Power of first three harmonics of periodic signal

In summary, periodic functions can be represented by complex Fourier coefficients, with a DC component denoted as F_n0. The power spectrum of a signal is defined as the squared modulus of the complex Fourier coefficient, denoted as S11(nω0). In a book, there is a periodic signal given and the task is to calculate the power of the first three harmonics. The correct expression for this power is not simply the sum of the squared modulus of the first three complex Fourier coefficients, but rather involves equating the trigonometric series coefficients to the complex Fourier coefficients and performing an integration.
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etf
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We know that periodic function can be written in terms of complex Fourier coefficients:
$$f(t)=Fn0+\sum_{n=-\infty,n\neq 0}^{n=\infty}F_ne^{jnw_0t}$$, where $$Fn=\frac{1}{T}\int_{\tau}^{\tau+T}f(t)e^{-jnw_0t}dt$$ and $$Fn0$$ is DC component. Power spectrum of signal is defined as $$S11(nw_0)=\left | F_n \right |^{2}$$, where $$\left | F_n \right |$$ is modulus of complex Fourier coefficient $$F_n$$.
In book, they gave us some periodic signal to write it in terms of complex Fourier coefficients and calculate power of first three harmonics. What is power of first three harmonics? Is it $$\left | Fn1 \right |^{2}+\left | Fn2 \right |^{2}+\left | Fn3 \right |^{2}$$?
 
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etf said:
Power spectrum of signal is defined as $$S11(nw_0)=\left | F_n \right |^{2}$$, where $$\left | F_n \right |$$ is modulus of complex Fourier coefficient $$F_n$$.

In book, they gave us some periodic signal to write it in terms of complex Fourier coefficients and calculate power of first three harmonics. What is power of first three harmonics? Is it $$\left | Fn1 \right |^{2}+\left | Fn2 \right |^{2}+\left | Fn3 \right |^{2}$$?
Not right. Start with the trigonometric series coefficients for which you hopefully know the power expression, then equate those coefficients to the Fn.

Or, perform (1/T) ∫T f2(t)dt given f(t) = Σ Fnexp(jnω0)t,
T = 2π/ω0.
 

1. What are harmonics in a periodic signal?

Harmonics in a periodic signal refer to the frequencies that are integer multiples of the fundamental frequency of the signal. These frequencies are also known as overtones and are created by the vibrations of the different components of the signal.

2. Why are the first three harmonics of a periodic signal important?

The first three harmonics are important because they determine the overall shape and characteristics of the signal. They can affect the amplitude, frequency, and phase of the signal, and are crucial in understanding the behavior of the signal over time.

3. How can the power of the first three harmonics be calculated?

The power of the first three harmonics can be calculated by squaring the amplitude of each harmonic and adding them together. This will give the total power of the signal. Alternatively, the power of each harmonic can be calculated separately and then summed up.

4. What is the significance of the power of the first three harmonics in a periodic signal?

The power of the first three harmonics can indicate the overall strength and dominance of the fundamental frequency in the signal. It can also provide information about any distortions or noise present in the signal, as well as the overall quality of the signal.

5. How can the power of the first three harmonics be used in practical applications?

The power of the first three harmonics can be used in various practical applications, such as in analyzing and filtering signals in communication systems, detecting and correcting distortions in audio and video signals, and in designing efficient and high-quality electrical circuits and equipment.

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