EECE 3620: Linear Time-Invariant Systems: Chapter 2

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1 EECE 3620: Linear Time-Invariant Systems: Chapter 2 Prof. K. Chandra ECE, UMASS Lowell September 7, Continuous Time Systems In the context of this course, a system can represent a simple or complex entity that processes input signals and generates output signals. Mathematically, one can represent the relationship between the input and output signals using some form of equations. Recalling, that the notation for input signals is x(t) and that for output signals is y(t), examples of mathematical models of the system behavior are: y(t) = 3x(t) y(t) = x(t) dy + 2y(t) dt = x(t) d 2 y dt + 2dy + y(t) 2 dt = x(t)

2 y(t) = t 0 x(τ)dτ Essentially, the mathematical model of the system must provide a generative model for the output y(t) given the system input x(t). Given such a representation, one has to determine if the system is linear, non-linear, time invariant or time-varying. Additional properties that become important are whether the system is causal, stable or unstable. We focus attention on one particular class of systems that are known as linear, time-invariant (LTI) systems. Characterizing systems as LTI is often an approximation of their behavior. Most systems exhibit some form of non-linearity and time varying behavior. However, the LTI characterization allows us to analyze the behavior of the system for a variety of input signals in a very insightful way. It also enables an efficient characterization in the frequency domain. 2 Linear Time-Invariant Systems A LTI system has to satisfy properties of linearity and time-invariance. Let us denote the relationship between output and input by the notation y(t) = S[x(t)] where S denotes the system operator. A system is said to be linear if it satisfies the properties of scaling and superposition. Scaling refers to the case where changing the amplitude of the input by a certain amount, results in a change in amplitude of the output by the same amount. To test this, consider two input signals x 1 (t) = x(t) and x 2 (t) = αx(t) = αx 1 (t) where α is the amplitude scaling parameter. Let the corresponding outputs be, y 1 (t) = 2

3 S[x 1 (t)] and y 2 (t) = S[x 2 (t)]. Substituting for x 2 (t), y 2 (t) = S[αx 1 (t)]. If the system is such that y 2 (t) = αs[x 1 (t)] = αy 1 (t), then the system satisfies the scaling property. The second requirement for a system to be linear is that it should satisfy the superposition property. Consider two inputs x 1 (t) and x 2 (t) producing as outputs y 1 (t) and y 2 (t) respectively. The system satisfies the superposition property if the input x 3 (t) = αx 1 (t) + βx 2 (t) produces an output y 3 (t) = αy 1 (t) + βy 2 (t). Consider the system y(t) = ax(t) + b. Is this system linear? The answer is no. Notice that the scaling property is not satisfied. If the input is changed from x(t) to αx(t), the output does not change from y(t) to αy(t). Rather the output is y(t) = a αx(t) + b. Only if b is zero, will this system be linear. 2.1 Time Invariance In order to determine if the system is time invariant, you have to show that an input that is either delayed or advanced in time will create an output that is correspondingly delayed or advanced. For example. if x 1 (t) = x(t) and x 2 (t) = x 1 (t t 0 ), then y 1 (t) = S[x 1 (t)] and y 2 (t) = S[x 2 (t)] = S[x 1 (t t 0 )]. If you can show that y 2 (t) = y 1 (t t 0 ) for any t 0 then the system is time invariant. Consider a system defined by y(t) = 3tx(t) + 2. Given x 1 (t) and x 2 (t) as defined above, the corresponding responses are y 1 (t) = 3tx 1 (t) + 2 and y 2 (t) = 3tx 2 (t)+2 = 3tx 1 (t t 0 )+2. Note that y 1 (t t 0 ) = 3(t t 0 )x(t t 0 )+2. Therefore y 2 (t) y 1 (t t 0 ) and the system is not time invariant. Observe that in the system definition, y(t) = 3tx(t) + 2, the coefficient 3

4 of x(t) is a time-varying variable. The coefficients are determined by the system and in this case they are time-varying. Can you verify that the system y(t) = sin(3x(t)) is in fact time-invariant, whereas the system y(t) = e tx(t) is time-varying? 3 Convolution Integral In the rest of this course, we will be working with LTI systems. We derive next a mathematical model that can describe a linear time-invariant (LTI) system. Consider as an input the unit delta function, i.e. x(t) = δ(t). Let the corresponding output of the LTI system be denoted as y(t) = h(t). Therefore, h(t) = S[δ(t)]. We call h(t) as the impulse response of the system. If the impulse is shifted in time as δ(t τ) then the output is h(t τ) since the system is time invariant. Now consider that the input δ(t τ) is scaled by x(τ), creating a new input x(τ)δ(t τ), resulting in an output x(τ)h(t τ). Thus far we have applied the time-invariance and the scaling property. To apply the superposition principle, consider as input x(τ)δ(t τ)dτ. Due to the linearity property being satisfied by the system, the output is y(t) = x(τ) h(t τ)dτ. By the property of integration across delta functions, we can also see that x(t) = x(τ)δ(t τ)dτ. Therefore, in conclusion we see that an input x(t) applied to a LTI system produces an output y(t) that is expressed as the integral, y(t) = x(τ) h(t τ)dτ (1) This integral, relating the input and output of the LTI system through its impulse response h(t) is known as the convolution integral. Note that the 4

5 left hand side is the output y(t) determined at time t and the right hand side is an integral involving known functions which are the input x(τ) and the system impulse response h(t τ). The integration variable is τ and the time of observation t is a constant. Observe that the function h(t τ) represents the impulse response that has been first reflected, that is h(τ) h( τ) and then shifted so that the origin is located at the observation point t, that is h(t τ) The convolution integral given in Eqn. 1 can be alternately represented by transforming the integration variable t τ = ζ, leading to dτ = dζ which results in y(t) = x(t ζ) h(ζ)dζ = x(t τ)h(τ)dτ (2) The convolution integral completely characterizes the LTI system if the system impulse response h(t) is known. The integral can determine the system response for any input signal x(t) given h(t). The impulse response can be determined several ways, either by measurement or by using the mathematical model of the system (differential or integral equations) that defines the input-output relationship. Two other properties of systems are important to consider. They are causality and stability. 4 Causal Systems A system is said to be causal if it produces an output at times equal to or greater than the time at which the input is applied. In other words, the 5

6 system cannot produce an output before an input is applied. Therefore if the input is delayed to begin at time t 0 = 10, i.e. x(t 10), then the output y(t) = 0 t < 10 for a causal system and the response begins for t 10. So how do we determine if the system is causal? We take a look at the system impulse response h(t) which is the response to the impulse input x(t) = δ(t). Note that the input is applied at t = 0. Therefore, for the system to be causal, the response (i.e. its output ) should occur at times greater than or equal to t = 0. This implies that the system impulse response h(t) = 0 for t < 0. A causal system is characterized by an impulse response function h(t) = 0 for t < 0. In such a case, the limits of integration of the convolution integral given in Eqn. 2 can be modified as, y(t) = 0 x(t τ)h(τ)dτ (3) 5 System Stability System stability refers to the property of the system not being able to generate unbounded outputs when driven by bounded inputs. Here boundedness refers to the bounds on the amplitude of the signal. Bounded input bounded output stability implies that for inputs x(t) <, outputs y(t) <. The convolution integral given in Eqn. 2 can be applied to show that the condition for system stability can be given in terms of the system impulse response h(t) as follows. Here we assume that the input is bounded as x(t) M y(t) = x(t τ)h(τ)dτ 6

7 M x(t τ)h(τ) dτ y(t) ML < h(τ) dτ where h(τ) dτ L. Therefore, the condition for system to be stable is that its impulse response be absolutely integrable. 6 Summary In this chapter, you were introduced to a particular class of systems that can be approximated by properties of linearity and time-invariance (LTI). The system impulse response function h(t) was shown to be fundamentally important in the characterization of LTI systems. With the knowledge of h(t), the response of the LTI system to any input signal x(t) can be determined using the convolution integral. The impulse response can also be used to determine if the system is causal and stable. 7