Periodic signals

Statement that a periodic signal must be sampled at least twice each period to avoid a determinate error in measuring its frequency. 

Nyquist theorem statement that a periodic signal must be... 

M. Maeda, S. Lu, G. Shaulsky, Y. Miyazaki, H. Kuwayama, Y. Tanaka, A. Kuspa, and W. F. Loomis, Periodic signaling controlled by an oscillatory circuit that includes protein kinases ERK2 and PKA. Science 304, 875-878 (2004). 

Y. X. Li and A. Goldbeter, Erequency specihcity in intercellular communication The influence of patterns of periodic signalling on target cell responsiveness. Biophys. J. 55, 125-145 (1989). 

The autocorrelation function of a periodic signal is itself periodic with a frequency equal to that of the original expression. It is to be noticed, however, that all information concerning the phase of the original signal is lost illustrating the nonunique description aspect. 

Equation III is merely stating that a correlation function of a periodic signal is also periodic while Equation IV states that the maximum amount of correlation of a periodic function occurs when the same point of that signal is compared with itself and that the correlation is diminished as one compares two points that are farther and farther apart (up to a difference of T). Equation V states that the autocorrelation function is symmetric about t = 0. 

The onset of this peak is closely related to stochastic resonance, which can occur if a weak periodic signal is added to the input. 

In WMS, phase-sensitive (i.e. lock-in) amplification demodulates the periodic signal with a very narrow electrical bandwidth to precisely measure the RMS amplitudes of the fundamental sinusoid and its second harmonic component, averaged over a period of time equal to the inverse of the electrical bandwidth. These are called the If and 2/ signals and each amplitude is expressed by the following equations, respectively,... 

In the case of quasi-periodic sinusoidal signals, the buzziness can often be linked to the fact that the phase coherence between sinusoidal components is not preserved. Shape invariant modification techniques for quasi-periodic signals are an attempt to tackle this problem. As explained in 9.4.2, quasi-periodic signals such as speech voiced segments or sounds of musical instruments can be thought of as sinusoidal signals whose frequencies are multiples of a common fundamental COo(x), but with additional, slowly varying phases 0 (/) ... 

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. 

The amplitudes and phases in Equation (9.10) can correspond to physically meaningful parameters for quasi-periodic signals typical of speech and music [Flanagan and Golden, 1966, Rabiner and Schafer, 1978a], In order to see this, the STFT is first written as... 

Tsong, T. Y. (1992) Molecular Recognition and Processing of Periodic Signals in Cells Study of Activation of Membrane ATPases by Alternating Electric Fields, Biochem. Biophys. Acta, 1113, pp. 53-70. 

As already noted, the properties of convolution and correlation are the same, whether or not a continuous or discrete transformation is used, but because of the cyclic nature of sampled sequences discussed previously, the mechanics of calculating correlation and convolution of functions are somewhat different. The discrete convolution property is applied to a periodic signal 5 and a finite, but periodic, sequence r. The period of 5 is N, so that 5 is completely determined by the N samples s0, Sj,. .., %. The duration of the finite sequence r is assumed to be the same as the period of the data N samples. Then, the convolution of 5 and r is... 

Wavelet analysis is a rather new mathematical tool for the frequency analysis of nonstationary time series signals, such as ECN data. This approach simulates a complex time series by breaking up the ECN data into different frequency components or wave packets, yielding information on the amplitude of any periodic signals within the time series data and how this amplitude varies with time. This approach has been applied to the analysis of ECN data [v, vi]. Since electrochemical noise is 1/f (or flicker) noise, the new technique of -> flicker noise spectroscopy may also find increasing application. 

This concept is one of the most difficult to quantitate. There are some relatively explicit definitions of information content for electronic communications. (For example, the Nyquist theorem defines the minimum sampling rate required in order to preserve the maximum frequency information in a periodic signal. And, the relationships between digital encoding formats and information content of a data base can be quantitated.) However, for the general problem of evaluating the results of instrumental measurements of chemical systems, the definitions for information content of data are very clear. 

Another type of feedback electrometer works in a quite different manner, aiid Fig. 7.5 illustrates the method of using it to measure charge density on the surface of an insulating sheet or film lying on an earthed base. The probe peeps at the surface in question through a hole in a metal box, and, in addition, the field to the probe is interrupted by a vibrating shutter. The periodic signal on the probe is amplified and phase-sensitively detected, and the output used to drive an amplifier whose own output in turn raises the potential of the metal box and the shutter. 

---