Lecture Two

Question 1

Review Trig:

Answer 1

X = A Cos(2piFT + Phase)
Y = A Sin (2piFT + Phase)

Question 2

What is the wave equation?

Answer 2

The value at any point in a sinusoid (v) can be given by;

V(t) = C (amplitude) Cos (2piFT - phase)

Question 3

What is fourier synthesis?

Answer 3

Construction of a (complex) periodic signal from summation (superposition) of two or more (simple) sinusoids

Question 4

In fourier synthesis or transformation, what is the first frequency?

Answer 4

1st frequency is the fundamental frequency and is equal to the frequency of the constructed (summed) wave

Question 5

What is unique about frequency 2 in fourier synthesis?

Answer 5

It is twice that of a the fundamental frequency.

BUT then the third one will be 3x the fundamental i.e increments?

Question 6

What is fouriers theorem?

Answer 6

A periodic signal (waveform) can only contain ‘harmonically related’ sinusoids–sinusoids that have frequencies that are integral (whole number) multiplies of the Fundamental frequency

Question 7

In what domain is the initial analogue signal?

Answer 7

Time domain

Question 8

What can a signal in the time domain be transformed into and whats it called when this is reversed?

Answer 8

Frequency domain

Fourier transformation

Inverse Fourier transformation to reverse this

Question 9

What is the wave equation for a fourier series wave?

Answer 9

V(t) = C cos(2piFT + phase)

Question 10

What is the wave equation that sums the entire fourier series?

Answer 10

The same but includes summation notation so look it up

Also includes V(o) added on the end for offset signals

V(t) = C cos(2piFT + phase) + V(o)

This is for rectangular amplitude and phase components

Question 11

What is the equation for the value at a point in a composite complex signal in real and imaginary components?

Answer 11

V(t)= Acos2PiFT + B Sin (2iFT) + V(o)

Polar components

Question 12

What is polar vs rectangular components?

Answer 12

Rectangular:
- Real and imaginary components

Polar:
- Phase and Amplitude

Question 13

What is a discrete fourier transform?

Answer 13

Taking the time domain and converting it into the frequency domain

Question 14

What is the sampling period?

Answer 14

Period that was recorded (Ts)

Question 15

What is the sampling duration?

Answer 15

Total length of time spent sampling

duration of recording

Question 16

What is sampling frequency?

Answer 16

1/Ts

Question 17

How many real and imaginary components are there?

Answer 17

The number of real (cosine) and imaginary (sine) components is the same as in the time domain.

Question 18

What is observed in the real and imaginary components?

Answer 18

Symmetry about the central frequency component.

Question 19

Mathmatically on the frequency spectrum where are the lowest frequencies?

Answer 19

At 1/Ts

Question 20

Whats the highest frequency proportional to?

Answer 20

Corresponds to sampling frequency

Question 21

How many unique components are there?

Answer 21

(n-1)/2 unique components

Question 22

What is the lowest frequency independent from?

Answer 22

Independent from sampling frequency

Question 23

What is the resolution determined by?

Answer 23

Frequency res olution is determined by the lowest frequency component (the Fundamental frequency f1) which itself is determined by the period of s ampling (‘duration of recording’).

Question 24

Does increasing the sampling frequency help?

Answer 24

Providing that sampling theorem is s atis fied, increasing sampling frequency does not alter the number of (useful) spectral components

there will be more components in the frequency domain but they will have an amplitude of zero

Question 25

What does padding with zeros do?

Answer 25

Padding with zeros (zero amplitude components) at the centre of the real and imaginary spectrums is a convenient way of interpolating when the invers e Fourier trans form is applied Allowing the original signal to be reconstructed at a higher sampling rate.