Aperiodic waveforms

These are the waveforms of bioelectricity, because the repetition rate of, for example, the respiration or the heart is not perfectly periodic. 

A single pulse or a step function excitation is the basis of relaxation theory. Power dissipation and temperature rise may for instance impede the use of repeated waveforms, and single pulse excitation is necessary. A single pulse is a pulse waveform with repetition interval oo, it has a continuous frequency spectrum as opposed to a line spectrum. The unit impulse (delta function) waveform is often used as excitation waveform. It is obtained with the pulse width 0 and the pulse amplitude oo, keeping the product = 1. The frequency spectrum consists of equal contributions of all frequencies. In that respect, it is equal to white noise (see the following section). Also, the infinite amplitude of the unit pulse automatically brings the system into the nonlinear region. The unit impulse is a mathematical concept a practical pulse applied for the examination of a system response must have limited amplitude and a certain pulse width. 

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  

Like the sine wave containing only one frequency, white noise signal is the other extreme containing all frequencies of equal amplitudes. As with file sine wave or unit impulse, white noise is an ideal concept. It is a fractal curve any enlargement will just bring up similar curves. It is an interesting excitation waveform, because file system is examined at all 

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Fig. 1.18. A typical period-doubling sequence leading ultimately to aperiodic waveforms...

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. 

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

The total energy in an aperiodic waveform can be found in one of two ways, with the equality noted commonly attributed to Parseval,... 

The DTFT of a waveform is a continuous function of the discrete or sample frequency co. The DFT can be used to compute or approximate the DTFT of an aperiodic waveform at uniformly separated sample frequencies ka>o = k2n/N, where k is any integer between 0 and N — 1. If N samples fully describe a finite duration x(n), then... 

Aperiodic waveform This phrase is used to describe a waveform that does not repeat itself in a uniform, periodic manner. Compare with periodic waveform. 

An impulse is an intentionally applied voltage or current in a laboratory. It is in the form of an aperiodic and unidirectional waveform (Figure 17.2). It rises rapidly without appreciable oscillations to a maximum value and then falls, usually less rapidly, to zero, with small, if any, loops of opposite polarity (Figure 17.4). The parameters... 

With a recorded waveform, we generally must assume that it represents the sum of nonsynchronized aperiodic signals, for example, from exogenic sources and endogenic activities such as respiration, peristaltic movements, heart beats, and nerve activities. In addition, there may be wideband noise and noise at discrete frequencies, for example, from the power line 50- or 60-Hz fundamentals. 

Fourier Analysis The Fourier Series for Continuous-Time Periodic Functions The Fourier Transform for Continuous-Time Aperiodic Functions The Fourier Series for Discrete-Time Periodic Functions The Fourier Transform for Discrete-Time Aperiodic Functions Example Apphcations of Fourier Waveform Techniques... 

Most if not all periodic and aperiodic continuous-time waveforms x t), of practical interest, have Fourier series or Fourier transform counterparts. However, to aid in the existence question, the following sufficient technical constraints on the function x t), known generally as the Dirichlet conditions, guarantee convergence of the Fourier technique ... 

The Fourier transform (FT) for an aperiodic, continous-time waveform x t) is given by... 

In a similar fashion, the continuous-time FT can be approximated using the DFT. If an aperiodic, continuous-time waveform x t) is sampled at a rate of Ts samples per second to obtain x( ), in accordance with the Nyquist sampling theorem, then uniformaUy separated samples of the FT are approximated by... 

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