slightly modified model in which the sink of the product becomes Michaelian - i.e. saturable - instead of linear, showed (see section 2.7) that oscillations can occur even if the allosteric enzyme contains a single subunit existing in two conformational states. Enzyme cooperativity is therefore not a condition sine qua non for oscillations to occur weaker nonlinearities, of the Michaelian type, distributed over several reactions of the system, can thus cooperate to raise its global... 

The solubility parameter values of the polyesterimide (10.76 [182]), the novolae (10.7), and the resole (11.1), imply that these polymers should be thermodynamieally miseible beeause the differenee of these parameters between the polyesterimide and a phenolie resin is a small number. The theoretieally predieted miseibility of polyesterimide with novolae or resole appears to be bom out experimentally sinee differential thermal analysis shows a single glass transition temperature for all the blends, and seanning eleetron... 

The next step is the analysis of a single, sinusoidal modulation in the DSC environment. In the top line of Fig. 4.91 Eq. (2) of Fig. 4.69 is repeated, the equation for the measurement of heat capacity in a standard DSC. The second line shows the needed insertions for T and AT for the case of modulation. When referred to T, the phase difference of AT is equal to 6 and its real part is the cosine, the derivative of the sine, as given by the top equation. Next, the insertion and simplification of the resulting equation are shown. By equating the real and imaginary parts of the equations separately, one finds the equations listed at the bottom of Fig. 4.91. These equations suggest immediately the boxed expression for C - C, in Fig. 4.92. 

Figure 3 Flow injection analysis of 1 nM of a 9.5 kilobase oligonucleotide at a Cu electrode with sinusoidal voltammetry. Upper graph shows the signal at different frequencies after Fourier transform of the voltammograms for single-stranded (ss) and double-stranded (ds) DNA. Lower trace shows a flow injection trace for the ds DNA at the sixth harmonic. The excitation sine was from 0.05 to 0.55 V at 2 Hz. The FIA flow rate was 0.5 mL/min and the electrolyte was 0.1 M NaOH. (Adapted from Ref. 23.)...

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