The z-Transform: Introduction
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Why z-Transform?
1. Many of signals (such as x(n)=u(n), x(n) = (0.5)nu(n), x(n) = sin(nω) etc. ) do not have a DTFT.
2. Advantages like Fourier transform provided:
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Solution process reduces to a simple algebraic procedures
The temporal domain sequence output y(n) = x(n)*h(n) can
be represent as Y(z)= X(z)H(z)
Properties of systems can easily be studied and
characterized in z – domain (such as stability..)
Topics:
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Definition of z –Transform
Properties of z- Transform
Inverse z- Transform
Definition of the z-Transform
1. Definition:The z-transform of a discrete-time signal x(n) is defined by
where z = rejw is a complex variable. The values of z for which the
sum converges define a region in the z-plane referred to as the
region of convergence (ROC).
2. Notationally, if x(n) has a z-transform X(z), we write
3. The z-transform may be viewed as the DTFT or an exponentially
weighted sequence. Specifically, note that with z = rejw, X(z) can be
looked as the DTFT of the sequence r--nx(n) and ROC is determined
by the range of values of r of the following right inequation.
ROC & z-plane
• Complex z-plane
z = Re(z)+jIm(z) = rejw
• Zeros and poles of X(z)
Many signals have z-transforms that are
rational function of z:
Factorizing it will give:
The roots of the numerator polynomial, βk,are
referred to as the zeros (o) and αk are referred
to as poles (x). ROC of X(z) will not contain
poles.
ROC properties
• ROC is an annulus or disc in the z-plane centred at the
origin. i.e.
• A finite-length sequence has a z-transform with a region
of convergence that includes the entire z-plane except,
possibly, z = 0 and z = . The point z = will be included if
x(n) = 0 for n < 0, and the point z = 0 will be included if x(n) = 0
for n > 0.
• A right-sided sequence has a z-transform with a region
of convergence that is the exterior of a circle:
ROC: |z|>α
• A left-sided sequence has a z-transform with a region of
convergence that is the interior of a circle:
ROC: |z|<β
• The Fourier Transform of x(n) converges absolutely if
and only if ROC of z-transform includes the unit circle
Properties of Z-Transform
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Linearity
If x(n) has a z-transform X(z) with a region of convergence Rx,
and if y(n) has a z-transform Y(z) with a region of convergence
Ry,
Z
w ( n )  ax ( n )  by ( n )  W ( z )  aX ( z )  bY ( z )
and the ROC of W(z) will include the intersection of Rx and Ry,
that is, Rw contains .R x  R y
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Shifting property
If x(n) has a z-transform X(z),
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x ( n  n 0 )  z
Z
 n0
X (z)
Time reversal
If x(n) has a z-transform X(z) with a region of convergence Rx
that is the annulus   z   , the z-transform of the timereversed sequence x(-n) is x (  n )  X ( z )
and has a region of convergence1   z  1 , which is denoted by 1 R
Z
1
x
Properties of Z-Transform
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Multiplication by an exponential
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If a sequence x(n) is multiplied by a complex exponential αn.
 x ( n )  X (
Z
n
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Convolution theorm
1
z)
If x(n) has a z-transform X(z) with a region of convergence Rx, and if
h(n) has a z-transform H(z) with a region of convergence Rh,
y ( n )  x ( n )  h ( n )  Y ( z )  X ( z ) H ( z )
Z
The ROC of Y(z) will include the intersection of Rx and Rh, that is,
Ry contains Rx ∩ Rh .
With x(n), y(n), and h(n) denoting the input, output, and unit-sample
response, respectively, and X(z), Y(x), and H(z) their z-transforms.
The z-transform of the unit-sample response is often referred to as
the system function.
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Conjugation
If X(z) is the z-transform of x(n), the z-transform of the complex
conjugate of x(n) is
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x ( n )  X ( z )
Z
Properties of Z-Transform
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Derivative
– If X(z) is the z-transform of x(n), the ztransform of is
nx ( n )   z
Z
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dX ( z )
dz
Initial value theorem
If X(z) is the z-transform of x(n) and x(n) is
equal to zero for n<0, the initial value, x(0),
maybe be found from X(z) as follows:
x ( 0 )  lim X ( z )
z