If F(j) = 10 V/(rad/s) for 4 << 2 rad/s, +10 V/(rad/s) for 2 << 4 rad/s, and 0 for all

Let a third-harmonic voltage be added to the fundamental to yield \(v = 2\ cos \omega_0 t + V_{m3}\ sin\ 3 \omega_0 t\), the waveform shown in Fig. 18.1c for \(V_{m3} = 1\). (a) Find the value of \(V_{m3}\) so that v(t) will have zero slope at \(\omega_0 t = 2 \pi / 3\). (b) Evaluate v(t) at \(\omega_0 t = 2 \pi /3\).

A periodic waveform f (t) is described as follows: f (t) = ?4, 0 < t < 0.3; f (t) = 6, 0.3 < t < 0.4; f (t) = 0, 0.4 < t < 0.5; T = 0.5. Evaluate (a) \(a_0\); (b) \(a_3\); (c) \(b_1\).

Write the Fourier series for the three voltage waveforms shown in Fig. 18.4.

Sketch each of the functions described; state whether or not even symmetry, odd symmetry, and half-wave symmetry are present; and give the period: (a) v = 0, ?2 < t < 0 and 2 < t < 4; v = 5, 0 < t < 2; v = ?5, 4 < t < 6; repeats; (b) v = 10, 1 < t < 3; v = 0, 3 < t < 7; v = ?10, 7 < t < 9; repeats; (c) v = 8t , ?1 < t < 1; v = 0, 1 < t < 3; repeats.

Determine the Fourier series for the waveforms of Practice Problem 18.4a and b.

Use the methods of Chap. 8 to determine the value of the current sketched in Fig. 18.9 at t equal to (a) \(\pi / 2\); (b) \(\pi\); (c) \(3 \pi / 2\).

Determine the general coefficient \(\mathbf{c}_{n}\) in the complex Fourier series for the waveform shown in (a) Fig. 18.4a; (b) Fig. 18.4c.

If f(t) = ?10 V, ?0.2 < t < ?0.1 s, f(t) = 10 V, 0.1 < t < 0.2 s, and f(t) = 0 for all other t, evaluate \(\mathbf{F} (j \omega)\) for \(\omega\) equal to (a) 0; (b) \(10 \pi\ rad/s\); (c) \(?10 \pi\ rad/s\); (d) \(15 \pi\ rad/s\); (e) \(?20 \pi\ rad/s\).

If \(\mathbf{F}( j \omega) = ?10\ V/(rad/s)\) for \(?4 < \omega < ?2\ rad/s\), +10 V/(rad/s) for \(2 < \omega < 4\ rad/s\), and 0 for all other \(\omega\), find the numerical value of f(t) at t equal to (a) \(10^{?4}\ s\); (b) \(10^{?2}\ s\); (c) \(\pi / 4\ s\); (d) \(\pi / 2\ s\); (e) \(\pi\ s\).

18.10 If \(i(t) = 10e^{20t} [u(t + 0.1) ? u(t ? 0.1)]\ A\), find (a) \(\mathbf{F}_{i}(j 0)\); (b) \(\mathbf{F}_{i}(j 10)\); (c) \(A_{i}(10)\); (d) \(B_{i}(10)\); (e) \(\phi_i (10)\).

Find the \(1\ \Omega\) energy associated with the current \(i(t) = 20e^{?10t} u(t)\ A\) in the interval (a) ?0.1 < t < 0.1 s; (b) \(?10 < \omega < 10\ rad/s\); (c) \(10 < \omega < \infty\ rad/s\).

Evaluate the Fourier transform at \(\omega = 12\) for the time function (a) \(4u(t) ? 10 \delta (t)\); (b) \(5e^{?8t} u(t)\); (c) \(4\ cos\ 8tu(t)\); (d ) \(?4\ sgn(t)\).

Find f (t) at t = 2 if \(\mathbf{F}(j \omega)\) is equal to (a) \(5e^{?j3 \omega} ? j (4 / \omega)\); (b) \(8[\delta (\omega ? 3) + \delta (\omega + 3)]\); (c) \((8 / \omega)\ sin\ 5 \omega\).

Find (a) \(\mathcal{F}\left\{5 \sin ^{2} 3 t\right\}\); (b) \(\mathcal{F}\left\{A \sin \omega_0 t\right\}\); (c) \(\mathcal{F}\{6 \cos (8 t+0.1 \pi)\}\).

The impulse response of a certain linear network is \(h(t) = 6e^{?20t} u(t)\). The input signal is \(3e^{?6t} u(t)\ V\). Find (a) \(\mathbf{H}(j \omega)\); (b) \(\mathbf{V}_{i} (j \omega)\); (c) \(V_o( j \omega)\); (d ) \(v_o (0.1)\); (e) \(v_o (0.3)\); \(( f ) v_{o,\ max}\).

Use Fourier transform techniques on the circuit of Fig. 18.23 to find \(i_1(t)\) at t = 1.5 ms if \(i_{s}\) equals (a) \(\delta (t)\ A\); (b) \(u(t)\ A\); (c) \(cos\ 500t\ A\).

Determine the fundamental frequency, fundamental radian frequency, and period of the following: (a) \(5\ sin\ 9t\); (b) \(200\ cos\ 70t\); (c) \(4\ sin (4t - 10^{\circ})\); (d) \(4\ sin (4t + 10^{\circ})\).

Plot multiple periods of the first, third, and fifth harmonics on the same graph of each of the following periodic waveforms (three separate graphs in total are desired): (a) \(3\ sin\ t\); (b) \(40\ cos\ 100t\); (c) \(2\ cos (10t - 90^{\circ})\).

Calculate \(a_0\) for the following: (a) \(4\ sin\ 4t\); (b) \(4\ cos\ 4t\); (c) \(4 + cos\ 4t\); (d) \(4\ cos (4t + 40^{\circ})\).

Compute \(a_0\), \(a_1\), and \(b_1\) for the following functions: (a) \(2\ cos\ 3t\); (b) \(3 ? cos\ 3t\); (c) \(4\ sin (4t - 35^{\circ})\).

(a) Calculate the Fourier coefficients \(a_0,\ a_1,\ a_2, a_3, b_1, b_2\), and \(b_3\) for the periodic function \(f(t)=2 u(t)-2 u(t+1)+2 u(t+2)-2 u(t+3)+\cdots\). (b) Sketch f (t) and the Fourier series truncated after n = 3 over 3 periods.

(a) Compute the Fourier coefficients \(a_0,\ a_1,\ a_2,\ a_3,\ a_4,\ b_1,\ b_2,\ b_3\), and \(b_4\) for the periodic function g(t) partially sketched in Fig. 18.28. (b) Plot g(t) along with the Fourier series representation truncated after n = 4.

For the periodic waveform f (t) represented in Fig. 18.29, calculate \(a_1,\ a_2,\ a_3\) and \(b_1,\ b_2,\ b_3\).

With respect to the periodic waveform sketched in Fig. 18.29, let \(g_n (t)\) represent the Fourier series representation of f (t) truncated at n. [For example, if n = 1, \(g_1 (t)\) has three terms, defined through \(a_0\), \(a_1\) and \(b_1\).] (a) Sketch \(g_2 (t)\), \(g_3 (t)\), and \(g_5 (t)\), along with f (t). (b) Calculate f (2.5), \(g_2 (2.5)\), \(g_3 (2.5)\), and \(g_5 (2.5)\).

With respect to the periodic waveform g(t) sketched in Fig. 18.28, define \(y_n (t)\) which represents the Fourier series representation truncated at n. (For example, \(y_2 (t)\) has five terms, defined through \(a_0,\ a_1,\ a_2,\ b_1\), and \(b_2\).) (a) Sketch \(y_3 (t)\) and \(y_5 (t)\) along with g(t). (b) Compute \(y_1 (0.5)\), \(y_2 (0.5)\), \(y_3 (0.5)\), and \(g(0.5)\).

Determine expressions for \(a_n\) and \(b_n\) for \(g(t ? 1)\) if the periodic waveform g(t) is defined as sketched in Fig. 18.28.

Plot the line spectrum (limited to the six largest terms) for the waveform shown in Fig. 18.4a.

Plot the line spectrum (limited to the five largest terms) for the waveform of Fig. 18.4b.

Plot the line spectrum (limited to the five largest terms) for the waveform represented by the graph of Fig. 18.4c.

State whether the following exhibit odd symmetry, even symmetry, and/or half-wave symmetry: (a) \(4\ sin\ 100t\); (b) \(4\ cos\ 100t\); (c) \(4\ cos (4t + 70^{\circ})\); (d) \(4\ cos\ 100t + 4\); (e) each waveform in Fig. 18.4.

Determine whether the following exhibit odd symmetry, even symmetry, and/or half-wave symmetry: (a) the waveform in Fig. 18.28; (b) g(t ? 1), if g(t) is represented in Fig. 18.28; (c) g(t + 1), if g(t) is represented in Fig. 18.28; (d) the waveform of Fig. 18.29.

The non periodic waveform g(t) is defined in Fig. 18.30. Use it to create a new function y(t) such that y(t) is identical to g(t) over the range of 0 < t < 4 and also is characterized by a period T = 8 and has (a) odd symmetry; (b) even symmetry; (c) both even and half-wave symmetry; (d) both odd and half-wave symmetry.

Calculate \(a_0\), \(a_1\), \(a_2\), \(a_3\) and \(b_1\), \(b_2\), \(b_3\) for the periodic waveform v(t) represented in Fig. 18.31.

The waveform of Fig. 18.31 is shifted to create a new waveform such that \(v_{new} (t) = v(t + 1)\). Calculate \(a_0\), \(a_1\), \(a_2\), \(a_3\) and \(b_1\), \(b_2\), \(b_3\).

Design a triangular waveform having a peak magnitude of 3, a period of 2 s, and characterized by (a) half-wave and even symmetry; (b) half-wave and odd symmetry.

Make use of symmetry as much as possible to obtain numerical values for \(a_0\), \(a_n\), and \(b_n\), \(1\ \leq\ n\ \leq\ 10\), for the waveform shown in Fig. 18.32.

For the circuit of Fig. 18.33a, calculate v(t) if \(i_s (t)\) is given by Fig. 18.33b and v(0) = 0.

If the waveform shown in Fig. 18.34 is applied to the circuit of Fig. 18.8a, calculate i(t).

The circuit of Fig. 18.35a is subjected to the waveform depicted in Fig. 18.35b. Determine the steady-state voltage v(t).

If the current waveform of Fig. 18.36 is applied to the circuit of Fig. 18.33a, calculate the steady-state voltage v(t).

Let the function v(t) be defined as indicated in Fig. 18.10. Determine \(\mathbf{c}_{n}\) for (a) v(t + 0.5); (b) v(t–0.5).

Calculate \(\mathbf{c}_{0}\), \(\mathbf{c}_{\pm 1}\), and \(\mathbf{c}_{\pm 2}\) for the waveform of Fig. 18.36.

Determine the first five terms of the exponential Fourier series representation of the waveform graphed in Fig. 18.33b.

For the periodic waveform shown in Fig. 18.37, determine (a) the period T; (b) \(\mathbf{c}_{0}\), \(\mathbf{c}_{\pm 1}\), \(\mathbf{c}_{\pm 2}\), and \(\mathbf{c}_{\pm 3}\).

For the periodic waveform represented in Fig. 18.38, calculate (a) the period T; (b) \(\mathbf{c}_{0}\), \(\mathbf{c}_{\pm 1}\), \(\mathbf{c}_{\pm 2}\), and \(\mathbf{c}_{\pm 3}\).

A pulse sequence has a period of 5 ?s, an amplitude of unity for \(?0.6\ <\ t\ <\ ?0.4\ \mu s\) and for \(0.4\ <\ t\ <\ 0.6\ \mu s\), and zero amplitude elsewhere in the period interval. This series of pulses might represent the decimal number 3 being transmitted in binary form by a digital computer. (a) Find \(\mathbf{c}_{n}\). (b) Evaluate \(\mathbf{c}_{4}\). (c) Evaluate \(\mathbf{c}_{0}\). (d) Find \(|\mathbf{c}_{n}|_{max}\). (e) Find N so that \(|\mathbf{c}_{n}|\ \leq\ 0.1|\mathbf{c}_{n}|_{max}\) for all n > N. (f) What bandwidth is required to transmit this portion of the spectrum?

Let a periodic voltage \(v_s (t)\) is equal to 40 V for \(0\ <\ t\ <\ \frac{1}{96}\ s\), and to 0 for \(\frac{1}{96}\ <\ t\ <\ \frac{1}{16}\ s\). If \(T = \frac{1}{16}\ s\), find (a) \(\mathbf{c}_{3}\); (b) the power delivered to the load in the circuit of Fig. 18.39.

Given \(g(t)=\left\{\begin{array}{ll} 5 & -1<t<1 \\ 0 & \text { elsewhere } \end{array}\right\}\) sketch (a) g(t); (b) \(\mathbf{G} (\ j \omega)\).

For the function \(v(t) = 2u(t) ? 2u(t + 2) + 2u(t + 4) ? 2u(t + 6)\ V\), sketch (a) v(t); (b) \(\mathbf{V} (\ j \omega)\).

Employ Eq. [46a] to calculate \(\mathbf{G} (\ j \omega)\) if g(t) is (a) \(5e^{?t} u(t)\); (b) \(5te^{-t} u(t)\).

Obtain the Fourier transform \(\mathbf{F} (\ j \omega)\) of the single triangle pulse plotted in Fig. 18.40.

Determine the Fourier transform \(\mathbf{F} (\ j \omega)\) of the single sinusoidal pulse waveform shown in Fig. 18.41.

For \(g(t) = 3e^{?t} u(t)\), calculate (a) \(\mathbf{G} (\ j \omega)\); (b) \(\mathbf{A}_{g} (1)\); (c) \(\mathbf{B}_{g} (1)\); (d) \(\phi (\omega)\).

Given that \(v(t) = 4e^{?|t|}\ V\), calculate the frequency range in which 85% of the \(1\ \Omega\) energy lies.

Calculate the \(1\ \Omega\) energy associated with the function \(f (t) = 4te^{?3t} u(t)\).

Use the definition of the Fourier transform to prove the following results, where \(\mathcal{F}\{f(t)\}=\mathbf{F}(j \omega)\): (a) \(\mathcal{F}\left\{f\left(t-t_{0}\right)\right\}=e^{-j \omega t_{0}{ }^{2}} \mathcal{F}\{f(t)\}\); (b) \(\mathcal{F}\{d f(t) / d t\}=j \omega \mathcal{F}\{f(t)\}\); (c) \(\mathcal{F}\{f(k t)\}=(1 /|k|) \mathbf{F}(j \omega / k)\); (d) \(\mathcal{F}\{f(-t)\}=\mathbf{F}(-j \omega)\); (e) \(\mathcal{F}\{t f(t)\}=j d[\mathbf{F}(j \omega)] / d \omega\).

Determine the Fourier transform of the following: (a); \(5u(t) ? 2 sgn(t)\); (b) \(2\ cos\ 3t ? 2\); (c) \(4e^{?j3t} + 4e^{j3t} + 5u(t)\).

Find the Fourier transform of each of the following: (a) \(85u(t + 2) ? 50u(t ? 2)\); (b) \(5 \delta (t) ? 2\ cos\ 4t\).

Sketch f (t) and \(|\mathbf{F}( \ j \omega)|\) if f (t) is given by (a) \(2\ cos\ 10t\); (b) \(e^{?4t} u(t)\); (c) \(5\ sgn(t)\).

Determine f (t) if \(\mathbf{F}( \ j \omega)\) is given by (a) \(4 \delta (\omega)\); (b) \(2/(5000 + j \omega)\); (c) \(e^{?j120 \omega}\).

Obtain an expression for f (t) if \(\mathbf{F}( \ j \omega)\) is given by (a) \(-j \frac{231}{\omega}\); (b) \(\frac{1+j 2}{1+j 4}\) (c) \(5 \delta(\omega)+\frac{1}{2+j 10}\).

Calculate the Fourier transform of the following functions: (a) \(2\ cos^2\ 5t\); (b) \(7\ sin\ 4t\ cos\ 3t\); (c) \(3\ sin (4t - 40^{\circ})\).

Determine the Fourier transform of the periodic function g(t), which is defined over the range 0 < t < 10 s by g(t) = 2u(t) ? 3u(t ? 4) + 2u(t ? 8).

If \(\mathbf{F}(\ j \omega)=20 \sum_{n=1}^{\infty}[1 /(|n| !+1)] \delta(\omega-20 n)\), find the value of f(0.05).

Given the periodic waveform shown in Fig. 18.42, determine its Fourier transform.

If a system is described by transfer function \(h(t) = 2u(t) + 2u(t ? 1)\), use convolution to calculate the output (time domain) if the input is (a) \(2u(t)\); (b) \(2te^{?2t} u(t)\).

Given the input function \(x(t) = 5e^{?5t} u(t)\), employ convolution to obtain a time-domain output if the system transfer function h(t) is given by (a) \(3u(t + 1)\); (b) \(10te^{?t} u(t)\).

(a) Design a noninverting amplifier having a gain of 10. If the circuit is constructed using a \(\mu A741\) op amp powered by \(\pm 15\ V\) supplies, determine the FFT of the output through appropriate simulations if the input voltage operates at 1 kHz and has magnitude (b) 10 mV; (c) 1 V; (d) 2 V.

(a) Design an inverting amplifier having a gain of 5. If the circuit is constructed using a \(\mu A741\) op amp powered by \(\pm 10\ V\) supplies, perform appropriate simulations to determine the FFT of the output voltage if the input voltage has a frequency of 10 kHz and magnitude (b) 500 mV; (c) 1.8 V; (d) 3 V.

With respect to the circuit of Fig. 18.43, calculate \(v_o (t)\) using Fourier techniques if \(v_i (t) = 2te^{?t} u(t)\ V\).

After the inductor of Fig. 18.43 is surreptitiously replaced with a 2 F capacitor, calculate \(v_o (t)\) using Fourier techniques if \(v_i (t)\) is equal to (a) \(5u(t)\ V\); (b) \(3e^{?4t} u(t)\ V\).

Employ Fourier-based techniques to calculate \(v_C (t)\) as labeled in Fig. 18.44 if \(v_i (t)\) is equal to (a) \(2u(t)\ V\); (b) \(2 \delta (t)\ V\).

Employ Fourier-based techniques to calculate \(v_o (t)\) as labeled in Fig. 18.45 if \(v_i (t)\) is equal to (a) \(5u(t)\ V\); (b) \(3 \delta (t)\ V\).

Employ Fourier-based techniques to calculate \(v_o (t)\) as labeled in Fig. 18.45 if \(v_i (t)\) is equal to (a) \(5u(t ? 1)\ V\); (b) \(2 + 8e^{?t} u(t)\ V\).

Apply the pulse waveform of Fig. 18.46a as the voltage input \(v_i (t)\) to the circuit shown in Fig. 18.44, and calculate \(v_C (t)\).

Apply the pulse waveform of Fig. 18.46b as the voltage input \(v_i (t)\) to the circuit shown in Fig. 18.44, and calculate \(v_C (t)\).

Apply the pulse waveform of Fig. 18.46a as the voltage input \(v_i (t)\) to the circuit shown in Fig. 18.44, and calculate \(i_C (t)\), defined consistent with the passive sign convention.

Apply the pulse waveform of Fig. 18.46b as the voltage input \(v_i (t)\) to the circuit shown in Fig. 18.45, and calculate \(v_o (t)\).

Apply the pulse waveform of Fig. 18.46b as the voltage input \(v_i (t)\) to the circuit shown in Fig. 18.45, and calculate \(v_o (t)\).

Apply the waveform of Fig. 18.36 to the circuit of Fig. 18.35b, and calculate the steady-state current \(i_L (t)\).

The voltage pulse \(2e^{?t} u(t)\ V\) is applied to the input of an ideal bandpass filter. The passband of the filter is defined by \(100\ <\ | f |\ <\ 500\ Hz\). Calculate the total output energy.