SimReal / Python / Matlab: Programming - Exercise - Mathematics - Fourier UiA Logo



SimReal - Python - Matlab 		001 - Fourier Transform - Detect frequencies.

We have the follwing signal function:

y(x) = 1.0·sin(ω1x) + 0.5·sin(ω2x)
where
ω1 = 2πf1 and ω2 = 2πf2
f1 = 50 Hz and f2 = 100 Hz

Write an application that use Fast Fourier Transform (FFT) to extract the two frequencies f1 and f1.
Draw the graph of the original signal function and the Fourier transformed function.

SimReal - Python - Matlab 		002 - Fourier Transform - Detect frequencies.

We have the follwing complex signal function:

y(x) = 1.0·e(jω1x) + 0.5·e(jω2x)
where
ω1 = 2πf1 and ω2 = 2πf2
f1 = 50 Hz and f2 = -100 Hz

Write an application that use Fast Fourier Transform (FFT) to extract the two frequencies f1 and f1.
Draw the graph of the original signal function and the Fourier transformed function.

SimReal - Python - Matlab 		003 - Fourier Transform / Inverse Fourier Transform.

We have the follwing complex signal function:

y(x) = sin(ω1x) + sin(ω2x)
where
ω1 = 2πf1 and ω2 = 2πf2
f1 = 10 Hz and f2 = 20 Hz

Write an application that use Fast Fourier Transform (FFT) to extract the two frequencies f1 and f1.
Then use the Inverse Fourier Transform (IFFT) to transform back to the original signal function. Draw the graph of the original signal function, the Fourier transformed function and the backward transformed function.

SimReal - Python - Matlab 		004 - Fourier Transform / Remove frequency / Inverse Fourier Transform.

We have the follwing complex signal function:

y(x) = sin(ω1x) + sin(ω2x)
where
ω1 = 2πf1 and ω2 = 2πf2
f1 = 10 Hz and f2 = 20 Hz

Write an application that use Fast Fourier Transform (FFT) to extract the two frequencies f1 and f1.
Remove the frequency f2 = 20 Hz in the frequency domain.
Remove by cutting off frequencies.
Then use the Inverse Fourier Transform (IFFT) to transform back to the original signal function, but now without the frequency f2 = 20 Hz.
Draw the graph of the original signal function, the Fourier transformed function both before and after the removal of the frequency and the backward transformed function.

SimReal - Python - Matlab 		005 - Fourier Transform / Remove frequency / Inverse Fourier Transform.

We have the follwing complex signal function:

y(x) = sin(ω1x) + 0.5·sin(ω2x)
where
ω1 = 2πf1 and ω2 = 2πf2
f1 = 10 Hz and f2 = 20 Hz

Write an application that use Fast Fourier Transform (FFT) to extract the two frequencies f1 and f1.
Remove the frequency f2 = 20 Hz in the frequency domain.
Remove by cutting off low frequency amplitudes.
Then use the Inverse Fourier Transform (IFFT) to transform back to the original signal function, but now without the frequency f2 = 20 Hz.
Draw the graph of the original signal function, the Fourier transformed function both before and after the removal of the frequency and the backward transformed function.

SimReal - Python - Matlab 		006 - Fourier Transform / Detect frequencies - Time varying signals.

Write an application to study some of the problems using fourier transform (FFT) with timevarying signals.

We have the follwing two signal functions:

y1(t) = sin(ω1t) + sin(ω2t)
y2(t) = sin(ω1t) t є [0, 0.5>, y2(x) = sin(ω2t) t є [0.5, 1]
where
ω1 = 2πf1, ω2 = 2πf2, f1 = 10 Hz, f2 = 40 Hz

SimReal - Python - Matlab 		007 - Fourier Transform / Inverse Fourier Transform - Noise removing.

Write an application that remove noise from a signal by the help of Fourier transformation and inverse Fourier transformation.
Start with an 'clean' signan with two frequencies.
Then add random high-frequency noise to the signal to form a mixed signal.
Compute the Fourier transfrom of the mixed signal.
Romove the high-frequency part from the Fourier transformed signal.
Re-construct the clean signal by inverse Fourier transformation.

SimReal - Python - Matlab 		008 - Fourier Transform - 2 Dim I.

Write an application that shows how to find details in a picture.
1. Start with a white picture, then draw the graph of the Fourier transform.
2. Start with a white picture with a dark grey point in the middle, then draw the graph of the Fourier transform.
3. Start with a white picture with a dark grey point in the upper left corner, then draw the graph of the Fourier transform.

Notice how the Fourier transform changes when pollutions (here a point) enter the picture.

SimReal - Python - Matlab 		009 - Fourier Transform - 2 Dim II.

Write an application that shows how to find details in a picture.
1. Start with a white picture, then draw the graph of the inverse Fourier transform.
2. Start with a white picture with a dark grey point in the middle, then draw the graph of the inverse Fourier transform.
3. Start with a white picture with a dark grey point in the upper left corner, then draw the graph of the inverse Fourier transform.

Notice how the inverse Fourier transform changes when pollutions (here a point) enter the picture.

SimReal - Python - Matlab 		010 - Fourier Serie - f(x) = x - Without SymPy.

Exercise 010 and 011 is the same, both about Fourier serie.
In exercise 010 we are going to program a Fourier serie using a loop.
In exercise 011 we are going to use Python library SymPy, where built-in Fourier serie.

Write an application to compute the Fourier Serie for the functioin f(x) = x.
Plot this function together with the serie for n=1, n=5 and n=9.

The Fourier serie for f(x) = x is equal to:
s(x) = 2·Σ(-1)n+1sin(nx)

SimReal - Python - Matlab 		011 - Fourier Serie - f(x) = x - SymPy.

Same as exercise 010, but now we are going to use built-in Fourier serie in the Python library SymPy.

Write an application to compute the Fourier Serie for the function f(x) = x.
Plot this function together with the serie for n=1, n=5 and n=9.

Use the Python library SymPy.

SimReal - Python - Matlab 		011 - Fourier Serie - f(x) = x - Different periods.

Same as exercise 011, but now we are going to use built-in Fourier serie in the Python library SymPy to compute the same compute the Fourier Serie for the function f(x) = x for different periods.
Plot this function together with the serie for periods [-π,π], [-1,1] and [0,1].

Use the Python library SymPy.

SimReal - Python - Matlab 		012 - Fourier Serie - Pulse Train.

Compute the Fourier Serie for the Pulse Train function.
Plot this function together with the serie for number of serie elements equal 20.

SimReal - Python - Matlab 		013 - Fourier Serie - Pulse Train.

Extend the exercise 012.
Compute the Fourier Serie for the Pulse Train function.
Plot this function together with the serie for number of serie elements equal 2, 5 and 20.

SimReal - Python - Matlab 		014 - Fourier Serie - f(x) = x2.

Compute the Fourier Serie for the function f(x) = x2.
Use the Python library SymPy.

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