Saikat Mukherjee

Now I ask my name….which I found at first in my way… –A sweet smell of desire…… it has things to say.

Tag: Electrical Engineering

Fourier Analysis for a Continuous Time Non-periodic Signal

Fourier Series is defined in trigonometric terms but cosine and sine can be defined with complex terms also-

cos{\theta} + isin{\theta} = e^{i\theta}
cos{\theta} - isin{\theta} = e^{-i\theta}

First of all before we go further one must know that along with defining any periodic function by the sum of various harmonics of sine and cosine, these can be defined by the following as well-

f(x) = \sum_{n=-\infty}^{\infty} C_{n}e^{\frac{in\pi\omega t}{l}} ;[Let the period of any periodic function is 2l ]
where angular velocity {\omega} = \frac{2\pi}{T} ; [T is the time period; i.e time required to pass 2\pi angle once]

{\omega} = {2\pi f} [f is the frequency]
Here {\omega} is considered as the frequency.

Here one should know that C_{n}=\frac{1}{2l}\int_{-l}^{l}f(x)e^{\frac{-in\pi x}{l}}\,\mathrm{d}x
Now defining f(x) in terms of \omega is called frequency domain expression and defining in terms of time (where laplace transformation is used to express any system) is called time domain expression.
A case can arise when f(x) is not periodic,this means f(x) has no period and time period T is tending to infinity (T \rightarrow \infty ) and so \omega\rightarrow 0 . As it’s non-periodic function, so obviously its frequency can’t be defined.
Say-

1-0

 

 

This is a non-periodic function where -a\leq\omega t\leq a ,means this function doesn’t repeat after a fixed period of time.
Now f(t) =\sum_{n=-\infty}^{\infty}C_{n}e^{jn\omega t} ; [ l term only comes when period is not 2\pi ,say here time period is 2\pi ]

\therefore C_{n}=\frac{1}{T}\int_{\frac{-T}{2}}^{\frac{T}{2}}f(t)e^{-jn\omega t}\,\mathrm{d}t
As here f(t) is non-periodic function, t \rightarrow \infty and \omega\rightarrow 0 ,so 'n' becomes meaningless in the above expression.
Let n\omega \rightarrow \omega
\omega \rightarrow \triangle \omega

\therefore T \rightarrow \frac{2\pi}{\triangle \omega}

\therefore f(t) becomes-

f(t)=\sum_{\omega =-\infty}^{\infty}C_{\omega}e^{j\omega t} [\forall \omega = 0,\pm \triangle \omega , \pm 2\triangle \omega ... ] \longrightarrow (1)

\therefore C_{\omega} = \frac{\triangle \omega}{2\pi}\int_{\frac{-T}{2}}^{\frac{T}{2}}f(t)e^{-j\omega t}\,\mathrm{d}t

\therefore Substituting C_{\omega} in equation (1)

f(t) = \frac{1}{2\pi}[\sum_{\omega =-\infty}^{\infty}\int_{\frac{-T}{2}}^{\frac{T}{2}}f(t)e^{-j\omega t}\,\mathrm{d}t]e^{j\omega t}\triangle \omega

\because T \rightarrow \infty ,\triangle \omega \rightarrow \,\mathrm{d}\omega and \sum \rightarrow \int

\therefore f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}[\int_{-\infty}^{\infty}f(t)e^{-j\omega t}\,\mathrm{d}t]e^{j\omega t}\,\mathrm{d}\omega

\therefore f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}\,\mathrm{d}\omega \Longrightarrow [Inverse Fourier Transform]

F({\omega}) = \mathscr{F}[f(t)]= \int_{-\infty}^{\infty}f(t)e^{-j\omega t}\,\mathrm{d}t \Longrightarrow [Fourier Transform]
[\mathscr{F} denotes fourier transform ]

So it has been shown that for a non-periodic function we have to first get the fourier trasform of that function (Of course firstly by verifying Dirichlet conditions) to get the fourier series of that function.

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The Engineering era………1

I was to read science in the Higher Secondary,that was my as well my father’s choice,obviously…why not..I used to have damn good concept in maths and physical sciences and I feel proud of me,and a total thanks to my father,he is the only reason behind my in depth knowledge and whatever marks I got.
I had always been in Jodhpur Park Boys School from class II(1996) to Class XII(2007) and 2007 was one of the best year of my life. I got a good marks in the science group(physics,chemistry,maths)…there was  a massacre in physics in our section-A,I was one of the heroes in physics while everyone was smashed by the f*cking examiner…I doubt whether he knows any physics or not..becuase I was flawless in my writings in the exam. I could have challenged and pull that bustard’s pant down. Anyway I didn’t do so because WBJEE result had got out before that,and I had to go to the counselling with all my genuine marksheets and certificates,So I was happily dissatisfied with my 82% marks in physics.
I and DK(saikat gupta) decided to choose the same college and same department of Engineering,that is the Electrical Engineering. We chose N.S.E.C as our college.
I was quite excited about the start of my college life and moreover my father was very happy as I got into a core stream of reputed Engineering college of Kolkata.
Our admission was 20th July,2007…and for that we had to cancel our Ladakh tour 😦 ,though we made it in 2008 🙂 . Anyway our first day of college was 2nd August,2007. I was in casuals,like a red T-shirt and a jeans,some Seniors of 2nd year were there inside the class room where we were asked to sit. They were just talking to us and just some interactions,nothing of ragging kinda. On that day,after some time director of our college came and were talking craps,anyway that was called as lecture! I have a very good habit of not following ‘lectures’,this my instinct actually. And for that,school teachers used to curse me a lot! 😛 For that,teachers used to throw their grievances to my father. And from the very first day of my school I used to care a damn on these issues! 😀
Now one can imagine how I can be in my college days! 😛
Anyway I was saying my first day at college. It was a duration of 1 to 2hours  in the college. While going out of the college our 2nd year seniors told us not to wear casuals before the freshers’ welcome. So we had to wear formal shirts/Tshirts  and trousers,I had only one or two trousers,so I had to use them in turns. 9th september was our freshers’ welcome,it was really well organized and I participated in a quiz,and was representing Electrical department with a team of three members. We came out third amongst five departments 😛 . Apart from this the food was good,a packet of biriyani for everyone. Unluckily on that particular day I did not carry any money sufficient to buy a bottle of water,I was dying to drink,D.K helped me out,and we enjoyed a lot,and returned to home on 10:30pm  by local train,and we were really damn tired!!
will be continued……