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Introduction & Overview
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Quick Overview
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This section elaborates on Parseval's theorem, which relates the average power of a continuous-time periodic signal to the sum of the squared magnitudes of its Fourier coefficients. It emphasizes the significance of this theorem in understanding energy conservation within the signal's representation and its applications in analyzing power components in various signal processing contexts.
Detailed
Parseval's Theorem (Power Relation)
Parseval's theorem provides a fundamental connection between the time and frequency domains for continuous-time periodic signals. It states that the average power of a signal over one period can be computed in two equivalent ways:
- In the time domain, it is expressed as the average of the squared magnitude of the signal.
- In the frequency domain, it is captured through the sum of the squares of the magnitudes of its Fourier series coefficients.
Mathematical Expression
The theorem is mathematically represented as:
Average Power = (1 / T_0) * Integral from 0 to T_0 of |x(t)|^2 dt = Sum from k = -infinity to infinity of |c_k|^2.
Proof Overview
The proof involves substituting the Fourier series representation of the signal into the time-domain integral, leveraging the orthogonality of complex exponentials to eliminate all cross-terms, ultimately leading to the expression for the average power in terms of coefficients.
Physical Interpretation
The average power derived from the time domain signifies the energy consumed (or delivered) over one period, crucial in electrical engineering applications. The sum of squares of the coefficients in the frequency domain identifies the power contributed by individual harmonic components.
Applications
Parseval's theorem finds its applications in computing average power in circuits, analyzing power distribution in signals, evaluating
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Parseval's Theorem Statement
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For a continuous-time periodic signal x(t) with fundamental period T_0, Parseval's theorem relates the average power of the signal in the time domain to the sum of the squared magnitudes of its Fourier series coefficients in the frequency domain.
Average Power = (1 / T_0) * Integral over one period T_0 of [|x(t)|^2 dt] = Sum from k = -infinity to infinity of [|c_k|^2]
Detailed Explanation
Parseval's theorem states that the average power of a periodic signal in the time domain can be calculated either by integrating the square of the signal over one period or by summing the squares of its Fourier coefficients in the frequency domain. The equation combines both perspectives, showing that both approaches yield the same result, reinforcing the idea that energy is conserved whether we view it in time or frequency.
Examples & Analogies
Think of a basketball game where the total points scored (average power) can be calculated based on various scoring methods β either counting the points made in each quarter (time domain) or by summing the points made by each player during the game (frequency domain). No matter how you calculate it, the total score remains the same, just like Parseval's theorem shows for average power.
Proof Sketch of Parseval's Theorem
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The proof involves substituting the exponential Fourier series representation of x(t) into the integral on the left side. Then, by exploiting the orthogonality property of the complex exponentials, all cross-terms integrate to zero, leaving only the sum of the squared magnitudes of the coefficients.
Detailed Explanation
The proof begins with expressing the signal in terms of its Fourier series using exponential functions. By substituting this series into the average power formula, the integral simplifies because of the orthogonality of complex exponentials: when you multiply different frequencies together, their integral over one period is zero. Only terms where the frequencies match contribute to the sum, which leads to the conclusion that the total power in the time domain equals the sum of the squared Fourier coefficients.
Examples & Analogies
Imagine trying to calculate the total weight of fruit in a basket. If you examine each type of fruit separately and note their weights (Fourier coefficients), when you add together the individual contributions, it corresponds to weighing the entire basket (the integral). The key is that different types of fruit donβt mix their weights together, similar to how different frequency components do not cross-multiply in the integral.
Physical Interpretation of Parseval's Theorem
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The left-hand side, (1 / T_0) * Integral [|x(t)|^2 dt], represents the average power of the signal over one period. For voltage or current signals across a 1-ohm resistor, |x(t)|^2 is instantaneous power, and its average is the average power.
The right-hand side, Sum of [|c_k|^2], represents the sum of the average powers contributed by each individual harmonic component. Each |c_k|^2 is the average power of the k-th harmonic.
Detailed Explanation
On the left side of Parseval's theorem, we calculate the average power of the signal during one complete cycle, which involves measuring the signal's strength over that time β akin to measuring how much energy it delivers. The right side tells us how much power is associated with each harmonic in the Fourier series, emphasizing that the overall power is made up of contributions from each frequency. Essentially, each harmonic carries its own weight in defining the signal's energy.
Examples & Analogies
Imagine a symphony orchestra where different instruments produce different notes (harmonics). Each instrument contributes its unique sound to create the overall music (average power). If you were to measure how much sound energy each instrument contributes to the whole symphony, you could use Parseval's theorem to show that all those contributions add up to the total loudness of the music, much like the theorem shows how energy is summed across frequency components.
Applications of Parseval's Theorem
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Parseval's theorem is extremely useful for:
- Calculating the total average power delivered by a periodic voltage or current source in a circuit.
- Assessing how power is distributed among different harmonic components of a signal.
- Quantifying the "energy" or "strength" of a signal in the frequency domain.
- As a basis for calculating Total Harmonic Distortion (THD), a common metric in audio and power systems to quantify the amount of unwanted harmonic content.
Detailed Explanation
Parseval's theorem finds its utility in various areas such as electrical engineering and signal processing. By allowing engineers to easily compute average power from either time or frequency perspectives, it aids in circuit design, analysis of signal quality, and identification of distortion levels in audio systems. The theorem simplifies complex calculations of power across different frequency components and is critical in ensuring efficient circuit operation and performance.
Examples & Analogies
Consider an electricity utility company determining how much power their generators produce. They could calculate power by either measuring how much energy is supplied overall (time domain) or by looking at the energy contributions from each generator's output (frequency domain). Parseval's theorem allows them to use either method with confidence that the information would match, ensuring consistent reporting and efficient generation plans.