Signals and systems schaum series solution manual

It also describes various types of systems. This feature is not available right now. Yes, Oppenheim is a little complicated to understand. Definitions of a signal and a system, Classification of signals, Basic operations on signals, Elementary signals, Systems viewed as interconnections of operations, Properties.

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 a 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 interconnec- tion 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. 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. The step response s t of a continuous-time LTI system is given by Find the impulse response h r of the system. Consider an integrator whose input x t and output y t are related by a Find the impulse response h t of the integrator. No, the system has memory. 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. Is the system described by the differential equation linear? 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 variable , and 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 t , the 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 t , the 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, X s 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 read 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 s , that is, all zeros of D s , are simple or distinct , then 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 constant- coefficient 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 integrodif- ferential 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. 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 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 A , without resorting to differentiation. This is shown as follows: First find c , and A, according to the regular procedure. Then substituting the values of c , and 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.

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. 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. Using the transform network technique, redo Prob.

In the circuit 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 i , t and i 2 t. Supplementary Problems 3. Show that if X I is a left-sided signal and X s converges for some value of s, then the ROC of X s is of the form equals the minimum real part of any of the poles of X s.

Hint: a Use Eqs. Hint: Use Eq. Using the Laplace transform, show that a 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 capacitor voltage before the switch closing is u,. Find the capacitor voltage for t 2 0. Before the switch closing, the capacitor C , is charged to u, V and the capacitor C , is not charged.

In this chapter we present the z-transform, which is the discrete-time counterpart of the Laplace transform. The z-trans- form is introduced to represent discrete-time signals or sequences in the z-domain z is a complex variable , and 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 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 r , and 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. Table Verifica- tion of these properties is given in Probs.

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. See Prob. Definition: The unilateral or one-sided z-transform X, z of a sequence x[n] is defined as [Eq.

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. Show that if x [ n ] is a right-sided sequence and X z converges for some value of z, then the ROC of X z is of the form where r,, is the maximum magnitude of any of the poles of X z.

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. 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: Proceed in a manner similar to Prob. Given a State all the possible regions of convergence. Verify the time-reversal property 4. Show the following properties for the z-transform. Find the z-transform of x [ n ]. 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. Consider a discrete-time LTI system whose system function H z is given by a Find the step response s[n]. Using the unilateral z-transform, solve the following difference equations with the given initial conditions.

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 transfor- mations 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. Periodic Signals: In Chap. 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 function of w , and the phase spectrum is an odd function of o for a real periodic signal.

ISBN pbk. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. Our interactive player makes it easy to find solutions to Signals and Systems problems you're working on - just go to the chapter for your book. Systems are operators that accept a given signal the input signal and produce a new signal the output signal.

The These are the download links for Ebooks Service Manual. 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 eJ"l ' 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. Differentiation in the Frequency Domain: -P x t - Equation 5.

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 t , and 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. Frequency Response: In Sec. Relationships repre- sented 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.

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. This form of distortion is called phase distortion. The result 5. Filtering is the process by which the relative amplitudes of the frequency components in a signal are changed or perhaps some frequency components are suppressed.

As we saw in the preceding section, for continuous-time LTI systems, the spectrum of the output is that of the input multiplied by the frequency response of the system. Therefore, an LTI system acts as a filter on the input signal. Here the word "filter" is used to denote a system that exhibits some sort of frequency-selective behavior. Ideal Frequency-Selective Filters: An ideal frequency-selective filter is one that exactly passes signals at one set of frequencies and completely rejects the rest.

The band of frequencies passed by the filter is referred to as the pass band, and the band of frequencies rejected by the filter is called the stop band. The most common types of ideal frequency-selective filters are the following. The frequency o, is called the cutoff frequency. In the above discussion, we said nothing regarding the phase response of the filters. T o avoid phase distortion in the filtering process, a filter should have a linear phase characteristic over the pass band of the filter, that is [Eq.

Note that all ideal frequency-selective filters are noncausal systems. Nonideal Frequency-Selective Filters: As an example of a simple continuous-time causal frequency-selective filter, we consider the RC filter shown in Fig. The output y t and the input x t are related by Prob.

Thus, the amplitude response H w l and phase response OJw are given by b Fig. There are many different definitions of system bandwidth. In this case W, is called the absolute bandwidth.

No bandwidth is defined for a high-pass or a bandstop filter. The 3-dB bandwidth is also known as the half-power bandwidth because a voltage or current attenuation of 3 dB is equivalent to a power attenuation by a factor of 2. This definition of W , ,, is useful for systems with unimodal amplitude response in the positive frequency range and is a widely accepted criterion for measuring a system's bandwidth, but it may become ambiguous and nonunique with systems having multiple peak amplitude responses.

Note that each of the preceding bandwidth definitions is defined along the positive frequency axis only and always defines positive frequency, or one-sided, bandwidth only. Signal Bandwidth: The bandwidth of a signal can be defined as the range of positive frequencies in which "most" of the energy or power lies. This definition is rather ambiguous and is subject to various conventions Probs.

Band-Limited Signal: A signal x t is called a band-limited signal if Thus, for a band-limited signal, it is natural to define o, as the bandwidth.

Using the orthogonality condition 5. That is, 5. Derive the trigonometric Fourier series Eq. Rearranging the summation in Eq. Consider the periodic square wave x t shown in Fig. Note also that x t in Fig. Now comparing Fig. Consider the periodic impulse train S G , t shown in Fig. Consider the triangular wave x t shown in Fig.

Using the differentiation technique, find a the complex exponential Fourier series of d t , and 6 the trigonometric Fourier series of x t 1. The derivative x l t of the triangular wave x t is a square wave as shown in Fig. Thus, we can find ck k 0 if the Fourier coefficients of x l t are known. The term c, cannot be determined by Eq.

Hence, from Eq. Using the differentiation technique, find the triangular Fourier series of x t. Find and sketch the magnitude spectra for the periodic square pulse train signal x t shown in Fig. Let Then since and the term in brackets is equal to e,-,. Let x l t and x2 t be the two periodic signals in Prob.

Show that Equation 5. Verify Parseval's identity 5. If d, and en are the complex Fourier coefficients of x, r and x2 t , respectively, then show that the complex Fourier coefficients ck of f t are given by where To is the fundamental period common to x, t , x2 t , and f t.

Verify the frequency-shifting property 5. Verify the duality property 5. Find the Fourier transform of the rectangular pulse signal x t [Fig. Find the Fourier transform of the signal [Fig. Find the Fourier transform of a periodic signal x t with period T o.

We express x t as Taking the Fourier transform of both sides and using Eq. Find the Fourier transform of the periodic impulse train [Fig. The Fourier transform of a signal x t is given by [Fig.

Verify the differentiation property 5. Thus, 5. Find the Fourier transform of the signum function, sgn t Fig. Prove t h e time convolution theorem 5. Using the time convolution theorem 5.

Verify the integration property 5. Using the integration property 5. Prove the frequency convolution theorem 5. Using the frequency convolution theorem 5. Verify Parseval's relation 5. Prove Parseval's identity [Eq. Show that Eq. By definition 5. From the inverse Fourier transform definition 5.

Find the Fourier transform of a gaussian pulse signal By definition 5. Thus, we get dX w -- w - --X w dw 2a Solving the above separable differential equation for X w , we obtain where A is an arbitrary constant. To evaluate A we proceed as follows. Figure shows the relationship in Eq. Using t h e Fourier transform, redo Prob. Thus, We observe that the Laplace transform method is easier in this case because of the Fourier transform of d t.

Consider the LTI system in Prob. If the input x t is the periodic square waveform shown in Fig. Note that x t is the same x t shown in Fig. By Eqs. The quantity glolH o l is referred to as the magnitude expressed in decibel!

This value of o is called the comer frequency. The plot of BH o is sketched in Fig. Note that the dotted lines represent the straight-line approximation of the Bode plots. The plot of OH w is sketched in Fig. Each term contributing to the overall amplitude is also indicated. Thus, from Eq.

Then from Eqs. Find the frequency response H o and the impulse function h t of the system. Consider an ideal low-pass filter with frequency response The input to this filter is the periodic square wave shown in Fig. Find the output y t 1. T h e equivalent bandwidth of a filter with frequency response H o is defined by where IH w lm, denotes the maximum value of the magnitude spectrum. Consider the low-pass RC filter shown in Fig.

Now The amplitude spectrum lH w l is plotted in Fig. Rewriting H w as and using Eq. The risetime t , of the low-pass RC filter in Fig. Another definition of bandwidth for a signal x t is the 90 percent energy containment bandwidth W,, defined by where Ex is the normalized energy content of signal x t.

Let x t be a real-valued band-limited signal specified by [Fig. The signal x, t is called the ideal sampled signal, T, is referred to as the sampling interr. The Fourier spectrum X, w is shown in Fig. This effect is known as aliasing. Let From Eq.

This is known as the uniform sampling theorem for low-pass signals. The frequency response H w of the ideal low-pass filter is given by [Fig. Note from the sampling theorem Probs. If this condition on the bandwidth of x t is not satisfied, then y t z x t. Using the result from Prob. Derive the harmonic form Fourier series representation 5. Show that the mean-square value of a real periodic signal x r is the sum of the mean-square values of its harmonics. Show that if then Hint: Repeat the time-differentiation property 5.

Using the differentiation technique, find the Fourier transform of the triangular pulse signal shown in Fig. Find the inverse Fourier transform of Hint: Differentiate Eq. Verify the frequency differentiation property 5. Let x t be a real signal with the Fourier transform X w. Consider a continuous-time LTI system with frequency response H w. Find the Fourier transform S w of the unit step response s t of the system. Consider the RC filter shown in Fig. Find the frequency response H w of this filter and discuss the type of filter.

Determine the 99 percent energy containment bandwidth for the signal Ans. The sampling theorem in the frequency domain states that if a real signal x t is a duration- limited signal. Hint: Expand x t in a complex Fourier series and proceed in a manner similar to that for Prob. The Fourier analysis plays the same fundamental role in discrete time as in continuous time.

As we will see, there are many similarities between the techniques of discrete-time Fourier analysis and their continuous-time counterparts, but there are also some important differences. Periodic Sequences: In Chap. As we saw in Sec. As we discussed in Sec. Discrete Fourier Series Representation: The discrete Fourier series representation of a periodic sequence x[n] with fundamen- tal period No is given by where c, are the Fourier coefficients and are given by Prob.

The Fourier coefficients c, are often referred to as the spectral coefficients of x[n]. Convergence of Discrete Fourier Series: Since the discrete Fourier series is a finite series, in contrast to the continuous-time case, there are no convergence issues with discrete Fourier series. Properties of Discrete Fourier Series: I. Periodicity of Fourier Coeficients: From Eqs. Duality: From Eq. Thus, writing c, as c[k], Eq. Other Properties: When x[n] is real, then from Eq.

Let x[n] S c k Then xe[n] Re[ck] 6. Parseval's Theorem: If x[n] is represented by the discrete Fourier series in Eq. That is, for some positive integer N , , Such a sequence is shown in Fig. Let x,Jn] be a periodic sequence formed by repeating x [ n ] with fundamental period No as shown in Fig. As shown in Fig.

Furthermore, since the summation in Eq. Fourier Spectra: The Fourier transform X R of x[n] is, in general, complex and can be expressed as As in continuous time, the Fourier transform X R of a nonperiodic sequence x[n] is the frequency-domain specification of x[n] and is referred to as the spectrum or Fourier spectrum of x[n].

The quantity IX R I is called the magnitude spectrum of x[n], and dR is called the phase spectrum of x[n]. Furthermore, if x[n] is real, the amplitude spectrum IX R I is an even function and the phase spectrum 4 n is an odd function of R.

Convergence of X R : Just as in the case of continuous time, the sufficient condition for the convergence of X R is that x[n] is absolutely summable, that is, m C Ix[n]kw 6. Connection between the Fourier Transform and the z-Transform: Equation 6. If x [ n ] is absolutely summable, that is, if x [ n ] satisfies condition 6. This is not generally true of sequences which are not absolutely summable. Consider the unit impulse sequence 6 [ n l.

Note that 6 [ n ] is absolutely summable and that the ROC of the z-transform of 6 [ n l contains the unit circle. That is, Next, by definition 6. Consider the unit step sequence u[nl. Note that the unit step sequence u[n] is not absolutely summable. The Fourier transform of u[n] is given by Prob.

There are many similarities to and several differences from the continuous-time case. Many of these properties are also similar to those of the z-transform when the ROC of X z includes the unit circle. Periodicity: As a consequence of Eq. Linearity: C. Time Shifting: D. Time Scaling: In Sec. Thus, time scaling in discrete time takes on a form somewhat different from Eq.

It states again the inverse relationship between time and frequency. Duality: In Sec. However, there is a duality between the discrete-time Fourier transform and the continuous-time Fourier series. Let From Eqs. Differentiation in Frequency: J. Differencing: The sequence x [ n ] - x [ n - 11 is called the firsf difference sequence. Equation 6. Accumulation: Note that accumulation is the discrete-time counterpart of integration.

The impulse term on the right-hand side of Eq. Convolution: As in the case of the z-transform, this convolution property plays an important role in the study of discrete-time LTI systems. Additional Properties: If x[n] is real, let where x,[n] and xo[n] are the even and odd components of x[n], respectively. From Eqs. Parseval's Relations: Equation 6. Some common sequences and their Fourier transforms are given in Table Let As in the continuous-time case, the function H R is called the frequency response of the system, I H R l the magnitude response of the system, and BH R the phase response of the system.

Furthermore, by the linearity property 6. Increasing w results in a sinusoid of ever-increasing frequency. Therefore, a sinusoid with any value of R outside the range 0 to 7r is identical to a sinusoid with R in the range 0 to 7r.

This result is a special case of the sampling theorem we discussed in Prob. Let Then from Eqs. Therefore, Eq. If the input x t is band-limited [Eq. However, from Eqs. However, there are methods for determining a discrete-time system so as to satisfy Eq.

It should be noted that the DFT should not be confused with the Fourier transform. There is a one-to-one correspondence between x [ n ] and X [ k ]. There is an extremely fast algorithm, called the fast Fourier transform FFT for its calculation.

The DFT is the appropriate Fourier representation for digital computer realization because it is discrete and of finite length in both the time and frequency domains. Note that the choice of N in Eq.

This addition of dummy samples is known as zero padding. Then the resultant x [ n ] is often referred to as an N-point sequence, and X [ k ] defined in Eq. By a judicious choice of N, such as choosing it to be a power of 2, computational efficiencies can be gained. Properties of the D m Because of the relationship 6. Basic properties of the DFT are the following: 2.

Time ShifCing: 3. Frequency Shifiing: 4. Time Reversal: 6. Duality: 7. Circular Convolution: where The convolution sum in Eq. Multiplication: where 9. Additional Properties: When x [ n ] is real, let Then x[n] - where x,[n] and xo[n] are the even and odd components of x [ n ] , respectively.

Parseval's Relation: Equation 6. Show that the set of complex exponential sequences is orthogonal on any interval of length N. Since each of the complex exponentials in the summation in Eq.