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Hint: Use Eq. (6.156) in Prob. 6.37. in .NET
Hint: Use Eq. (6.156) in Prob. 6.37. Denso QR Bar Code Reader In .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. QR Code 2d Barcode Maker In .NET Using Barcode generator for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications. CHAP. 61 FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
QR Code JIS X 0510 Scanner In Visual Studio .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Drawing Barcode In Visual Studio .NET Using Barcode generation for VS .NET Control to generate, create bar code image in .NET framework applications. Consider a continuoustime LTI system with the system function
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Code128 Recognizer In VB.NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Linear Barcode Encoder In Java Using Barcode printer for Java Control to generate, create 1D Barcode image in Java applications. ( b ) Let R,, and R , be the specified frequencies ( < T ) of the prototype lowpass filter and the new lowpass filter, respectively. Then show that Hint: ein~ a Set einpl = 1  a e i n ~and solve for a.
Consider a discretetime prototype lowpass filter with system function
( a ) Find the 3dB bandwidth of the prototype filter. (6) Design a discretetime lowpass filter from this prototype filter so that the 3dB bandwidth of the new filter is 2 1 ~ / 3 . FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
Hint: Ans.
Use the result from Prob. 6.77. (a) a,,,= Determine the DFT of the sequence
I aN Ans. X [ k ] = 1  a e  i ( 2 r / N ) k
k = 0 . 1 , ..., N  1 Evaluate the circular convolution where
( a ) Assuming N = 4.
( b ) Assuming N = 8 . Ans. (a) y [ n ] = ( 3 , 3 , 3 , 3 ) ( b ) y[nI=~l,2,3,3,2,l,O,O) Consider the sequences x[nl and h[nl in Prob. 6.80. ( a ) Find the 4point DFT of x[nl, hln], and y [ n ] . ( b ) Find y [ n ] by taking the IDFT of Y [ k ] . Ans. ( a ) [ X[Ol, X[11, X[21, X[311 = [4,O,0,Ol [H[Ol,H[11, HI21, H[311= [3, j , 1 , jl [ Y[Ol,Y [11, Y P l , Y[311= [ 12,0,0,01 (6) y [ n I = { 3 , 3 , 3 , 3 ) Consider a continuoustime signal A t ) that has been prefiltered by a lowpass filter with a cutoff frequency of 10 kHz. The spectrum of x ( t ) is estimated by use of the Npoint DFT. The desired frequency resolution is 0.1 Hz. Determine the required value of N (assuming a power of 2 ) and the necessary data length T I . Ans. 2'' and T , = 13.1072 s
7
State Space Analysis
7.1 INTRODUCTION
So far we have studied linear timeinvariant systems based on their inputoutput relationships, which are known as the external descriptions of the systems. In this chapter we discuss the method of state space representations of systems, which are known as the internal descriptions of the systems. The representation of systems in this form has many advantages: 1. It provides an insight into the behavior of the system. 2. It allows us to handle systems with multiple inputs and outputs in a unified way. 3. It can be extended to nonlinear and timevarying systems. Since the state space representation is given in terms of matrix equations, the reader should have some familiarity with matrix or linear algebra. A brief review is given in App. A. 7.2 THE CONCEPT OF STATE A.
Definition: The state of a system at time to (or n o ) is defined as the minimal information that is sufficient to determine the state and the output of the system for all times t 2 to (or n 2 n o ) when the input to the system is also known for all times t 2 to (or n 2 n o ) . The variables that contain this information are called the state variables. Note that this definition of the state of the system applies only to causal systems. Consider a singleinput singleoutput LTI electric network whose structure is known. Then the complete knowledge of the input x ( t ) over the time interval  m to t is sufficient to determine the output y ( t ) over the same time interval. However, if the input x ( t ) is known over only the time interval t o to t , then the current through the inductors and the voltage across the capacitors at some time to must be known in order to determine the output y ( t ) over the time interval to to t . These currents and voltages constitute the "state" of the network at time t o . In this sense, the state of the network is related to the memory of the network. Selection of State Variables: Since the state variables of a system can be interpreted as the "memory elements" of the system, for discretetime systems which are formed by unitdelay elements, amplifiers, and adders, we choose the outputs of the unitdelay elements as the state variables of the system (Prob. 7.1). For continuoustime systems which are formed by integrators, amplifiers, and adders, we choose the outputs of the integrators as the state variables of the system (Prob. 7.3). For a continuoustime system containing physical energystoring ele

