Frequency Domain Analysis - The Z transform

Question 1

What are the main reason for using Z Transforms

Answer 1

1. compact and convient way of describing DSP systems 2. widely used by DSP engineers 3. pole-zero great for vizualising stablility and characteristics

Question 2

What is the definition of the Z transform

Answer 2

Question 3

Describe what “recovering” a signal means when dealing with Z transforms and the two ways it may be done…

Answer 3

going from X(z) to the signal x[n]. Using partial fractions to put it in the form it can be used (via the table), producing the inverse z transform function an then superimposing 3 components before performing long division on the to get the required power series..

The second method is called - using a recursive algorithm….

Question 4

Describe how partical fractions are performed

Answer 4

Question 5

Describe how to do long division

Answer 5

Question 6

What is the symbol for the transfer function

Answer 6

H(z)

Question 7

What is the transfer function equivalent of?

Answer 7

The transfer function H(z) is equivalent of the frequency response function H(Omega)

Question 8

What is the definition of the inverse function?

Answer 8

Question 9

What is the inverse transfer function used to work out?

Answer 9

The inverse transform of X(z)H(z) giving an output of y[n]

Question 10

What are the two more easy to use altneratives to the formal inverse defintion to calculate the inverse transfom of X(z)H(z)

Answer 10

1. Express X(z) as a power series
2. Use the look up table of z-transform pairs

Question 11

Unilateral Z transforms table pairs

waveform | signal x[n] | Spectrum X[z] | Z planes and zeros | - 8 rows

Answer 11

Question 12

Unilateral Z transform properties

property or operation | Signal | Z Transform | - 8 rows

Answer 12

Question 13

What can a H(z) function be equivalent to?

Answer 13

An eqivilent time domain function (a difference equation)

Question 14

Describe the final value theorem and the equations that support it (Picture of Pge 106 in book)

Answer 14

Question 15

Describe what an argand diagram is..

Answer 15

It a circular diagram that both poles and circles can be put n in order to visualise the stablity and frequency response characteristics

Question 16

What does substituting exp(jΩ) do for z give?

Answer 16

The frequency response of the system i.e H(z) becomes H(Ω)

Question 17

Describe what operations are being performed to get the weird spiking graph

Answer 17

This graph represents fourier transform in the z-plane giving |H(Ω)| (the magnitude of the spectra signal)

Question 18

What is the magnitude of the spectra signal eqaul to {|H(Ω)|}

Answer 18

The product of all the zero-vector lengths divided by the product of all of the pole vector lengths

Question 19

Visualose the freqeuncy response of an LTI processor - That is |H(Ω)| which is equal to the sum of the zero-vector lengths divided by the pole vector lengths - where gain may be afactor but will not alter te shape.

Answer 19

Question 20

Describe why first and second order z- transforms are useful

Answer 20

It is done to summarise the performance of a pole-zero, frequency response and impulse response

Question 21

Defintion of first order and second order z transform

Answer 21

Question 22

Characteristic of a first order z transform

Answer 22

Question 23

Characteristic of a second order system

Answer 23

Question 24

Important First and second order equations

Answer 24