3 The Fourier series

Question 1

Define frequency (wave number)?

Answer 1

The no. of waveforms that fit in that interval

Question 2

Linear freq:

Answer 2

No. of waveforms per unit time interval

Question 3

Angular freq:

Answer 3

No. of waveforms per 2 pi time interval

Question 4

Whats Aperiodic signals?

Answer 4

NOT periodic
EG: Where there’s discontinuity
- exp are aperiodic

Question 5

What do we use for temporal signals?

Answer 5

Angular frequency: omega

Question 6

What do we use for spatial systems?

Answer 6

Angular spatial frequency: K

Question 7

What does synthesis mean?

Answer 7

Term used to express adding together cosine and sine functions during Fourier analysis

Question 8

Why do we use cosine and sine in Fourier series?

Answer 8

• Adding cosines and sines, makes it simpler. Like its quite simple to differentiate exponentials
• Sinusoidal functions makes life simpler for LSI and LTI systems
• Fourier series allows us to represent periodic signals as sums of sinusoidal functions.
• We can do this by adding together sinusoids of different frequencies

Question 9

How to synthesise a function?

Answer 9

• Adding together sinusoids of varying frequencies until the error is 0

Question 10

What is key to Fourier analysis?

Answer 10

Orthogonality

Question 11

What is key to Fourier analysis?

Answer 11

Orthogonality

• Useful for decomposing into orthogonal basis functions and vectors
• Pivital on how Fourier analysis works

Question 12

Whats the continuous case of orthogonality?

Answer 12

Multiplying 2 continuous functions, then integrating, gives 0 for functions which are orthogonal to each other.

Question 13

When is something strictly not orthogonal?

Answer 13

• When 2 functions have matching frequencies.
• This is since it’ll become sine or cosine squared which is strictly positive + integrating that will give a finite result so not orthogonal.
• Delta functions when perfectly aligned are not orthogonal
• exp(ix) and exp(-ix) are not orthogonal.

Question 14

What is orthogonal?

Answer 14

• 2 functions of different frequencies
• 2 functions of cosine and sine
• complex exponentials
• when we multiply and when added equal 0, its othogonal

Question 15

What is orthogonal?

Answer 15

• 2 functions of different frequencies
• 2 functions of cosine and sine
• complex exponentials
• when we multiply and when added equal 0, its othogonal
• cos and sin multiplied

Question 16

When is it not a linear time invariant system?

Answer 16

The sinusoid changes frequency in the output - this is not a linear system

Question 17

When is it not a linear time invariant system?

Answer 17

The sinusoid changes frequency in the output - this is not a linear system
-can only change amp

Question 18

What happens to integral?

Answer 18

-Integral of delta function is 1, so just end up with (t-kT)

Question 19

Whats Fourier series?

Answer 19

-A summation of weighted cosines and sign functions

Question 20

Whats an eigen function?

and Eigen value?

Answer 20

If we have an eigen function, if we put this through a system the only thing that changes is the amplitude by an eigenvalue

Question 21

What can we get from periodic signals?

Answer 21

Fourier series

Question 22

What do we get from aperiodic signals?

Answer 22

Fourier transform

Question 23

Whats the amplitude of the Dirac delta function?

Answer 23

Amplitude is ∞ (infinitesimally narrow)

Question 24

Whats the amplitude of the Kronecker function?

Answer 24

Amplitude is equal to 1

Question 25

What do we know about convolution?

Answer 25

It’s commutative

Question 26

What should we always do when we have to find imaginary numbers or addition?

Answer 26

We need to convert to cartesian coordinates

And solve for sin(angle)

Question 27

Basic analysing

Answer 27

So as P increases, so the amplitude of the spectrum increases.
As A increases (the width of the pulse in the time domain), so the width of the main spectral peak decreases.
There is an inverse relationship in terms of width of functions in the time (or space) domain and the frequency domain.
Clearly, as A and B increase (i.e. x(t) becomes a wider signal), the spectrum contains sines and cosines which are sharper, of reduced width.