Common Z-transform pairs in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Z-transform of the unit impulse sequence \( \delta[n] \)?

The Z-transform of \( \delta[n] \) is 1. This is because \( \delta[n] \) is 1 at \( n=0 \) and 0 elsewhere, so \( X(z) = \sum_{n=-\infty}^{\infty} \delta[n] z^{-n} = 1 \).

beginner

What is the Z-transform of the unit step sequence \( u[n] \)?

The Z-transform of \( u[n] \) is \( \frac{1}{1 - z^{-1}} \) for \( |z| > 1 \). This comes from summing the geometric series \( \sum_{n=0}^\infty z^{-n} \).

intermediate

What is the Z-transform of the sequence \( a^n u[n] \) where \( |a| < |z| \)?

The Z-transform is \( \frac{1}{1 - a z^{-1}} \) for \( |z| > |a| \). This is a geometric series with ratio \( a z^{-1} \).

intermediate

What is the Z-transform of the sequence \( n u[n] \)?

The Z-transform of \( n u[n] \) is \( \frac{z^{-1}}{(1 - z^{-1})^2} \) for \( |z| > 1 \). This is derived by differentiating the Z-transform of \( u[n] \) with respect to \( z^{-1} \).

beginner

What is the Z-transform of the delayed sequence \( x[n-k] \) in terms of \( X(z) \)?

The Z-transform of \( x[n-k] \) is \( z^{-k} X(z) \). This means delaying the sequence by \( k \) shifts the Z-transform by multiplying by \( z^{-k} \).

What is the region of convergence (ROC) for the Z-transform of \( a^n u[n] \)?

A |z| = |a|

B |z| < |a|

C All z

D |z| > |a|

The ROC for \( a^n u[n] \) is outside the circle of radius \( |a| \), so \( |z| > |a| \).

What is the Z-transform of the sequence \( \delta[n-3] \)?

A 1

B z^{-3}

C z^{3}

D 0

Delaying the impulse by 3 multiplies the Z-transform by \( z^{-3} \).

Which sequence has the Z-transform \( \frac{1}{1 - z^{-1}} \)?

A Unit step \( u[n] \)

B Unit impulse \( \delta[n] \)

C Sequence \( n u[n] \)

D Sequence \( a^n u[n] \)

The unit step \( u[n] \) has Z-transform \( \frac{1}{1 - z^{-1}} \).

What operation in time domain corresponds to multiplying by \( z^{-k} \) in Z-domain?

A Delaying the sequence by k

B Differentiating the sequence

C Scaling the sequence

D Reversing the sequence

Multiplying by \( z^{-k} \) corresponds to delaying the sequence by \( k \) steps.

What is the Z-transform of the sequence \( n u[n] \)?

A \( \frac{1}{1 - z^{-1}} \)

B \( z^{-1} \)

C \( \frac{z^{-1}}{(1 - z^{-1})^2} \)

D 1

The Z-transform of \( n u[n] \) is \( \frac{z^{-1}}{(1 - z^{-1})^2} \).

Explain the Z-transform of the unit step sequence and its region of convergence.

Think about summing a geometric series starting at n=0.

Describe how delaying a sequence by k affects its Z-transform.

Consider the shift property of the Z-transform.