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2d barcode generator vb.net Consider a discretetime LTI system with impulse response h [ n ]given by in .NET framework
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[CHAP. 2
2.57. Consider a discretetime LTI system with impulse response h [ n ] given by
h [ n ]= (  ; ) l 1 u [ n  I ] ( a ) Is the system causal
( b ) Is the system stable
Ans.
( a ) Yes; ( b ) Yes
2.58. Consider the RLC circuit shown in Fig. 232. Find the differential equation relating the output current y ( t ) and the input voltage x ( t ) . Ans.
d2y(t) R dy(t) + y(t) =  1 &([I dl
Fig. 232 2.59. Consider the RL circuit shown in Fig. 233. Find the differential equation relating the output voltage y ( t ) across R and the input voltage x( t 1. Find the impulse response h ( t ) of the circuit. Find the step response d t ) of the circuit. Fig. 233 CHAP. 21
LINEAR TIMEINVARIANT SYSTEMS
Consider the system in Prob. 2.20. Find the output y ( t ) if x ( t ) = e"'u(t) and y(0) = 0. te"u(t) Is the system described by the differential equation
linear
Am. No, it is nonlinear
Write the inputoutput equation for the system shown in Fig. 234. 2 y [ n ]  y[n  11 = 4 x [ n ] 2 x [ n  11 Fig. 234 Consider a discretetime LTI system with impulse response
h[n]= n=0,1 otherwise
Find the inputoutput relationship of the system.
Am. y [ n ] = x [ n ] x[n  11 Consider a discretetime system whose input x [ n ] and output y [ n ] are related by
y[n] iy[n 11 =x[n] with y [  l ] = 0. Find the output y [ n ] for the following inputs: ( a ) x[nl = (f )"u[nl; )"u[nl ( b ) xtnl= (f
( a ) y [ n ] = 6[(;)"+'  ( f ) " + ' ] u [ n ] ( b ) y [ n l = ( n + lX;)"u[n] Consider the system in Prob. 2.42. Find the eigenfunction and the corresponding eigenvalue of the system. zn,A =
3
Laplace Transform and ContinuousTime LTI Systems
3.1 INTRODUCTION
A basic result from 2 is that the response of an LTI system is given by convolution of the input and the impulse response of the system. In this chapter and the following one we present an alternative representation for signals and LTI systems. In this chapter, the Laplace transform is introduced to represent continuoustime signals in the sdomain ( s is a complex variable), and the concept of the system function for a continuoustime LTI system is described. Many useful insights into the properties of continuoustime LTI systems, as well as the study of many problems involving LTI systems, can be provided by application of the Laplace transform technique. 3.2 THE LAPLACE TRANSFORM
In Sec. 2.4 we saw that for a continuoustime LTI system with impulse response h(t), the output y ( 0 of the system to the complex exponential input of the form e" is where
Definition: The function H ( s ) in Eq. (3.2) is referred to as the Laplace transform of h(t). For a general continuoustime signal x(t), the Laplace transform X(s) is defined as The variable s is generally complexvalued and is expressed as
The Laplace transform defined in Eq. (3.3) is often called the bilateral (or twosided) Laplace transform in contrast to the unilateral (or onesided) Laplace transform, which is defined as CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUSTIME LTI SYSTEMS
where 0  = lim,,,(O  E ) . Clearly the bilateral and unilateral transforms are equivalent only if x(t) = 0 for t < 0. The unilateral Laplace transform is discussed in Sec. 3.8. We will omit the word "bilateral" except where it is needed to avoid ambiguity. Equation (3.3) is sometimes considered an operator that transforms a signal x ( t ) into a function X(s) symbolically represented by and the signal x ( t ) and its Laplace transform X(s) are said to form a Laplace transform pair denoted as B. The Region of Convergence: The range of values of the complex variables s for which the Laplace transform converges is called the region of convergence (ROC). To illustrate the Laplace transform and the associated ROC let us consider some examples.

