Hwei hsu biography examples

Professional Experience Open Close. Research Interest Open Close. Selected Publication Open Close. Amartuvshin, O. Aging Cell, doi. Lin, K-Y. Nature Communications, 11, article number: Tseng, C-Y. Her research output in this area is encapsulated in the publication titled "Automatic Detecting Unethical Behavior in Open-source Software Projects.

For instance, she teaches the concept of "Scope of Requirement" by using the familiar "Buying food" example, emphasizing the importance of understanding customer needs before product development. As an online Teaching Assistant, she overcomes technical and participation challenges by providing GitHub backups for materials, enabling Live Chatting for prompt feedback, and using project progress charts to boost student engagement.

Hsu's teaching approach combines practicality with innovation, making her an effective educator. Hsu worked as a teaching assistant for the following subjects at UTS. Hsu worked in the following IT research roles after obtaining her PhD. Hsu worked in the following IT professional roles before joining the Ph. Bio In AugustHsu obtained her Ph.

Australian Joint Conference on Artificial Intelligence Verify the B I B 0 stability condition [Eq. Assume that the input x t of a continuous-time LTI system is bounded, that is, Ix t ll k, all t 2. The system shown in Fig. Then Thus, by definition 2. Consider a stable continuous-time LTI system with impulse response h t that is real and even.

Show that cos w t and sin w t are eigenfunctions of this system with the same real eigenvalue. Thus, H jw converges for any w. Then by Eqs. Let the system be represented by T. The continuous-time system shown in Fig. Write a hwei hsu biography examples equation that relates the output y t and the input x t 1. Then the input-output relation of the integrator is given by Differentiating both sides of Eq.

T h e continuous-time system shown in Fig. Write a differential equation that relates the output y t and the input x t. Note that, in general, the order of a continuous-time LTI system consisting of the interconnection of integrators and scalar multipliers is equal to the number of integrators in the system. Consider a continuous-time system whose input x t and output y t are related by where a is a constant.

Consider the system in Prob. This is shown as follows. Let x J t and x, t be two input signals and let yt and y, r be the corresponding outputs. Multiplying Eq. Find the impulse response h r of the system. The impulse response h t should satisfy the differential equation The homogeneous solution h h t to Eq. Otherwise, h t would have a derivative of S t that is not part of the right-hand side of Eq.

Substituting Eq. Substituting these values in Eq. The input x [ n ] and the impulse response h [ n ] of a discrete-time LTI system are given by a Compute the output y[n] by Eq. For n r 0, we have Changing the variable of summation k to m a Fig. For n 2 0, we have Thus, we obtain the same result as shown in Eq. I36bwe obtain 2. Thus, summing x [ k ] h [ n- k ] for 0 s n 2 5, we obtain which is plotted in Fig.

If x[ n ] and x 2 [ n ] are both periodic sequences with common period N,the convolution of x[ n ] and x 2 [ n ] does not converge. In this case, we define the periodic convolution of x[ n ] and x 2 [ n ] as Show that f [ n ] is periodic with period N. Since x[ n ] is periodic with period N, we have Then from Eq. Show that if the input x[n] to a discrete-time LTI system is periodic with period Nthen the output y[n] is also periodic hwei hsu biography examples period N.

Let h[n] be the impulse response of the system. The impulse response h[n] of a discrete-time LTI system is shown in Fig. Determine and sketch the output y[n] of this system t o the input x[n] shown in Fig. From this definition derive the causality condition 2. Consider a discrete-time LTI system whose input x [ n ] and output y [ n ] are related by Is the system causal?

By definition 2. Thus, the system is not causal. Assume that the input x [ n ] of a discrete-time LTl system is bounded, that is, Ix[n]l l k l all n 2. Consider a discrete-time LTI system with impulse response h [ n ] given by Is this system causal? Write a difference equation that relates the output y [ n ] and the input x [ n ]. Thus, from Fig.

Write a difference equation that relates the output y[n] and the input x [ n ]. Note that, in general, the order of a discrete-time LTI system consisting of the interconnection of unit delay elements and scalar multipliers is equal to the number of unit delay elements in the system. Consider the discrete-time system in Fig. Unit delay I qb- 11 Fig.

Then from Fig. Setting n and 2. Thus, Combining Eqs. Note also that Eq. Consider the discrete-time system in Prob. We can solve Eq. Find the impulse response h [ n ] for each of the causal LTI discrete-time systems satisfying the following difference equations and indicate whether each system is a FIR or an IIR system. Supplementary Problems 2. Show that for an arbitrary starting point no.

Hint: See Probs. Consider an integrator whose input x t and output y t are related by a Find the impulse response h t of the integrator. Consider the RLC circuit shown in Fig. Find the differential equation relating the output current y t and the input voltage x t. Consider the RL circuit shown in Fig. 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. No, it is nonlinear 2. Write the input-output equation for the system shown in Fig. Find the eigenfunction and the corresponding eigenvalue 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 continuous-time signals in the s-domain s is a complex variableand the concept of the system function for a continuous-time LTI system is described. Many useful insights into the properties of continuous-time LTI systems, as well as the study of many problems involving LTI systems, can be provided by application of the Laplace transform technique.

Definition: The function H s in Eq. For a general continuous-time signal x tthe Laplace transform X s is defined as The variable s is generally complex-valued and is expressed as The Laplace transform defined in Eq. The unilateral Laplace transform is discussed in Sec. We will omit the word "bilateral" except where it is needed to avoid ambiguity.

Equation 3. 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. Thus, the ROC for this example is specified in Eq. In Laplace transform applications, the complex plane is commonly referred to as the s-plane.

The horizontal and vertical axes are sometimes referred to as the a-axis and the jw-axis, respectively. Comparing Eqs. Therefore, in order for the Laplace transform to be unique for each signal x tthe ROC must be specified as par1 of the transform. Poles and Zeros of X s 1: Usually, X s will be a rational function in s, that is, The coefficients a, and b, are real constants, and m and n are positive integers.

Similarly, the roots of the denominator polynomial, p, are called the poles of X s because X s is infinite for those values of s. Therefore, the poles of X s lie outside the ROC since X s does not converge at the poles, by definition. The zeros, on the other hand, may lie inside or outside the ROC. Thus, a very compact representation of X s in the s-plane is to show the locations of poles and zeros in addition to the ROC.

Traditionally, an " x " is used to indicate each pole location and an " 0 " is used to indicate each zero. The properties of the ROC are summarized below. We assume that X s is a rational function of s. Property 1: The ROC does not contain any poles. Property 5: If x t is a two-sided signal, that is, x t is an infinite-duration signal that is neither right-sided nor left-sided, then the ROC is of the form where a, and a, are the real parts of the two poles of X s.

Note that Property 1 follows immediately from the definition of poles; that is, infinite at a pole. For verification of the other properties see Probs. Instead of having to reevaluate the transform of a given signal, we can simply refer to such a table and hwei hsu biography examples out the desired transform. Verification of these properties is given in Probs.

Thus, Eq. The corresponding effect on the ROC is illustrated in Fig. Time Reversal: If Fig. Integration in the Time Domain: If then Equation 3. Table summarizes the properties of the Laplace transform presented in this section. The evaluation of this inverse Laplace transform integral requires an understanding of complex variable theory.

From the linearity property 3. Partial-Fraction Expansion: If X s is a rational function, that is, of the form a simple technique based on partial-fraction expansion can be used for the inversion of Xb. Simple Pole Case: If all poles of X sthat is, all zeros of D sare simple or distinctthen X s can be written as where coefficients ck are given by If D s has multiple roots, that is, if it contains factors of the form s -pi ', we say that pi is the multiple pole of X s with multiplicity r.

The inverse Laplace transform of Q s can be computed by using the transform pair 3. The System Function: In Sec. The system function H s completely characterizes the system because the impulse response h t completely characterizes the system. Figure illustrates the relationship of Eqs. Stabilio: In Sec. Similarly, the impulse response of a parallel combination of two LTI systems [Fig.

Definitions: The unilateral or one-sided Laplace transform X, s of a signal x t is defined as [Eq. Since x t in Eq. Basic Properties: Most of the properties of the unilateral Laplace transform are the same as for the bilateral transform. The unilateral Laplace transform is useful for calculating the response of a causal system to a causal input when the system is described by a linear constantcoefficient differential equation with nonzero initial conditions.

The basic properties of the unilateral Laplace transform that are useful in this application are the time-differentiation and time-integration properties which are different from those of the bilateral transform. They are presented in the following. Repeated application of this property yields where 2. Integration in the Time Domain: C.

Transform Circuits: The solution for signals in an electric circuit can be found without writing integrodifferential equations if the circuit operations and signals are represented with their Laplace transform equivalents. In order to use this technique, we require the Laplace transform models for individual circuit elements. These models are developed in the following discussion and are shown in Fig.

Applications of this transform model technique to electric circuits problems are illustrated in Probs. Signal Sources: where u t and i t are the voltage and current source signals, respectively. Thus, the ROC of X s includes the entire s-plane. Note that from Eq. But this is not the case. Show that if x t is a right-sided signal and X s converges for some value of s, then the R O C of X s is of the form equals the maximum real part of any of the poles of X s.

The signal x t is sketched in Figs. From Table H - 1 s-a c Fig. Verify the time-shifting property 3. Verify the time-scaling property 3. Verify the time differentiation property 3. Verify the differentiation in s property 3. Verify the integration property 3. Using the various Laplace transform properties, derive the Laplace transforms of the following signals from the Laplace transform of u t.

HS all s c Using the differentiation in s property 3. Verify the convolution property 3. If a zero of one transform cancels a pole of the other, the ROC of Y s may be larger. Thus, we conclude that 3. Using the convolution property 3. Thus, x t is a double-sided signal and from Table we obtain 3. Thus, x t is a right-sided signal and from Table we obtain into the above expression, after simple computations we obtain Alternate Solution: We can write X s as As before, by Eq.

Thus, x t is a right-sided signal and from Table we obtain Note that there is a simpler way of finding Awithout resorting to differentiation. This is shown as follows: First find cand A, according to the regular procedure. Then substituting the values of cand A, into Eq. Thus, x t is a right-sided signal and from Table and Eq. Using the differentiation in s property 3.

Find the system function H s and the impulse response h t of the RC circuit in Fig. Using the Laplace transform, redo Prob. From Prob. The output y t of a continuous-time LTI system is found to be 2e-3'u t when the input x t is u t.

Hwei hsu biography examples: More than 40 million

Using the Laplace transfer, redo Prob. Then h r is noncausal that is, a left-sided signal and from Table we get The feedback interconnection of two causal subsystems with system functions F s and G s is depicted in Fig. Find the overall system function H s for this feedback system. Thus, taking the unilateral Laplace transform of the above equation and using Eq.

Again using Eq. This definition is sometimes referred to as the 0 definition. Using the unilateral Laplace transform, redo Prob. Consider t h e RC circuit shown in Fig. Assume a Find the current i t. When the current i t is the output and the input is r, tthe differential equation governing the circuit is 1 3. Using the transform network technique, redo Prob.

In the hwei hsu biography examples in Fig. Find the inductor current i t for t 2 0. When the switch is in the closed position for a long time, the capacitor voltage is charged to 10 V and there is no current flowing in the capacitor. Next, using Fig. Thus, we have 3. Consider the circuit shown in Fig. The voltages on capacitors C, and C, before the switches are closed are 1 and 2 V, respectively.

This step change in voltages will result in impulses in it and i 2 t. Circuits having a circuits. Proceed in a manner similar to Prob. Differentiate both sides of Eq. If c Hint: a Use Eqs. Find the Laplace transform of x t Ans. Hint: Use Eq. Using the Laplace transform, show that Use Eq. Hint: a Find the system function H s by Eq. The step response of an continuous-time LTI system is given by 1 - e-' u t.

Find the input x t. Hint: Use the result from Prob. Find the unilateral Laplace transforms of the periodic signals shown in Fig. Consider the RC circuit in Fig. The switch is closed at t the switch closing is u. Find the capacitor voltage for t 2 0. Before the switch closing, the capacitor Cis charged to u, V and the capacitor Cis not charged. In this chapter we present the z-transform, which is the discrete-time counterpart of the Laplace transform.

The z-transform is introduced to represent discrete-time signals or sequences in the z-domain z is a complex variableand the concept of the system function for a discrete-time LTI system will be described. The Laplace transform converts integrodifferential equations into algebraic equations. In a similar manner, the z-transform converts difference equations into algebraic equations, thereby simplifying the analysis of discrete-time systems.

The properties of the z-transform closely parallel those of the Laplace transform. However, we will see some important distinctions between the z-transform and the Laplace transform. Definition: T h e function H z in Eq. The z-transform defined in Eq. The unilateral z-transform is discussed in Sec. As in the case of the Laplace transform, Eq.

The Region of Convergence: As in the case of the Laplace transform, the range of values of the complex variable z for which the z-transform converges is called the region of convergence. T o illustrate the z-transform and the associated R O C let us consider some hwei hsu biographies examples. Then Alternatively, by multiplying the numerator and denominator of Eq.

Consequently, just as with rational Laplace transforms, it can be characterized by its zeros the roots of the numerator polynomial and its poles the roots of the denominator polynomial. The ROC and the pole-zero plot for this example are shown in Fig. In z-transform applications, the complex plane is commonly referred to as the z-plane. Thus, as in the Laplace Fig.

We assume that X Z is a rational function of z. Property 5: If x [ n ] is a two-sided sequence that is, x [ n ] is an infinite-duration sequence that is neither right-sided nor left-sided and X z converges for some value of z, then the ROC is of the form where rand r, are the magnitudes of the two poles of X z. Note that Property 1 follows immediately from the definition of poles; that is, X z is infinite at a pole.

For verification of the other properties, see Probs. Unit Impulse Sequence 61 nl: From definition 1. Unit Step Sequence d n l : Setting a C. Table Accumulation Convolution H. Summary of Some z-transform Properties For convenient reference, the properties of the z-transform presented above are summarized in Table Inversion Formula: As in the case of the Laplace transform, there is a formal expression for the inverse z-transform in terms of an integration in the z-plane; that is, where C is a counterclockwise contour of integration enclosing the origin.

Formal evaluation of Eq. From the linearity property 4. Power Series Expansion: The defining expression for the z-transform [Eq. Thus, if X z is given as a power series in the form we can determine any particular value of the sequence by finding the coefficient of the appropriate power of 2 - '. This approach may not provide a closed-form solution but is very useful for a finite-length sequence where X z may have no simpler form than a polynomial in z - ' see Prob.

For rational r-transforms, a power series expansion can be obtained by long division as illustrated in Probs. Partial-Fraction Expansion: As in the case of the inverse Laplace transform, the partial-fraction expansion method provides the most generally useful inverse z-transform, especially when X t z is a rational function of z. Equation 4.

The system function H z completely characterizes the system. Similarly, if the system is anticausal, that is, then h[n] is left-sided and the ROC of H z must be of the form That is, the ROC is the interior of a circle containing no poles of H z in the z-plane. Stability: In Sec. Thus, the unilateral z-transform of x[n] can be thought of as the bilateral transform of x[n]u[n].

Since x[n]u[n] is a right-sided sequence, the ROC of X, z is always outside a circle in the z-plane. Basic Properties: Most of the properties of the unilateral z-transform are the same as for the bilateral z-transform. The unilateral z-transform is useful for calculating the response of a causal system to a causal input when the system is described by a linear constant-coefficient difference equation with nonzero initial conditions.

The basic property of the unilateral z-transform that is useful in this application is the following time-shifting property which is different from that of the bilateral transform. Find the z-transform of a From Eq. Note that X z includes both positive powers of z and negative powers of z. The remaining zeros of X z are at The pole-zero plot is shown in Fig.

Then from Eq. Note that this is not the case for the Laplace transform. Verify t h e time-shifting property 4. Thus, we have 4. Verify the multiplication by n or differentiation in z property 4. Verify the convolution property 4. If a zero of one transform cancels a pole of the other, the ROC of Y z may be larger. Thus, we conclude that 4. Verify the accumulation property 4.

Thus, we must divide to obtain a series in the power of z - '. Thus, we must divide to obtain a series in power of z. Using partial-fraction expansion, redo Prob. Find the inverse t-transform of Note that X Z is an improper rational function; thus, by long division, we have Let Then where Thus. Using the z-transform, redo Prob. Using t h e z-transform, redo Prob.

Let x[nl and y[nl be the input and output of the system.

Hwei hsu biography examples: Follow Hwei P. Hsu and explore

The output y [ n ] of a discrete-time LTI system is found to be 2 f "u[n]when the input x [ n ] is u [ n ]. Thus, by the result from Prob. Thus, the system is causal. As in Prob. Consider the discrete-time system shown in Fig. For what values of k is the system BIB0 stable? Thus, as shown in Prob. Using the unilateral z-transform, redo Prob. From the time-shifting property 4.

Hint: 4. Given a State all the possible regions of convergence. Show the following properties for the z-transform. Hint: a Use Eqs. Find the z-transform of x [ n ]. Use Eq. Differentiate both sides of the given relation consecutively with respect to a. Using the z-transform, verify Eqs. Using the method of partial-fraction expansion, redo Prob.

Find the system function H z and its impulse response hbl. In addition, greater insights into the nature and properties of many signals and systems are provided by these transformations. In this chapter and the following one, we shall introduce other transformations known as Fourier series and Fourier transform which convert time-domain signals into frequency-domain or spectral representations.

In addition to providing spectral representations of signals, Fourier analysis is also essential for describing certain types of systems and their properties in the frequency domain. In this chapter we shall introduce Fourier analysis in the context of continuous-time signals and systems. When x t is real, then from Eq. Harmonic Form Fourier Series: Another form of the Fourier series representation of a real periodic signal x t with fundamental period To is Equation 5.

The term Co is known as the d c component, and the term C, cos kwot - 0, is referred to as the kth harmonic component of x t. Although the latter two are common forms for Fourier series, the complex form in Eq. Convergence of Fourier Series: It is known that a periodic signal x t has a Fourier series representation if it satisfies the following Dirichlet conditions: 1.

Note that the Dirichlet conditions are sufficient but not necessary conditions for the Fourier series representation Prob. Since the index k assumes only integers, the amplitude and phase spectra are not continuous curves but appear only at the discrete frequencies k o. They are therefore referred to as discrete frequency spectra or line spectra.

Thus, Hence, the amplitude spectrum is an even hwei hsu biography examples of wand the phase spectrum is an odd function of o for a real periodic signal. Therefore, we obtain which is the Fourier representation of a nonperiodic x t. Symbolically they are denoted by and we say that x t and X w form a Fourier transform pair denoted by X 4 5.

Convergence of Fourier Transforms: Just as in the case of periodic signals, the sufficient conditions for the convergence of X o are the following again referred to as the Dirichlet conditions : 1. Although the above Dirichlet conditions guarantee the existence of the Fourier transform for a signal, if impulse functions are permitted in the transform, signals which do not satisfy these conditions can have Fourier transforms Prob.

Note that since the integral in Eq. Thus, in the remainder of this book both X o and X j w mean the same thing whenever we connect the Fourier transform with the Laplace transform. If x t is absolutely integrable, that is, if x r satisfies condition 5. This is not generally true of signals which are not absolutely integrable. The following examples illustrate the above statements.

Consider the unit impulse function S t. Consider the exponential signal From Eq. Consider the unit step function u t. Note that the unit step function u t is not absolutely integrable. Many of these properties are similar to those of the Laplace transform see Sec. Time Shifting: Equation 5. This is known as a linear phase shift of the Fourier transform X w.

Frequency Shifting: The multiplication of x t by a complex exponential signal is sometimes called complex modulation. Note that the frequency-shifting property Eq. Time Scaling: where a is a real constant. This property follows directly from the definition of the Fourier transform. Equation 5. Thus, the scaling property 5. Time Reversal: Thus, time reversal of x t produces a like reversal of the frequency axis for X o.

Duality or Symmetry : The duality property of the Fourier transform has significant implications. This property allows us to obtain both of these dual Fourier transform pairs from one evaluation of Eq. Differentiation in the Time Domain: Equation 5.

Hwei hsu biography examples: About the Author​​ Hwei

Integration in the Time Domain: Since integration is the inverse of differentiation, Eq. As in the case of the Laplace transform, this convolution property plays an important role in the study of continuous-time LTI systems Sec. Multiplication: The multiplication property 5. Thus, multiplication in the time domain becomes convolution in the frequency domain Prob.

Additional Properties: If x t is real, let where x, t and xo t are the even and odd components of x t 1, respectively. Let Then Equation 5. Equations 5. Note that the quantity on the left-hand side of Eq. Parseval's identity says that this energy content E can be computed by integrating Ix w 12 over all frequencies w. For this reason Ix w l2 is often referred to as the energy-density spectrum of x tand Eq.

Table contains a summary of the properties of the Fourier transform presented in this section. Some common signals and their Fourier transforms are given in Table Common Fourier Transforms Pairs sin at 5. Relationships represented by Eqs. Consider the complex exponential signal with Fourier transform Prob. Furthermore, by the linearity property 5.

The magnitude response IH o l is sometimes referred to as the gain of the system. Distortionless Transmission: For distortionless transmission through an LTI system we require that the exact input signal shape be reproduced at the output although its amplitude may be different and it may be delayed in time. Therefore, if x t is the input signal, the required output is 5.

This is illustrated in Figs. Taking the Fourier transform of both sides of Eq. Amplitude Distortion and Phase Distortion: When the amplitude spectrum IH o of the system is not constant within the frequency band of interest, the frequency components of the input signal are transmitted with a different amount of gain or attenuation. This effect is called amplitude distortion.