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Forward Z-Transform
The Fourier transform of a discrete signal
is defined as:
provided
is absolutely summable:
Obviously some signals may not satisfy this condition and their Fourier transform
do not exist. To overcome this difficulty, we can multiply the given
by an exponential function
so that
may be forced to be
summable for certain values of the real parameter
. Now the discrete
time Fourier transform becomes:
The result of this summation is a function of a complex variable defined as:
This is the forward Z-transform of the discrete signal
:
Inverse Z-Transform
Given the Z transform
, the original time signal can be obtained by
the inverse Z transform, which can be derived from the corresponding Fourier
transform. As shown above, we have:
Now
can be obtained by the inverse Fourier transform:
Multiplying both sides by
, we get:
To represent the inverse transform in terms of
(instead of
), we note
and the inverse Z transform can be obtained as:
Note that the integral with respect to
from
to
becomes
an integral with respect to
in the complex z-plane,
along a circle with a fixed radius
and a varying angle
from
to
. Now we have the z-transform pair:
The forward and inverse z-transform pair can also be represented as
In particular, if we let
, i.e.,
, then the Z
transform becomes the discrete-time Fourier transform:
This is the reason why sometimes the discrete Fourier spectrum is expressed
as a function of
.
Different from the discrete-time Fourier transform which converts a 1-D signal
in time domain to a 1-D complex spectrum
in frequency
domain, the Z transform
converts the 1D signal
to a complex
function defined over a 2-D complex plane, called z-plane, represented
in polar form by radius
and angle
.
In particular, when this 2D function
is evaluated
along the unit circle
corresponding to
, it becomes a
1D periodic function
, the discrete Fourier transform of
.
Graphically, the periodic spectrum of the signal can be found as the cross
section of the 2D function
along the unit circle
.
Transfer Function of LTI system
The output
of a discrete LTI system with input
can be found
by convolution
where
is the impulse response function of the system. In
particular, if the input is a complex exponential
then the output
can be found to be:
This is the eigenequation with the complex exponential
being
the eigenfunction of any discrete LTI system, corresponding to its
eigenvalue defined as:
which is the z-transform of its impulse response
, called the
transfer function of the LTI system. In particular, when
,
i.e.,
, the transfer function
becomes the
frequency response function, the Fourier transform of the impulse
response:
Next: Conformal Mapping between S-Plane
Up: Z_Transform
Previous: Z_Transform
Ruye Wang
2014-10-28