3 The Fourier series Flashcards
Define frequency (wave number)?
The no. of waveforms that fit in that interval
Linear freq:
No. of waveforms per unit time interval
Angular freq:
No. of waveforms per 2 pi time interval
Whats Aperiodic signals?
NOT periodic
EG: Where there’s discontinuity
- exp are aperiodic
What do we use for temporal signals?
Angular frequency: omega
What do we use for spatial systems?
Angular spatial frequency: K
What does synthesis mean?
Term used to express adding together cosine and sine functions during Fourier analysis
Why do we use cosine and sine in Fourier series?
- 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
How to synthesise a function?
- Adding together sinusoids of varying frequencies until the error is 0
What is key to Fourier analysis?
Orthogonality
What is key to Fourier analysis?
Orthogonality
- Useful for decomposing into orthogonal basis functions and vectors
- Pivital on how Fourier analysis works
Whats the continuous case of orthogonality?
Multiplying 2 continuous functions, then integrating, gives 0 for functions which are orthogonal to each other.
When is something strictly not orthogonal?
- 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.
What is orthogonal?
- 2 functions of different frequencies
- 2 functions of cosine and sine
- complex exponentials
- when we multiply and when added equal 0, its othogonal
What is orthogonal?
- 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
When is it not a linear time invariant system?
The sinusoid changes frequency in the output - this is not a linear system
When is it not a linear time invariant system?
The sinusoid changes frequency in the output - this is not a linear system
-can only change amp
What happens to integral?
-Integral of delta function is 1, so just end up with (t-kT)
Whats Fourier series?
-A summation of weighted cosines and sign functions
Whats an eigen function?
and Eigen value?
If we have an eigen function, if we put this through a system the only thing that changes is the amplitude by an eigenvalue
What can we get from periodic signals?
Fourier series
What do we get from aperiodic signals?
Fourier transform
Whats the amplitude of the Dirac delta function?
Amplitude is ∞ (infinitesimally narrow)
Whats the amplitude of the Kronecker function?
Amplitude is equal to 1
What do we know about convolution?
It’s commutative
What should we always do when we have to find imaginary numbers or addition?
We need to convert to cartesian coordinates
And solve for sin(angle)
Basic analysing
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.
