Fourier Series

Question 1

What is a Fourier Series?

Answer 1

Allows us to represent any periodic function as an infinite sum of sine and cosine.

Question 2

What does the Fourier Series tell us?

Answer 2

How much of each frequency is present.

Question 3

If f(x) has even symmetry(mirror about y axis) what does this mean for the Fourier series?

Answer 3

an= (some are) non-zero
a0=non-zero( a0 = 0 if the average value of the function is zero)
bn=0 for all n ( f(x) is even)

Question 4

If f(x) has odd symmetry what does this mean for the Fourier series?

Answer 4

an= 0 for all n (f(x) is odd)
a0=0 (function is odd)
bn= (some are) non-zero

Question 5

What functions have neither odd nor even symmetry?

Answer 5

Exponential, logs and compound.
Will contain sine and cosine components
some an and bn is non-zero
a0=non-zero(a0=0 if average of the function is 0)

Question 6

If f(x) is even what does it mean for the integral?

Answer 6

even function x even function = even function

Question 7

If f(x) is odd what does it mean for the integral?

Answer 7

odd function x odd function = even function

Question 8

How can we simplify an even integral?

Answer 8

If we have an integral over an even function over symmetrical limits
∫(- L/2 to L/2) = 2∫(0 to L/2)

Question 9

How do you describe a function that is only valid over a limited range?

Answer 9

• Find the Fourier series by assuming the function is periodic. (Justification: If we only evaluate the result over the range given we can ignore everything else)
• Using half range series we expand the original function and define the periodicity as 2 x its range

EITHER
Even extension - series only has cosine terms
Odd extension - series only has sine terms

Question 10

What do the values (an,bn) of the coefficients of a Fourier Series tell us?

Answer 10

Tells us which frequency components are present in the function and their amplitudes

Question 11

How do you describe the frequency present and relative amplitude?

Answer 11

Using F(f) = Frequency function/Frequency spectra ( f(t) and F(f) or f(x) of F(k))

X-axis: (f) frequency as integer multiple of fundamental frequency(f_0)

Y-axis: (F(f)) relative amplitudes (you can write the pre-factor apart of the axis)