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physics
principles communications systems

Principles of Communications Systems, Modulation and Noise 7th edition Rodger E. Ziemer, William H. Tranter - Solutions

(a) Plot the single-sided and double-sided amplitude and phase spectra of the square wave shown in Figure 2.35(a).(b) Obtain an expression relating the complex exponential Fourier series coefficients of the triangular waveform shown in Figure 2.35(b) and those of Xa (t) shown in Figure 2.35(b).Plot
Rewrite the MATLAB simulation of Example 5.8 for the case of an absolute-value type of non-linearity. Is the spectral line at the bit rate stronger or weaker than for the square-law type of non-linearity?
(a) For ISI-free signaling using pulses with raised cosine spectra, give the relation of the roll-off factor, β, to data rate, R = 1/T, and channel bandwidth, fmax (assumed to be ideal low pass).(b) What must be the relationship between R and fmax for realizable raised-cosine spectra pulses?
(a) Show by a suitable sketch that the trapezoidal spectrum given below satisfies Nyquist’s pulse shaping criterion:P(f) = 2Λ (f / 2W) - Λ(f / W)(b) Find the pulse-shape function corresponding to this spectrum.
Given the following channel pulse response samples:pc (-3T) = 0.001 pc(-2T) = -0.01 pc(-T) = 0.1 pc(0) = 1.0pc (T) = 0.2 pc(2T) = -0.02 pc(3T) = 0.005(a) Find the tap coefficients for a three-tap zero
Repeat Problem 5.15 for a five-tap zero-forcing equalizer.Data From Problem 5.15Given the following channel pulse response samples:pc (-3T) = 0.001 pc(-2T) = -0.01 pc(-T) = 0.1 pc(0) = 1.0pc (T) = 0.2 pc(2T) = -0.02
A simple model for a multi path communications channel is shown in Figure 5.20(a).Figure 5.20 (a)(a) Find Hc (f) = Y(f) / X (f) for this channel and plot |Hc(f)| for Î² = 1 and 0.5.(b) In order to equalize, or undo, the channel-induced distortion, an equalization filter is used. Ideally,
Given the following channel pulse response:pc (-4T) = -0.01; pc (-3T) = 0.02; pc (-2T) = -0.05; pc (-T) = 0.07; pc (0) = 1;pc (T) = -0.1; pc (2T) = 0.07; pc (3T) = -0.05; pc (4T) = 0.03;(a) Find the tap weights for a three-tap
Repeat Problem 5.18 for a five-tap zero-forcing equalizer.Data From Problem 5.18Given the following channel pulse response:pc (-4T) = -0.01; pc (-3T) = 0.02; pc (-2T) = -0.05; pc (-T) = 0.07; pc (0) = 1;pc (T) = -0.1; pc (2T) = 0.07; pc (3T)
In a certain digital data transmission system the probability of a bit error as a function of timing jitter is given by where z is the signal-to-noise ratio, |Î”T|, is the timing jitter, and T is the bit period. From observations of an eye diagram for the system, it is determined
(a) Using the superposition and time-in-variance properties of a linear time-invariant system find the response of a low pass RC filter to the inputx(t) = u(t) - 2u(t - T) + 2u(t - 2T) - u(t - 3T)Plot for T / RC = 0.4, 0.6,1,2 on separate axes. Use MATLAB to do so.(b) Repeat for -x(t). Plot
It is desired to transmit data ISI free at 10 kbps for which pulses with a raised-cosine spectrum are used. If the channel bandwidth is limited to 5 kHz, ideal low pass, what is the allowed roll-off factor, β?
Two semi-infinite filaments on the z axis lie in the regions − ∞ < z < −a and a < z < ∞. Each carries a current I in the az direction.(a) Calculate H as a function of ρ and ϕ at z = 0.(b) What value of a will cause the magnitude of H at ρ = 1, z = 0, to be one-half the value
By appropriate solution of Laplace’s and Poisson’s equations, determine the absolute potential at the center of a sphere of radius a, containing uniform volume charge of density ρ0. Assume permittivity ∈0 everywhere.
The two conducting planes illustrated in Figure 6.14 are defined by 0.001 < Ï < 0.120 m, 0 < z < 0.1 m, Ï• = 0.179 and 0.188 rad. The medium surrounding the planes is air. For Region 1, 0.179 < Ï• < 0.188; neglect fringing and find(a)
Coaxial conducting cylinders are located at ρ = 0.5 cm and ρ = 1.2 cm. The region between the cylinders is filled with a homogeneous perfect dielectric. If the inner cylinder is at 100 V and the outer at 0 V, find(a) The location of the 20 V equipotential surface;(b) Eρ max;(c) r if the charge
The derivation of Laplace’s and Poisson’s equations assumed constant permittivity, but there are cases of spatially varying permittivity in which the equations will still apply. Consider the vector identity, ∇ · (ψG) = G·∇ψ + ψ∇ ·G, where ψ and G are scalar and vector functions,
The functions V1(ρ, ϕ, z) and V2(ρ, ϕ, z) both satisfy Laplace’s equation in the region a < ρ < b, 0 ≤ v < 2π, −L < z < L; each is zero on the surfaces ρ = b for −L < z < L; z = −L for a < ρ < b; and z = L for a < ρ < b; and each is 100 V on the
Show that in a homogeneous medium of conductivity σ, the potential field V satisfies Laplace’s equation if any volume charge density present does not vary with time.
Two conducting plates, each 3 × 6 cm, and three slabs of dielectric, each 1 × 3 × 6 cm, and having dielectric constants of 1, 2, and 3, are assembled into a capacitor with d = 3 cm. Determine the two values of capacitance obtained by the two possible methods of assembling the capacitor.
A continuous data signal is quantized and transmitted using a PCM system. If each data sample at the receiving end of the system must be known to within ±0.20% of the peak-to-peak full-scale value, how many binary symbols must each transmitted digital word contain? Assume that the message signal
The imperfect second-order PLL is defined as a PLL with the loop filterF(s) = s + α / s + λαin which λ is the offset of the pole from the origin relative to the zero location. In practical implementations λ is small but often cannot be neglected. Use the linear model of the PlLL and derive the
In this problem we wish to develop a base band (low pass equivalent model) for a Costas PLL. We assume that the loop input is the complex envelope signal x̃(t) = Ac m(t) ejϕ(t)and that the VCO output is ejϕ(t). Derive and sketch the model giving the signals at each point in the model.
A Costas PLL operates with a small phase error so that sin ψ ≈ ψ ≈ and cos ψ ≈ 1 Assuming that the low pass filter preceding the VCO is modeled as a/(s + α), where α is an arbitrary constant, determine the response to m(t) = u(t - t0).
Verify (4.120) by showing that Kte - Kttu(t) satisfies all properties of an impulse function in the limit as Kt → ∞.
Using a single PLL, design a system that has an output frequency equal to 7/3 f0, Where f0 is the input frequency. Describe fully, by sketching, the output of the VCO for your design. Draw the spectrum at the VCO output and at any other point in the system necessary to explain the operation of
By adjusting the values of R,L, and C in Figure 4.38, design a discriminator for a carrier frequency of 100 MHz, assuming that the peak frequency deviation is 4 MHz. What is the discriminator constant KDfor your design?Figure 4.38 L = 10-3 H C = 10-9 F Envelope e(t) R = 103 Q x,(t) Ур)
Consider the FM discriminator shown in Figure 4.38. The envelope detector can be considered ideal with an infinite input impedance. Plot the magnitude of the transfer function E(f) / Xr (f). From this plot, determine a suitable carrier frequency and the discriminator constant KD, and estimate
A narrow band FM signal has a carrier frequency of 110 kHz and a deviation ratio of 0.05. The modulation bandwidth is 10 kHz. This signal is used to generate a wide band FM signal with a deviation ratio of 20 and a carrier frequency of 100 MHz. The scheme utilized to accomplish this is illustrated
A sinusoidal message signal has a frequency of 250 Hz. This signal is the input to an FM modulator with an index of 8. Determine the bandwidth of the modulator output if a power ratio, Pr, of 0.8 is needed. Repeat for a power ratio of 0.9.
An FM modulator is followed by an ideal band pass filter having a center frequency of 500 Hz and a bandwidth of 70 Hz. The gain of the filter is 1 in the pass band. The unmodulated carrier is given by 10 cos(1000πt), and the message signal is m(t) = 10 cos (20πt). The transmitter
Prove that Jn(Î²) can be expressed as and use this result to show thatJ-n(Î²) = (-1)n Jn(Î²) л cos(B sin x — пх)dx J„(B) = п
By making use of (4.30) and (4.39), show that ο0 Σ)-1 n=-00
An audio signal has a bandwidth of 15 kHz. The maximum value of |m(t)| is10 V. This signal frequency modulates a carrier. Estimate the peak deviation and the bandwidth of the modulator output, assuming that the deviation constant of the modulator is(a) 20 Hz/V(b) 200 Hz/V(c) 2
An FM modulator has fc = 2000 Hz and fd = 20 Hz/V. The modulator has input m(t) = 5 cos[2π(10)t].(a) What is the modulation index?(b) Sketch, approximately to scale, the magnitude spectrum of the modulator output. Show all frequencies of interest.(c) Is this narrow band FM? Why?(d) If
An FM modulator with fd= 10 Hz/V. Plot the frequency deviation in Hz and the phase deviation in radians for the three message signals shown in Figure 4.37.Figure 4.37 т(0) 4 3 1 3 4 т() т() 3 3 2 2.5 2 3 -1 -1 -2 -2 -3 -3 1. 2. 1.
Repeat the preceding problem assuming that m(t) is the triangular pulse 4Î›[1/3(t - 6)].Data From Problem 11An FM modulator has outputwhere fd = 20 Hz/V. Assume that m(t) is the rectangular pulse m(t) = 4II [1/8(t - 4)](a) Sketch the phase deviation in radians.(b) Sketch the
An FM modulator has outputwhere fd = 20 Hz/V. Assume that m(t) is the rectangular pulse m(t) = 4II [1/8(t - 4)](a) Sketch the phase deviation in radians.(b) Sketch the frequency deviation in hertz.(c) Determine the peak frequency deviation in hertz.(d) Determine the peak phase
A transmitter uses a carrier frequency of 1000 Hz so that the unmodulated carrier is Ac cos(2πfct). Determine both the phase and frequency deviation for each of the following transmitter outputs:(a) xc (t) = cos[2π(1000)t + 40 sin (5t2)](b) xc (t) = cos[2π(600)t]
Determine and sketch the spectrum (amplitude and phase) of an angle-modulated signal assuming that the instantaneous phase deviation is ϕ(t) = β sin(2πfmt). Also assume β = 10, fm = 30 Hz, and fc = 2000 Hz.
Given that J0(5) = -0.178 and that J1(5) = -0.328, determine J3(5) and J4(5).
Modify the simulation program given in Computer Example 4.4 by replacing the trapezoidal integrator by a rectangular integrator. Show that for sufficiently high sampling frequencies the two PLLs give performances that are essentially equivalent. Also show that for sufficiently small sampling
The power of an un-modulated carrier signal is 50 W and the carrier frequency is fc = 40 Hz. A sinusoidal message signal is used to FM modulate it with index β = 10. The sinusoidal message signal has a frequency of 5 Hz. Determine the average value of xc(t). By drawing appropriate spectra, explain
Compute the single-sided amplitude and phase spectra of xc3 (t) = A sin[2Ï€fct + Î² sin(2Ï€fmt)]and xc4 (t) = Ac sin[2Ï€fct + Î² cos(2Ï€fmt)]Compare the results with Figure 4.5.Figure 4.5 AL2(B)| AHo($)] AV¿(B[ AV4B)|
We previously computed the spectrum of the FM signal defined by xc1 (t) = Ac cos[2πfct + β sin(2πfmt)]Now assume that the modulated signal is given byxc2 (t) = Ac cos[2πfct + β cos(2πfmt)]Show that the amplitude spectrum of xc1 (t) and xc2 (t) are identical. Compute the phase
Redraw Figure 4.4 assuming m(t) = A sin (2Ï€fmt + Ï€/6)Figure 4.4 R(t) fm fm fm Ф() R(t) Re Re Ac Ac fm (a) if fc +fm fe - fm fc fe-fm fe fe+ fm (b) fe +fm .f fe-fm fe fe-fm fe fe+ fm -f (c) Phase Amplitude Phase Amplitude
Repeat the preceding problem for kp-1/2Ï? and 3/8Ï?. Data From problem 1 Let the input to a phase modulator be m(t) = u(t - t0), as shown in Figure 4.1(a). Assume that the unmodulated carrier is Ac cos(2Ï?fct) and that fct0 = n, where n is an integer. Sketch accurately the phase modulator
The purpose of this exercise is to demonstrate the properties of SSB modulation. Develop a computer program to generate both upper-side band and lower-side band SSB signals and display both the time-domain signals and the amplitude spectra of these signals. Assume the message signal m(t) = 2
Using the same message signal and value for fm used in the preceding computer exercise, show that carrier reinsertion can be used to demodulate an SSB signal. Illustrate the effect of using a demodulation carrier with insufficient amplitude when using the carrier reinsertion technique.
Assume that a DSB signal xc (t) = Acm(t) cos (2πfct + ϕ0) is demodulated using the demodulation carrier 2 cos [2πfct + θ(t)]. Determine, in general, the demodulated output yD (t). Let Ac = 1 and θ(t) = θ0, where θ0 is a constant, and determine the mean-square error between
A message signal is given byand the carrier is given by c(t) = 100 cos(200Ï€t)Write the transmitted signal as a Fourier series and determine the transmitted power. 5 5 m(t) = > 10 sin(2¤kf„t) Σ т k k=1
Design an envelope detector that uses a full-wave rectifier rather than the half-wave rectifier shown in Figure 3.3. Sketch the resulting wave forms, as was done in for a half-wave rectifier. What are the advantages of the full-wave rectifier?Figure 3.3 -Envelope ze,(t) x,(1) cc R eo(t) (b) (a) RC
In this computer exercise we investigate the properties of VSB modulation. Develop a computer program (using MATLAB) to generate and plot a VSB signal and the corresponding amplitude spectrum. Using the program, show that VSB can be demodulated using carrier reinsertion.
Three message signals are periodic with period T, as shown in Figure 3.32. Each of the three message signals is applied to an AM modulator. For each message signal, determine the modulation efficiency for Î± = 0.2, Î± = 0.3, Î± = 0.4, Î± = 0.7,
Using MATLAB simulate delta modulation. Generate a signal, using a sum of sinusoids, so that the bandwidth is known. Sample at an appropriate sampling frequency (no slope overload). Show the stair step approximation. Now reduce the sampling frequency so that slope overload occurs. Once again, show
In Example 3.1 we determined the minimum value of m(t) using MATLAB. Write a MATLAB program that provides a complete solution for Example 3.1. Use the FFT for finding the amplitude and phase spectra of the transmitted signal xc (t).
The positive portion of the envelope of the output of an AM modulator is shown in Figure 3.33. The message signal is a waveform having zero DC value. Determine the modulation index, the carrier power, the efficiency, and the power in the side bands.Figure 3.33 40 25 10 37 2 EIN
Sketch Figure 3.20 for the case where fLO= fc- fIF. Desired signal Local oscillator fi+ f2=f LO Signal at mixer ini output 2=fF 2f +f2 Image signal fi + 2f2 = fe+ f IF Image signal at mixer output Д! f2 = f F 2f1 + 3f 2 Passband of IF filter
Using a sum of sinusoids as the sampling frequency, sample and generate a PAM signal. Experiment with various values of τfs. Show that the message signal is recovered by low pass filtering. A third-order Butter worth filter is suggested.
A message signal is a square wave with maximum and minimum values of 8 and -8 V, respectively. The modulation index α = 0.7 and the carrier amplitude Ac = 100 V. Determine the power in the side bands and the efficiency. Sketch the modulation trapezoid.
In this problem we examine the efficiency of AM for the case in which the message signal does not have symmetrical maximum and minimum values. Two message signals are shown in Figure 3.34. Each is periodic with period T, and Ï„ is chosen such that the DC value of m(t) is zero. Calculate
An AM modulator operates with the message signalm(t) = 9 cos (20πt) - 8 cos (60πt)The unmodulated carrier is given by 110 cos(200πt) and the system operates with an index of 0.8.(a) Write the equation for mn(t), the normalized signal with a minimum value of -1.(b) Determine {m2n(t)}, the power
Rework Problem 3.8 for the message signalm(t) = 9 cos(20πt) + 8 cos(60πt)Data from Problem 3.8(a) Write the equation for mn(t), the normalized signal with a minimum value of -1.(b) Determine {m2n(t)}, the power in mn(t).(c) Determine the efficiency of the modulator.(d) Sketch the double-sided
An AM modulator has outputxc (t) = 40 cos [20π(200)t] + 5 cos[2π(180)t] + 5 cos[2π(220)t]Determine the modulation index and the efficiency.
An AM modulator has outputxc(t) = A cos[2π(200)t] + B cos[2π(180)t] +B cos[2π(220)t]The carrier power is P0 and the efficiency is Eff. Derive an expression for Eff in terms of P0, A, and B. Determine A,B, and P0 = 200 W and Eff = 30%.
An AM modulator has outputxc(t) = 25 cos[2π(150)t] + 5 cos[2π(160)t] + 5 cos[2π(140)t]Determine the modulation index and the efficiency.
An AM modulator is operating with an index of 0.8.The modulating signal ism(t) = 2 cos(2πfmt) + cos(4πfmt) + 2 cos(10πfmt)(a) Sketch the spectrum of the modulator output showing the weights of all impulse functions.(b) What is the efficiency of the modulation process?
Consider the system shown in Figure 3.35. Assume that the average value of m(t) is zero and that the maximum value of |m(t)| is M. Also assume that the square-law device is defined by y(t) = 4x (t) + 2x2(t).Figure 3.35(a) Write the equation for y(t).(b) Describe the filter that yields an AM signal
Assume that a message signal is given bym(t) = 4 cos(2πfmt) + cos(4πfmt)Calculate an expression forxc(t) = 1/2 Acm(t) cos(2πfmt) ± 1/2 Acm̂ (t) sin (2πfct)for Ac = 10. Show, by sketching the spectra, that the result is upper-side band or lower-side band SSB depending upon the choice of the
Redraw Figure 3.10 to illustrate the generation of upper-side band SSB. Give the equation defining the upper side band filter. Complete the analysis by deriving the expression for the output of an upper-side band SSB modulator.Figure 3.10 HL f) .f fe fe DSB spectrum Л .f fe fe SSB spectrum (a) sgn
Squaring a DSB or AM signal generates a frequency component at twice the carrier frequency. Is this also true for SSB signals? Show that it is or is not.
Prove analytically that carrier reinsertion with envelope detection can be used for demodulation of VSB.
Figure 3.36 shows the spectrum of a VSB signal. The amplitude and phase characteristics are the same as described in Example 3.3. Show that upon coherent demodulation, the output of the demodulator is real.Figure 3.36 + fe *fc
A mixer is used in a short-wave superheterodyne receiver. The receiver is designed to receive transmitted signals between 10 and 30 MHz. High-side tuning is to be used. Determine an acceptable IF frequency and the tuning range of the local oscillator. Strive to generate a design that yields the
A superheterodyne receiver uses an IF frequency of 455 kHz. The receiver is tuned to a transmitter having a carrier frequency of 1100 kHz. Give two permissible frequencies of the local oscillator and the image frequency for each. Repeat assuming that the IF frequency is 2500 kHz.
A DSB signal is squared to generate a carrier component that may then be used for demodulation. (A technique for doing this, namely the phase-locked loop, will be studied in the next chapter.) Derive an expression that illustrates the impact of interference on this technique.
A continuous-time signal is sampled and input to a holding circuit. The product of the holding time and the sampling frequency is τ fs. Plot the amplitude response of the required equalizer as a function of τ fs. What problem, or problems, arise if a large value of τ is used while the sampling
A continuous data signal is quantized and transmitted using a PCM system. If each data sample at the receiving end of the system must be known to within ±0.25% of the peak-to-peak full-scale value, how many binary symbols must each transmitted digital word contain? Assume that the message signal
Five messages band limited to W, W, 2W, 5W and 7W, Hz, respectively, are to be time-division multiplexed. Devise a sampling scheme requiring the minimum sampling frequency.
A delta modulator has the message signalm (t) = 3 sin 2π(10)t + 4 sin 2π(20)tDetermine the minimum sampling frequency required to prevent slope overload, assuming that the impulse weights δ0 are 0.05π.
Five messages band limited to W, W, 2W, 4W, and 4W Hz,are to be time-division multiplexed. Devise a commutator configuration such that each signal is periodically sampled at its own minimum rate and the samples are properly interlaced. What is the minimum transmission bandwidth required for this
Repeat the preceding problem assuming that the commutator is run at twice the minimum rate. What are the advantages and disadvantages of doing this?Repeat the preceding problemFive messages band limited to W, W, 2W, 4W, and 4W Hz, are to be time-division multiplexed. Devise a commutator
In an FDM communication system, the transmitted base band signal isx(t) = m1(t) cos(2πf1t) + m2(t) cos(2πf2t)This system has a second-order non-linearity between transmitter output and receiver input. Thus, the received base band signal y(t) can be expressed as y(t) = a1x(t) +
Write a computer program to evaluate the coefficients of the complex exponential Fourier series of a signal by using the fast Fourier transform (FFT). Check it by evaluating the Fourier series coefficients of a square waveand comparing your results with Computer Exercise 2.2.
Generalize the computer program of Computer Example 2.1 to evaluate the coefficients of the complex exponential Fourier series of several signals. Include a plot of the amplitude and phase spectrum of the signal for which the Fourier series coefficients are evaluated. Check by evaluating the
Sketch the single-sided and double-sided amplitude and phase spectra of the following signals:(a) x1(t) = 10 cos(4πt + π/8) + 6 sin(8πt + 3π/4)(b) x2(t) = 8 cos(2πt + π/3) + 4 cos(6πt + π/4)(c) x3(t) = 2 sin(4πt + π/8) + 12 sin(10πt)(d) x4(t) = 2 cos(7πt + π/4) + 3 sin(18πt + π/2)(e)
Writea computer program to sum the Fourier series for the signals given in Table 2.1. The number of terms in the Fourier sum should be adjustable so that one may study the convergence of each Fourier series. Table 2.1 Fourier Series for Several Periodic Signals Signal (one period) Coefficients for
A signal has the double-sided amplitude and phase spectra shown in Figure 2.33. Write a time-domain expression for the signal. |Phase |Amplitude 4 -4 -2 4 4
The sum of two or more sinusoids may or may not be periodic depending on the relationship of their separate frequencies. For the sum of two sinusoids, let the frequencies of the individual terms be f1 and f2, respectively. For the sum to be periodic, f1 and f2 must be commensurable; i.e., there
Sketch the single-sided and double-sided amplitude and phase spectra of (a) x1 (t) = 5 cos (12πt - π/6)(b) x2 (t) = 3 sin(12πt) + 4 cos (16πt) (c) x3 (t) = 4 cos (8πt) cos (12πt) (d) x4 (t) = 8 sin (2πt) cos (5πt) (e) x5 (t) = cos (6πt) + 7 cos (30πt) (f) x6 (t)
(a) Show that the function Î´Îµ(t) sketched in Figure 2.4(b) has unity area.Figure 2.4 (b)(b) Show thatÎ´Îµ(t) = e-1e-t/Îµu(t)has unity area. Sketch this function for E = 1,1/2, and 1/4. Comment on its suitability as an approximation for
Write a computer program to find the bandwidth of a lowpass energy signal that contains a certain specified percentage of its total energy, for example, 95%. In other words, write a program to find
Use the properties of the unit impulse function given after (2.14) to evaluate the following relations.(a)(b) 10+ means just to the right of 10; -10- means just to the left of -10(c) (d) (e) L[ + exp(-2t)]s(2t – 5) dt 00 L- ô (t – 5n)] di (f² + 1) [E -10+ 00 –10- n=-00
Which of the following signals are periodic and which are aperiodic? Find the periods of those that are periodic. Sketch all signals.(a) xa (t) = cos (5Ï€t) + sin (7Ï€t) (b) (c)(d) xd (t) = sin(3t) + cos (2Ï€t)(e) (f) X, (t) = E, A(t – 2n) 00 %3D
Write the signal x(t) = cos(6πt) + 2 sin(10πt) as(a) The real part of a sum of rotating phasors.(b) A sum of rotating phasors plus their complex conjugates.(c) From your results in parts (a) and (b), sketch the single-sided and double-sided amplitude and phase spectra of x(t).
Find the normalized power for each signal below that is a power signal and the normalized energy for each signal that is an energy signal. If a signal is neither a power signal nor an energy signal, so designate it. Sketch each signal (α is a positive constant).(a) x1 (t) = 2 cos(4πt + 2π/3)(b)
Classify each of the following signals as an energy signal or as a power signal by calculating E, the energy, or P, the power (A, B, θ, ω, and τ are positive constants).(a) x1(t) = A| sin (ωt + θ)|(b) x2(t) = Aτ / √τ + jt, j = √-1(c) x2(t) = Ate -t/τ u(t) (d) x4(t) = II(t/τ) +
Find the powers of the following periodic signals. In each case provide a sketch of the signal and give its period.(a) x1 (t) = 2 cos (4Ï€t - Ï€/3)(b) (c) (d) t-4η x, ()-Σ3Π n=-00 2 t-6η x, ()ΣοΛ 1-0ο
For each of the following signals, determine both the normalized energy and power. Tell which are power signals, which are energy signals, and which are neither.(Note: 0 and ∞ are possible answers.)(a) x1(t) = 6e(-3+j4π)t u(t) (b) x2(t) = II[(t - 3) /2] + II(t - 3/6)(c) x3(t) =
Show that the following are energy signals. Sketch each signal.(a) x1(t) = II(t/12) cos (6Ï€t)(b) x2(t) = e-|t|/3(c) x3(t) = 2u(t) - 2u(t - 8)(d) r1-10 u (2) da u(1) dà + S x4(1) = [Lo u(a) da – 2 L t-20 %3D -0- -00 -00

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