Fourier Circuit Analysis

What is the relationship between the period T of a periodic function's fundamental frequency and its radian frequency ω0?

What does the Fourier series coefficient a0 represent?

For the half-sinusoidal waveform shown in Figure 18.2, which has a period of 0.4 seconds, what is the value of the dc component a0 in terms of the peak voltage Vm?

If a periodic function f(t) possesses even symmetry, where f(t) = f(-t), which coefficients in its trigonometric Fourier series are guaranteed to be zero?

If a periodic function f(t) possesses odd symmetry, where f(t) = -f(-t), which coefficients in its trigonometric Fourier series are guaranteed to be zero?

A periodic function f(t) has half-wave symmetry if it satisfies the condition f(t) = -f(t - T/2). What is the primary characteristic of the Fourier series for such a function?

In the circuit of Figure 18.8a, a periodic voltage source with a dc component of 5 V is applied to a series RL circuit with R=4 ohms and L=2 H. What is the dc component of the forced current response?

How is the complex Fourier coefficient cn related to the trigonometric Fourier coefficients an and bn for n = 1, 2, 3, ...?

How is the amplitude of the nth sinusoidal component of a Fourier series related to the magnitude of the complex coefficients |cn|?

For the square wave in Figure 18.10 with T=2 and V=1, the complex coefficient is given by cn = (1/nπ) * [2sin(nπ/2) - sin(nπ)]. What is the value of c1?

The sampling function is defined as Sa(x) = sin(x)/x. What is its value at x = 0?

What is the Fourier transform of a unit impulse function located at t = t0?

What is the Fourier transform of a constant function of time, f(t) = K?

According to Parseval's theorem, the total 1-ohm energy delivered by a signal f(t) is found by integrating the squared magnitude of its Fourier transform, |F(jω)|^2, over all radian frequency ω. What is the correct scaling factor for this integral?

In the frequency domain, how is the Fourier transform of the output of a linear system, F0(jω), related to the Fourier transform of the input, Fi(jω), and the system function, H(jω)?

What is the relationship between a linear system's impulse response h(t) and its system function H(jω)?

What is the relationship between the system function H(jω) derived from Fourier transform analysis and the sinusoidal steady-state transfer function G(ω)?

A circuit has a system function H(jω) = j2ω / (4 + j2ω). An input voltage vi(t) = 5e^(-3t)u(t) is applied. What is the Fourier transform of this input voltage?

A voltage pulse is given by v(t) = 4e^(-3t)u(t) volts. What is the total 1-ohm energy in this signal?

What is the frequency of the first harmonic of a sinusoid also known as?

The Fourier theorem states that a periodic function f(t) can be represented by an infinite series. Which of the following is the trigonometric form of the Fourier series?

What is the integral expression used to calculate the cosine coefficient 'an' for the trigonometric Fourier series of a function f(t) with period T?

In a line spectrum for a Fourier series, if a harmonic component has both a sine term (coefficient bn) and a cosine term (coefficient an), what represents the total amplitude of that component?

For a function with both half-wave and even symmetry, how can the coefficient 'an' for odd harmonics be calculated more efficiently?

For a function with both half-wave and odd symmetry, which set of coefficients is guaranteed to be zero for all n?

The exponential Fourier series represents a periodic function f(t) as a sum of complex exponentials. What is the integral used to calculate the coefficient cn?

For a periodic function f(t) that has even symmetry, how does the calculation of its complex Fourier coefficient cn simplify?

For a periodic function f(t) that has odd symmetry, how does the calculation of its complex Fourier coefficient cn simplify?

What is a sufficient condition for the Fourier transform F(jω) of a function f(t) to exist?

If f(t) is an even function of time, what are the characteristics of its Fourier transform F(jω)?

If f(t) is an odd function of time, what are the characteristics of its Fourier transform F(jω)?

For a voltage v(t) with Fourier transform Fv(jω), what physical quantity does |Fv(jω)|^2 represent?

What is the Fourier transform of cos(ω0t)?

What is the Fourier transform of the signum function, sgn(t)?

What is the Fourier transform of the unit-step function, u(t)?

The Fourier transform of a periodic function f(t) with complex Fourier coefficients cn and fundamental radian frequency ω0 is a series of impulses. What is the correct expression for this transform?

What is the Fourier transform of the convolution of two time functions, f(t) * g(t)?

In the analysis of the half-wave rectified sinusoid in Example 18.1, with peak voltage Vm, what is the value of the coefficient a1?

For the half-wave rectified sinusoid in Example 18.1, the coefficient an for n > 1 is given by an = (2Vm/π) * cos(nπ/2) / (1 - n^2). What is the value of a2?

For the square wave in Fig. 18.10, the complex Fourier coefficient for n odd is cn = (2/nπ) * sin(nπ/2). What is the value of c3?

For a periodic train of rectangular pulses of duration τ and period T, the 'width' of the envelope of its discrete spectrum is primarily determined by which parameter?

The frequency-shift theorem for Laplace transforms states that replacing 's' with '(s+a)' in the frequency domain corresponds to what operation in the time domain?

Differentiating a Laplace transform F(s) with respect to s corresponds to what operation in the time domain?

Integrating a Laplace transform F(s) from s to infinity corresponds to what operation in the time domain?

The time-scaling theorem for Laplace transforms relates the transform of f(at) to the transform of f(t), which is F(s). What is the correct relationship?

In Example 18.2, a periodic signal is applied to a series RL circuit with R=4 ohms and L=2 H. The fundamental radian frequency is 2 rad/s. What is the impedance Zn of the circuit at the nth harmonic frequency?

Examine the square wave in Figure 18.4b. What type of symmetry does this waveform possess?

What type of symmetry does the triangular waveform shown in Figure 18.4c possess?

A fundamental voltage is v1(t) = 2 cos(ω0t). A third-harmonic voltage v3a(t) = cos(3ω0t) is added to it. What is the period of the resultant waveform v(t) = v1(t) + v3a(t)?

Using the summary of Fourier Transform pairs in Table 18.2, what is the Fourier transform of e^(-αt)u(t)?