Z-Transform

Question 1

What is the value of Z in the Z-transform?

Answer 1

z = e^sT

where T is the sampling period and s = σ + jω

Question 2

What is the Z-transform used for?

Answer 2

Discrete-time system and signal analysis.

Question 3

How do we derive x[n]?

Answer 3

Assume that the x[n] results from sampling a continuous-time signal x_c(t) with a sampling period of T.

This is equivalent to multiplying a x_c(t) by a train of impulses at t - nT, n = 1,2,3,…∞.

Question 4

Define the general form of X[z] as a train of impulses in the Z-domain

Answer 4

X[z] = x0 + x₁z^-1 + x₂z^-2 + x₃z^-3 + … + z_nz^-n

Question 5

What operation is represented with z and z^-1 ?

Answer 5

Time shift

• z = time advance by one sample
• z^-1 = time delay by one sample

Question 6

What is the Z-transform of an impulse function?

Answer 6

Question 7

What is the Z-transform of a unit step function?

Answer 7

Question 8

How do we represent translation from the time-domain into the Z-domain?

Answer 8

Question 9

How do we represent Z-domain scaling?

Answer 9

Question 10

How do we represent time reversal?

Answer 10

Question 11

How can we use the infinite geometric series to compute the Z-transform?

Answer 11