• Writing explicitly a few of the terms: • Z-transform is an infinite power series, it exists only for those values of z for this series converges.. Z - transform • The z-transform of th
Trang 1Xử lý tín hiệu số
Z - transform
Ngô Quốc Cường
Ngô Quốc Cường
ngoquoccuong175@gmail.com
sites.google.com/a/hcmute.edu.vn/ngoquoccuong
Trang 34.1 Z - transform
• Given a discrete-time signal x(n), its z-transform is defined as the following series:
where z is a complex variable
• Writing explicitly a few of the terms:
• Z-transform is an infinite power series, it exists only for those values of z for this series converges
• The region of convergence (ROC) of X(z) is the set of all
values of z for which X(z) attains a finite value
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Trang 44.1 Z - transform
• Example: Determines the z-transform of the following finite
duration signals
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Trang 54.1 Z - transform
• Solution
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Trang 64.1 Z - transform
• Example
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Trang 74.1 Z - transform
Recall that
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Trang 84.1 Z - transform
• Example
• Solution
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Trang 114.1 Z - transform
• We have identical closed-form expressions for the z
transform
• A closed-form expressions for the z transform does not
uniquely specify the signal in time domain
• The ambiguity can be resolved if the ROC is specified
• Z – transform = closed-form expressions + ROC
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Trang 124.1 Z - transform
• Example
• Solution
– The first power series converges if |z| > |a|
– The second power series converges if |z| < |b|
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Trang 13• Case 1
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Trang 14• Case 2
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Trang 15• Characteristics families of signals with their corresponding
ROC
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Trang 174.1 Z - transform
• The z-transform of the impulse response h(n) is called the
transfer function of a digital filter:
• Determine the transfer function H(z) of the two causal filters
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Trang 184.2 Properties of Z-transform
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Trang 19• Example
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Trang 21• Example
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Trang 24Exercise
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Trang 264.2 Properties of Z-transform
• The ROC of z-k X(z) is the same as that of X(z) except for z=0
if k>0 and z=∞ if k<0
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Trang 27• Solution
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Trang 284.2 Properties of Z-transform
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Trang 29• Example
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Trang 314.2 Properties of Z-transform
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Trang 32• Example
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Trang 334.2 Properties of Z-transform
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Trang 344.2 Properties of Z-transform
• Example
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Trang 354.2 Properties of Z-transform
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Trang 364.2 Properties of Z-transform
• Example
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Trang 384.2 Properties of Z-transform
• Convolution in Z domain
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Trang 394.3 RATIONAL Z-TRANSFORM
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Trang 404.3.1 Poles and Zeros
• An important family of z-transforms are those for which X(z)
is a rational function
• Poles and Zeros
– Zeros: value of z for which X(z) = 0;
– Poles: value of z for which X(z) = ∞
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Trang 414.3.1 Poles and Zeros
• Example
• Solution
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Trang 424.3.2 Causality and Stability
• A causal signal of the form
will have z-transform
• the common ROC of all the terms will be
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Trang 434.3.2 Causality and Stability
• if the signal is completely anticausal
• the ROC is in this case
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Trang 444.3.2 Causality and Stability
• Causal signals are characterized by ROCs that are outside the maximum pole circle
• Anticausal signals have ROCs that are inside the minimum pole circle
• Mixed signals have ROCs that are the annular region between two circles—with the poles that lie inside the inner circle contributing causally and the poles that lie outside the outer circle contributing anticausally
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Trang 454.3.2 Causality and Stability
• Stability can also be characterized in the z-domain in terms of the choice of the ROC
• A necessary and sufficient condition for the stability of a signal x(n) is that the ROC of the corresponding z-transform contain the unit circle
• A signal or system to be simultaneously stable and causal, it
is necessary that all its poles lie strictly inside the unit circle in the z-plane
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Trang 464.3.2 Causality and Stability
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Trang 474.3.3 System function of LTI
• System function
• From a linear constant coefficient equation
• We have,
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Trang 484.3.3 System function of LTI
• Or equivalently
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Trang 494.3.3 System function of LTI
• Example
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Trang 504.3.3 System function of LTI
• Solution
• The unit sample response
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Trang 514.4 Inverse Z- transform
• By contour integration
• By power series expansion
• By partial fraction expansion
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Trang 52• The partial fraction expansion method can be applied to
z-transforms that are ratios of two polynomials
• The partial fraction expansion of X(z) is given by
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Trang 53• Example
• The two coefficients are obtained as follows:
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Trang 54• If the degree of the numerator polynomial N(z) is exactly equal to the degree M of the denominator D(z), then the PF expansion must be modified
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Trang 55• Example
• Compute all possible inverse z-transforms of
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Trang 57• Example
• Determine all inverse z-transforms of
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Trang 58• Solution
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Trang 59• there are only two ROCs I and II:
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Trang 60MORE ABOUT INVERSE Z-TRANSFORM
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Trang 61• Distinct poles
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Trang 62• Multiple order poles
• Solution
• In such a case, the partial fraction expansion is:
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Trang 64Exercise
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Trang 65• a)
• b)
• c)
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