General periodic signals

Many non-sinusoidal waveforms are periodic the test for a periodic waveform is that there exists some period T, for whieh the following is true  

That is, at multiples of T the value is always the same. The lowest value for which this is true is called the fundamental period, denoted Tq. The fi equency Fq = /Tq is called the fundamental frequency, and its angular equivalent is (Oo = l/2jiro. A harmonic frequency is any integer multiple of the fundamental frequency, 2Fo,3Fq,. 

It can be shown that any periodic signal can be created by summing sinusoids which have frequencies that are harmonics of the fundamental. If we have a series of harmonically related sinusoids numbered 0,1,2,. ..jA, then the complete signal is given by  

As 0 X of = 0 and oico5 (( )o) is a constant, we can combine ( ((lio) and gq into a new constant Aq, leaving the first term as simply this new Aq. This can be thought of as an amplitude offset (or DC offset in electrical terminology), so that the waveform does not have to be centred on the y-axis. In the most general case, the number of terms can be infinite, so the general form of 

Equation 10.6 is a specific form of the Fourier series, one the most important concepts in signal processing. The process as just described is known as Fourier synthesis, that is the process of synthesising signals by using appropriate values and applying the Fourier series. It should be noted lhat while the sinusoids must have frequencies that are harmonically related, the amplitudes and phases of each sinusoid can take on any value. In other words, the form of the generated waveform is determined solely by the amplitudes and phases. 

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Imagine we have a signal x t) = of known fundamental period but unknown harmonic (that is, we know coo but not /). We can calculate the iimer product of this signal with each harmonic k = 1,2,3... of the fundamental. All the answers will be zero except the one for when the harmonics are the same (k = /). As a general periodic signal is just the sum of a set of harmonics, we can use this method to find the value of every harmonic in a signal. Mathematically, we show this by starting with the Fourier series ... 

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 possibility to quantify the response of a stochastic system to periodic signals is to generalize the notion of synchronization, which is known from deterministic nonlinear oscillators. We will pursue this idea in what follows. To this end we review in section 2.2 the notion of effective synchronization in stochastic systems. The mean number of synchronized system cycles turns out to be an appropriate quantity to characterize the synchronization properties of the system to the periodic signal. However the task remains to calculate this quantity. This calculation will be based on discrete renewal models for bistable and excitable dynamics. These discrete models are introduced in section 2.3. We first recapitulate the well known two state model for the stochastic dynamics of an overdamped particle in a doublewell system [10] and afterwards introduce a phenomenological discrete model for excitable dynamics. In section 2.4 a theory to calculate the mean frequency and effective diffusion coefficient in periodically driven renewal processes is presented. These two quantities allow to calculate the mean number of synchronized cycles. Finally in section 2.5 we apply this theory to investigate synchronization in bistable and excitable systems. 

We observe that, although Fourier is suitable for periodic signals, it is not particularly efficient for sharp edges. It is obvious why not the basis functions themselves, the sines/cosines are smooth. This leads to a general observation something sharp, sudden or narrow in the time domain will be something wide in the frequency domain. 

Figure 7-42 is a lime-domain signal from a source that contains many wavelengths. It is considerably more complex than those shown in Figure 7-41. Because a large number of wavelengths arc involved, a full cycle is not completed in the time period shown. A pailcrii of beats appears as certain wavelengths pass in and out of phase. In general, the signal power de-...

In general, we need to analyse non-periodic signals as well as periodic ones. Fourier synthesis and analysis can not deal with non-periodic signals, but a closely related operation, the Fourier Transform is designed to do just this. By definition, the Fourier transform is ... 

U-shaped clevis to which a vaned tailpiece is attac-hed. The wheel rotates because of the difference in drag for the two sides of the cup, and a signal proportional to the revolutions of the wheel is generated. The velocity is determined from the count over a period or time. The current meter is generally usebil in the range of 0.15 to 4.5 m/s (about 0.5 to 15 ft/s) with an accuracy of 2 percent. For additional information see Creager and Justin, Hydroelectric Handbook, 2d ed., Wiley, New York, 1950, pp. 42-46. 

Many events signaled the beginning of a lengthy transition period for the utility industry in general, and more specifically, for the facility planning activity. The timeline m many instances was blurred, with some of the blurring attributable to a reluctance to change. 

There are many industrial applications in which permanent records (extending over long periods of time) of the instrument readings are required. Chart recorders of various forms are available for this purpose. The most common general-purpose unit is the digital strip chart recorder, in which the input signal is used to drive the movement of a recording arm that passes over a paper chart in the y-direction. At the same time, the chart is... 

There are actually two independent time periods involved, t and t. The time period ti after the application of the first pulse is incremented systematically, and separate FIDs are obtained at each value of t. The second time period, represents the detection period and it is kept constant. The first set of Fourier transformations (of rows) yields frequency-domain spectra, as in the ID experiment. When these frequency-domain spectra are stacked together (data transposition), a new data matrix, or pseudo-FID, is obtained, S(absorption-mode signals are modulated in amplitude as a function of t. It is therefore necessary to carry out second Fourier transformation to convert this pseudo FID to frequency domain spectra. The second set of Fourier transformations (across columns) on S (/j, F. produces a two-dimensional spectrum S F, F ). This represents a general procedure for obtaining 2D spectra. 

In three-dimensional experiments, two different 2D experiments are combined, so three frequency coordinates are involved. In general, the 3D experiment may be made up of the preparation, evolution (mixing periods of the first 2D experiment, combined with the evolution t ), mixing, and detection ( ) periods of the second 2D experiment. The 3D signals are therefore recorded as a function of two variable evolution times, t and <2, and the detection time %. This is illustrated in Fig. 6.1. 

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