Fourier Series Representation of Signals (Proof)

[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. (:

Thank you so much for your help in advance guys!

 

[MathematicalPhysicist]

It follows from:

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

 

[Icetray]

It follows from:

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

Thank you so much! :D

 

[Icetray]

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

 

[Nickie]

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!