206554: Digital Signal Processing

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20-6554 Digital Signal Processing

• Chapter 4
• Z-Transforms

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The z-transform is closely related to
frequency-domain analysis. Definition
The summation assumes that the signal starts at
n0 this is OK for causal processors.
X(z) is a power series in z-1, with coefficients
equal to successive values of the time-domain
signal xn. Expressing X(z) as a power series
therefore allows us to regenerate the signal.
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Note the examples on p99 of the book.
End of page
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Substitute
for z -
- clear link between this and the Fourier
Transform
Can also think of z as a time-shift operator
Multiplying by z advances time by one step.
Dividing by z delays time by one step.
e.g. a unit impulse delayed by k sampling
intervals has the transform
The z-transform brings the benefits of a
frequency-domain approach to signal and system
analysis.
Time domain convolution frequency domain
multiplication
End of page
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Convolution in the time domain
X(z) is the transform of xn and H(z) is the
transform of hn
Compare with yn 2 3 3 3 6 0 1 0 0
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Derivation of xn using recursive algorithm
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Assume z-transform represents an LTI system
rather than a signal, and denote it by H(z). The
corresponding time function must be the impulse
response hn. In general
Suppose a particular system has a z-transform of
This gives
Since multiplication by z is equivalent to a
time-domain advance by one sampling interval,
then -
i.e.
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The same technique is used for much more complex
examples. Suppose we need to find the inverse of
the following z-transform -
Multiply out as before
and obtain the corresponding difference equation
End of section
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z-plane Poles and Zeros

• Describing systems and signals

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A z-transform can always be expressed in the form
Allowing for a gain factor, the transform can
therefore completely be represented by the roots
of N(z) and D(z), so
The zi are called the ZEROS of X(z) and the pi
are called the POLES.
If the time function is real, then the poles and
zeros are either also real, or occur in complex
conjugate pairs. Can be represented using an
Argand diagram called the z-plane.
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Example 4.3(a) Plot the z-plane poles and zeros
of
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Example 4.3(b) Plot the z-plane poles and zeros
of
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Consider a single real pole at za
Stability
Cross-multiplying
Taking the inverse transform, the processors
difference equation is
or
The corresponding impulse response is
Successive terms in the signal are therefore
System is therefore stable if
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Poles give important information about STABILITY
of a system see p108
A system is only stable if all the poles lie
inside the unit circle
Since the radius of the complex number
representing poles is therefore important, it
helps to represent poles in polar form zr exp(j?)
For a single complex pole
No restrictions on zero locations. Zeros at the
origin introduce a time advance.
A minimum delay system can be obtained by
ensuring an equal number of poles and zeros
i.e. that the order of the numerator that of
the denominator.
End of section
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Converting quadratic pole/zero terms between
Cartesian and polar form
So, for example, for a pole pair expressed as
we can write
and
and
so
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Geometrical Evaluation of the Fourier Transform
in the z-plane
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• Z can be substituted by exp(j?), for which all
  values lie on the unit circle.
• 0 corresponds to the real axis and as ?
  increases we move around the unit circle,
  anti-clockwise, until reaching 2?.
• Repeated rotations represent sampling effect
  (spectrum repeats)

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The numerator of the transfer function can be
represented as vectors from the origin to the
zeros the denominator by vectors from the origin
to the poles.
The amplitude can therefore be represented by the
product of the lengths of the zero vectors,
divided by the product of the lengths of the pole
vectors.
End of page
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Use GeomFT.exe to explore the geometrical
representation of the digital Fourier transform.
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The program can represent pole and zero vectors,
and show the corresponding amplitude and phase
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This is the example from page 114 (note that
their figure is only a sketch!)
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First Second-order LTI systems
Can build higher order systems by cascading first
and second-order systems
Can achieve frequency-selectivity with poles
Zeros used to ensure no time delay consider
systems with zeros at the origin
First-order systems Single real pole (inside
unit circle, for stability) on positive axis
gives lowpass filter, peaking at ?0 On negative
axis gives highpass filter, peaking at ??
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• Moving the pole closer to the unit circle causes
• The peak gain to increase
• The bandwidth to decrease
• The impulse response to decay more slowly

Second-order systems More complex
response. Second-order zero at the origin, and a
complex-conjugate pole pair at (r, ? ). The
ANGLE governs the frequency of peak gain The
RADIUS determines the bandwidth
End of presentation