Question about Fourier Series/Transform

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In summary, the Fourier series and transform can be used to represent signals in the frequency domain. The Fourier series is for discrete-time signals, and the Fourier transform is for continuous-time signals. The Fourier series and transform can be used to represent signals in the frequency domain.
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Joseph Yellow
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Hi guys, I'm now studying Fourier series/transform for representing signals in the frequency domain.

I'm having a bit of a hard time getting the gist of it. Right now I'm using the book "signals and systems" (oppenheim) because that's the one my teacher uses.

My problem is this: both the book and my teacher divide the subject into a lot of segments:

  • Fourier series for discrete time periodic signals.
  • Fourier series for continuous-time periodic signals.
  • Fourier transform for discrete-time aperiodic signals.
  • Inverse Fourier transform for discrete-time aperiodic signals.
  • Fourier transform for continuous-time aperiodic signals.
  • Inverse Fourier transform for continuous-time aperiodic signals.
  • Fourier transform for discrete-time periodic signals.
  • Inverse Fourier transform for discrete-time periodic signals.
  • Fourier transform for continuous-time periodic signals.
  • Inverse Fourier transform for continuous-time periodic signals.
Although the formulas are pretty similar, the method for each one differs a little bit (as far as I understood it at least), and I'm getting a little bit overwhelmed as I'm trying to get it all.

My question is if need to remember every single little detail about each case or if there's something I'm not getting and there is a more intuitive/general way to think and solve problems involving any of the cases listed above.

If anyone has a good alternative resource (video/text) to Oppenheim's book I would be very glad to hear about it too.

Thanks in advance.
 
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As you suspect, they can all be done in a fairly unified way. But doing that may require studying concepts (generalized functions, test functions, etc.) that may take some patience and some more mathematical theory. There is an excellent Stanford lecture series on youtube that presents a unified theory
(see ).
It consists of 30 lectures, but you may satisfy your intellectual curiosity with fewer. I found all 30 to be worth while.
 
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FactChecker said:
...There is an excellent Stanford lecture series on youtube that presents a unified theory
(see ).


This video lecture looks interesting. I will check it out.
 
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Related to Question about Fourier Series/Transform

What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions with different frequencies and amplitudes. It allows us to break down a complex function into simpler components and analyze it in the frequency domain.

What is a Fourier transform?

A Fourier transform is a mathematical operation that converts a function from the time domain to the frequency domain. It allows us to analyze the frequency components of a non-periodic function and identify its dominant frequencies.

What is the difference between a Fourier series and a Fourier transform?

A Fourier series is used for periodic functions, while a Fourier transform can be applied to both periodic and non-periodic functions. Additionally, a Fourier series gives a discrete representation of a function, while a Fourier transform gives a continuous representation.

What is the practical application of Fourier series and transform?

Fourier series and transform are widely used in various fields such as signal processing, image processing, data compression, and solving differential equations. They are also used in audio and video compression, medical imaging, and speech recognition.

What are the limitations of Fourier series and transform?

Fourier series and transform assume that the function being analyzed is periodic or has a finite number of oscillations. They also assume that the function is continuous and has a well-defined Fourier transform. These assumptions may not hold for some real-world applications, leading to inaccurate results.

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