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.
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.
