[PROBLEMS](#page-13-0)

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PROBLEMS

• 7.1-1 Suppose signal x(t) = t ² [u(t)−u(t −2)] has Fourier transform X(ω). Define a 3-periodic replication of ^x(t) as ^y(t) ⁼ %^{∞ n}=−∞ 2x(t − 1 − 3n). Determine Yk, the Fourier series of y(t), in terms of the Fourier transform X(·).
• 7.1-2 Show that for a real x(t), Eq. (7.10) can be expressed as

$$x(t) = \frac{1}{\pi} \int_0^\infty |X(\omega)| \cos[\omega t + \angle X(\omega)] d\omega$$

This is the trigonometric form of the Fourier integral. Compare this with the compact trigonometric Fourier series.

7.1-3 Show that if x(t) is an even function of t, then

$$X(\omega) = 2 \int_0^\infty x(t) \cos \omega t \, dt$$

and if x(t) is an odd function of t, then

$$X(\omega) = -2j \int_0^\infty x(t) \sin \omega t \, dt$$

Hence, prove that if x(t) is a real and even function of t, then X(ω) is a real and even function of ω. In addition, if x(t) is a real and odd function of t, then X(ω) is an imaginary and odd function of ω.

7.1-4 A signal x(t) can be expressed as the sum of even and odd components (see Sec. 1.5-2):

x(t) = xe(t) +xo(t)

(a) If x(t) ⇐⇒ X(ω), show that for real x(t),

xe(t) ⇐⇒ Re[X(ω)]

and

$$x_o(t) \Longleftrightarrow j \operatorname{Im}[X(\omega)]$$

• (b) Verify these results by finding the Fourier transforms of the even and odd components of the following signals: (i) u(t) and (ii) e−atu(t).
• 7.1-5 Using Eq. (7.9), find the Fourier transforms of the signals x(t) in Fig. P7.1-5.
• 7.1-6 Using Eq. (7.9), find the Fourier transforms of the signals depicted in Fig. P7.1-6.
• 7.1-7 Use Eq. (7.10) to find the inverse Fourier transforms of the spectra in Fig. P7.1-7.

Figure P7.1-6

Figure P7.1-8

• 7.1-8 Use Eq. (7.10) to find the inverse Fourier transforms of the spectra in Fig. P7.1-8.
• 7.1-9 If x(t) ⇐⇒ X(ω), then show that

$$X(0) = \int_{-\infty}^{\infty} x(t) dt$$

and

$$x(0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) d\omega$$

Also show that

$$\int_{-\infty}^{\infty} \operatorname{sinc}(x) dx = \int_{-\infty}^{\infty} \operatorname{sinc}^{2}(x) dx = \pi$$

Figure P7.2-4

• 7.2-1 Sketch the following functions: (a) rect(t/2)
  • (b) (3ω/100)
  • (c) rect((t −10)/8)
  • (d) sinc(πω/5)
  • (e) sinc ((ω/5)−2π)
  • (f) sinc (t/5)rect(t/10π )
• 7.2-2 Using Eq. (7.9), show that the Fourier transform of rect(^{t − 5}) is sinc(ω/2)e−j5ω. Sketch the resulting amplitude and phase spectra.
• 7.2-3 Using Eq. (7.10), show that the inverse Fourier transform of rect((ω ⁻10)/2π ) is sinc(πt) ej10*^t* .
• 7.2-4 Find the inverse Fourier transform of X(ω) for the spectra illustrated in Fig. P7.2-4. [Hint:

^X(ω) = |X(ω)|ej ^X(ω). This problem illustrates how different phase spectra (both with the same amplitude spectrum) represent entirely different signals.]

• 7.2-5 (a) Can you find the Fourier transform of eatu(t) when ^a>1 by setting ^{s = jω} in the Laplace transform of eatu(t)? Explain.
  • (b) Find the Laplace transform of x(t) shown in Fig. P7.2-5. Can you find the Fourier transform of x(t) by setting s = jω in its Laplace transform? Explain. Verify your answer by finding the Fourier and the Laplace transforms of x(t).

Figure P7.2-5

• 7.3-1 Apply the duality property to the appropriate pair in Table 7.1 to show that

  • (a) ^{1 2} [δ(t)+j/πt] ⇐⇒ u(ω)
  • (b) δ(t +T) +δ(t −T) ⇐⇒ 2 cos Tω
  • (c) δ(t +T)−δ(t −T) ⇐⇒ 2jsin Tω

• 7.3-2 A signal x(t) has Fourier transform X(ω). Determine the Fourier transform Y(ω) in terms of X(ω) for each of the following signals y(t): (a) ^y(t) ^{= 1 5} x(−2t +3) (b) ^y(t) ⁼ ej2*^t x*∗(−3t −6)

• 7.3-3 A signal x(t) has Fourier transform X(ω). Determine the inverse Fourier transform y(t) in terms of x(t) for each of the following spectra Y(ω),

(a)

$$Y(\omega) = \frac{4}{3}e^{-j2\omega/3}X(-\omega/3)$$

(b)

$$Y(\omega) = \frac{1}{3}e^{j2(\omega - 2)}X^*\left(\frac{\omega - 2}{3}\right)$$

7.3-4 The Fourier transform of the triangular pulse x(t) in Fig. P7.3-4 is expressed as

$$X(\omega) = \frac{1}{\omega^2} (e^{j\omega} - j\omega e^{j\omega} - 1)$$

Use this information, and the time-shifting and time-scaling properties, to find the Fourier transforms of the signals xi(t)(i = 1, 2, 3, 4, 5) shown in Fig. P7.3-4.

• 7.3-5 Using only the time-shifting property and Table 7.1, find the Fourier transforms of the signals depicted in Fig. P7.3-5.
• 7.3-6 Consider the fact that the τ -duration triangle function ^ω τ has inverse Fourier transform

Figure P7.3-4

2 0 234 0 -

(a) (b)

4 -3 -

Figure P7.3-7

-4 -3 -

τ ^4π sinc2 ^tτ 4 . Use the duality property to determine the Fourier transform Y(ω) of signal y(t) = (t).

7.3-7 Use the time-shifting property to show that if x(t) ⇐⇒ X(ω), then

$$x(t+T) + x(t-T) \Longleftrightarrow 2X(\omega)\cos T\omega$$

This is the dual of Eq. (7.32). Use this result and Table 7.1 to find the Fourier transforms of the signals shown in Fig. P7.3-7.

7.3-8 Prove the following results, which are duals of each other:

$$x(t)\sin \omega_0 t \Longleftrightarrow \frac{1}{2j}[X(\omega - \omega_0) - X(\omega + \omega_0)]$$ $$\frac{1}{2j}[x(t+T) - x(t-T)] \Longleftrightarrow X(\omega)\sin T\omega$$

Use the latter result and Table 7.1 to find the Fourier transform of the signal in Fig. P7.3-8.

2 234

• 7.3-9 The signals in Fig. P7.3-9 are modulated signals with carrier cos 10t. Find the Fourier transforms of these signals by using the appropriate properties of the Fourier transform and Table 7.1. Sketch the amplitude and phase spectra for Figs. P7.3-9a and P7.3-9b.
• 7.3-10 Use the frequency-shifting property and Table 7.1 to find the inverse Fourier transform of the spectra depicted in Fig. P7.3-10.
• 7.3-11 Let X(ω) = rect(ω) be the Fourier transform of a signal x(t).
  • (a) For ya(t) = x(t) ∗ x(t), sketch Ya(ω).
  • (b) For yb(t) = x(t) ∗ x(t/2), sketch Yb(ω).

(a) (b)

Figure P7.3-10

• (c) For yc(t) = 2x(t), sketch Yc(ω).
• (d) For yd(t) = x2(t), sketch Yd(ω).
• (e) For ye(t) = 1−x2(t), sketch Ye(ω).
• 7.3-12 Use the time-convolution property to prove pairs 2, 4, 13, and 14 in Table 2.1 (assume λ < 0 in pair 2, λ¹ and λ² < 0 in pair 4, λ¹ < 0 and λ² > 0 in pair 13, and λ¹ and λ² > 0 in pair 14). These restrictions are placed because of the Fourier transformability issue for the signals concerned. For pair 2, you need to apply the result in Eq. (1.10).
• 7.3-13 A signal x(t) is bandlimited to B Hz. Show that the signal xn(t) is bandlimited to nB Hz.
• 7.3-14 Find the Fourier transform of the signal in Fig. P7.3-5a by three different methods:
  • (a) By direct integration using Eq. (7.9).
  • (b) Using only pair 17 (Table 7.1) and the time-shifting property.
  • (c) Using the time-differentiation and time-shifting properties, along with the fact that δ(t) ⇐⇒ 1.

7.3-15 (a) Prove the frequency-differentiation property (dual of the time-differentiation property):

$$-jtx(t) \Longleftrightarrow \frac{d}{d\omega}X(\omega)$$

• (b) Use this property and pair 1 (Table 7.1) to determine the Fourier transform of te−atu(t).
• 7.3-16 Adapt the method of Ex. 7.17 and use the frequency-differentiation (see Prob. 7.3-15) and other properties to find the inverse Fourier transform x(t) of the triangular spectrum X(ω) = (ω/2).
• 7.3-17 Adapt the method of Ex. 7.17 and use the frequency-differentiation (see Prob. 7.3-15) and other properties to find the inverse Fourier transform x(t) of the spectrum X(ω) = π ω 2 rect ^ω 4 .
• 7.4-1 For a stable LTIC system with transfer function

$$H(s) = \frac{1}{s+1}$$

find the (zero-state) response if the input x(t) is

• (a) e−2*^t u*(t)
• (b) e−^t u(t)
• (c) et u(−t)
• (d) u(t)
• 7.4-2 A stable LTIC system is specified by the frequency response

$$H(\omega) = \frac{-1}{j\omega - 2}$$

Find the impulse response of this system and show that this is a noncausal system. Find the (zero-state) response of this system if the input x(t) is

• (a) e−^t u(t)

• (b) et u(−t)

• 7.4-3 A periodic signal x(t) = 1 + 2 cos(5πt) + 3 sin(8πt) is applied to an LTIC system with impulse response h(t) = 8sinc(4t) cos(2πt) to produce output y(t) = x(t) ∗ h(t).

  • (a) Determine ω0, the fundamental radian frequency of x(t).
  • (b) Determine X(ω), the Fourier transform of x(t).
  • (c) Sketch the system’s magnitude response |H(ω)| over −10π ≤ ω ≤ 10π.
  • (d) Is the system h(t) distortionless? Explain.
  • (e) Determine y(t).

• 7.4-4 A periodic delta train ^x(t) ⁼ %^{∞ n}=−∞ δ(t − πn) is applied to an LTIC system with impulse response ^h(t) ⁼ sin(3t)sinc2 ^t π to produce zero-state output y(t) = x(t) ∗ h(t).

  • (a) Determine ω0, the fundamental radian frequency of x(t).
  • (b) Determine X(ω), the Fourier transform of x(t).
  • (c) Sketch the system’s magnitude response |H(ω)| over −10π ≤ ω ≤ 10π.
  • (d) Is the system h(t) distortionless? Explain.
  • (e) Determine y(t).

• 7.4-5 Signals x1(t)=104rect(104t) and x2(t) = δ(t) are applied at the inputs of the ideal lowpass filters H1(ω) = rect(ω/40,000π ) and H2(ω) = rect(ω/20,000π ) (Fig. P7.4-5). The outputs y1(t) and y2(t) of these filters are multiplied to obtain the signal y(t) = y1(t)y2(t). (a) Sketch X1(ω) and X2(ω).

• (b) Sketch H1(ω) and H2(ω).

• (c) Sketch Y1(ω) and Y2(ω).

• (d) Find the bandwidths of y1(t), y2(t), and y(t).

• 7.4-6 A lowpass system time constant is often defined as the width of its unit impulse response h(t) (see Sec. 2.6-2). An input pulse p(t) to this system acts like an impulse of strength equal to the area of p(t) if the width of p(t) is much smaller than the system time constant, and provided p(t) is a lowpass pulse, implying that its spectrum is concentrated at low frequencies. Verify this behavior by considering a system whose unit impulse response is h(t) = rect(t/10−3). The input pulse is a triangle pulse p(t) = (t/10−6). Show that the system response to this pulse is very nearly the system response to the input Aδ(t), where A is the area under the pulse p(t).

• 7.4-7 A lowpass system time constant is often defined as the width of its unit impulse response h(t) (see Sec. 2.6-2). An input pulse p(t) to this system passes practically without distortion if the width of p(t) is much greater than the system time constant, and provided p(t) is a lowpass pulse, implying that its spectrum is concentrated at low frequencies. Verify this behavior by considering a system whose unit impulse response is h(t) = rect(t/10−3). The input pulse is a triangle pulse p(t)=(t). Show that the system output to this pulse is very nearly kp(t), where k is the system gain to a dc signal, that is, k = H(0).

• 7.4-8 A causal signal h(t) has a Fourier transform H(ω). If R(ω) and X(ω) are the real and the imaginary parts of H(ω), that is, H(ω) = R(ω)+ jX(ω), then show that

$$R(\omega) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{X(\omega)}{\omega - y} d\omega$$

and

$$X(\omega) = -\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{R(\omega)}{\omega - y} d\omega$$

assuming that h(t) has no impulse at the origin. This pair of integrals defines the Hilbert transform. [Hint: Let he(t) and ho(t) be the even and odd components of h(t). Use the results in Prob. 7.1-4. See Fig. 1.24 for the relationship between he(t) and ho(t).]

This problem states one of the important properties of causal systems: that the real and imaginary parts of the frequency response of a causal system are related. If one specifies the real part, the imaginary part cannot be specified independently. The imaginary part is predetermined by the real part, and vice versa. This result also leads to the conclusion that the magnitude and angle of H(ω) are related, provided all the poles and zeros of H(ω) lie in the LHP.

7.5-1 Consider a filter with the frequency response

$$H(\omega) = e^{-(k\omega^2 + j\omega t_0)}$$

Show that this filter is physically unrealizable by using the time-domain criterion [noncausal h(t)] and the frequency-domain (Paley–Wiener) criterion. Can this filter be made approximately realizable by choosing t⁰ sufficiently large? Use your own (reasonable) criterion of approximate realizability to determine t0. [Hint: Use pair 22 in Table 7.1.]

7.5-2 Show that a filter with frequency response

$$H(\omega) = \frac{2(10^5)}{\omega^2 + 10^{10}} e^{-j\omega t_0}$$

is unrealizable. Can this filter be made approximately realizable by choosing a sufficiently large t0? Use your own (reasonable) criterion of approximate realizability to determine t0.

• 7.5-3 Determine whether the filters with the following frequency response H(ω) are physically realizable. If they are not realizable, can they be realized approximately by allowing a finite time delay in the response?

  • (a) 10−⁶ sinc (10−6ω)
  • (b) 10−⁴ (ω/40,000π)
  • (c) 2π δ(ω)

• 7.5-4 Consider signal x1(t), its Fourier transform X1(f), and several other signals, as shown in Fig. P7.5-4. Notice, spectra are drawn as a function of hertzian frequency f rather than radian frequency ω.

  • (a) Accurately sketch X2(f), the Fourier transform of x2(t).
  • (b) Accurately sketch x3(t), the inverse Fourier transform of X3(f).
  • (c) The signal x4(t) = x1(t) + x2(t) is passed through an ideal lowpass filter with 3 Hz cutoff to produce output y4(t). Accurately sketch y4(t).

Figure P7.5-4

• 7.6-1 Define ^x(t) ^{= 1 2π} sinc(t/2) with Fourier transform X(ω) = rect(ω). Use Parseval’s theorem to determine $ ^∞ −∞ sinc2(^{t −}2)dt.
• 7.6-2 Show that the energy of a Gaussian pulse

$$x(t) = \frac{1}{\sigma\sqrt{2\pi}}e^{-t^2/2\sigma^2}$$

is 1/(2σ ^√π ). Verify this result by using Parseval’s theorem to derive the energy Ex from X(ω). [Hint: See pair 22 in Table 7.1. Use the fact that $ ∞ −∞ ^e−x2/² dx ^{= √}2π.]

7.6-3 Use Parseval’s theorem of Eq. (7.45) to show that

$$\int_{-\infty}^{\infty} \operatorname{sinc}^2(kx) dx = \frac{\pi}{k}$$

• 7.6-4 A lowpass signal x(t) is applied to a squaring device. The squarer output x2(t) is applied to a lowpass filter of bandwidth f (in hertz) (Fig. P7.6-4). Show that if f is very small (f → 0), then the filter output is a dc signal y(t) ≈ 2Exf . [Hint: If x2(t) ⇐⇒ A(ω), then show that Y(ω) ≈ [4πA(0)f]δ(ω) if f → 0. Now, show that A(0) = Ex.]
• 7.6-5 Generalize Parseval’s theorem to show that for real, Fourier-transformable signals x1(t) and x2(t)

$$\int_{-\infty}^{\infty} x_1(t) x_2(t) dt$$

= $\frac{1}{2\pi} \int_{-\infty}^{\infty} X_1(-\omega) X_2(\omega) d\omega$
= $\frac{1}{2\pi} \int_{-\infty}^{\infty} X_1(\omega) X_2(-\omega) d\omega$

7.6-6 Show that

$$\int_{-\infty}^{\infty} \operatorname{sinc} (Wt - m\pi) \operatorname{sinc} (Wt - n\pi) dt$$ $$= \begin{cases} 0 & m \neq n \\ \frac{\pi}{W} & m = n \end{cases}$$

( )2 Lowpass filter x(t) x y(t) 2Exf 2(t)

Figure P7.6-4

[Hint: Recognize that

$$\operatorname{sinc}(Wt - k\pi) = \operatorname{sinc}\left[W\left(t - \frac{k\pi}{W}\right)\right] \\ \Longleftrightarrow \frac{\pi}{W} \operatorname{rect}\left(\frac{\omega}{2W}\right) e^{-jk\pi\omega/W}$$

Use this fact and the result in Prob. 7.6-5.]

• 7.6-7 (a) What does it mean to compute the 95% essential bandwidth B of a signal x(t) with Fourier transform X(ω)?
  • (b) Determine the 95% essential bandwidth B of a signal with spectrum X(ω) = rect(ω).
  • (c) Determine the 95% essential bandwidth B of a signal with spectrum X(ω) = (ω).
• 7.6-8 Using a 95% energy criterion, determine the essential bandwidth B of a signal that has a Fourier transform given by X(ω) = e−|ω^| .
• 7.6-9 For the signal

$$x(t) = \frac{2a}{t^2 + a^2}$$

determine the essential bandwidth B (in hertz) of x(t) such that the energy contained in the spectral components of x(t) of frequencies below B Hz is 99% of the signal energy Ex.

• 7.7-1 For each of the following baseband signals (i) m(t) = cos 1000t, (ii) m(t) = 2 cos 1000t + cos 2000t, and (iii) m(t) = cos 1000t cos 3000t:

  • (a) Sketch the spectrum of m(t).
  • (b) Sketch the spectrum of the DSB-SC signal m(t) cos 10,000t.
  • (c) Identify the upper sideband (USB) and the lower sideband (LSB) spectra.
  • (d) Identify the frequencies in the baseband, and the corresponding frequencies in the DSB-SC, USB, and LSB spectra. Explain the nature of frequency shifting in each case.

• 7.7-2 A message signal ^m(t) with spectrum ^M(ω) ^{= 1} 1000 ^ω 2π6000 is to be transmitted using a communication system. Assume all single-sideband systems have suppressed carriers.

  • (a) What is the hertzian bandwidth of m(t)?

• (b) Sketch the spectrum of the transmitted signal if the communication system is DSB-SC with ω*^c* = 2π 100,000.

• (c) Sketch the spectrum of the transmitted signal if the communication system is AM with ω*^c* = 2π 100,000 and a modulation index of μ = 1. What is the corresponding carrier amplitude A?

• (d) Sketch the spectrum of the transmitted signal if the communication system is USB with ω*^c* = 2π 100,000.

• (e) Sketch the spectrum of the transmitted signal if the communication system is LSB with ω*^c* = 2π 100,000.

• (f) Suppose we want to transmit m(t) on each of an FDM system’s four channels: DSB-SC at carrier ω1, AM (μ = 1) at carrier ω2, USB at carrier ω3, and LSB at carrier ω4. Determine carrier frequencies ω¹ < ω² < ω³ < ω⁴ so that the FDM spectrum begins at a frequency of 100,000 Hz with 5,000 Hz deadbands separating adjacent messages. What is the end hertzian frequency of the FDM signal?

• 7.7-3 You are asked to design a DSB-SC modulator to generate a modulated signal km(t) cosωct, where m(t) is a signal bandlimited to B Hz (Fig. P7.7-3a). Figure P7.7-3b shows a DSB-SC modulator available in the stockroom. The bandpass filter is tuned to ω*^c* and has a bandwidth of 2B Hz. The carrier generator available generates not cosωct, but cos³ ωct.

  • (a) Explain whether you would be able to generate the desired signal using only this equipment. If so, what is the value of k?

• (b) Determine the signal spectra at points b and c, and indicate the frequency bands occupied by these spectra.

• (c) What is the minimum usable value of ωc?

• (d) Would this scheme work if the carrier generator output were cos2 ωct? Explain.

• (e) Would this scheme work if the carrier generator output were cos*ⁿ* ωct for any integer n ≥ 2?

• 7.7-4 In practice, the analog multiplication operation is difficult and expensive. For this reason, in amplitude modulators, it is necessary to find some alternative to multiplication of m(t) with cosωct. Fortunately, for this purpose, we can replace multiplication with a switching operation. A similar observation applies to demodulators. In the scheme depicted in Fig. P7.7-4a, the period of the rectangular periodic pulse x(t)

Figure P7.7-4

Figure P7.7-3

shown in Fig. P7.7-4b is T⁰ = 2π/ωc. The bandpass filter is centered at ±ω*^c* and has a bandwidth of 2B Hz. Note that multiplication by a square periodic pulse x(t) in Fig. P7.7-4b amounts to periodic on-off switching of m(t), which is bandlimited to B Hz. Such a switching operation is relatively simple and inexpensive. Show that this scheme can generate an amplitude-modulated signal k cos ωct. Determine the value of k. Show that the same scheme can also be used for demodulation, provided the bandpass filter in Fig. P7.7-4a is replaced by a lowpass (or baseband) filter.

7.7-5 Figure P7.7-5a shows a scheme to transmit two signals m1(t) and m2(t) simultaneously on the same channel (without causing spectral interference). Such a scheme, which transmits more than one signal, is known as signal multiplexing. In this case, we transmit multiple signals by sharing an available spectral band on the channel; hence, this is an example of the frequency-division multiplexing. The signal at point b is the multiplexed signal, which now modulates a carrier of frequency 20,000 rad/s. The modulated signal at point c is now transmitted over the channel.

• (a) Sketch the spectra at points a, b, and c.
• (b) What must be the minimum bandwidth of the channel?
• (c) Design a receiver to recover signals m1(t) and m2(t) from the modulated signal at point c.
• 7.7-6 The system shown in Fig. P7.7-6 is used for scrambling audio signals. The output y(t) is the scrambled version of the input m(t).

Figure P7.7-6

• (a) Find the spectrum of the scrambled signal y(t).
• (b) Suggest a method of descrambling y(t) to obtain m(t).

A slightly modified version of this scrambler was first used commercially on the 25-mile radio-telephone circuit connecting Los Angeles and Santa Catalina Island.

• 7.7-7 Figure P7.7-7 presents a scheme for coherent (synchronous) demodulation. Show that this scheme can demodulate the AM signal [A + m(t)] cos ωct regardless of the value of A.
• 7.7-8 Sketch the AM signal [A + m(t)] cos ωct for the periodic triangle signal m(t) illustrated in Fig. P7.7-8 corresponding to the following modulation indices:
  • (a) μ = 0.5
  • (b) μ = 1
  • (c) μ = 2
  • (d) μ = ∞

How do you interpret the case μ = ∞?

7.9-1 Consider the signal x(t) defined as

$$x(t) = \begin{cases} 1 - |t| & -\frac{1}{2} \le t \le \frac{1}{2} \\ 0 & \text{otherwise} \end{cases}$$

• (a) Sketch the signal x(t) over −2 ≤ t ≤ 2.
• (b) Use time-differentiation and other Fourier transform properties to determine X(ω). The only integration you should use is to determine the dc component X(0).

(c) Using MATLAB, verify the correctness of X(ω) by synthesizing a 3-periodic replication of the original time-domain signal x(t).

[Hint: Follow the approach taken in Ex. 7.17.]

• 7.9-2 Consider the signal ^x(t) = |t|rect ^t−¹ 3 .
  • (a) Sketch the signal x(t) over −5 ≤ t ≤ 5.
  • (b) Use time-differentiation and other Fourier transform properties to determine X(ω). The only integration you should use is to determine the dc component X(0).
  • (c) Use MATLAB to plot the magnitude spectrum |X(ω)| and the phase spectrum X(ω) over suitable ranges of ω.
  • (d) Using MATLAB, verify the correctness of X(ω) by synthesizing a 10-periodic replication of the original time-domain signal x(t).

[Hint: Follow the approach taken in Ex. 7.17.]

• 7.9-3 Consider the continuous-time aperiodic signal x(t) = rect(t) with Fourier transform X(ω) = sinc(ω/2). Furthermore, let y(t) = (1 − |t − 1|)(u(t) − u(t − 2)) with Fourier transform Y(ω).
  • (a) Express the Fourier transform Y(ω) in terms of X(ω).
  • (b) Suppose we create Fourier series coefficients Vk by sampling Y(ω) according to Vk = Y(2πk/3). Sketch the corresponding time-domain signal v(t) over a suitable range of time t.
  • (c) Use MATLAB to synthesize and plot v(t) using the Fourier series coefficients Vk =

Y(2πk/3). Verify that the synthesized waveform matches the result of part (b).

• (d) Suppose we again create Fourier series coefficients Vk according to Vk = Y(2πk/3). Next, we upsample Vk by factor 2 to create Wk. Sketch the time domain signal p(t) that has Fourier series coefficients Pk = Vk +Wk.
• (e) Use MATLAB to synthesize and plot p(t) using the Fourier series coefficients Pk = Vk + Wk defined in part (d). Verify that the synthesized waveform matches the result of part (d).
• 7.9-4 Consider the signal ^x(t) ^{= e}−atu(t). Modify CH7MP2 to compute the following essential bandwidths.
  • (a) Setting a=1, determine the essential bandwidth W¹ that contains 95% of the signal energy. Compare this value with the theoretical value presented in Ex. 7.20.
  • (b) Setting a=2, determine the essential bandwidth W² that contains 90% of the signal energy.
  • (c) Setting a=3, determine the essential bandwidth W³ that contains 75% of the signal energy.
• 7.9-5 A unit amplitude pulse with duration τ is defined as

$$x(t) = \begin{cases} 1 & |t| \le \tau/2 \\ 0 & \text{otherwise} \end{cases}$$

• (a) Determine the duration τ¹ that results in a 95% essential bandwidth of 5 Hz.

• (b) Determine the duration τ² that results in a 90% essential bandwidth of 10 Hz.

• (c) Determine the duration τ³ that results in a 75% essential bandwidth 20 Hz.

• 7.9-6 Consider the signal ^x(t) ^{= e}−atu(t).

• (a) Determine the decay parameter a¹ that results in a 95% essential bandwidth of 5 Hz.

• (b) Determine the decay parameter a² that results in a 90% essential bandwidth of 10 Hz.

• (c) Determine the decay parameter a³ that results in a 75% essential bandwidth 20 Hz.

• 7.9-7 Use MATLAB to determine the 95, 90, and 75% essential bandwidths of a one-second triangle function with a peak amplitude of 1. Recall that a triangle function can be constructed by the convolution of two rectangular pulses.

• 7.9-8 A 1/3 duty-cycle square-pulse T0-periodic signal x(t) is described as

$$x(t) = \begin{cases} 1 & -T_0/6 \le t \le T_0/6 \\ 0 & T_0/t \le |t| \le T_0/2 \\ x(t+T_0) & \forall t \end{cases}$$

• (a) Use spectral sampling to determine the Fourier series coefficients Dn of x(t) for T⁰ = 2π. Evaluate and plot Dn for (0 ≤ n ≤ 10).
• (b) Use spectral sampling to determine the Fourier series coefficients Dn of x(t) for T⁰ = π. Evaluate and plot Dn for (0 ≤ n ≤ 10). How does this result compare with your answer to part (a)? What can be said about the relation of T⁰ to Dn for signal x(t), which has fixed duty cycle of 1/3?
• 7.9-9 Determine the Fourier transform of a Gaussian pulse defined as ^x(t) ^{= e}−^t 2 . Plot both x(t) and X(ω). How do the two curves compare? [Hint:

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-(t-a)^2/2}dt=1$$

for any real or imaginary a.]