DSP (Spring, 2007)
The z-Transform
The z-Transform
— Introduction
z
Why do we study them?
„ A generalization of DTFT.
Some sequences that do not converge for DTFT have valid z-transforms.
„ Better notation (compared to FT) in analytical problems (complex variable theory)
„ Solving difference equation. Æ algebraic equation.
z Fourier Transform, Laplace Transform, DTFT, & z-Transform
Fourier Transform
∞
ℑ{x (t )} = ∫ x(t )e − jΩt dt
−∞
To encompass a broader class of signals:
∫
∞
−∞
∞
( x (t )e −σt )e − jΩt dt ≡ ∫ x (t )e − st dt ≡ L{x (t )}
−∞
Laplace Transform
jΩ
S-domain
σ
Region of Convergence
NCTU EE
1
DSP (Spring, 2004)
The z-Transform
jw
x[n]
X(e )
DTFT
⇔
n
w
2π
⇑
⇓
x(t)
X(jΩ)
F.T.
⇒
t
Ω
2π/T
T
∞
∑ x[k ]δ (t − kT )
x (t ) =
k = −∞
Similarly,
∞
∞
∞
L{x (t )} = L{ ∑ x[k ]δ (t − kT )} = ∫ { ∑ x[k ]δ (t − kT )}e − st dt =
−∞
k = −∞
=
∞
∑ x[k ]e
k = −∞
− skT
≡
∞
∑ x[ k ] z
−k
k = −∞
∞
∑ x[k ]∫
k = −∞
∞
−∞
δ (t − kT )e − st dt
≡ Z{x[n ]} ≡ X ( z )
k = −∞
z-Transform
z Eigenfunctions of discrete-time LTI systems
zn
If x[ n ] = z 0
n
z 0n : some complex constant
y[n ] = x[n ] ∗ h[n ] =
∞
∑ x[n − k ]h[k ] =
k = −∞
Remark:
H ( z) z n
DiscreteTime LTI
X ( z ) z = e jw =
∞
∞
∞
k = −∞
k = −∞
∑ z 0n −k h[k ] = { ∑ h[k ]z 0−k }z 0n = H ( z 0 ) z 0n
∑ x[n ]e
− jnw
n = −∞
DTFT can be viewed as a special case:
z = e jω
2
DSP (Spring, 2004)
The z-Transform
— z-Transform
z (Two-sided) z-Transform (bilateral z-Transform)
∞
∑ x[n]z − n ≡ X ( z )
Forward: Z {x[n]} =
n = −∞
From DTFT viewpoint:
Z {x[n ]} = F {r − n x[n ]} re jω = z
(Or, DTFT is a special case of z-T when z = e jω , unit circle.)
Inverse: x[n] = 1
X ( z) z
2πj ∫
n −1
dz ≡ Z −1[ X ( z )]
Γ
Note: The integration is evaluated along a counterclockwise circle on the complex z plane
with a radius r. (A proof of this formula requires the complex variable theory.)
z
Single-sided z-Transform (unilateral) – for causal sequences
∞
X ( z ) = ∑ x[n]z − n
n=0
z Region of Convergence (ROC)
The set of values of z for which the z-transform converges.
„ Uniform convergence
If z = re jω (polar form), the z-transform converges uniformly if x[n]r − n is absolutely
summable; that is,
∞
∑ | x[n]r − n | < ∞
n = −∞
„ In general, if some value of z, say z = z1 , is in the ROC, then all values of z on the circle
defined by | z |=| z1 | are also in the ROC.
Î ROC is a “ring”.
„ If ROC contains the unit circle, |z| =1, then the FT of this sequence converges.
„ By its definition, X(z) is a Laurent series (complex variable)
Î X(z) is an analytic function in its ROC
Î All its derivatives are continuous (in z) within its ROC.
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DSP (Spring, 2004)
The z-Transform
„ DTFT v.s. z-Transform
-- x [n] = sin ω c n ,
1
πn
−∞ < n< ∞
Not absolutely summable; but square summable
Æ z-transform does not exist; DTFT (in m.s. sense) exists.
-- x2 [ n] = cos ω 0 n,
−∞ < n<∞
Not absolutely summable; not square summable
Æ z-transform does not exist; “useful” DTFT (impulses) exists.
-- x3 [ n] = a n u[ n],
| a |> 1, − ∞ < n < ∞
Æ z-transform exist (a certain ROC); DTFT does not exists.
z Some Common Z-T Pairs
„ δ [n] ↔ 1 ,
δ [n − m ] ↔ z −m ,
δ [n + m ] ↔ z m ,
„ u[n ] ↔
1
1 − az
− a n u [ − n − 1] ↔
„ r
n
rn
m > 0, z < ∞
1
,
1 − z −1
„ a nu[n ] ↔
m > 0, z > 0 ,
z > 1 , − u [ − n − 1] ↔
−1
,
1
1 − az
1
,
1 − z −1
z <1
z > a ,
−1
,
z < a
1 − [r cos ω 0 ]z − 1
cos [ω 0 n ]u [ n ] ↔
, z > r
1 − [2 r cos ω 0 ]z − 1 + r 2 z − 2
1 − [r sin ω 0 ]z − 1
[
]
sin ω 0 n u [ n ] ↔
, z > r
1 − [2 r sin ω 0 ]z − 1 + r 2 z − 2
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DSP (Spring, 2004)
The z-Transform
— Properties of ROC for z-Transform
z Rational functions
X ( z) =
P( z )
Q( z )
Poles – Roots of the denominator; the z such that X (z ) → ∞
Zeros – Roots of the numerator; the z such that X ( z ) = 0
„ Properties of ROC
(1) The ROC is a ring or disk in the z-plane centered at the origin.
(2) The F.T. of
x[n ] converges absolutely ⇔ its ROC includes the unit circle.
(3) The ROC cannot contain any poles.
(4) If x[n] is finite-duration, then the ROC is the entire z-plane except possibly
z = 0 or
z = ∞.
(5) If x[n] is right-sided, the ROC, if exists, must be of the form
bly z
z > rmax except possi-
= ∞ , where rmax is the magnitude of the largest pole.
(6) If x[n] is left-sided, the ROC, if exists, must be of the form z < rmin except possi-
bly z
= 0 , where rmin is the magnitude of the smallest pole.
(7) If x[n] is two-sided, the ROC must be of the form r1 < z < r2 if exists, where r1 and
r2 are the magnitudes of the interior and exterior poles.
(8) The ROC must be a connected region.
In general, if
X (z ) is rational, its inverse has the following form (assuming N poles: {d k } )
N
x[n] = ∑ Ak (d k ) n . For a right-sided sequence, it means n ≥ N1 , where N1 is the first
k =1
nonzero sample.
N
The nth term in the z-transform is x[n]r − n = ∑ A (d r −1 ) n .
k
k
k =1
This sequence converges if
∞
∑ | d k r −1 |n < ∞ for every pole k = 1,K , N .
In order to
n = N1
be so,
| r |>| d k |, k = 1,K, N .
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DSP (Spring, 2004)
The z-Transform
— Pole Location and Time-Domain Behavior for Causal
Signals
Reference: Digital Signal Processing by Proakis & Manolakis
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DSP (Spring, 2004)
The z-Transform
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DSP (Spring, 2004)
The z-Transform
— The Inverse z-Transform
Inverse formula: x[n] = 1
X ( z) z
2πj ∫
n −1
dz
Γ
This formula can be proved using Cauchy integral theorem (complex variable theory).
z
Methods of evaluating the inverse z-transform
(1) Table lookup or inspection
(2) Partial fraction expansion
(3) Power series expansion
z
Inspection (transform pairs in the table) – memorized them
z
Partial Fraction Expansion
b + b z −1 + L + bM z − M Æ
X ( z ) = 0 1 −1
a0 + a1 z + L + a N z − N
Hence, it has M zeros (roots of
X ( z) =
∑ bk z M − k
z N (b0 z M + L + bM )
z M ( a0 z N + L + a N )
), N poles (roots of
∑ ak z N −k ), and (M-N)
poles at zero if M>N (or (N-M) zeros at zero if N>M).
−1
−1
Æ X ( z ) = b0 (1 − c1 z ) L (1 − cM z ) ; ck , nonzero zeros;
a0 (1 − d1 z −1 ) L (1 − d N z −1 )
d k , nonzero poles.
„ Case 1: M < N , strictly proper
Simple (single) poles:
X ( z) =
AN
A1
A2
+
+L+
−1
−1
(1 − d1 z ) (1 − d 2 z )
(1 − d N z −1 )
where Ak = (1 − d k z −1 ) X ( z ) | z = d
k
Multiple poles: Assume
X ( z) =
di
is the sth order pole. (Repeated s times)
N
Ak
Cs
C1
C2
+
+
+L+
−1
−1
−1 2
(1 − d i z ) (1 − d i z )
(1 − d i z −1 ) s
k =1, k ≠ i (1 − d k z )
∑
single-pole terms
where
1
Cm =
( s − m)!(− d i )
multiple-pole terms
s−m
s−m
⎧d
s
−1 ⎫
⎨ s − m [(1 − d i w) X ( w )]⎬
⎩ dw
⎭ w = d i−1
„ Case 2: M ≥ N
X ( z) =
M −N
N
s
Ak
Cm
+∑
−1
−1 m
k =1, k ≠ i (1 − d k z ) m =1 (1 − d i z )
∑ Br z − r + ∑
r =0
impulses
single-poles
multiple-pole
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DSP (Spring, 2004)
z
The z-Transform
Power Series Expansion
X ( z) =
∞
∑ x[n]z
−n
n = −∞
„ Case 1: Right-sided sequence, ROC:
It is expanded in powers of
Ex. X ( z ) =
z −1 .
1
, | z |>| a |
1 − az −1
„ Case 2: Left-sided sequence, ROC:
It is expanded in powers of
Ex. X ( z ) =
z > rmax
z < rmin
z.
1
, | z |<| a |
1 − az −1
„ Case 3: Two-sided sequence, ROC: r1 < z < r2
X ( z) =
X + ( z)
+
converges for
| z |> r1
Æ x[ n] =
x+ [ n ]
causal sequence
+
X − ( z)
converges for
| z |< r2
x− [ n ]
anti-causal sequence
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DSP (Spring, 2004)
The z-Transform
— z-Transform Properties
If x[ n] ↔ X [ z ] and y[ n] ↔ Y [ z ] , ROC: R X , RY
„ Linearity: ax[n] + by[n] ↔ aX ( z ) + bY ( z )
ROC:
R' ⊃ RX ∩ RY -- At least as large as their intersection; larger if pole/zero can-
cellation occurs
„ Time Shifting: x[ n − n0 ] ↔ z − n 0 X ( z )
ROC: R ' = R X ± { 0 or ∞ }
„ Multiplication by an exponential seqence:
a n x[n] ↔ X ( z a)
ROC: R' = a R X -- expands or contracts
„ Differentiation of X(z): nx[n] ↔ − z dX ( z ) ,
dz
ROC: R ' = R X
„ Conjugation of a complex sequence: x * [n] ↔ X * ( z*) ,
ROC: R ' = R X
„ Time reversal: x * [−n] ↔ X * (1 / z*) ,
ROC: R ' = 1 / R X (Meaning: If R X : rR <| z |< rL , then R ': 1 / rL <| z |< 1 / rR .
Corollary: x[−n] ↔ X (1 / z )
„ Convolution: x[n] ∗ y[n] ↔ X ( z )Y ( z )
ROC: R ' ⊃ R X ∩ RY (=, if no pole/zero cancellation)
„ Initial Value Theorem:
If x[n]=0, n<0, then x[0] = lim X ( z )
z →∞
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DSP (Spring, 2004)
The z-Transform
„ Final Value Theorem:
If (1) x[n]=0, n<0, and
(2) all singularities of (1 − z −1 ) X ( z ) are inside the unit circle,
then x[∞] = lim(1 − z −1 ) X ( z )
z →1
Remarks: (1) If all poles of X(z) are inside unit circle, x[ n] → 0 as n → ∞
(2) If there are multiple poles at “1”, x[ n ] → ∞ as n → ∞
(3) If poles are on the unit circle but not at “1”, x[n] ≈ cos ω 0 n
<Supplementary>
z-Transform Solutions of Linear Difference Equations
Use single-sided z-transform:
Z { y[n − 1]} = z −1Y ( z ) + y[−1]
Z { y[n − 2]} = z −2Y ( z ) + z −1 y[−1] + y[−2]
Z { y[ n − 3]} = z −3Y ( z ) + z −2 y[−1] + z −1 y[ −2] + y[ −3]
For causal signals, their single-sided z-transforms are identical to their two-sided
z-transforms.
Ex., Find y[n] of the difference eqn.
y[ n ] − 0.5 y[ n − 1] = x[ n ] with x[ n ] = 1, n ≥ 0 , and y[ −1] = 1
(Sol) Take the single-sided z-transform of the above eqn.
Æ Y ( z ) − 0.5{z −1Y ( z ) + y[−1]} = X ( z ) =
⎧
1
⎫⎧
1
1
1 − z −1
⎫
Æ Y ( z ) = ⎨⎩1 − 0.5 z −1 ⎬⎭⎨⎩0.5 + 1 − z −1 ⎬⎭
=
0.5
1
+
1 − 0.5 z −1 (1 − 0.5 z −1 )(1 − z −1 )
2
0.5
−
−1
1− z
1 − 0.5 z −1
Take the inverse z-transform
Æ Y ( z) =
Æ y[ n] = 2 − 0.5(0.5) n , n ≥ 0
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