![]() Let’s say that our system is a 50 Ω resistor the input signal is a current and the output signal is a voltage. In an additive system, if the input is x C(t), the output y C(t) is the addition of the two original output signals: y C(t) = y 1(t) + y 2(t). ![]() Now, we create a composite signal x C(t) by adding the original two input signals: x C(t) = x 1(t) + x 2(t). Let’s say that the input signals x 1(t) and x 2(t) produce output signals y 1(t) and y 2(t). Another way to think about this is that the system treats the signals that preceded the summation as though they are separate, despite the fact that they are delivered and processed as one signal.Īdditivity might be more easily explained via math. If the input delivered to an additive system is a composite signal created by adding separate signals, the output will also be a composite signal containing modified versions of these added signals. When we’re interested in the relationship between current and power (or voltage and power), the resistor is a nonlinear system. But it’s not really the component itself that is linear-rather, the relationship between voltage and current, which we call Ohm’s law, is linear. The example of the resistive heater is interesting because most of us, I think, would naturally identify a resistor as a linear component. In this case, kx(t) ≠ ky(t), and the system is not homogeneous. Thus, if we increase the current by a factor of ten (i.e., k = 10), the output increases by a factor of one hundred. ![]() The power dissipated as heat is proportional to the square of the amount of current flowing through the resistor. The input signal, x(t), is a current produced by a current source, and the output signal, y(t), is the heat generated by the resistive element. Now let’s imagine that our system is a resistive heater. The output response of a low-pass filter, which is a linear system, scales with the input signal. You can multiply the input signal by any factor and the output signal will be multiplied by the same factor. The amplitude of the output signal depends on the frequency of the input signal, but scaling the input signal will always result in equal scaling of the output signal, regardless of frequency. The fundamental idea is that scaling the input signal (i.e., multiplying it by a constant) will scale the output in the same way. The principle of homogeneity is also called the scalar rule or the scaling property. If this same thing happens for any input signal and any constant, the system exhibits homogeneity. If the system is linear, the new output signal will be ky(t). Then, we apply a signal kx(t), where k is a constant. Let’s say we apply an input signal x(t) to a system, and it produces an output signal y(t). To determine if a system is linear, we need to answer the following question: When an input signal is applied to the system, does the output response exhibit homogeneity and additivity? If a system is both homogeneous and additive, it is a linear system. Linearity is the key to mathematical analysis and manipulation in signal-processing applications: a concept known as superposition is the foundation of digital signal processing, and superposition is applicable only when we’re working with linear systems. For example, linear audio amplifiers produce sound with less distortion, and linear operation in an RF signal chain helps a receiver to correctly decode quadrature-amplitude-modulated (QAM) waveforms. In circuit design, we often strive for linearity because it leads to desirable output characteristics. We have powerful mathematical tools that help us to analyze systems that are both linear and time-invariant, and many physical phenomena can be accurately modeled as linear systems-even though these phenomena are usually not perfectly linear when we account for all the details. Systems that fall into the “linear” category are of special interest for engineers. ![]() A system can be something as simple as an RC low-pass filter or as complex as a microprocessor. These components can be physical devices, such as resistors and transistors, or computational processes, such as addition and multiplication. For our purposes, a system is a component or collection of components that accepts an input signal and produces an output signal.
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