Z-Transform

Filter order in a FIR filter =

number of zeros = number of delays

An FIR filter has poles, zeros or both

only zeros

If x[n] is causal, then the ROC is...

outside a disk

For BIBO stability, what is required in the zeros and poles of the LTI system?

poles must be within the unit circle, zeros don't effect BIBO

why is the ROC important?

• A z-transform converges only for certain values of z and does not exist for other values of z • z-transforms are non-unique without it

Geometric formula for |H(z)| =

(product of distances from z to all zeros )/ (product of distances from z to all poles)

Convolution in the time domain =

multiplication in the z-transform domain

what is the region of convergence for a causal signal?

|z| > |a|

Interpret.... X(z) = [(z − ζ1)(z − ζ2)· · ·(z − ζM)] / [(z − p1)(z − p2)· · ·(z − pN )] = z ^(N−M)[ (1 − ζ1z −1 )(1 − ζ2z −1 )· · ·(1 − ζMz −1 )]/ [(1 − p1z−1)(1 − p2z−1)· · ·(1 − pN z−1)]

A rational z-transform X(z) factored into its zeros and poles

DTFT is a special case of the z-transform or z-transform is a special case of the DTFT?

DTFT is a special case of the z-transform

What ways have we learned to calculate the Z-transform?

Partial fraction expansion and factorization

The DTFT is a special case of the z-transform or the z-transform is a special case of the DTFT?

The DTFT is a special case of the z-transform

Explain the ROC and it's importance in the z-transform

The ROC is the set of complex numbers for which the z-transform is absolutely summable or stated another way, all values for which the z-transform converges

Transfer function in the z domain H(z) =

Y(z)/X(z)

What is the z transform and what is it's main use?

The z-transform generalizes the DTFT for analyzing infinite-length signals and systems

What is inverse z-transform? How is it achieved?

When the analysis is needed in discrete format rather than the frequency domain of the Z transform

What is the order of the transfer function?

Whichever is larger, M or N

What is the denominator of the transfer function in the Z domain?

X(Z) = recursive average = poles

Interpret the equation for a z-transform

X(z) is a complex-valued function of a complex variable

What does diagonalization by eigendecomposition imply?

Y (z) = X(z) H(z)

What is the numerator of the transfer function in the Z domain?

Y(Z) = moving average = zeros

The z-transform is a _________ function of a __________ variable

a complex function of a complex variable

What are the 'core' basis functions of the z transform?

complex exponentials z^n with arbitrary z ∈ C

What is the interpretation of the 'core' basis functions of each system we have studied?

eigenvectors of LTI systems

If x[n] is two-sided (neither causal nor anti-causal), then either the ROC is

donut shaped OR the z-transform does not converge

A causal LTI system is BIBO stable iff

ff all of its poles are inside the unit circle

If x[n] is anti-causal, then the ROC is

inside a disk

If x[n] has finite duration, then the ROC is....

the entire z-plane (except z = 0 or z = ∞)

An LTI system is BIBO stable iff the ROC of H(z) includes

the unit circle |z| = 1

Transfer function H(z) equals

the z-transform of the impulse response h[n]

causal signal

x[n] = α^(n)*u[n]

anti-causal signal

x[n] = −α ^(n)*u[−n − 1]

z-transform pair

x[n] ←→ X(z)

How does poles and zeros affect BIBO stability?

zeros don't affect BIBO stability Poles do affect BIBO stability, if they are inside the unit circle then stable, if they are outside the unit circle then unstable

what is the region of convergence for an anti-causal signal?

|z| < |a|