Understanding Fourier Series, Transform and DFT

[Jag1972]

Hello All,
I am little confused with the Fourier family of transforms. I would really appreciate it if someone could have a look at them
My understanding is as follows:

Fourier Series: Only used for Peiodic continuous signals.

Fourier Transform: Can be used for periodic or aperiodic signals, I think. This transform to me seems
to look the same as the Fourier Series its just more compact as it uses Eulers formulae and embeds the average
into the 1 summation (n=0). If signal is aperiodic then is is just assummed to be periodic.

Discrete Fourier Transform: Always finite as data is bieng sampled.

In some of the books I am reading there are 4 versions in the Fourier family, I don't quite know why.

Thank you in advance for taking the time to help in advance.

Jag.

 

[Ele38]

Fourier Transform: Can be used for periodic or aperiodic signals, I think. This transform to me seems
to look the same as the Fourier Series its just more compact as it uses Eulers formulae and embeds the average
into the 1 summation (n=0). If signal is aperiodic then is is just assummed to be periodic.

The Fourier transform requires a continuous function as the input and its output is another continuous function (so the spectrum of the signal is NOT discrete). Unlike Fourier transform, Fourier serie gives a discrete spectrum, not a continuous one.

In some of the books I am reading there are 4 versions in the Fourier family

Maybebecause there is DFT, Discrete Fourier Transform, and FFT, Fast Fourier Transform. They are similar, but the second one is faster.