Fourier Methods Flashcards
Integrate f(x) over a whole period.
State the trigonometric form of fourier series.
Calculate for r=p, r=p=0, r /= p
Shown in lecture 2
May need to use sin(a)sin(b) = 1/2(cos(a-b) - cos(a+b)
State the results for the following.
r = p = 0
r = p
r does not = p
State the equations for the coefficients of a trigonometric fourier series.
Derive the equations for the fourier coefficients.
Covered in lecture 2.
For ar term multiply f(x) by cos(2pix/L) and integrate between x0 and (x0 + L).
For br term multiply f(x) by sin(2pix/L) and integrate between the same bounds.
Rember orthonormality rules for combinations of sin and cos.
What do the Dirichlet B.C tell you?
What are the Dirichlet boundary conditions?
Read green. They tell you that a function can be expanded as a fourier series.
Which fourier coefficients are 0 for odd or even functions?
1/2 is added to make the square wave function shown from a function that can be expressed as an odd function f(x) -1/2.
Do before looking at answers/sketches.
What happens at discontinuities?
Fourier series overshoots at discontinuities, this overshoot NEVER dissapears or changes in size but the position of the overshoot does move CLOSER to the discontinuity as the number of terms included increases.
Try the question.
What is the complex form of the FOURIER SERIES
Note that c-r = cr.
What is the equation for the coefficients of the COMPLEX FOURIER SERIES?
What are the results for the orthogonality of exponentials when r=p, r=p=0, r does not =p?
Complete then look at answers (task is asking for complex fourier series coefficients)
Derive the equation for the complex coefficients of a fourier series- look at image after.
What is Parseval’s theorem? State the equation for complex and trigonometric form.
Proof Parseval’s theorem in complex form. State final answer in terms of trigonometric fourier coefficients.
Use Parseval’s theorem to estimate the rms voltage given the equation V(t). DO NOT LOOK PAST EQ UNTIL ATTEMPTED.
(i) on Image
(ii) for what x is sinx(x)= 0?
sinc(x) also approaches 0 as x tends to infinity. sinc(x) —> 0 as x—> inf
What is the definition of the dirac-delta function?
What is the maxima (at x=0)?
F(w) –> ? as x—> 0
State integral result.
If this is one form of the delta function, what is its form as a fourier transform of another function? DO NOT DO DERIVATION, is on a later slide.
VERY USEFUL IN CALCULATIONS
What would the resultant function of a PRODUCT of g(x) and delta look like?
What is the result of the integral of a function and the dirac-delta function?
Do the derivation.
Hint:
What are the bounds of integration (look at width and centre of delta function)?
Take limits of delta. MEAN VALUE THEOREM FOR INTEGRATION. Limits again for g(c)
Try then look at answer. FT of an infinite mono-chromatic wave. Hint: use delta-function, don’t forget it is an even function.
Try then look at answer. Fourier transform of an exponential.
State the equations for the fourier transform and inverse fourier transform, in terms of t and w, then in terms of x and k.
What does this equal?
State the result.
Show if have time.
What is the result of this?
What is the result of this?
Prove this.
The result is a translation by frequency w0.
What is a finite train wave?
Try the example
What is the fourier transform of a translation?
Hint: Substitution
What is Parserval’s Theorem for a fourier series?
How does it relate the fourier series to the fourier transform?
What is convolution?
State the convolution of 2 functions f(x) and g(x)?
The result should have:
1. Blurred edges (sharp edges become blurred diagonal slopes)
2. Peaks are smoothed.
3. If noise is present, noise is filtered.
Special case: convolution with a delta-function
What does finite experimental resolution make it impossible to do?
Give an example?
READ FOR ANSWERS AFTER
How can convolution be applied/what does its application allow us to achieve?
What would the convolution theorem and the point spread function allow us to do to a blurred image?
Read image after.
On top of undoing experimental response and recovering the original data with the convolution theorem, we can also apply fourier transforms and a low-pass filter to remove noise from the image. Both make a clearer image.
Knowing the experimental response (i.e. the point spread function), we can reconstruct the original data (and restore/deblur images).
What is the convolution THEOREM? (Hint: Fourier Transform).
I.e. State or derive the formula.
The fourier transform of a convolution is equal to the product of the seperate fourier transforms multiplied by sqrt(2pi).
Time shifting property
Calculate the fourier transform for the following top-hat function
Given the fourier transform for a discrete time intervals, state the period, Omega_p, p of the fourier transform signal.
What is the Nyqist frequency?
Find the Nyqist frequency and state which wave will have aliasing.
(a) 10 Hz cosine wave.
(b) 90 Hz cosine wave.
Sketch he FT + Replicas of the digitized 10 Hz and 90 Hz waves.
The sample rate is 100 Hz.
Red- Main frequency, Green- Replicas.
For 10 Hz cosine (no aliasing):
Original impulses at ±10 Hz
Replicas every 100 Hz due to sampling:
… -190, -90, -10, 90, 190 … Hz
For 90 Hz cosine (aliasing happens):
Original impulses at ±90 Hz
After sampling, 90 Hz aliases:
f(alias) =
=∣90−100∣=10 Hz
Therefore, 90 Hz appears as 10 Hz in the digital domain.
Result:
Both 10 Hz and 90 Hz cosines have the same FT after digitization:
Delta spikes at ±10 Hz and all their periodic replicas every 100 Hz.
How can you avoid aliasing with a function that is not bandwith limited?
Solution: Pass through a low-pass filter before sampling.
Sum is from 0 to N-1 for N points of data.
What are the issues that arise due to dealing with discrete data (aliasing)?
- Causes signals to be indistinguishable.
- Distortions arise when signal is reconstructed from samples- i.e. wagon wheel effect.
Solve this Ordinary Differential Equation (find I(t)).
Find the form of p to satisfy the equation, using the results given in the image.
Hint: f(p) must satisfy the equation for all functions.
Only one potential solution is approporiate.
Constant solution for p is not appropriate and is boring, so ignore.
Starting from the wave equation, look for solutions of the form:
Hint: Use seperation of variables.
How does the solution look like, if a string which is fixed at 𝑥 = 0 and
𝑥 = 𝑙, and is plucked at 𝑡 = 0?
Only deterine D, C, A values for this part, the rest will be done later.
How does the solution look like, if a string which is fixed at 𝑥 = 0 and
𝑥 = 𝑙, and is plucked at 𝑡 = 0?
What can fourier transforms be used for (in terms of PDEs and ODEs)?
Solve the following to find u(x, t).
Fourier transforms can be used to turn a PDE in real space into an ODE in Fourier space.
Explanation of the Two Graphs:
Top Graph (10 Hz Signal)
Red Dots: The original 10 Hz signal at ±10 Hz.
Green Dots: Replicas of 10 Hz appearing at multiples of the sampling frequency
f s=100 Hz (e.g.,±110 ,±210,±110,±210 Hz).
Blue Dashed Lines: Indicate the Nyquist Frequency (50 Hz).
Bottom Graph (90 Hz Signal)
Red Dots: The true 90 Hz signal at ±90 Hz (before sampling).
Magenta Dots: The 90 Hz signal aliases to 10 Hz because it’s above the Nyquist limit.
Green Dots: Replicas of the 90 Hz signal (and its aliased version) appear at multiples of 100 Hz.
Blue Dashed Lines: Show the Nyquist Frequency.
Key Takeaways
The 90 Hz signal aliases to 10 Hz, making it indistinguishable from an actual 10 Hz wave after sampling.
The Fourier Transform of the sampled signals looks identical for both cases.
Avoid aliasing by ensuring signals stay below the Nyquist limit before sampling.
