6 Assessing Aliasing

Question 1

what are mirror frequencies?

Answer 1

The calculation of the Discrete Fourier Transform (DFT) results in a spectrum which exhibits mirror frequencies

Question 2

Definition of mirror frequencies?

Answer 2

Question 3

Example of mirror frequencies?

Answer 3

Question 4

The equation for mirror frequency?

Answer 4

The mirror frequency f_m depends on the sampling frequency fs and
the (harmonic) signal frequency f_k

Question 5

Equation for the spectral amplitude of the mirror frequency?

Answer 5

The spectral amplitude at the mirror frequency is the same as the
spectral amplitude at the signal frequency

Question 6

The spectral coefficient at the mirror frequency?

Answer 6

The spectral coefficient at the mirror frequency is the complex conjugate of the spectral coefficient at the signal frequency

Question 7

Details regarding the spectral coefficients at the mirror frequencies?

Answer 7

As a result, the spectral coefficients at the mirror frequencies do not add any new information to the spectrum, because they can be calculated directly from the first half of the spectrum k < N/2

Question 8

Normal spectrum limitations?

Answer 8

The spectrum is therefore normally limited to one half of the
harmonic frequencies f_k

Question 9

The smallest and largest frequencies in a spectrum:

Answer 9

The smallest frequency in a spectrum is f0 = 0 Hz and
the largest frequency in a spectrum is f_N/2 = f_s/2

Question 10

Nyquist frequency?

Answer 10

The largest frequency in a spectrum is called the Nyquist frequency f_NYQ = f_s/2 = f_N/2

Note that the Nyquist frequency does not have any mirror frequency, or it is the mirror frequency to itself, such that the corresponding spectral coefficient is a real number

Question 11

Nyquist frequency spectral coefficient?

Answer 11

Question 12

The spectral coefficient at f₀= 0Hz

Answer 12

Question 13

How can the spectral coefficient of the Nyquist frequency be specifically calculated:

Answer 13

Question 14

How are mirror frequencies avoided in engineering applications?

Answer 14

Question 15

What is Aliasing?

Answer 15

• Aliasing is a mathematical consequence of the calculation of the spectrum, i.e., the periodicity of the rotating phasor.
• When the Nyquist theorem is not observed, aliasing can occur
• Aliasing is the appearance of a signal in the spectrum at the wrong frequency

Question 16

How can Aliasing be detected?

Answer 16

Aliasing can be detected by changing the sampling frequency and noting the change of the apparent signal frequency.

Question 17

How to tell when no aliasing has occurred?

Answer 17

If the apparent signal frequency is independent of the sampling frequency, no aliasing occurred

Question 18

How to tell when aliasing has occurred?

Answer 18

If the apparent signal frequency depends on the sampling frequency, aliasing has occurred

Question 19

Aliasing example?

Answer 19

Aliasing results in an apparent signal with the wrong frequency

Question 20

The equation used to calculate the aliasing frequency?

Answer 20

Question 21

The number of aliasing frequencies per signal?

Answer 21

Each signal only has a single aliasing frequency

Question 22

How can aliasing be avoided?

Answer 22

• observing the Nyquist theorem
• increasing the sampling frequency to at least two times the signal frequency
• analogue filtering prior to digital recording

Question 23

Uses of aliasing?

Answer 23

Aliasing can also be put to good use, for example to economise the resources required for recording a signal, if the signal has a limited bandwidth -> advanced digital signal processing