Simple periodic signals sinusoids

The sinusoid signal forms the basis of many aspects of signal processing. A sinusoid can be represented by either sine function or a cosine function 

It may not be immediately clear why the sine and cosine function, which we probably first en-coimtered in trigonometry, have an ihing to do with waveforms or speech. In fact it turns out that the sinusoid function has important interpretations beyond trigonometry and is found in many places in the physical world where oscillation and periodicity are involved. For example, bolli llie movement of a simple pendulum and a bouncing spring are described by sinusoid functions. 

We define frequency, F, as the number of times the signal repeats in unit time, and this is clearly the reciprocal of the period  

To save us writing 271 everywhere, a quantity called angular frequency is normally used, which is denoted by (O and has units radians per second  

The parameters (o and ( ) can thus be used to create a sinusoid with any frequency or phase. A final parameter, A is used the scale the sinusoid and is called the amplitude. This gives the general sinusoid function of  

Rgure 10.3 (a) The cosine function with phase shifts —1, —2, —3 and —4. (b) The cosine function with frequency fOO Hz. 

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Kinds oi Inputs Since a tracer material balance is represented by a linear differential equation, the response to anv one kind of input is derivable from some other known input, either analytically or numerically. Although in practice some arbitrary variation of input concentration with time may be employed, five mathematically simple input signals supply most needs. Impulse and step are defined in the Glossaiy (Table 23-3). Square pulse is changed at time a, kept constant for an interval, then reduced to the original value. Ramp is changed at a constant rate for a period of interest. A sinusoid is a signal that varies sinusoidally with time. Sinusoidal concentrations are not easy to achieve, but such variations of flow rate and temperature are treated in the vast literature of automatic control and may have potential in tracer studies. 

Inputs Although some arbitrary variation of input concentration with time may be employed, five mathematically simple tracer input signals meet most needs. These are impulse, step, square pulse (started at time a, kept constant for an interval, then reduced to the original value), ramp (increased at a constant rate for a period of interest), and sinusoidal. Sinusoidal inputs are difficult to generate experimentally. 

For a numerical demonstration, consider the case of the DWT of a simple signal using the Daubechies-4 wavelet filter (N/ = 4). The periodic input signal is one period of a sinusoidal waveform (N = 2 = 16), with matched end-points ... 

Plots of these functions against time are shown in Figure 10.2, and from these it should be clear that both are periodic, that is that they exactly repeat over time. The period T is the length of time between any two equivalent points on successive repetitions. In the simple sinusoid, the period is T = 27t that is, for any give point, the signal has the same value 271 seconds later. 

Again, as we would expect, because the periods were not sinusoidal, this has harmonics at multiples of the main spike. The amplitude effect will be present in the cepstrum also, but as this is varying much more slowly than the periods , it will be in the lower range of the cepstrum. As the amplitude effect is the spectral envelope and the spikes represent the harmonics (ie the pitch), we see that these operations have produced a representation of the original signal in which the two components lie at different positions. A simple filter can now separate them. 

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