Fiber Optic Tutorials

This is a continuation from the previous tutorial - causal LTI systems described by differential and difference equations.   In this tutorial, we take another look at the continuous-time unit impulse function in order to gain additional intuitions about this important idealized signal and to introduce a set of related signals known collectively as singularity functions. In particular, in the unit impulse and unit step functions tutorial we suggested that a continuous-time unit impulse could be viewed as the idealization of a pulse that is "short enough" so that its shape and duration is of no practical consequence—i.e., so that as...

This is a continuation from the previous tutorial - properties of linear time-invariant (LTI) systems.   An extremely important class of continuous-time systems is that for which the input and output are related through a linear constant-coefficient differential equation. Equations of this type arise in the description of a wide variety of systems and physical phenomena. For example, as we illustrated in the continuous-time and discrete-time signals tutorial, the response of the RC circuit in Figure 1.1 and the motion of a vehicle subject to acceleration inputs and frictional forces, as depicted in Figure 1.2, can both be described through...

This is a continuation from the previous tutorial - continuous-time LTI systems and convolution integral.   In the preceding two tutorials, we developed the extremely important representations of continuous-time and discrete-time LTI systems in terms of their unit impulse responses. In discrete time the representation takes the form of the convolution sum, while its continuous-time counterpart is the convolution integral, both of which we repeat here for convenience: \[\tag{2.39}y[n]=\sum_{k=-\infty}^{+\infty}x[k]h[n-k]=x[n]*h[n]\] \[\tag{2.40}y(t)=\displaystyle\int\limits_{-\infty}^{+\infty}x(\tau)h(t-\tau)\text{d}\tau=x(t)*h(t)\] As we have pointed out, one consequence of these representations is that the characteristics of an LTI system are completely determined by its impulse response. It is important to emphasize...

This a continuation from the previous tutorial - discrete-time LTI systems - the convolution sum.   In analogy with the results derived and discussed in the discrete-time LTI systems and convolution sum tutorial, the goal of this tutorial is to obtain a complete characterization of a continuous-time LTI system in terms of its unit impulse response. In discrete time, the key to our developing the convolution sum was the sifting property of the discrete-time unit impulse — that is, the mathematical representation of a signal as the superposition of scaled and shifted unit impulse functions. Intuitively, then, we can think of...

This is a continuation from the previous tutorial - continuous-time and discrete-time systems.   Introduction In previous tutorials we introduced and discussed a number of basic system properties. Two of these, linearity and time invariance, play a fundamental role in signal and system analysis for two major reasons. First, many physical processes possess these properties and thus can be modeled as linear time-invariant (LTI) systems. In addition, LTI systems can be analyzed in considerable detail, providing both insight into their properties and a set of powerful tools that form the core of signal and system analysis. Our goal is to develop an understanding...