Electrical Engineering: Ch 18: Fourier Series (13 of 35) Even Periodic Functions

Summary

The video script discusses simplifying the process of finding Fourier series coefficients for even periodic functions. It explains that even functions, which are symmetrical about the vertical axis, can be mirrored to produce duplicates on the other side. For such functions, the Fourier series involves only cosine terms, as sine terms become zero. The coefficients are calculated by integrating over half the period, doubling the amplitude, which simplifies the process significantly.

Takeaways

• 🔢 Recognizing the type of function (even or odd) simplifies finding the Fourier series.
• 🔄 Even periodic functions resemble cosine functions with symmetry about the vertical axis.
• ↔️ An even function remains the same if you replace the independent variable T with -T.
• 🔍 Examples of even periodic functions show identical behavior on both sides of the vertical axis.
• ✖️ When calculating Fourier coefficients for even functions, integrate only over half the period.
• 🧮 The coefficient a₀ is calculated as 2/T, with integration over half the period.
• 💡 The aₙ coefficients are found using a 4/T factor, only integrating over half the period.
• 🚫 For even functions, the Bₙ coefficients always equal zero, making the calculations simpler.
• ⏩ No need to compute Bₙ terms when working with even periodic functions, saving time.
• 📈 The symmetry of even functions helps streamline Fourier series calculations.

Q & A

What is an even function in the context of Fourier series?

An even function in the context of Fourier series is a function that is symmetric about the vertical axis. If you take one side of the function and flip it over, it should look exactly the same on the other side.

How does the property of an even function relate to cosine functions?

Even functions have properties similar to cosine functions because they both exhibit symmetry about the vertical axis. This means that if you replace every independent variable T with -T, you get the same function.

What is the significance of even periodic functions in Fourier series?

Even periodic functions are significant in Fourier series because they allow for a simplification in calculating coefficients. The symmetry means that only cosine terms (a_n) need to be calculated, and sine terms (b_n) are zero.

How does the integration process differ for even functions when calculating Fourier coefficients?

For even functions, the integration for calculating coefficients a_n only needs to be done over half the period instead of the entire period, and the amplitude is doubled. This is due to the symmetry of the function.

Why are the b_n coefficients zero for even functions?

The b_n coefficients are zero for even functions because the function does not have any odd symmetry, which is required for sine terms in the Fourier series. Since even functions are symmetric about the vertical axis, there are no sine components.

Can you provide an example of an even periodic function?

Yes, an example of an even periodic function could be a cosine function itself, like cos(ωt), or any function that is symmetric about the y-axis and repeats its values over a period.

What is the formula for calculating a_n coefficients for even functions?

The formula for calculating a_n coefficients for even functions in the Fourier series is \( a_n = \frac{2}{T} \int_{0}^{T/2} f(t) \cos(n\omega t) dt \), where T is the period of the function.

How does the script suggest simplifying the process of finding Fourier coefficients for even functions?

The script suggests that when dealing with even functions, you can ignore the calculation of b_n coefficients altogether since they are zero, and only calculate the a_n coefficients.

What is the implication of not having to calculate b_n coefficients for even functions?

Not having to calculate b_n coefficients for even functions simplifies the process of finding the Fourier series representation of the function, as it reduces the number of integrals that need to be computed.

How does the script explain the process of flipping a function to determine if it's even?

The script explains that to determine if a function is even, you can visualize flipping one side of the function over the vertical axis and see if it matches the other side, or mathematically replace T with -T and check if the function remains the same.

What is the role of the vertical axis in determining the evenness of a function?

The vertical axis plays a crucial role in determining the evenness of a function because it serves as the line of symmetry. A function is even if it is symmetric about the vertical axis.

Outlines

00:00

🔍 Understanding Even and Periodic Functions in Fourier Series

This section introduces the concept of identifying whether a function is even or odd to simplify the process of finding the Fourier series. An even function, resembling a cosine function, has symmetry around the vertical axis. By flipping the function along this axis, it appears identical on both sides. In mathematical terms, substituting the independent variable \( T \) with \( -T \) results in the same function. The periodic nature of the function is emphasized, and the examples provided show how even periodic functions mirror on both sides of the axis.

🧠 Simplifying Fourier Coefficients for Even Functions

This part explains how to calculate Fourier coefficients when dealing with even periodic functions. Due to the function's symmetry, only half the period needs to be integrated, simplifying the process. The coefficient \( A_0 \) is found using \( 2/T \) instead of the standard \( 1/T \) because of the halved period. Similarly, for \( A_n \), the integral is taken over half the period, with the amplitude doubled. It also highlights the fact that all \( B_n \) coefficients, which relate to sine terms, are automatically zero, eliminating the need to calculate them, thus saving time.

⚡ Efficiency in Fourier Series Calculation for Even Functions

This section emphasizes how recognizing a function as even simplifies the Fourier series calculation. Since all \( B_n \) coefficients are zero for even functions, only the \( A_0 \) and \( A_n \) coefficients need to be computed. This reduction in calculations makes the process more efficient, especially when handling even periodic functions. The takeaway message is to always check for symmetry in the function to streamline the analysis and avoid unnecessary steps in future problems.

Keywords

💡Fourier series

The Fourier series is a mathematical tool used to represent a function as a sum of sine and cosine functions. It's central to the video's theme as it discusses how to simplify the process of finding the Fourier series for certain types of functions. The script mentions that recognizing the type of function, such as even or odd, can make finding the Fourier series easier.

💡Even function

An even function is symmetrical with respect to the y-axis, meaning that f(x) = f(-x). In the context of the video, even functions are discussed as a type of function that simplifies the Fourier series calculation because they do not have odd harmonics (B_n terms). The script uses the example of cosine functions to illustrate even functions.

💡Odd function

An odd function is symmetrical with respect to the origin, meaning that f(-x) = -f(x). Although not explicitly defined in the script, the concept is implied when discussing the properties of even functions and how they contrast with odd functions in the context of Fourier series.

💡Periodic function

A periodic function is one that repeats its values at regular intervals or periods. The script mentions that even functions can be considered periodic, and this property is used to simplify the Fourier series calculation for such functions.

💡Cosine function

The cosine function is a type of even function that repeats its values in a regular pattern, making it periodic. The video uses cosine functions as an analogy for even periodic functions, explaining how their properties can be used to simplify the Fourier series.

💡Vertical axis

The vertical axis, or y-axis, is a reference point for discussing symmetry in functions. The script mentions that even functions have a mirror image about the vertical axis, which is a key property used to identify and work with even functions in Fourier series.

💡Coefficients

In the context of Fourier series, coefficients are the constants that multiply the sine and cosine functions. The script explains how to determine these coefficients for even periodic functions, emphasizing that they are crucial for constructing the series.

💡A_n

A_n represents the coefficients for the cosine terms in the Fourier series. The script discusses how to calculate A_n for even functions, noting that the integration is done over half the period, which simplifies the process.

💡B_n

B_n represents the coefficients for the sine terms in the Fourier series. The script explains that for even functions, all B_n coefficients are zero, which means they do not need to be calculated, simplifying the process.

💡Integration

Integration is a mathematical operation used to find the coefficients in the Fourier series. The script describes how integration is applied differently for even functions, only over half the period, due to their symmetry.

💡Omega (ω)

Omega (ω) represents the angular frequency in the context of the Fourier series. The script mentions it in the formula for calculating A_n and B_n, where ωT is part of the cosine function integrated over the period.

Highlights

The process of finding the Fourier series can be simplified by recognizing the type of function involved.

Even functions can be treated as even periodic functions similar to cosine functions.

Cosine functions are not periodic but share properties with even functions, such as symmetry about the vertical axis.

An even function can be identified by its mirror image about the vertical axis.

Replacing the independent variable T with -T in an even function yields the same function.

Periodic functions that are even can be visually identified by their symmetry when flipped.

For even periodic functions, the coefficients can be determined with specific formulas.

The coefficient A_n can be found using 2/T instead of the usual 1/T due to the function's symmetry.

Integration for A_n only needs to be performed over half the period for even functions.

The coefficient A_0 is calculated by doubling the amplitude and integrating over half the period.

For even functions, all the B_n coefficients are zero, simplifying the Fourier series calculation.

There is no need to calculate B_n coefficients for even functions as they are guaranteed to be zero.

This property of even functions saves time in Fourier series calculations.

Only the A_n coefficients need to be calculated for even periodic functions.

The method simplifies the process of finding Fourier series for even periodic functions.

This approach is practical for functions that exhibit even periodicity.

The significance of this method is its efficiency in handling even periodic functions in Fourier series.

Transcripts

welcome to electron line our job in

finding the Fourier series can be made

easier if we recognize what kind of

function we're dealing with if we're

dealing with an even or not function if

we're dealing with an even function and

we should call it an even periodic

function then it's kind of like a cosine

function of course the cosine functions

not a periodic function per se but it

has the same properties as a cosine

function but other words it has a mirror

image about the vertical axis but other

words if we take this side of the

function and we flip it over here we

have an exact duplicate on the other

side another way of thinking about an

even function is if we replace every

independent variable T by a negative T

we get the exact same function that's

what that really means and so we have

some periodic functions here that can be

considered even again if you think about

taking this and flipping it over it

looks exactly the same on both sides of

the vertical axis same over here I can

flip this over same over here we can

flip this over so therefore these are

considered even periodic functions and

when you're dealing with even periodic

functions there's something very

specific about the coefficients we can

determine the coefficients as follows

Asaph not can be found to be 2 over T

remember normally we say 1 over T but

then we integrate over the entire period

we only have to integrate over half the

period because of the symmetry on both

sides of vertical axis and therefore we

doubled the magnitude so it's 2 over T

but then integrating only over half the

period of the f of T DT when we find the

in the a sub ends again we only have to

integrate from zero to half the period

but instead of having 2 over T there we

now have a 4 over T we double the

amplitude but we only integrate over

half the period the rest is the same f

of T times the cosine of n Omega T DT

but what's nice is you can be assured

that when you deal with an even function

you do not have any beasts event terms

in other words all the B sub n factors

become equal to 0 and you don't have to

try them out you could simply not bother

with those integrals so

makes the job a lot easier you only have

to try to find the ace if not and you

only need to try to find the ace of ends

you don't even have to try to be Saban's

because now you know for sure they're

going to be equal to zero

as long as the function is an even

periodic function so remember that it'll

save you time in the future