Hint: Use Eq. (6.156) in Prob. 6.37. in .NET

Make QR Code ISO/IEC18004 in .NET Hint: Use Eq. (6.156) in Prob. 6.37.

Hint: Use Eq. (6.156) in Prob. 6.37.
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CHAP. 61 FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
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Consider a continuous-time LTI system with the system function
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Determine the frequency response H d ( R ) of the discrete-time system designed from this system based on the impulse invariance method. -in , where T, is the sampling interval of h c ( t ) . Am. H ( n ) = T, e-Ts ( 1 - e-T3e-ia )
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Consider a continuous-time LTI system with the system function
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H A S )=
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Determine the frequency response H d ( R ) of the discrete-time system designed from this system based on the step response invariance, that is,
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where s c ( t ) and s d [ n ] are the step response of the continuous-time and the discrete-time systems, respectively.
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Let H p ( z ) be the system function of a discrete-time prototype low-pass filter. Consider a new discrete-time low-pass filter whose system function H ( z ) is obtained by replacing z in H p ( z ) with ( z - a ) / ( l - a z ) , where a is real.
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( a ) Show that
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( b ) Let R,, and R , be the specified frequencies ( < T ) of the prototype low-pass filter and the new low-pass filter, respectively. Then show that
Hint:
ein~ a Set einpl = 1 - a e i n ~and solve for a.
Consider a discrete-time prototype low-pass filter with system function
( a ) Find the 3-dB bandwidth of the prototype filter. (6) Design a discrete-time low-pass filter from this prototype filter so that the 3-dB bandwidth of the new filter is 2 1 ~ / 3 .
FOURIER ANALYSIS OF DISCRETE-TIME 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 4-point 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 continuous-time signal A t ) that has been prefiltered by a low-pass filter with a cutoff frequency of 10 kHz. The spectrum of x ( t ) is estimated by use of the N-point 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 time-invariant systems based on their input-output 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 time-varying 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 single-input single-output 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 discrete-time systems which are formed by unit-delay elements, amplifiers, and adders, we choose the outputs of the unit-delay elements as the state variables of the system (Prob. 7.1). For continuous-time 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 continuous-time system containing physical energy-storing ele-
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