Periodic waveforms

Most of the types described above have the facility for single-shot operation if it is necessary to measure single events (i.e. transients). In these cases, the timebase is triggered by the start of the transient. The limitation is the persistence of the screen luminescence since the event only occurs once, rather than a repetitive series of events, as happens with a periodic waveform where the trace is, in effect, overwritten during each operation of the timebase. 

Research into the acoustics of musical instruments has revealed considerable evidence that aperiodicity and noise play an important role in the sound quality of a musical instrument. This research reinforces the justifications for using more than one period for looping in sampling. Since the loop is actually a periodic waveform, the number of samples in that loop of course determines the number of spectral components that can be present in the spectrum, and their frequency spacing. N samples are completely specified by N/2 complex Fourier components. At 44100 Hz sample rate, for a 256 sample loop, the spacing between frequencies would be 44100/256 = 172 Hz. Noise or other aperiodic components would be forced to fall on one of these bins. The longer the loop, the closer that spectral components can become, and the more aperiodic they can become. A truly aperiodic waveform would need an infinite loop, but our perception mechanism can be fooled into perceiving aperiodicity with a much shorter loop. 

Schemes to introduce instabilities and irregularities into purely periodic waveforms have tended to be less successful than simply using a loop with enough periods of a natural waveform to seem non-stationary. For many instruments, a loop of a less than...

Crest factor — Ratio between the peak value and the root mean square (RMS) value of a periodic waveform. Figure FI indicates the crest factor of two periodic waveforms. Crest factor is one indication of the distortion of a periodic waveform from its ideal characteristics. 

Form factor — Ratio between the RMS value and the average value of a periodic waveform. Form factor is another indicator of the deviation of a periodic waveform from the ideal characteristics. For example, the average value of a pure sinusoidal wave averaged over a cycle is 0.637 times the peak value. The RMS value of the sinusoidal wave is 0.707 times the peak value. The form factor, FF, is calculated as FF = 0.707/0.637 = 1.11. 

The Fourier expression is an infinite series. In this equation, V, represents the constant or the DC component of the waveform. Vu V2, V3,..., Vn are the peak values of the successive terms of the expression. The terms are known as the harmonics of the periodic waveform. The fundamental (or first harmonic) frequency has a... 

FIGURE 4.2 Nonsinusoidal voltage waveform Fourier series. The Fourier series allows expression of nonsinusoidal periodic waveforms in terms of sinusoidal harmonic frequency components. 

The waveform-type 3, which had a longer negative pressure period than waveform-type 1, resulted in more than 90% reduction in filtration resistance with a minimum TMP of about 40 kPa, implying the effect of duration of the backflushing. Waveform-types 3 and 4 were long and short sinusoidal forms, providing a continuous variation of pressure. There seemed to be less effective reduction in cake resistance with the shorter negative pressure period (waveform-type 4). 

Any periodic waveform, such as the square wave in Figure A.6.la, can be represented as a superposition of sinusoidal components (11-14) comprising a fundamental frequency /o = l/To, where is the period of the waveform, plus the harmonics of/o. That is. 

The addition of surfactants is not always recommended in normal and differential pulse polarography, because their presence may reduce the sensitivity of these methods in certain circumstances. Many other forms of polarography have been suggested and tested. Prominent among these are AC polarographic methods, which use sinusoidal and other periodic waveforms. Most modern instruments offer a range of techniques to the electroanalytical chemist. 

We saw in Section 4.2 that the plucked string supports certain spatial vibrations, called modes. These modes have a very special relationship in the case of the plucked string (and some other limited systems) in that their frequencies are all integer multiples of one basic sinusoid, called thefundamental. This special series of sinusoids is called a harmonic series, and lies at the basis of the Fourier series representation of shapes, waveforms, oscillations, etc. The Fourier series solves many types of problems, including physical problems with boundary constraints, but is also applicable to any shape or function. Any periodic waveform (repeating over and over again), can be transformed into a Fourier series, written as ... 

Frequency modulation relies on modulating the frequency of a simple periodic waveform with another simple periodic waveform. When the frequency of a sine wave of average frequency(called the carrier) is modulated by another sine wave of frequency (called the modulator), sinusoidal sidebands are created at frequencies equal to the carrier frequency plus and minus integer multiples of the modulator frequency. FM is expressed as ... 

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 ... 

Discrete Frequency Discrete Fourier Transform (DFT) Fourier analysis, periodic waveform... 

A harmonic is any integer multiple of the fundamental Ifequenr of a periodic waveform. 

Any periodic waveform can be made from a sum of suitably scaled and phase shifted sinusoids. This is the Fourier series. 

Fourier analysis can be used to find the amplitudes and phases of the component harmonics in a periodic waveform. 

Discrete frequency Discrete Fourier transform (DFT) Fourier analysis, periodic waveform Continuous frequency Discrete Fourier transform (DTFT) Fourier transform Continuous variable z-transform Laplace transform... 

When eyes are closed the sine-wave-shaped large-amplitude alpha rhythm appears in the scalp potentials, of which frequency fluctuates within a frequency range 8-13 Hz, and this almost periodic waveform is clearly seen in the spontaneous electroencephalogram. The instantaneous frequency is defined from the interval between consecutive zerocrossing points. Its power spectral density is found to be of 1// in a certain range of the Fourier frequency. It is another example of 1// fluctuations of biological rhythm phenomena. When a subject is in a... 

A periodic waveform repeats itself exactly at regular time intervals (the period T). It is predictive at any moment in the future we can foresee the exact value. According to Fourier, any periodic waveform can be considered to be the sum of a. fundamental sine wave of frequency fi = 1/T, and sine waves at certain discrete frequencies, the harmonics (2fi, 3fi, 4fi, and so on). A periodic waveform is an idealized concept the waveform is to have lasted and to last forever. At the time we start and stop it, other frequency components than the harmonics appear as transients. 

The sine wave is a very special periodic waveform in the sense fliat it is the only waveform containing just one frequency the fundamental frequency. Why does just the sine wave have such special qualities ... 

The only waveform containing just one frequency is the sine wave. A periodic waveform can be created by a sum of sine waves, each being a harmonic component of the sine wave at the fundamental frequency determined by the period. This is illustrated in Figure 8.6(a), showing the sum of a fundamental and its third and fifth harmonic... 

According to the Fourier series Eq. 8.25, any periodic waveform is the sum of a fundamental sinusoid and a series of its harmonics. Notice that, in general, each harmonic component consists of a sine and cosine component. Of course, either of them may be zero for a given waveform in the time domain. Such a waveform synthesis (summation) is done in the time domain, but each wave is a component in the frequency domain. The frequency spectrum of a periodic function of time f(t) is therefore a line spectrum. The amplitudes of each discrete harmonic frequency component is ... 

Figure 8.7 Square, ramp, and pulse periodic waveforms.

What then is the frequency content, for example, of a single rectangular pulse It can be found from the periodic waveform by letting the period < . The frequency spectrum F(w) of a positive pulse of amplitude A and duration T is ... 

A periodic waveform occupies a line spectrum an aperiodic waveform occupies a continuous frequency spectrum. 

Figure 8.9 Top Periodic waveform with a line harmonic frequency spectrum. Bottom Nonperiodic waveform has a continuous frequency spectrum. Line spectrum amplitude [volt]. Continuous spectrum amplitude [volt y s ].

According to Eq. 8.33, there may be a non-zero signal density at any frequency, thus a continuous spectrum is possible. With periodic waveforms, only line spectra were possible this is illustrated in Figure 8.9. 

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