Discrete Time Systems: Impulse responses and convolution; An - - PowerPoint PPT Presentation

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Discrete Time Systems: Impulse responses and convolution; An introduction to the Z-transform

Lecture 5

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Impulse response and system output

Using impulse response, output can be calculated as: y[k] = h[k] * u[k]

• Proof:
• Conclusion
• The impulse of a system describes the input/output behavior

completely.

Definition of impulse response Time-invariance Linearity Definition of convolution

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Impulse response and system output

Visually:

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Z-transform

• Discrete equivalent to the Laplace-transform
• Converts time dependent descriptions of systems to the time-

independent Z-domain.

• Simplifies many calculations
• Convolution theorem → convolution becomes multiplication
• Linear difference equations become simple algebraic expressions
• ...

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Z-transform

• 2 forms
• Bilateral:
• Requires knowledge of h for all values of k, including negative

values

• Can be used for non-causal systems
• Unilateral:
• Only requires knowledge of h for positive values of k
• Can only be used for causal systems without loss of information

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Z-transform: properties

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Transfer function (DT)

The transfer function of a discrete system is the Z-transform of the impulse response.

• H(z) = Z{h[k]}
• Recall: y[k] = h[k] * u[k]
• Let

U(z) = Z{u[k]} Y(z) = Z{y[k]}

• Thus, applying the convolution theorem:

Y(z) = H(z) . U(z)

• BUT: Only applies when system starts from a null state

(Reason: impulse response itself starts from a null state)

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Getting rid of convolutions (DT)

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List of common Z-transform pairs

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List of common Z-transform pairs

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• For the Z-transform to converge the following must hold:
• We will look at convergence separately for positive and negative k,

splitting the convergence criterion in 2:

• Using z = r e

jϴ

with R+ as small as possible and R- as large as possible we get:

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Region of convergence

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Region of convergence

• The sums are finite if and

Region of convergence:

• R+< R-: Ring

R-< R+: No ROC

• Causal system, for negative k:

cannot contain any poles of the system

• ROC of a stable system always contains

the unit circle

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R+ R-

Source: http://www.expertsmind.com/learning/z-transform-and-realization-of-digital-filters-assignment-help-7342873888.aspx

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Region of convergence

• System 1: Causal: h[k] = { …, 0, 0, 0, 1, 1/3, (1/3)

2

, (1/3)

3

, … } System 2: Anti-causal: h[k] = { …, 3

3

, 3

2

, 3, 1, 0, 0, 0, … }

• Analytical representation: h[k] = (1/3)

k

• After Z-transform: H(z) = z / (z – 1/3)
• 2 systems with very different behaviors, but the same transfer

function?

• Answer: different ROC:
• System 1: |z| > 1/3
• System 2: |z| < 1/3

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Inverse Z-transform

• Split the transfer function up in partial fractions
• This is done by first factorizing the denominator
• If all poles have multiplicity 1 then the following can be used:
• The coefficients can be calculated by:

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Inverse Z-transform

• If there are poles with multiplicity higher than 1 then the following

approach is needed:

• Where the highest coefficient for each pole can be calculated by:

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Inverse Z-transform

Any remaining coefficients can be found by evaluating the equation:

• for a number of values of z.

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Inverse Z-transform

• Because of the linearity of the inverse Z-transform, each partial

fraction can be transformed individually and the results can be added together afterwards.

• The individual inverse Z-transforms can be found with the following:

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Example

• Transfer function:
• Partial fraction decomposition:
• Using the given formula’s:
• This gives:
• By evaluating the transfer function for z=1 we get:

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Example

The resulting transfer function is:

• We can now find the inverse Z-transform for each individual fraction:

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Inverse Z-transform

• Another technique for calculating the inverse Z-transform is direct

division

• The numerator of the transfer function is divided by the denominator

via long division.

• Example:
• ⇒Z
• 1

{F(z)} = 1, 3, 12, 25, ...

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Solving difference equations with the Z-transform

• A system is described by a difference equation of the following form:
• After the Z-transform:
• Rearranged:

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Solving difference equations with the Z-transform

lWe’ll apply the following transformation of the double summations:

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Solving difference equations with the Z-transform

• The final simplified result is:
• With this result it is easy to find the resulting output from a given

input or vice-versa given a difference equation.

• Right-hand fraction = output resulting from starting conditions: will

vanish with time = transient behavior

• Left-hand fraction = output resulting from input: will remain = steady

state response

• is the “transfer function” of the system

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Steady state behavior via Z-transform

Starting from the previous result:

• We wish to find the resulting output from the input:
• To simplify derivation, we use:

u[k] = e

j(kα + θ)

• With Z-transform:

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Steady state behavior via Z-transform

Filling in U(z) and splitting into partial fractions:

• Calculating the coefficient c:
• After the inverse Z-transform:

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Steady state behavior via Z-transform

Because of linearity we can ignore the imaginary component, leading to the result:

• In most applications (= stable system) we can ignore transient

behavior as it will quickly die out

• Using the transfer function steady state behavior can easily be

determined by converting sinusoidal signals to phasors The input: cos(kα + θ) Will produce steady state: |H(ejα )| cos(kα + θ + ∠H(ejα))

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Example

• We’ll have a look at the steady state response to the input

for the system:

• Evaluating the transfer function for the exponential with pulsation 3

gives:

• The resulting output has been reduced to a third in amplitude and

has undergone a small phase shift.

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Interpreting the Z-transform

• Z-domain = frequency domain, similar to Laplace
• The results used for calculating steady state behavior can be used to

give a more concrete interpretation to the Z-domain

• A z-value is the phasor representation of a sinusoidal signal in the k-

domain

• Every signal in the k-domain can be decomposed into a sum of

sinusoidal signals.

• Every signal in the k-domain can be analyzed in the Z-domain as a

sum of sinusoidal signals.

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State space to transfer function

• The transfer function can be derived directly from the state space

model of a system:

• The Z-transform gives:
• Rearranged to have X(z) in explicit form:

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State space to transfer function

• The result for Y(z):
• Left hand term = effect on output from starting conditions = transient

behavior

• Right hand term = effect on output from input = steady state

behavior

• If all previous effects have died out we can equate the starting

conditions to 0: Y(z) = H(z) U(z) H(z) = C (zI – A )-1 B + D

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Link between Eigenvalues and poles

• Are Eigenvalues of A poles of H(z)?
• As z approaches an Eigenvalue of A, is no longer defined.
• may still be defined depending on the values of C

and B.

• An Eigenvalue of A will sometimes, but not always, be a pole of H(z).
• If every Eigenvalue of A is also a pole of H(z) then a minimal number
• f internal states has been achieved

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Link between Eigenvalues and poles

• Are poles of H(z) Eigenvalues of A?
• B, C and D are matrixes with properly defined values
• If H(z) is undefined then must be the cause
• z must be an Eigenvalue of A
• Poles of H(z) are always Eigenvalues of A

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Stability (DT)

• BIBO-Stability (Bounded-Input Bounded-Output)
• Every bounded input results in a bounded output
• Internal Stability
• Stricter than BIBO-Stability
• All possible internal states return to zero after a finite time in the

absence of an input.

• All Eigenvalues of the matrix A are contained within the a circle of

radius 1 around zero in the complex plane.

• BIBO-Stability follows from Internal Stability, but the inverse is not

necessarily true.

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Stability (DT)

• A discrete system is BIBO-Stable if all poles of H(z) are within a circle
• f radius 1 around the origin.

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Overview

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