Fourier Series

Question 1

fourier series is very useful for

Answer 1

describing phenomena with any kind of repeating, periodic behaviour.

Question 2

Almost any kind of
periodic signal can be written as

Answer 2

sums of sines and cosines, or alternatively as exponentials.

Question 3

for understanding fourier series, it is helpful to think of them as

Answer 3

a decomposition in
terms of orthogonal sets of vectors.

Question 4

in the case of fourier series, the vectors are

Answer 4

functions

Question 5

in the case of fourier series, the orthogonality is defined in terms of

Answer 5

a generalisation of the inner product of two vectors

Question 6

inner product of vectors

Answer 6

<a|b>=a^cross b = sum for i=1 to N of ai^*bi

where cross denotes the hermitian conjugate and we have made us of Dirac notation

Question 7

inner product of functions f and g

Answer 7

<f|g>= integral dx f(x)^*g(x)

Question 8

for a scalar, the hermitian conjugate and the complex conjugate are

Answer 8

identical

Question 9

proof for the functions sin(2πrx/L) and cos(2πrx/L) being the orthogonal basis vectors in the space of periodic functions with period L.

Answer 9

recognising odd and even functions
integrating an odd function gives zero
(one uses a trig identity)

Question 10

fourier coefficients

Answer 10

a0, ar and br

depend on what specific function f(x) is being considered

Question 11

how to obtain the cosone fourier coefficients

Answer 11

1. using basis vector cos(2pipx/L) take the inner product with fourier series expansion
2. consider each term and rearrange

Question 12

the sine and cosine terms in a fourier series span

Answer 12

a vector space of functions

Question 13

the fourier series for any function f(x) in the space, converges to

Answer 13

f(x) wherever f(x) is continuous

Question 14

the fourier series of any function f(x) satisfies

Answer 14

the dirichlet conditions

Question 15

the dirichlet conditions

Answer 15

1. f(x) must be periodic
2. f(x) must be single-valued and either continuous or with a finite number of finite discontinuities
3. f(x) must be of bounded variation
4. the integral over |f(x)| must converge

Question 16

what does the dirichlet condition ‘must be of bounded variation’ mean

Answer 16

essentially the distance travelled by walking along the curve for one period is finite

Question 17

example of a function with discontinuities

Answer 17

square wave

Question 18

the fourier series of a function with a discontinuity at x=xd converges to

Answer 18

thehalf-way value

(between left hand limit and right hand limit)

Question 19

Gibb’s phenomenon

Answer 19

overshooting near each discontinuity remaining the same size despite more and more terms being added

Question 20

what other functions can be sued as an orthogonal basis

Answer 20

e^irkx and e^-ipkx where k=2pi/L

Question 21

complex fourier series

Answer 21

f(x)= sum to infinity of c_r e^irkx

Question 22

for real functions,c_-r=

Answer 22

cr*

Question 23

how to prove the relationship between real trigonometric and complex fourier series

Answer 23

splitting the definition of the complex fourier series expansion into real and imaginary parts

Question 24

fourier series of a non-periodic function

Answer 24

need to continue the functions outside the given range to make it periodic (eg use sawtooth function or triangle wave)

Question 25

Parseval’s theorem

Answer 25

for a function f(x) with complex fourier series coefficients cr and period L

can calculate the inner product of a function with itself

Question 26

frequency or power spectrum

Answer 26

distribution of the numbers |cr|^2