# Fourier Series Representation of Signals (Proof)

• Icetray
In summary, a Fourier series is a mathematical representation of a periodic signal as a sum of sine and cosine waves, which allows for easier analysis and understanding of the signal's properties. It is derived using the Fourier transform and can be represented in two forms: continuous for continuous signals and discrete for discrete signals. Any periodic signal can be represented by a Fourier series, and the accuracy of the representation depends on the number of terms used.
Icetray
Hi guys,

I was studying the proof below and just can't figure out the the first highlighted step leads to the second and I was wondering if you guys can help me to fill that in. (:

It follows from:

cos(a+b)= cos(a)cos(b)-sin(a)sin(b)

MathematicalPhysicist said:
It follows from:

cos(a+b)= cos(a)cos(b)-sin(a)sin(b)

Thank you so much! :D

-- it's okay, I for it. Really stupid question. hahaha ---

Hello,

Thank you for reaching out for help with understanding the proof for Fourier series representation of signals. Let me try to explain the first highlighted step and how it leads to the second.

The first highlighted step is using the definition of the Fourier series coefficients, which is given by:

cn = (1/T) ∫f(t)e^(-jωnt) dt

where T is the period of the signal, f(t) is the signal, ω is the angular frequency, and n is the harmonic number.

In the proof, we are trying to show that the Fourier series representation of a signal f(t) is given by:

f(t) = ∑ cn e^(jωnt)

To do this, we need to find the values of the Fourier coefficients cn. The first highlighted step is showing that the Fourier coefficient cn is equivalent to the complex exponential term e^(jωnt) multiplied by the inverse Fourier transform of f(t). This is because the inverse Fourier transform of the complex exponential term e^(jωnt) is a delta function, which is defined as:

δ(x) = (1/2π) ∫e^(jωx) dω

Substituting this into the definition of the Fourier series coefficients, we get:

cn = (1/T) ∫f(t)e^(-jωnt) dt = (1/T) ∫f(t)δ(ω-n) dω

Using the property of the delta function, we can rewrite this as:

cn = (1/T) f(n)

where f(n) is the value of the signal at time t=nT, which is the nth sample of the signal. This is how the first highlighted step leads to the second, by using the definition of the inverse Fourier transform and the property of the delta function.

I hope this explanation helps you understand the proof better. If you have any further questions, please don't hesitate to ask. Keep up the good work in your studies!

## 1. What is a Fourier series representation of signals and why is it important?

A Fourier series is a mathematical representation of a periodic signal as a sum of sine and cosine waves. It is important because it allows us to break down complex signals into simpler components, making it easier to analyze and understand the properties of the signal.

## 2. How is a Fourier series derived?

A Fourier series is derived using the Fourier transform, which is a mathematical tool that decomposes a signal into its frequency components. The Fourier transform is then used to calculate the coefficients of the sine and cosine waves that make up the Fourier series representation.

## 3. What is the difference between a continuous and discrete Fourier series?

A continuous Fourier series is used for signals that are continuous and have an infinite number of data points, while a discrete Fourier series is used for signals that are discrete, meaning they have a finite number of data points. The mathematical equations for calculating the coefficients differ between the two types of Fourier series.

## 4. Can any signal be represented by a Fourier series?

Yes, any periodic signal can be represented by a Fourier series. However, non-periodic signals cannot be represented by a Fourier series, as they do not have a repeating pattern.

## 5. How is the accuracy of a Fourier series representation determined?

The accuracy of a Fourier series representation is determined by the number of terms used in the series. The more terms that are included, the more accurate the representation will be. In practice, a finite number of terms is used to approximate the signal, rather than an infinite number of terms.

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