PG1995
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Q1: 2L is the period if f(x)=f(x+2*L), which they say it is.
Q2: Periodic functions are often described in terms of just the first period. The previous and next periods (cycles) are assumed to occur the same such that f(x)=f(x+T) where T is the period.
As far as the Fourier integral theorem, it is definitely helpful to memorize one form of the theorem. Personally I know it in the exponential form, and not in the form you show.
In my view, in an exam one is tested more for one's memory rather than one's understanding of the concepts. But very important factor which we haven't included is that this varies from one institution to another. Now I'm more of the opinion that one should have to maintain a balance between conceptual understanding and memorization.
In real work, concepts drive everything we do, and memory is based on the person (some people remember well and others dont) and our familiarity with the subject at hand (things we use often, we remember, and things we rarely use, we are more likely to forget).
You are right that testing strategy varies with institution and also with the subject and with the teacher. I've taken many courses where we were allowed to use all our notes and our books too. Most of my graduate level classes had take home exams where any (non-human) resources could be used. These types of tests heavily stress the creative use of concepts. Then there are other situations where no books or notes are allowed and you are expected to know by memory. I think for engineering/physics subjects, the former is better than the latter since it reflects what people do in the real world.
By the way, was it the same exponential form given under **broken link removed** "Equivalent Forms of Fourier's Integral Theorem" that you had in mind?
Kindly also help me with **broken link removed** when you have time. The Document #1 used in the attachment was taken from this PDF.
The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.
The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments.
The Wikipedia article says:
I had thought that along the x-axis frequency is plotted and y-axis is used for the representation of amplitude. Could you please tell me how the y-axis incorporates both amplitude and phase information simultaneously? I couldn't find any such graph on the google. I'm sure I was not using a proper search phrase.
I feel Fourier series and transform are intuitively easier to understand than the Laplace. What is this "moment" thing in this context? I know the 'moment' from physics as the force multiplied by perpendicular distance from the turning point. Kindly help me if possible. Thnaks
Overall it seems correct. I haven't checked every detail, but I trust you transferred the formulas correctly.Q1:
Do you find what I say here correct? The discrete version of the Fourier is comparatively difficult. I went through three different books and collected the formulas.
Q2:
In almost all the formulas the main variable is Ω which has units of rad/sample. When a discrete function is transformed into its Fourier form, the y-axis shows magnitude and x-axis is labelled with Ω. What does the graph say in this case? I don't get especially the labeling of x-axis. For example, in case of Fourier series for periodic continuous time functions, the y-axis shows magnitude/amplitude and x-axis is labelled with frequency. Suppose, the frequency is 50 Hz and magnitude is 10, then it means that you need a sinusoid with magnitude 10 and frequency 50 Hz. If you think my question is not clear, then please let me know. I will try to put it differently. Thank you.