[Math] How to visualise Fourier Transform of a function

fourier analysisfourier series

I solved many problems on Fourier series,transforms and inverse fourier transforms as part of my academics. And i am aware that FT converts a time domain signal to frequency domain and IFT is vice versa.

How to visualise that FT really does convert a time domain signal to frequency domain?

My Approach:

Actually when i first thought about this i started with Fourier series. A function is expressed as sum of sine and cosine functions.Then i thought why only sine,cosine? which made me realise its related to right angle triangle (to get x and y co-ordinates of a point) and angle is related to the distance of point from origin. This is where omega*t creeps into theta of sine and cosine. And as x axis is time domain and t creeps in here.

Am i in the right path? Please guide me through this…

Best Answer

A Fourier transform represents the amount of oscillation of a particular frequency $\omega$ in a function. A function having one frequency is represented by a spike at that frequency. A periodic function is represented by spikes at an arithmetic sequence of frequencies. In general, higher frequencies represent faster variations in the original function. By performing a low-pass filter, we are cutting off the higher frequencies and "smoothing" out the function.

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[Math] Interpreting the sign of fourier coefficients

Yes, you're correct that taking $z_n=a_n+ib_n$ will allow you to express your signal in terms of amplitude and phase using $|z|$ and $\operatorname{arg}(z)$ respectively.

You can always switch from $a_n$ and $b_n$ to $z_n$ and see how changing the former parameter set affects the latter. Namely, if you change sign of $a_n$but not $b_n$, or vice-versa, you effectively conjugate $z_n$ (and also multiply by $-1$ if it's $a_n$which you change). Then it's easy to see how $\operatorname{arg}(z)$ will change.

Note that if you use complex parameter $z_n$ instead of real parameters $a_n$ and $b_n$, then should still have a pair of parameters for each $|f|$ where $f$ is frequency (i.e. you have to work with signed frequencies): $a_n+ib_n$ and $a_n-ib_n$, which after doing inverse Fourier transform would cancel out the imaginary part of each other and give you real signal.

And, answering your comment, yes, $\operatorname{arg}(z_n)$ is the phase shift of the harmonic $n$ from its purely cosine version.

[Math] Interpreting Fourier transform frequency graph

If I remember things correctly, the interpretation would be that the temperature curve can be described as 4 functions:

T1 = 30*sin(2*pi * t)
T2 = 15*sin(pi * t)
T3 = 60*sin(2*pi/3 * t)
T4 = 15*sin(pi/2 * t)

and the total temperature is T = T1 + T2 + T3 + T4

this, however, does not mean that the temperature reaches 60 degrees, it could be canceled out by one of the other functions, or alternatively surpass 60 degrees should the waves coincide. So your interpretation is partially correct, if the temp only followed T3 then yes the temp would reach 60 degrees every 3rd hour, but is not necessarily true in the general case because of the summation of the other functions. Hope you understand :)

// Jonathan

Best Answer

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[Math] Interpreting the sign of fourier coefficients

[Math] Interpreting Fourier transform frequency graph

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