Unit impulse waveform

By definition, H gives the transfer function and frequency response for a unit impulse. In reality of course, the vocal tract input for vowels is the quasi-periodic glottal waveform. For demonstration piuposes, we will examine the effect of the /ih/ filter on a square wave, which we will use as a (very) approximate glottal source. We can generate the output waveforms y[n] by using the difference equation, and find the fi equency response of this vowel from //(e/ ). The input and output in the time domain and frequency domain are shown in figure 10.26. If the transfer function does indeed accurately describe ihe frequency behaviour of the filter, we should expect the spectra oiy[n, calculated by DFT to match H eJ )X(eJ ). We can see fiom figure 10.26 that indeed it does. 

Z(w) = 1/wC, and in the s-domain Z(s) = 1/sC. The Laplace transforms of some very important excitation waveforms are very simple for example, for a unit impulse it is 1, a unit step function 1/s, a ramp 1/s, etc. That is why the excitation with, for example, a unit impulse is of special interest examining the response of a system. In the extended immittance definition, calculations with some nonsinusoidal waveforms become very simple. Even so, Laplace transforms are beyond the scope of this book. 

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. 

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... 

The basic unit of the radar waveform is the impulse response h(n) of the transmit filter. It is expedient to view this response in the time domain as an FM chirp signal with envelope a(n) and phase ty(n)... 

---