Chapter 2 One Dimensional Continuous Time System

2.2 One Dimensional Continuous Time System Definition 1: An one dimensional analog system or continuous time system can be defined as a mapping function T which maps a real-valued analog signal f(t) to another real-valued g(t) such that

T f(t)T[f(t)] input system output

Definition 2: The mapping T is linear if it satisfies the following equations T[af 1 (t) + bf 2 (t)] = aT[f 1 (t)] + bT[f 2 (t)] (additivity) T[af(t)] = aT[f(t)] (homogeneity) for any If a system is not linear it is called a nonlinear system.

Example 1: A multiplier system is defined as The multiplier is an amplifier that converts a very small audio signal into a large audio to drive the speaker. The multiplier is a transformer that convert a low voltage sinusoidal wave into a high voltage sinusoidal wave or vice versa.

Example 2: A differentiator is defined as

Example 3: An integrator is defined as

Example 4: A delay system is defined as where D > 0 is a time delay.

Example 5: The following square wave if with the period 2  can be approximated by

f N (t) is the superposition (linear combination) of sine waves. It is easy to know that

Figure2.1: (a) N = 1 (b) N=3 (c) N = 5 (d) N = 10 (e) N=50 (f) N = 100

Figure 2.1 shows f 1 (t), f 3 (t), f 5 (t), f10(t), f 50 (t) and f 100 (t). As N increase There are overshoot and undershoot also increases at t = 0. the following MATLAB program show the plot of f 100 (t)

% % f(wc) = -1, -pi < wc < 0 % = 1, 0 <= wc < pi wc = 0.5 * pi; ws = 0.01* pi;

N = pi /ws; Nc = wc/ws; f = zeros(1, 2*N-1); f(1:N-1) = -1 * ones(1,N-1); f(N:2*N-1) = ones(1, N);

Nop = 100; w = (-pi + ws) : ws : (pi -ws); s = zeros(size(w) ); for i = 1:1:Nop s = s + 2/pi * (1 - (-1)^i )/i * sin(i*w); end; plot(w, f, w,s);

Example 6: Figure 2.2(a) shows f(t) = cos(200  t) cos(1000  t)). It is easy to know that

Figure 2.2(b) shows the analog signal Obviously the differentiation system attenuates low frequency signal and magnifies high frequency. It is a high pass system.

Figure 2.2: (a) f(t) = cos(200  t) cos(1000  t) (b) (c) (a)(b)(c)

Figure 2.2(c) show the analog signal The integration system attenuates high frequency and magnifies low frequency. it is a low pass system. Also, if f(t) is processed by an integration system then

Example 7: A square system is defined as

Example 8: A exponential system is defined as

Example 9: A natural logarithm system is defined as

Definition 3: The system T is linear time- invariant if

Example 10: The system g(t)  g(t  D) = f(t) is a linear time invariant system. Let h ’ (t) = T(  (t  a), is the output when  (t  a) Thus,

Assume that the system is linear time invariant.It is known that the output of the system is the impulse response h(t) when the input is  (t). At time t  a if the input is f(t  a) =  (t  a) then the output is

From the above two equations we can easily obtain The equality holds if Therefore, the system is time invariant.

Example 11: The system g(t)  tg(t  D) = f(t) is a time varying system Let h ’ (t) be the output when the input applies to the system. Thus, At time t  a we obtain

Assume that the system is linear time invariant. It is known that the output of the system is the impulse response h(t) when the input is d(t  a). Therefore, From the two previous equations we can easily obtain which implies

Thus, If the system is time invariant which is contraction to Equation h(t  a  D) = 0, Therefore,and the impulse response of the system is time varying.

Theorem 1: If h(t) is the impulse response of a linear invariant system and f(t) is the input of the system then the output is

h(t) f(t)

Definition 4: A system T is called causal if its present output does not depend on its future input

Definition 5: A system T is call BIBO stable if its input and output is bounded. That is, if then

Theorem 2: If the impulse response h(t) of the system is absolute integrable the system is BIBO stable.

Assume that the input f(t) is bounded. There exist a real number M 1 such that Since h(t) is absolute integrable there exists a real number M 2 such that

Then, the absolute output is The output g(t) is bounded.

Example 12: A multiplier system is defined as where k is a constant and is called the gain of the system.The system is an all pass filter.

Example 13: A time delay system is defined as where T is called the delay time. It can be easily that the system is linear time invariant. The system is casual since the impulse response h(t) =  (t  T) = 0 for t < 0. The output depends on its past input but does not depends on the future input.

is linear time invariant. However, it is not causal because the current output g(t) depends on future input f(t + T). In fact, its impulse response for t < 0. Note that the system

Example 14: A differentiation system is defined as It is a high pass filter. If f(t) = e j  t then g(t) =j  e j  t. For small , |g(t)| is very small while for large , |g(t)| is very large. It is a high pass filer.

Example 15: An integration system is defined as It is a low pass filter. If For small , |g(t)| is very large while for large , |g(t)| is very small. It is a low pass filter. The integration filter is linear time invariant.

Example 16: A fall-wave rectifier is defined as

Example 17: A half-wave rectifier is defined as if otherwise. Both the full wave rectifier and the half wave rectifier are nonlinear system.

Definition 6: the modulation of a signal f(t) is defined as where m(t) is called the modulating signal.

2.3 Linear Differential Equations Definition 7: an ordinary differential equation is an equation that has derivatives with respect to an independent variable only. If the equations has the derivatives with respect to at least two variables it is called an partial differential equation.

Definition 8: The ordinary differential equation is a linear differential equation of order N. If a differential equation can not be written as above equation it is nonlinear.

ak(t) ak(t) equation is said to be non- homogeneous; other- wise it it said to be homogeneous. If the initial conditions are given the differential equation is called the initial-value problem. Each Coefficient depends on on the variable t. If the

In practical systems, is usually assumed to be constant so that the system is LTI.

Example 18: The following equations are ordinary differential equations.

Example 19: The following equations are partial differential equations

Example 21: The following ordinary differential equations are linear.

Example 22: The following ordinary differential equations are nonlinear.

R +- Voltage across the resistor

L +- Voltage across the inductor

C +- Voltage across the capacitor

The Kirchhoff’s Current Law The sum of the currents at a node in a circuit is zero.

The Kirchhoff’s Voltage Law The sum of the voltages around a loop in a circuit is zero.

= 0

Example 23: A R-L series circuit shown in Figure 2.5 R L V +  i(t)i(t) Figure 2.5: The R-L series circuit.

The current i(t) that satisfies where V(t) is the input voltage. It is a first order differential equation.

Example 24: Figure 2.6(a) show a circuit where R and C in series. R C V +  i(t)i(t) Figure 2.6(a): The R-C series circuit.

The current flows in the circuit is i(t). The voltage across the capacitor is and the voltage across the resistor is Ri(t). Thus, The equation can be also written as where q(t) is the charge across the capacitor.

Example 25: A series R-L-C circuit shown in Figure 2.7 R L V(t) i(t)i(t) Figure 2.7: A R-L-C series circuit. C q(t)q(t)

The input voltage V(t) satisfies where q(t) denotes the charge in the capacitor and i(t) denotes the current of the circuit. It is a linear second differential equation.

The R, L, and C values in the circuit can change with time. However, they are assumed to be constant for analysis.