Week 2 SCM (Fourier series)

Question 1

what is a series estimate for constants and how does this relate to fourier series?

Answer 1

• Some constants can be expressed as a series of other numbers, with the estimation getting more accurate the more terms in the series
• The same can apply. Any periodic function can be approximated as an infinite series of trigonometric functions
• For example, you can create a square wave by adding lots of different sine functions of different frequencies
• You can also create a triangular wave with a function using the sum of sine waves

Question 2

what are 3 functions that are used in common fourier series?

Answer 2

Bo = Constant/DC
An= Odd coefficient (sine)
Bn= Even Coefficient (cos)

Any repeating function can be deconstructed into these three terms

Question 3

What is the difference between the time domain and the frequency domain?

Answer 3

A time domain graph displays the changes in signal over a span of time

A frequency domain graph displays how much of the signal exists within a given frequency band concerning a range of frequencies

You can convert between the two domains using Fourier series

Question 4

what does the fast fourier transform do?

Answer 4

It changes the domain of a signal from time to frequency

Question 5

why is fourier transform important?

Answer 5

• Signals are easier to break down and visualise in the fourier (frequency) domain than in the time domain
• In music it can be used to deconstruct the bass line from the higher pitches, or to separate different vocals

Question 6

What is the frequency of a fourier series?

Answer 6

The frequency is determined by the period (T) of the function that is considered

The lowest frequency (base) in the series is 1/T. The rest of the frequencies (harmonic frequencies) are multitudes of this lowest frequency e.g 2/T , 3/T, 4/T. However, this does not nessercarily imply that each of these frequencies are actually present in the series.

Question 7

What is Fourier tranform VS. inverse fourier transform?

Answer 7

FOURIER TRANSFORM:
from the function/signal to the sum of trigonometric functions

INVERSE FOURIER TRANSFORM:
from the sum of trigonometric functions to the function/signal

Question 8

in what situations would only sines be needed to represent the signal and in what situation would only cosines be needed to represent the signal?

Answer 8

• If the signal is symmetric around time 0 only cosines are needed to represent the signal
• If the signal is antisymmetric around 0 only antisines are needed to represent the signal

Question 9

Is the fourier transform to be used on periodic or non periodic signals?

Answer 9

The fourier transform can be used on non-periodic signals

In this sense we can think of the fourier transform as a generalization of the fourier series

This is good because in the real world signals are hardly ever periodic

However, the fourier transform can only be applied when the signal can be expressed as a function. This is rarely the case for real world signals. To obtain the frequency content of sampled signals we use the discrete fourier transform

Question 10

When would we use the discrete fourier transform and fast fourier transform

Answer 10

• The fourier transform can only be used when the signal can be expressed as a function
• When the signal isnt expressed as a function we use the discrete fourier transform
• This obtains the frequency content from sampled signals
• So it changes from the time domain into the frequency domain
• When calculating this on a computer we could use the fast fourier transform. To be fast, most implementations assume that the signal length is a power of 2. If this is the case, the fast fourier transform will return exactly the same value as the discrete fourier transform. If it is not the case, the algorithm will add values to the signal until its length has become the next power of two (zero padding)

Question 11

What would sine/cosine coefficients of zero mean

Answer 11

A sine or cosine coefficient of zero implies that the frequency is not present in the signal

Question 12

What is the fourier spectrum and what are its two parts?

Answer 12

The graph plotted between the fourier coefficients of a periodic function ( x(t) ) and the frequency

It has two parts:

• The amplitutde spectrum which is the plot of amplitude of fourier coefficients versus frequency
• The phase spectrum which is the plot of the phase of fourier coefficients versus frequency
• The amplitude spectrum and the phase spectrum together are known as the fourier frequency spectra of the periodic signal.
• This type of representation is known as the frequency domain representation

Question 13

How many coefficients for each frequency are shown on the fourier spectrum frequency domain

Answer 13

There are two for each frequency, one for the sine and one for the cosine

If one of the coefficients is zero it means that the related frequency is not in the spectrum

Question 14

what is the x and y axis of a fourier frequency spectra

Answer 14

on the horizonal axis is the frequency

A form of the coefficients for each frequency is set out vertically

• for a power spectrum the sum of the squared coefficients is displayed
• for an amplitude spectrum the square root of all the values on the power spectrum is displayed

Question 15

how would longer and shorter periods of a periodic signal be shown on the frequency domain spectra?

Answer 15

longer periods have lower frequency, and so are shown closer to zero on the left of the horizontal axis

shorter periods have higher frequency and so are shown as higher numbers to the right of the horizontal axis

Question 16

how are different values of B related to frequencies?

Answer 16

smaller values of B are related to smaller frequencies

Question 17

what is the highest frequency in the frequency spectrum called?

Answer 17

The nyquist frequency, which is equal to half the sampling frequency

This can be intuitively understood by realising we need at least two points to represent an oscillation

Question 18

what are some practical applications of Fourier analysis

Answer 18

• We can transform EEG signals from the time domain to the frequency domain. This is because in the frequency domain there is often relavent information to be obtained for clinical decision making
• For example in an EEG, the spectral power in the frequency band between 8 and 13 Hz (Alpha band) conveys information about how relaxed the patient is
• It is also used in the spectral analysis of EMG signals and accelerometry signals to assess tremor by measuring the electrical activity of the muscle. Translating to frequency spectrums can help the clinician distinguish between types of tremors