Signals and Systems - Lecture 3: Discrete-Time LTI Systems: … · 2018. 10. 10. · Complex Exponential Sequences Periodic Signals Eigenfunctions of LTI Systems The z-Transform Deﬁnition

Signals and Systems

Lecture 3: Discrete-Time LTI Systems: Frequency DomainConcepts

Dr. Guillaume Ducard

Fall 2018

based on materials from: Prof. Dr. Raffaello D’Andrea

Institute for Dynamic Systems and Control

ETH Zurich, Switzerland

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Outline

1 Complex Exponential SequencesDefinitionDiscrete-time frequency range

2 Periodic SignalsDefinitionPeriodicity constraint

3 Eigenfunctions of LTI Systems

4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality

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Complex Exponential SequencesPeriodic Signals

Eigenfunctions of LTI SystemsThe z-Transform

DefinitionDiscrete-time frequency range

Outline





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Recall: a complex number z ∈ C can be written in

Polar form

z = |z|ejΩ,

where j2 = −1,|z| ≥ 0 is the magnitude of z,

and Ω ∈ (−π, π] is the phase of z. Note thatΩ can be defined in any 2π interval, such as[0, 2π).

Cartesian form

z = a+ jb,

where a, b ∈ R.

Re(z)

Im(z)

Ω0

z0

|z0|

a0

b0

The two forms are related inthat |z| =

√a2 + b2 and

Ω = atan2(b, a), whereatan2 is the four-quadrantarc-tangent function.

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Complex exponential sequence

Consider the complex exponential sequence with

x[n] = zn for all n,where z ∈ C. (1)

This can be rewritten using Euler’s formula, ejΩ = cosΩ + j sinΩ,as:

x[n] = zn

= (|z|ejΩ)n

= |z|nejΩn

= |z|n(cos(Ωn) + j sin(Ωn)),

where Ω is the discrete-time frequency of the sequence’s oscillatorycomponent.

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Outline





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Discrete-time frequency range

CT sinusoids

CT sinusoids, for example cos(ωt), are distinct for all frequenciesω ≥ 0.

DT sinusoids

cos((Ω + 2πk)n) = cos(Ωn+ 2πkn) = cos(Ωn), for k ∈ Z.

Consequences

1 DT frequencies Ω and Ω+ 2πk give identical sequences for allintegers k.

2 DT frequency is therefore restricted to an interval oflength 2π.

3 Usually the intervals [0, 2π) or (−π, π] are chosen, howeverany interval of length 2π is appropriate.

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DefinitionPeriodicity constraint

Outline





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A sequence x is called periodic

if there exists an integer N > 0 such that x[n+N ] = x[n] for all n.

It follows that x[n+mN ] = x[n] for all m ∈ Z.

The smallest N for which x[n+N ] = x[n] for all n is calledthe fundamental period.

ExampleThe following signal is periodic with fundamental period 2.

n

x[n]

−1 0 1 2 39 / 33




Outline





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Periodicity constraint

A CT sinusoid cos(ωt) is always periodic with a period of T = 2π|ω| .

If this periodic CT signal is uniformly sampled with a samplingtime of Ts, the resulting DT signal x[n] = cos(Ωn), withfrequency Ω = ωTs, is periodic if and only if

Ω

2π=

m

N, for some integers m and N.

In other words, x[n] is periodic if and only if Ω/2π isrational. (You will prove this in the problem sets.)

Furthermore, if mN

is an irreducible fraction, then N is thefundamental period of the signal.

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Table of Content





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Eigenfunctions of LTI Systems

Let the LTI system G have input u[n] = zn, with z ∈ C. Thenthe output sequence y[n] is calculated using the convolutiony = h ∗ u to be:

y[n] = Gzn =∞∑

k=−∞

h[k]zn−k =∞∑

k=−∞

h[k]znz−k

=

∞∑

k=−∞

h[k]z−kzn

= H(z)zn,

where

H(z) :=

∞∑

k=−∞

h[k]z−k.

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Eigenfunctions of LTI Systems

It follows that:

zn is an eigenfunction of G

and H(z) is the corresponding eigenvalue.

Recall the CT counterpart:

y = Gu, with u(t) = est gives y(t) = H(s)est, where H(s) is theLaplace Transform of the impulse response of G.⇒ In CT, the Laplace Transform enables the analysis of CTsystems in the frequency domain.

The equivalent DT transformation

is known as the z-transform and allows DT systems to be analyzedin the frequency domain. We investigate this transformation next.

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DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality

Outline





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Given a sequence x[n], its z-transform X(z) is defined as

X(z) :=∞∑

n=−∞

x[n]z−n, z ∈ C.

The transform pair, used to indicate the relationship between thesequence x[n] and its z-transform X(z), is denoted asx[n] ←→ X(z).

Remark:This is a slight abuse of notation as we use X(z) to denote both thez-transform of x[n] and its value at a particular z; however, this notationhelps to distinguish between the z-transform and the DT Fourier transform(introduced in Lecture 4), and it should be clear from context whether thefunction or the function value is intended.

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Properties

Given that a sequence x[n] is related to its z-transform X(z) asx[n] ←→ X(z), the following properties hold:

• Linearity: a1x1[n]+ a2x2[n] ←→ a1X1(z) + a2X2(z)

• Time-shifting x[n − 1] ←→ z−1X(z)

x[n + 1] ←→ zX(z)

• Convolution: x1[n] ∗ x2[n] ←→ X1(z) ·X2(z)

• Accumulation: n∑

k=−∞

x[k] ←→z

z − 1X(z)

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Outline





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Recall: Geometric series

Sn = a+ a · r + a · r2 + · · ·+ a · rn−1 =

n−1∑

k=0

a · rk = a

n−1∑

k=0

rk = a

1− rn

1− r,

where the first term is a ∈ R and the term r ∈ R is called common ratio.

Proof

Sn = a+ a · r + a · r2 + · · ·+ a · rn−1

r · Sn = a · r + a · r2 + a · r3 + · · ·+ a · rn

Sn − r · Sn = a− a · rn

Sn · (1− r) = a(1− rn)

Sn = a1− rn

1− rwith r 6= 1 .

If the common ratio has a module strictly smaller than one (|r| < 1), the seriesconverges to

limn→∞

n−1∑

k=0

a rk = lim

n→∞a1− rn

1− r= a

1

1− r. (2)

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Consider the sequence x[n] defined by :

x[n] =

an n ≥ 0, a ∈ R, a 6= 0

0 otherwise

Its z-transform X(z) is given as:

X(z) =

∞∑

n=0

anz−n =

∞∑

n=0

(a

z

)n

.

It can be shown that the above sum converges iff |a/z| < 1, or|z| > |a|.In that case,

X(z) =1

1− az

=z

z − a(3)

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Now, consider a different sequence

x[n] =

−an n < 0, a ∈ R, a 6= 0

0 otherwise

X(z) = −−1∑

n=−∞

anz−n = −∞∑

n=1

a−nzn = −∞∑

n=1

(z

a

)n

.

The above sum converges iff |z/a| < 1, or |z| < |a|. In that case:

X(z) =− z

a

1− za

=−z

a− z=

z

z − a

The z-transforms of the two sequences are identical!

The difference is the region of convergence (ROC), the values of z for whichX(z) converges.Therefore, strictly speaking, the z-transform must also include the ROC touniquely specify the sequence in the time domain.

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Outline





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Transfer functions of LTI systems

For an LTI system with impulse response h[n] we have

y[n] = u[n] ∗ h[n] ←→ Y (z) = U(z)H(z)

∴H(z) =Y (z)

U(z).

H(z) is the transfer function of the system. It is the z-transform ofits impulse response.

1 Therefore, as previously discussed, we also need to know theROC of H(z) to uniquely capture the input-output behaviorof the system,

2 however, in practice this is not necessary (as shown later).

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Transfer function from LCCDE

Consider a system described by the LCCDE:

N∑

k=0

aky[n− k] =

M∑

k=0

bku[n− k]

Using the z-transform, we can write

N∑

k=0

aky[n− k] =M∑

k=0

bku[n− k]←→N∑

k=0

akz−kY (z) =

M∑

k=0

bkz−kU(z),

and can then write the system’s transfer function as

Y (z)

U(z)=

b0 + b1z−1 + . . . + bMz−M

a0 + a1z−1 + . . .+ aNz−N= H(z).

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Let us compare this with the shift operator z introduced in Lecture1:

N∑

k=0

akz−ky =

M∑

k=0

bkz−ku

y

u=

b0 + b1z−1 + . . . + bMz

−M

a0 + a1z−1 + . . .+ aNz−N

= H(z),

where H(z) can now be thought of as an operator acting onsequences. This is due to the relationship between z, whichoperates in the time domain and z, which operates in thefrequency domain:

x[n+ 1]←→ zX(z)

x[n+ 1] = (zx)[n].

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Transfer function from a state-space description

The transfer function H(z) can also be determined starting from astate-space description:

q[n+ 1] = Aq[n] +Bu[n]

y[n] = Cq[n] +Du[n]

l

zQ(z) = AQ(z) +BU(z)

→ Q(z) = (zI −A)−1BU(z)

Y (z) = CQ(z) +DU(z)

→ H(z) =Y (z)

U(z)= C(zI −A)−1B +D

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Outline





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Stability and causality

We will address the subtle aspects of stability and causality with anexample. Consider the following difference equation:

y[n] = ay[n− 1] + u[n].

We can readily calculate its impulse response as

h[n] =

an for n ≥ 0

0 otherwise

h = . . . , 0, 0, 1↑, a, a2, a3, . . . .

We therefore have that the system captured by the above differenceequation is stable if |a| < 1, since then

∑∞n=−∞ |h[n]| <∞.

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Let us now rewrite the above difference equation in a different way:

y[n− 1] =1

ay[n]−

1

au[n].

If we now go backwards in time, we observe that the following isalso a valid impulse response:

h[n] =

−an for n < 0

0 otherwise

h = . . . ,−1

a3,−

1

a2,−

1

a, 0↑, 0, . . . .

So in this instance, the system is stable if |a| > 1. The difference isthat in this interpretation of the difference equation, the system isnot causal (the impulse response is non-zero for negative timeindices).

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We have, in fact, seen this example before:

y[n] = ay[n− 1] + u[n]

←→ Y (z) = az−1Y (z) + U(z)

zY (z) = aY (z) + zU(z)

Y (z)

U(z)= H(z) =

z

z − a.

Recall that H(z) has two possible inverse z-transforms, onecausal and one non-causal, depending on its ROC.

In practice, we usually require the system to be causal, andthus it is not necessary to specify the ROC of H(z).

The conclusion is that stability and causality are intertwined.30 / 33




Therorem Given a transfer function H(z), there exists a stable and causalinterpretation for the underlying system iff all poles of H(z) are inside the unitcircle. That is, given pole p (a value p for which |H(p)| =∞), then |p| < 1.

Proof We will show this result for the special case that H(z) is a rationaltransfer function, as would arise from LCCDE and SS descriptions of a system.We also assume that the finite poles of H(z) are distinct. In that case, H(z)can be factored as:

H(z) =

Q∑

k=0

ckzk +

R∑

k=1

βk

z

z − pk.

We observe that in order for H(z) to have a causal interpretation, ck = 0 fork > 0. If this is not the case, y will always depend on future values of u; thiscorresponds to poles of H(z) at z =∞. Therefore, consider

H(z) = c0 +

R∑

k=1

βk

z

z − pk.

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We can write down a causal impulse response h using the above z-transformand the z-transform of the unit impulse sequence δ[n]←→ 1:

h[n] =

c0 +R∑

k=1

βk n = 0

R∑

k=1

βkpnk n > 0

0 otherwise

It can then be shown that

∞∑

n=0

|h[n]| <∞ iff |pk| < 1.

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Non-causal systems

Causality is, however, not always necessary. Consider

y[n] =1

3(u[n+ 1] + u[n] + u[n− 1]).

There is no causal interpretation of this LCCDE. But in fact, this is a simplemoving average filter that can be used to smooth signals in post-processingapplications. In other words, causality is not a requirement in non-real-timeapplications. We have the following result:

Theorem Given the transfer function H(z), there exists a stable, but notnecessarily causal, interpretation for the underlying system iff H(z) has nopoles on the unit circle. That is, if |H(p)| =∞, then |p| 6= 1.

Proof.

We now prove one direction of the above statement: If there is a stable interpretation for the system, then H(z)has no poles on the unit circle. Let |z| = 1. It follows that

|H(z)| =

∣

∣

∣

∣

∣

∣

∞∑

n=−∞

h[n]z−n

∣

∣

∣

∣

∣

∣

≤∞∑

n=−∞

|h[n]| < ∞.

As required, H(z) has no pole p with |p| = 1. The proof of the other direction, that if H(z) has no pole p with

|p| = 1 there exists a stable interpretation for the system, is similar to the proof of the previous theorem. You will

prove this in the problem sets. 33 / 33

Documents

Signals and Systems - Lecture 3: Discrete-Time LTI Systems: … · 2018. 10. 10. · Complex Exponential Sequences Periodic Signals Eigenfunctions of LTI Systems The z-Transform Deﬁnition