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|
