# Evaluate Fourier series coefficients and power of a signal

• gruba
In summary, the conversation discusses deriving the expression for coefficients of Fourier series in exponential form for a sequence of rectangular pulses with amplitude A, period T, and duration θ. It also touches upon evaluating the signal power depending on the coefficients of Fourier series and the relationship between θ and ωo. One approach mentioned is to derive a power series where the sum is A2θ/T, and to consider symmetry and integration over one period. It is also suggested to use τ/T = θ/(2π) to define the width of the pulse. Additionally, the concept of power in a given harmonic being independent of power in other harmonics is mentioned, as well as the use of Parseval's theorem.
gruba

## Homework Statement

Derive the expression for coefficients of Fourier series in exponential form for the sequence of rectangular pulses (with amplitude A, period T and duration θ) shown in this image:

Derive the expression for signal power depending on the coefficients of Fourier series.

-Fourier series

## The Attempt at a Solution

I don't know how to evaluate the signal power depending on the coefficients of Fourier series.

Average power is given by:

Average power is not dependent on Fourier series coefficients. Is this a mistake in evaluation?
Is the signal power depending on the coefficients of Fourier series equal to the average power?

How to evaluate the signal power depending on the coefficients of Fourier series?

A few minor points first: in the equation defining the Fourier series, x(t) should be s(t); and θ = ωot , so you should have sin(/2) etc.
What happens when k = 0? And in your equation defining ak, why do you restrict k to be positive?

Average power is not dependent on Fourier series coefficients. Is this a mistake in evaluation?

Reference https://www.physicsforums.com/threa...es-coefficients-and-power-of-a-signal.907514/

I don't have full answers to your questions, but you may be expected to derive a power series whose sum is A2θ/T.

It should be obvious from symmetry that your Fourier series will be a sum of cosines; you could show this by combining the +k and -k terms. When you square it, you'll get terms proportional to cos(ot).cos(ot), where k and l are integers. Most of these terms will vanish when you integrate over one period, T. Which ones will remain, and what will they look like after integration? Without working it out, I think from that point everything might reduce to a sensible answer.

John Park said:
[...] and θ = ωot , so you should have sin(/2) etc. [...]

According to the problem statement, θ represents the fixed pulse duration. It is not a function of time $t$, rather it is a constant. Why the variable θ was chosen to represent the pulse duration is anybody's guess. But in this case I do not think the claim that θ = ωot is correct.

Power is directly obtainable from the Fourier coefficients. Hint: power in a given harmonic is independent of power in any of the other harmonics.
Cf. also Parseval's theorem.

## 1. What is a Fourier series and what does it represent?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions with different amplitudes and frequencies. It represents the decomposition of a complex signal into its basic frequency components.

## 2. What is the significance of evaluating Fourier series coefficients?

Evaluating Fourier series coefficients allows us to determine the amplitudes and phases of the sinusoidal components that make up a signal. This can help us understand the behavior and characteristics of the signal, and also allows us to reconstruct the original signal from its frequency components.

## 3. How do you calculate the Fourier series coefficients?

The Fourier series coefficients can be calculated using the Fourier series formula, which involves integrating the product of the signal and a complex exponential function over one period of the signal. Alternatively, they can also be calculated using the discrete Fourier transform (DFT) algorithm.

## 4. What is the power of a signal and how is it related to Fourier series coefficients?

The power of a signal is a measure of the signal's strength or energy. It is related to the Fourier series coefficients through Parseval's theorem, which states that the total power of a signal is equal to the sum of the squares of its Fourier series coefficients.

## 5. How can evaluating Fourier series coefficients help in signal processing and analysis?

By evaluating Fourier series coefficients, we can analyze and manipulate signals in the frequency domain. This allows us to filter out unwanted frequency components, remove noise, and extract useful information from a signal. It also helps in understanding the behavior and characteristics of a signal, which can be useful in various fields such as telecommunications, image processing, and audio engineering.

• Calculus and Beyond Homework Help
Replies
3
Views
566
• Engineering and Comp Sci Homework Help
Replies
6
Views
4K
• Engineering and Comp Sci Homework Help
Replies
3
Views
5K
• Engineering and Comp Sci Homework Help
Replies
9
Views
2K
• Engineering and Comp Sci Homework Help
Replies
3
Views
1K
• Engineering and Comp Sci Homework Help
Replies
2
Views
1K
• General Math
Replies
1
Views
1K
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Engineering and Comp Sci Homework Help
Replies
12
Views
2K