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The Z-Transform
Let xn , n = 0, 1, 2, . . . be a sequence of numbers,
x 0 , x1 , x2 , . . .
The Z-Transform of xn is defined to be
X(z) =
+∞
X
xn z −n
n=0
This amounts to exchanging a sequence for a function of
z. It turns out that the process is generally reversible.
In fact, the sequence of numbers xn are the coefficients
of the Taylor series expansion of X(w−1 ) where w = z −1 .
We say that X(z) is the Z-transform of xn , and that xn
is the inverse Z-transform of X(z). We write X(z) =
X {xn } and xn = Z −1 {X(z)}.
We say that xn lives in the time domain and that X(z)
lives in the frequency domain.
Think of z as a complex variable.
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Transform 1
Next, we compute the transform of xn = rn where r is a
constant. Using the definition,
X(z) =
+∞
X
xn z
−n
=
n=0
+∞
X
rn z −n
n=0
Now, let
S=
N
X
rk z −k = 1 + rz −1 + r2 z −2 + . . . + rN z −N
k=0
and so
rz −1 S = rz −1 + r2 z −2 + . . . + rN +1 z −N −1
Subtraction then gives
S − rz −1 S = 1 − rN +1 z −N −1
= 1 − rz −1 S = 1 − rN +1 z −N −1
1 − rN +1 z −N −1
⇒ S=
1 − rz −1
−1 If |z| > |r|, then rz < 1, so that rN +1 z −N −1 → 0 as
N → ∞, and then
S→
1
1 − rz −1
as N → ∞
Hence,
X(z) =
+∞
X
rn z −n =
n=0
1
1 − rz −1
This proves that
Z {rn } =
2
1
1 − rz −1
Property 1 - Linearity
It’s clear that if X(z) = Z{xn } and Y (z) = Z{yn } then
Z{c1 xn + c2 yn } = c1 X(z) + c2 Y (z)
where c1 and c2 are any constants.
Property 2 - Right Shift
Suppose that the sequence xn is shifted right by one step.
Let the resulting sequence be denoted by yn , so that
y0 = 0, y1 = x0 , y2 = x1 etc. Then
Y (z) =
+∞
X
yn z −n
n=0
=
+∞
X
xn−1 z −n
n=1
=z
−1
+∞
X
xn−1 z −n+1
n=1
=z
−1
+∞
X
xk z −k
k=0
−1
= z X(z)
This shows that
Z {xn−1 } = z −1 X(z)
So a right shift in the time domain corresponds to multiplication by z −1 in the frequency domain.
Repeated use of the above property shows that
Z {xn−k } z −k X(z)
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Property 3 - Left Shift
Suppose that the sequence xn is shifted left by one step.
Let the resulting sequence be denoted by yn , so that
y0 = x1 , y1 = x2 , y2 = x3 etc, and x0 falls off the edge, so
to speak. Then
Y (z) =
+∞
X
yn z −n
n=0
=
+∞
X
xn+1 z −n
n=0
= +x0 z − x0 z +
+∞
X
xn+1 z −n
n=0
= −x0 z +
+∞
X
xn+1 z −n
n=−1
A change in variable k = n + 1 gives
Y (z) = −x0 z +
+∞
X
xk z −k+1
k=0
= −x0 z + z
+∞
X
xk z −k
k=0
= −x0 z + zX(z)
This shows that
Z {xn+1 } = zX(z) − zx0
So a left shift in the time domain corresponds to multiplication by z with an initial condition term in the frequency domain.
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Property 4 - Differentiation
By definition,
X(z) =
+∞
X
xn z −n
n=0
Differentiate with respect to z.
+∞
X
dX
xn (−n)z −n−1
(z) =
dz
n=0
Then,
+∞
X
dX
−1
(z) = −z
xn nz −n
dz
n=1
+∞
X
dX
⇒ −z
(z) =
xn nz −n
dz
n=1
Hence,
⇒ −z
dX
(z) = Z {yn }
dz
where
yn = nxn
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Difference Equations
The above transform and properties are enough to solve
a broad class of difference equations. Consider an example.
Find xn when
xn+1 + 3xn = 2−n , subjevtto x0 = 1
Take the z-transform,
Z {xn+1 + 3xn } = Z 2−n
By linearity,
Z {xn+1 } + 3Z {xn } = Z 2−n
By transform 1 and property 3 above,
zX(z) − zx0 + 3X(z) =
1
−1
1 − z2
Next, solve for X(z),
⇒ (z + 3) X(z) =
⇒ X(z) =
1−
1
z −1
2
1
−1 + zx0
1 − z2
(z + 3)
+ zx0
1
(z + 3)
Using the given initial condition and some algebra,
z
z
⇒ X(z) =
+
1
z − 2 (z + 3) (z + 3)
Finally, we get the inverse z-transform, for which we need
a partial fraction expansion (PFE). Thus,
⇒ X(z) =
6
z
6
1
+
+
1
(z + 3)
7 (z + 3) 7 z − 2
1
−n
1 = Z{2 }, and property 2 then gives
−1
1−z 2
−1
z −1
1
z
−n+1
=
=
= Z{2
}, and similarly
z + 3 1 + 3z −1
1 − z −1 21
Z{(−3)−n+1 }. Hence,
n−1
z
1 1
6
n−1
un−1 +
xn = (−3) un−1 +
7
7 2
(z + 3)
Now,
where un = 0 when n < 0 and un = 1 when n ≥ 0.
Notes
The z-transform is named after a famous Russian control
systems researcher Ya Tzypkin. In the German language,
his surname begins with the letter z.
You should know that the above notes deal with the
one-sided z-transform, and there is also a two-sided ztransform. The latter deals with two-sided sequences,
such as
1 1 1
. . . , −8, 4, −2, 1, − , , − , . . .
2 4 8
−n
In this example, xn = 2 , and n is not restricted to be
positive. The ideas above are sufficient to work out the
properties of the sided z-transform.
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