Fiber Optic Tutorials

This is a continuation from the previous tutorial - Fourier series and LTI systems.   In a variety of applications, it is of interest to change the relative amplitudes of the frequency components in a signal or perhaps eliminate some frequency components entirely, a process referred to as filtering. Linear time-invariant systems that change the shape of the spectrum are often referred to as frequency-shaping filters. Systems that are designed to pass some frequencies essentially undistorted and significantly attenuate or eliminate others are referred to as frequency-selective filters. As indicated by eqs. (3.124) and (3.131) [refer to the Fourier series...

This is a continuation from the previous tutorial - Fourier series representation of discrete-time periodic signals.   In the preceding few tutorials, we have seen that the Fourier series representation can be used to construct any periodic signal in discrete time and essentially all periodic continuous-time signals of practical importance. In addition, in the response of LTI systems to complex exponentials tutorial we saw that the response of an LTI system to a linear combination of complex exponentials takes a particularly simple form. Specifically, in continuous time, if \(x(t)=e^{st}\) is the input to a continuous-time LTI system, then the output is...

This is a continuation from the previous tutorial - properties of continuous-time Fourier series.   In this tutorial, we consider the Fourier series representation of discrete-time periodic signals. While the discussion closely parallels that of Fourier series representation of continuous-time periodic signals, there are some important differences. In particular, the Fourier series representation of a discrete-time periodic signal is a finite series, as opposed to the infinite series representation required for continuous-time periodic signals. As a consequence, there are no mathematical issues of convergence such as those discussed in the convergence of the Fourier series in the Fourier series representation...

This is a continuation from the previous tutorial - Fourier series representation of continuous-time periodic signals.   As mentioned in the Fourier series representation of continuous-time periodic signals tutorial, Fourier series representations possess a number of important properties that are useful for developing conceptual insights into such representations, and they can also help to reduce the complexity of the evaluation of the Fourier series of many signals. In Table 3.1 we have summarized these properties. In later tutorials, in which we develop the Fourier transform, we will see that most of these properties can be deduced from corresponding properties of...

This is a continuation from the previous tutorial - the response of LTI systems to complex exponentials.   1. Linear Combinations of Harmonically Related Complex Exponentials As defined in the exponential and sinusoidal signals tutorial, a signal is periodic if, for some positive value of \(T\), \[\tag{3.21}x(t)=x(t+T)\qquad\text{for all }t\] The fundamental period of \(x(t)\) is the minimum positive, nonzero value of \(T\) for which eq. (3.21) is satisfied, and the value \(\omega_0=2\pi/T\) is referred to as the fundamental frequency. In the exponential and sinusoidal signals tutorial we also introduced two basic periodic signals, the sinusoidal signal \[\tag{3.22}x(t)=\cos\omega_0t\] and the periodic complex exponential \[\tag{3.23}x(t)=e^{j\omega_0t}\]...