ANALOG SYSTEMS - nts.uni-duisburg-essen.dents.uni-duisburg-essen.de/downloads/ss1/SS1-K3.pdfANALOG...

Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 1

FachgebietNachrichtentechnische Systeme

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Chapter 3

ANALOG SYSTEMS

(Version 2.1)



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3.1 A Short introduction to Network Functions

Network function: the mathematical relation between the Laplace transform of the excitation and the answer, if one assumes:

• The energy-less initialization state of all the network elements (zero-state)

• All the network elements are linear and time-independent.

One speaks of:

• Two-terminal network function, impedance function, admittance function excitation and answer are applied to the same terminal.

• System function excitation and answer are applied to different terminal. Notation of system function: ( )LH p



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3.1 A Short introduction to Network Functions( )I p

( )U pNetwork with Linear time-Independentelements

( ) Excitation( ) Answer

( )( )( )

U pI p

U pZ pI p

==

=

1

2

2

1

( ) Excitation( ) Answer

( )( )( )L

U pU p

U pH pU p

==

=

Networkwith linear,time-independentelements

1U ( )p 2U ( )p

Impedance function of a network

Definition of system function of a network



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3.1 A short introduction to Network Functions

( )( )( )

P pN pQ p

=

Some important properties of network function:

• Generally a LTI network can be described in rational fractionalfunctions in p:

(all the coefficients of P(p) and Q(p)are real and constant)

( )( ) ( ) , where ( ) : magnitude and ( ) : phasej pN p N p e N p pϕ ϕ=

• Or by the representation of the network function by its magnitude and its phase results:



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3.1 A short introduction to Network Functions

( ) ( ) ( ) is an even function( ) ( ) ( ) is an odd function

N j N j N jω ω ωϕ ω ϕ ω ϕ ω

= − →

− = →

ii

2

1

The system function is derived from quantities of the same dimension:

( )LUH pU

=

i

( )( ) ( )

2 1

1 2

( ) = ( ) with

( ) ( )( ) ( ) 20 log is the damping ratio( ) ( )

and ( ) ( ) =- ( ) is the damping angle

Lj H ja jbL L

L

L

H j e H j e

U j U jH j a dBU j U jb H j

ωω ωω ω

ω ωω ωω ω

ω ϕ ω ω

− −=

= ⇒ =

= −

If is applied:p j ω= ⋅

Additional notes:



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Chapter 3

ANALOG SYSTEM

3.2 Basic Properties of a System



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3.2.1 Definition of a System and General Remarks

Source DrainTransmitter ReceiverTransmission Channel

Distortion n(t)

Input excitation Cause

Output answer effect

( )x t ( )y t

One shall speak of a “system” if:

• a mathematical representation of such a circuit is given as an “input-output relation”.

Some examples:

Example of General transmission system



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Example of electrical system (Four-terminal network)

RL

aR

( )ai t

( )2U t( )1U t C

( ) ( ) ( ) ( )1 2 1

Input-Output connections: or au t u t u t i t→ →



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( )1s t

( )2s t

n

m

Example of Mechanical system

Mechanical ExamplePathForce, AccelerationInput-Output Connection:

( ) ( )1 2s t s t→



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2u

( )2u t( )1u t1u

AB

( )1u t ( )2u tRD

Example of a half-way rectifier Rectifier input-output relation with an example input

( )1u t

( )2u t

t

t

A

A−

B



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• Now 2 questions occur:

Is a system by a special input-output relation uniquely and completely represented ?

Under which circumstances is a benefit given from using the systems point of view ?

The uniqueness of the input-output relation of a system is not always given. It depends on the properties of the chosen input signal Unipolar input signal and

appropriate output signal of the rectifier

( )1u t

( )2u t

A

B

t

t



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Within the system view, it will be important to use suitable input functions for uniquely specifying the system properties. Here is an example for that:

Characteristics curve of a rectifier with saturation

If a bipolar impulse chain is used as an input for the rectifier, it can not be figured out whether the system has a saturation or not !

( )1u t

( )2u t

A

B



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( )1u t ( )2u t

R

C

A

T

( )1u t

t

( )2u t

t

The systems which are set up completely different can generally exhibit the same mathematical description. Here is an example:

Different Circuits with the same Input-Output relation

( )1u t ( )2u t

L

R



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Benefits from the system theory:

1. Unique characterization of LTI system by analysis of the answer of the system at a certain standard input function is possible. Mathematically it leads to a LTI transform in time domain:

( ) ( ) or ( ) [ ( )]s t g t g t T s t⇒ =

2. Unique characterization of LTI system in frequency domain by thetransfer function is possible it helps in simplifying the calculation.



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3.2.2 Basic System Properties

A system ( ) ( ) is called time-invariant if it follows: ( ) ( )s t g t s t g tτ τ→ − → −

1 1( ) ( )s t g t→

2 2( ) ( )s t g t→

( ) ( ) ( ) ( )s t g t as t ag t→ ⇒ →

1 2 1 2( ) ( ) ( ) ( )s t s t g t g t+ → +

1. Time-invariance:

The system’s reactions are always the same, independent from any delays at the input.

2. Additivity:

A system is called additivity if:

3. Homogeneity:



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1( ) ( ) whereby ( ) ( )

n

i i ii

x t s t s t g t=

= →∑

1then ( ) ( ) because of the linearity

n

ii

y t g t=

= ∑

0 0( ) ( )

1 0t

s t tt

ε<⎧

= = ⎨ ≥⎩

0 0( )

1 0t

tg t

e tτ−

<⎧⎪= ⎨⎪ − ≥⎩

1 2 1 2( ) ( ) ( ) ( )as t bs t ag t bg t+ → +

4. Linearity:The combination of both additivity and homogeneity results linearity.

If the excitation x(t) of the system is known, then the wanted answer y(t) can be determined in the following way:

Example: let’s assume an LTI system with its answer to the unit-step as followings:

answer



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( )s t

t

( )g t

tAnswer of a system to an Unit-step function

0 0( ) ( )

1 0t

s t tt

ε<⎧

= = ⎨ ≥⎩

0 0( )

1 0t

tg t

e tτ−

<⎧⎪= ⎨⎪ − ≥⎩

( ) ( ) ( )x t t t Tε ε= − −

?( )x t

t

1

T



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The solution goes by exploiting the linearity and time-invariance of the system:

• First step: separation of the input function into 2 steps using rect-function

• Second step: find out the 2 answers for each rect-function

• Third step: superposition of the answer

In the following slide it shows the graphical determination of the system answer to a rectangular impulse



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1

0 0( ) ( )

1 0t

x t tt

ε<⎧

= = ⎨ ≥⎩

1

0 0( )

1 0t

ty t

e tτ−

<⎧⎪= ⎨⎪ − ≥⎩

2

0 0( ) ( )

1 0t

x t t Tt

ε<⎧

= − − = ⎨− ≥⎩2

0 0( )

1 0t T

ty t

e tτ−

−

<⎧⎪= ⎛ ⎞⎨− − ≥⎜ ⎟⎪

⎝ ⎠⎩

1 2( ) ( ) ( )0 t<01 0 t T0 t >T

x t x t x t= +

⎧⎪= ≤ ≤⎨⎪⎩

1 2( ) ( ) ( )

0 t<0

1 0 t T

1 t>T

t

t T

y t y t y t

e

e e

τ

τ τ

−

−

= +

⎧⎪⎪⎪= − ≤ ≤⎨⎪ ⎛ ⎞⎪ ⋅ −⎜ ⎟⎪ ⎝ ⎠⎩

( )1x t

t

1+

T

( )1y t

t

( )2x t

t

T

1−

( )2y t

t

A

T

( )x t

t

( )y t

t



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3.2.3 Realisable Systems

( ) 0 for s t t tν≡ < ( ) 0 for g t t tν≡ <

1. Real systems:

A system is called real when it follows:

Real input Real output

2. Causal system:

A system is causal if the output signal g(t) until an arbitrary time depends only the input signal s(t) until this time.

No effects before the cause

3. Stable system:

1 1

A system ( ) ( ) is stable in the BIBO sense (Bounded Input Bounded Output) if: ( ) , then ( )

s t g ts t M t g t M t

→

< < ∞ ∀ < < ∞ ∀



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( )s t ( )g t

( )s t

t

( )g t

t

3.2.3 Realisable SystemsAn example of non-stable system: the ideal integrator

( ) ( ) ( )t

s t g t s dτ τ−∞

→ = ∫

0

A system is asymptotically stable if a damped input signal causes a damped output signal

lim ( ) 0 lim ( ) 0t t

s t g t→+∞ →+∞

= → =



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3.2.3 Realisable Systems4. Memory less and dynamic systems:

If for anytime t, the value of the output signal g(t) of a system depends exclusively on the value of the input signal s(t) at the same time t, then the system is called memory-less



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Chapter 3

ANALOG SYSTEM

3.3 Analog Linear Time-Invariant System



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3.3.1 Convolution Integral and Impulse Response

[ ]( ) ( ) ( )s t g t T s t→ =

0 ( ) ts t rectT⎛ ⎞= ⎜ ⎟⎝ ⎠

A LTI system is denoted as:

Derivation of general form of T[s(t)]:

Let‘s assume: a LTI system reacts to a small rectangular impulse as:

0 0 0( ) ( ) [ ( )]

s t g t T s t→ =

⇒

0( )g t

t

0( )s t

t



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The answer of the system to an arbitrary input signal can be calculatedapproximately if the properties LTI are used:

Division of an input signal into a staircase signal

( )s t

0T− 0T 02T0

0iT

( )as t



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0 0 0 0 0 0( ) ( ) ( ) ( )s iT s t iT s iT g t iT⋅ − → ⋅ −

0 0 01

( ) ( ) ( ) ( )ai

g t g t s iT g t iT∞

=

≈ = ⋅ −∑

0, 1, 2,...,i = ± ± ±∞0 0 0

1

( ) ( ) ( ) ( )ai

s t s t s iT s t iT∞

=

≈ = −∑

Step 1: separate the input signal into staircase signal elements

with

Step 2: because the system is time-invariant, it follows:

Step 3: because the system is linear, one can use the superposition:

0The smaller is, the more accurate the approximation of ( ) is. aT g t



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Superposition of the single answers

( )ag t

( ) ( )00s g ti

( ) ( )0 0 0s T g t T−i

t00T 02T

( ) ( )0 0 0s iT g t iT−i

0a



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3.3.1 Convolution Integral and Impulse Response0Now a limiting 0 provides a perfect approximation: T →

01. becomes an infinitesimal T dτ

0 02. becomes the continous variable , so that ( ) ( )i T s i T sτ τ⋅ ⋅ →

0

( ) lim ( )aTs t s t

→∞=

0 00 0

0

( )( ) ( )ai

s t iTs t s iT TT

∞

=−∞

−= ⋅ ⋅∑ 0 0

0 00

( )( ) ( )ai

g t iTg t s iT TT

∞

=−∞

−= ⋅ ⋅∑

For this, an extension is made as followings:

(*)

The relation (*) is fulfilled if:

0 0

0

( )3. becomes Dirac's delta function ( )s t i T tT

δ τ− ⋅−

!0 0

0

( )4. ( ) where ( ) called the impulse response.g t i T h t h tT

τ− ⋅= −



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0

lim ( ) ( ) ( ) ( )aTs t s t s t dτ δ τ τ

∞

−∞→∞= = −∫

0

lim ( ) ( ) ( ) ( )aTg t g t s h t dτ τ τ

∞

−∞→∞= = −∫

( ) ( ) ( ) ( ) ( )t g t h t d h tδ δ τ τ τ∞

−∞

→ = − =∫

With the „extraction-property“ of Dirac‘s delta function, one obtains:

convolutional integral

Some remarks:

• convolutional integral is a general method for determing theresponse of an LTI system to any excitation.

• h(t), the impulse response, is required to evaluate the convolutionalintegral and can be determined as following:



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( ) ( ) ( ) with ( ) 0 for 0

( ) ( )t

g t s h t d h t t t

s h t d

τ τ τ τ τ τ

τ τ τ

+∞

−∞

−∞

= ⋅ − − ≡ − < ⇔ <

= −

∫

∫

( ) 0 for 0h t t≡ <

• For causal systems with „No effect before the cause“ follows:

The impulse response of a causal system is also a causal signal function

The convolutional integral for causal system can be described as:



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1 for 0( )0 for 0

tRCe th t RC

t

−⎧≥⎪= ⎨

⎪ <⎩

Example: given the following:

( )tδ

R

C ( )h t

( )h t

tRC

1RC

( )tδ

t0

Dirac's delta functionImpulse response

RC - circuit



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( ) ( ) ( )g t s h t dτ τ τ+∞

−∞

= ⋅ −∫

0

0 0

1 2( )2

Ttts t a rect a rectT T

⎛ ⎞−⎜ ⎟⎛ ⎞= ⋅ − = ⋅ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠

Question: Reaction of the system to a rectangular impulse at the input

?

Solution: Evaluation of the convolution integral:

( )s t

a

0T t



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1. ( ) 0 for 0g t t≡ <

00

2. ( ) ( ) ( ) for 0t

g t s h t d t Tτ τ τ= ⋅ − ≤ ≤∫

0 0

1( )

1 0

t tt tRC RC RC

t tRC RC RC

ag t a e d e e dRC RC

ta e RCe a eRC

τ τ

τ

τ τ−⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= ⋅ = ⋅

⎛ ⎞= ⋅ = −⎜ ⎟⎜ ⎟

⎝ ⎠

∫ ∫

It leads to the following relations and results:

0

00

3. ( ) ( ) ( ) for T

g t s h t d t Tτ τ τ= ⋅ − >∫0 0

0

0 0

0

1( )

1 0

t tT TRC RC RC

Tt tRCRC RC RC

ag t a e d e e dRC RC

Tae e ae e

τ τ

τ

τ τ−⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ +− − ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= ⋅ ⋅ = ⋅ ⋅

⎛ ⎞= = ⋅ − +⎜ ⎟⎜ ⎟

⎝ ⎠

∫ ∫

( )g t

t0T RC

a



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1RC

( )h τ

0 RCτ

( ) ( ) 1 RCh eRC

τ

τ τ−

=

( ) ( ) 1 RCh eRC

τ

τ τ− =1

RC

( )h τ−

RC− 0

1RC

( )h t τ−

( )0h t τ−

( )s τ

1F 2F

0t t= 1t t=0

a

2t t=

( )2h t τ−

( ) ( ) 1 tRCh t s t e

RC

τ

τ τ−

−− = −

τ

τ



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3.3.2 Step Response

( ) ( )t h tδ →

( ) ( ) ( ) ( )s t g t s t h t→ = ∗

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )t

s t t g t t h t t h d h dε ε ε τ τ τ τ τ+∞

−∞ −∞

= → = ∗ = − =∫ ∫

( ) ( )t

w t h dτ τ−∞

= ∫

The system response to any excitation s(t) can be determined as:

Example: Impulse response

Similarly, the response of an LTI-system to the unit step function gives:

In short: Step response

Moreover: ( ) ( )dh t w tdt

=



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3.3.2 Step Response

( ) ( ) ( ) ( )s t g t s h t dτ τ τ+∞

−∞

→ = −∫

. . .b b

a a

bu dv u v v du

a= −∫ ∫

( )( ) ( ) ( ) ( )dus u s du s ddττ τ τ τ ττ

′ ′= ⇒ = ⇒ =

Proof:

The partial integration gives:

( ) ( ) where dvh t d dv h t t x d dxd

τ τ τ τ ττ

− = ⇒ = − − = → − =

( ) ( ) ( ) ( ) ( )dv h x v x h x dx w x w tdx

τ= − ⇒ = − = − = − −∫

( ) ( ) ( ) ( ) ( ) ( )s t g t s w t w t s dτ τ τ τ τ+∞

−∞

+∞′→ = − − + −

−∞ ∫

Thus:



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3.3.3 The Transfer Function H(ω)

0 0 0( ) cos sinj ts t s e s t js tω ω ω= = +

( )0 0

0 0

( ) ( ) ( )

( ) ( )

j t j t

j t j j t

g t s e h t h s e d

s e h e d s e H

ω ω τ

ω ωτ ω

τ τ

τ τ ω

+∞−

−∞

+∞−

−∞

= ∗ = ⋅

= ⋅ = ⋅

∫

∫

( ) ( ) j tH h t e dtωω+∞

−

−∞

= ⋅∫

Let the exponential function be the excitation of a LTI-system:

The response of any LTI-system is given by the convolution of impulse responseand input signal:

where:Transfer function of the system: The Fourier transform of the systemsimpulse response



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( ) ( ) ( ) ( ) ( )g t s t h t s h t dτ τ τ+∞

−∞

= ∗ = −∫

( )( ) ( ) ( ) ( ) ( ) 0( )

GG S H H SSωω ω ω ω ωω

= ⇔ = ∀ ≠

By applying the Fourier transform to the convolution as shown before, one gets:

Effectless systems:

Cascade of two LTI systems

Let‘s assume a combined system as follows:

( )zg t( )1h t ( )2h t

( )s t ( )g t



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1

2 1 2

( ) ( ) ( )( ) ( ) ( ) [ ( ) ( )] ( )

( )z

z

g t s t h tt h t g t s t h t h tg t g

= ∗∗ ⇒ = ∗ ∗=

1 2( ) ( ) ( ) ( )G S H Hω ω ω ω=

1 2

1 2

( ) [ ( ) ( )] ( )( ) ( ) [ ( ) ( )]

g t s t h t h tg t s t h t h t

= ∗ ∗= ∗ ∗ 1 2( ) ( )[ ( ) ( )] ( ) ( )totG S H H S Hω ω ω ω ω ω= =

One obtains:

Theorem: When connecting effectless systems in a chain, the transfer functionsare multiplied.

Moreover:

The properties of the convolution product lead to the following equations:



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0

1 1

0

1( ) ( )

1 1 1 1 1 0 1

tj t j tRC

j t j tRC RC

H h t e dt e e dtRC

e dt eRC RC j RCj

RC

ω ω

ω ω

ω

ωω

+∞ +∞−− −

−∞

⎛ ⎞ ⎛ ⎞+∞ − + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

+∞−= = ⋅ =

++

∫ ∫

∫

1 0( ) ( )0 0

tRCe for tg t h t RC

for t

−⎧⋅ ≥⎪⇒ = ⎨

⎪ <⎩

Example of a simple filter as a LTI-system:

Given is the impulse response of the following system:

( )s t

R

C



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2

1

1( ) 1 ( )1( ) 1

U j C HU j RCR

j C

ω ω ωω ω

ω

= = =++

The same result can be obtained by using network analysis:

RC-Circuit as complex voltage divider

2

1

( )( )( )

UHU

ωωω

=

( )1U t ( )2U t

R

1j Cω

This method (determination of transfer function based on network analysis) works for any LTI network!



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1RC−

1 ( ) Re H ω

ω1RC

2RC− 2

RC

even in ω

odd in ω( ) Im H ω

1( )1

Hj RC

ωω

=+

2

1Re ( ) ( )1 ( ) RH H

RCω ω

ω= =

+ 2Im ( ) ( )1 ( ) X

RCH HRC

ωω ωω

= − =+

( ) ( )( ) ( ) ( )j jH H e A eϕ ω ϕ ωω ω ω= =

( )H ω

( )ϕ ω

Real and Imaginary part of the transfer function of an RC circuit:

Example:

Another representation for the transfer function:

Magnitude of the transfer function

Phase of the transfer function

Real and imaginary part of transfer function of an RC-circuit (low-pass filter)

In general: ( ) Re ( ) Im ( ) ( ) ( )R XH H j H H jHω ω ω ω ω= + = +



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2 2

2

1( ) (Re ( ) ) (Im ( ) )1 ( )

H H HRC

ω ω ωω

= + =+

Im ( )( ) arctan arctan( )

Re ( )H

RCH

ωϕ ω ω

ω⎛ ⎞

= = −⎜ ⎟⎜ ⎟⎝ ⎠

( ) ( )H Hω ω= −

( ) ( )H Hω ω∗− =

( ) ( )ϕ ω ϕ ω− = −

The magnitude and phase of the transfer function of an RC-Circuit:

Example:

Because h(t) is real-valued function, one gets:



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Magnitude and phase of an RC-Circuit (Low-pass filter)

1RC

−

1( )H ω

ω

( )ϕ ω

2π

0

2π

−

ω

1RC

1RC

−1

RC



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3.3.4 Analog LTI-Systems, Describable byDifferential Equations

Description of a LTI system with h(t) and H(ω)

( ) ( ) ( )t h t h tδ ∗ =

( ) ( ) ( )h t t d h tδ τ τ+∞

−∞

− =∫

( ) ( )1 H Hω ω⋅ =

( )h t

( )H ω

( )tδ

1



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3.3.4 Analog LTI-Systems, Describable byDifferential Equations

0 0

( ) ( )( ) ( )

m nd da s t c g tdt dt

ν µ

ν µν µν µ= =

=∑ ∑

the coefficThese kinds ient of system a and are constantsre LTI if .a cν µ

A certain class of analog LTI system can also be described by finite differential equations as follows:

The Fourier transform of both sides of the equation above yields:

0 0( ) ( ) ( ) ( )

m n

a j S c j Gν µν µ

ν µ

ω ω ω ω= =

=∑ ∑

0

0

( )( )( )( ) ( )

m

n

a jGHS c j

νν

ν

µµ

µ

ωωωω ω

=

=

= =∑

∑1 1 0

0

( )( )( )( ) ( )

m

n

a jGh t F FS c j

νν

ν

µµ

µ

ωωω ω

− − =

=

⎧ ⎫⎪ ⎪⎧ ⎫ ⎪ ⎪= =⎨ ⎬ ⎨ ⎬

⎩ ⎭ ⎪ ⎪⎪ ⎪⎩ ⎭

∑

∑or



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3.3.5 Causal, Analog Low-Pass and Band-Pass Systems

Short introduction about Low-pass and Band-pass system:

Most transmission channels are limited to a certain frequency range due to:

• natural property of the channel (cable, radio,...)

• usage of filters to separate the information content and the noise

A system view is useful to:• describe the basic properties of filters, transmission system ortheir effect on given signals

• simplify the design of transmission and other systems

Low-pass and band-pass filters are most important system for theory and practice



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( ) ( )( ) ( ) ( )j jbH H e A eϕ ω ωω ω ω −= =

( ) 20 log ( ) a A dBω ω= −

( ) ( )b ω ϕ ω= −

1( ) ( )Phτ ω ϕ ωω

= −

There are other ways of writing the transfer function of filters:

Hereby, it follows:

1. Damping ratio:

2. Damping angle:

3. Phase delay:

is the time delay of a cosine signal

corresponding to a specific phase φ



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0 00

1( ) ( )Phτ ω ϕ ωω

= −

0The signal output is amplified by the factor ( ) and is delayed about the phase delay:

A ω

00

00 0

( )( )

0 0 0 0( ) ( ) ( )j t

j j tg t s A e e s A eϕ ω

ωωϕ ω ωω ω

⎛ ⎞+⎜ ⎟

⎝ ⎠= =

So that the signal output holds:

To the explanation of the phase delay

( ) ( ) 00 0

j tg t s H e ωω= ⋅

( )h t

( )H ω

( )s t



N T S


( ) ( )Grd

dτ ω ϕ ω

ω= −

0( ) ( ) cosss t s t tω=

0( ) 0 for with s g gS ω ω ω ω ω• = >

4. Envelope delay or group delay:

For the explanation, a narrow-band signal is provided in term of amplitude modulated cosine

with the following characteristics is observed:

• This signal will be applied to a linear filter with constant amplitudeand linear damping angle:

Linear filter: ( )s t ( )g t( ) ( ) ( )jbH A e ωω ω −= ⋅



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0 0 0( ) ( ) for g gA Aω ω ω ω ω ω ω= − < < +

0 0 0 0 0 00

( ) ( ) ( ) ( ) .( ) ( ) ( ) ( )Ph Grdbdt

ϕ ω ω ϕ ω ϕ ω ω ω ω τ ω ω ω τ ωω

= − = + − = − − −

0( ) ( ) cos ( )s Gr Phg t s t tτ ω τ= − −

For this case, it can be shown:

Magnitude of an input signal and magnitude and damping angle of thetransmission system

( )b ω

( )A ω

0ω −Ω 0ω 0ω +ΩΩ−Ω 0

( )S ω



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Note: Negative phase retardation and envelope delays are no contradictionsto the reality because both are defined for the steady-state condition of thesystem.

Realizable system with partly negative envelope delay

Example:

2R

C

CR2U1U



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0 0 0 0( ) ( ) ( ) where and are real and constants t g t h s t t h t→ = −

Distortion-less system:

is a system or a filter which does not change the form of the input signal

Example: Input and output of a distortion-less system

( )s t

t

0s 0 0s h

( )g t

0t t



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0 01. ( ) ( ) ( ) ( ) ( )s t g t s t h t h s t t→ = ∗ = −

002. ( ) : transfer functionj tH h e ωω −=

0 0( ) and ( ) .A h tω ϕ ω ω= = −

0Ph Gr tτ τ= =

0

( ) const. : amplitude distortions( ) : phase distortions

At

ωϕ ω ω

≠≠ −

ii


Properties of a distortion-less system:

3. This means:

thus: frequency-independent

All deviations from these properties are called linear distortions:when



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00( )

2j t

g

H A rect e ωωωω

−⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

00( ) ( ( )) g

g

Ah t si t t

ωω

π= −

The ideal Low-pass filter:

The transfer function:

The impulse response:

A low-pass filter is a filter which has the property

( ) ( ) 0 for 2g gA H fω ω ω ω π= = > =



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The ideal Low-pass filter in the frequency and time domain

( )h t0 0

gh Aωπ

=

0 0tt

01

2 g

tf

− 01

2 g

tf

+

gω− gω0

( ) 0b tω ω=( ) ( )H Aω ω=

ω

0A

( ) 00 2

j t

g

H A rect e ωωω

ω−

⎛ ⎞= ⋅⎜ ⎟⎜ ⎟

⎝ ⎠

F



N T S


0

( ) ( )x

Si x si dξ ξ= ∫ ( ) ( )Si x Si x− = ( )2

Si π−∞ =

0 01 2( ) 1 ( ( ))2 gt A Si t tω ω

π⎡ ⎤= + −⎢ ⎥⎣ ⎦

Using integral sine Si(x):

and and

One gets:

The step response:

0

0

0 0

0 0 0

( ) ( ) ( ( ))

( ( )) ( ( )) ( Si func. is symmetric)

t t

g

t t

g gt

t h d h si t d

h si t d si t d

ω τ τ ω τ τ

ω τ τ ω τ τ

−∞ −∞

−∞

= = −

⎡ ⎤= − + −⎢ ⎥

⎢ ⎥⎣ ⎦

∫ ∫

∫ ∫



N T S


Step response of a Low-pass filter with Overshooting

t0t

12 gf

0

2A

0A

( )w tmax. overshooting 8.95%



N T S


00

0 0

( ) ( ) ge e e

AdA t w t t h t tt t t tdt

ωπ

= = == =

12e

g g

tf

πω

= =

• Overshooting: 8.95 % of the step amplitude.

• Rise time of the low-pass:

0Step size = rise time maximum gradienteA t ⋅

So the rise time becomes:

reverse proportional to cut-off frequency



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00 00( ) j tH A rect rect e ωω ω ω ωω

ω ω−⎡ − + ⎤⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎢ ⎥∆ ∆⎝ ⎠ ⎝ ⎠⎣ ⎦

0 0

0 0 0 0

0 0

( ) ( )00

00 0 0

( ) ( )2 2 2 2

( )2 2

( ) cos( ( ))2

j t j t

j t t j t t

h t A si t e si t e t t

A si t t e e

A si t t t t

ω ω

ω ω

ω ω ω ω δπ πω ωπω ω ω

π

−

− − −

⎡∆ ∆ ∆ ∆ ⎤⎛ ⎞ ⎛ ⎞= + ∗ −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦∆ ∆⎛ ⎞ ⎡ ⎤= − +⎜ ⎟ ⎣ ⎦⎝ ⎠∆ ∆⎛ ⎞= − −⎜ ⎟

⎝ ⎠

The ideal Band-pass filter:

is defined by the following relation:

Non-zero in a finite band This range does not include = 0

ωω

∆ii



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Impulse response of a Band-pass filter

02t πω

+02t πω

− 0t

0

( )h t0A Aωπ



N T S


0

0 0 0

0 0

2( ) 2 ( ) cos ( )2

( ) cos ( )T

Ah t si t t t t

h t t t

ωω ω

π

ω

∆∆⎡ ⎤= − −⎢ ⎥⎣ ⎦

= ⋅ −

00 00( ) 2 j tH A rect e ωω ωω

ω−−⎛ ⎞= ⎜ ⎟∆⎝ ⎠

00( ) 2 j t

TH A rect e ωωωω

−⎛ ⎞= ⎜ ⎟∆⎝ ⎠

If one compares the impulse response of the ideal band-pass and the ideal low-pass, one can gets:

( ) ( ) complex envelope with( ): impulse response of the band-pass( ) : equivalent low-pass impulse response

T

T

h t h th th t

= ×

with the following relations:

and



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Symmetrical Band-pass and appropriate equivalent Low-pass

02A

0A( ) 0b tω ω=

0ω+ω

0ω−2ω∆

2ω∆

−

ω∆

ω∆

gcritical frequency of the 2

equivalent low-pass

ωω ∆=

( )equiv.lpA ω



N T S


Re ( ) Re ( ) : is an even function in H Hω ω ω= −

Im ( ) Im ( ) : is an odd function in H Hω ω ω= − −

The Non-symmetrical Band-pass:

The impulse response of a band-pass (and all other LTI-Systems) must be a real-valued function of time and has the following properties:



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Spectrum of a Non-symmetrical Band-pass

ω∆

0ω0ω−

ω∆

( ) Im H ω

( ) Re H ω

( )BH ω− ( )BH ω+



N T S


( ) ( ) ( ) with ( ) ( ) ( ) and( ) ( ) ( )

H H HH HH H

ω ω ω

ω ω ε ω

ω ω ε ω

+ −

+

−

= +

= ⋅

= ⋅ −

0( ) 2 ( )TH Hω ω ω+= +

( ) Re ( ) Im ( )T T TH H j Hω ω ω= +

( ) ( ) ( )Th t u t j v t= + ⋅

For the following considerations, it gives:

Because of its properties:

Equivalent low-pass signal of the non-symmetrical band-pass

( ) Re TH ω

( ) ( )0T BH Hω ω ω+= +

0

( ) Im TH ω

ω∆



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0 0( ) 2 ( ) 2 ( )TH H Hω ω ω ω ω+ − ∗⎡ ⎤= + = − −⎣ ⎦

00

1 1( ) ( ) ( )2 2TH H Hω ω ω ω+ = − =

01( ) ( )2 TH Hω ω ω− = − −

( ) ( ) ( )H H Hω ω ω+ −= +

Based on the formulas above, one can get the following relations:

or

with:



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00 0

2 20 0

( ) Re ( ) ( ) cos ( )sin

( ) ( ) cos( ( )) ( ) cos( ( ))

j tT

T T T

h t h t e u t t v t t

u t v t t t h t t t

ω ω ω

ω ϕ ω ϕ

= = −

= + + = +

The impulse response of the non symmetrical band-pass is given by:

( ) : equivalen low-pass or complex envelope( ) : in-phase component( ) : quadrature component

Th tu t

tν

The impulse response of the general band-pass is an amplitude and angle-modulated cosine signal.

0

0

All relations between ( ), ( ), ( ) and its Fourier Transforms

also hold for ( ), ( ), ( ) and its Fourier Transforms! (see chap. 2, S.95-100)T

T

s t s t s t

h t h t h t



N T S


( ) ( ) ( ) ( )y t g t y t h t→ = ∗ ( ) ( ) ( )G Y Hω ω ω= ⋅

*0 0

0 0 0 0

0 0 0 0

1 1( ) ( ) ( )2 21 1 1 1 ( ) ( ) . ( ) ( )2 2 2 2

1 1 ( ) ( ) ( ) ( )4 4

T T

T T T T

T T T T

G G G

Y Y H H

Y H Y H

ω ω ω ω ω

ω ω ω ω ω ω ω ω

ω ω ω ω ω ω ω ω

∗ ∗

∗ ∗

= − + − −

⎡ ⎤ ⎡ ⎤= − + − − − + − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= − ⋅ − + − − ⋅ − −

Transmission of Band-pass signals through Band-pass filters:

For any LTI-system with an arbitrary input y(t), the following holds:

Representing band-pass signal by means of equivalent low-pass signal:



N T S


1( ) ( ) ( )2T T TG Y Hω ω ω=

1( ) ( ) ( )2T T Tg t y t h t= ∗

[ ]0 01( ) Re ( ) Re ( ) ( )2

j t j tT T Tg t g t e y t h t eω ω⎧ ⎫= = ∗⎨ ⎬

⎩ ⎭

0If one assumes, that ( ) 0 ( ) for (means: ( ) and ( ) are narrow-banded), then:

T TY and Hy t h t

ω ω ω ω= ≤ −

and thus:



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0( ) ( )H A Aω ω= =

00( ) j tH A e ωω −= 0 0( ) ( )h t A t tδ= −

0 0 0 0( ) ( ) ( ) ( ) ( )s t g t s t A t t A s t tδ→ = ∗ − = −

All-pass filters:

is an LTI system with the property:

All-pass filters with linear phase:

are:

0

0

ideal delay elements for 0 ideal predictors for 0 , because of:

tt>

<

ii

Usage: often used to correct the phase of transmission channels and filters.



N T S


00 00

0 0

00

0

( ) ( ). ( ( ))2 2

( ) ( )2

gk g

gg

At t t th t h t rect si t t rectt t

A tsi t rect t tt

ωω

π

ωω δ

π

⎛ ⎞ ⎛ ⎞− −= = −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛ ⎞

= ∗ −⎜ ⎟⎝ ⎠

0 00 0 0

1( ) 2 . ( ) ( )2 2

j t j tK K

g

H A rect t si t e H eω ωωω ω ωπ ω

− −⎛ ⎞

= ∗ =⎜ ⎟⎜ ⎟⎝ ⎠

Causal Low-pass filters:

can be derived from the non-causal, ideal low-pass by multiplying theimpulse response with a suitable rectangular function:



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Multiplication of the impulse response of the ideal low-pass with a rectangular function

( )h t0

02t trect

t⎛ ⎞−⎜ ⎟⎝ ⎠

t0t0

0A ωπ∆

02t



N T S


Magnitude and phase of the causal low-pass filter

The causal low-pass filter lost the shape of the ideal low-pass filter

( )gH ω

( ) 0b tω ω=

gωgω− 0 ω



N T S

3.3.6 The System function HL(p)

0 0( ) where pt t j ts t s e s e e p jσ ω σ ω= = = +

( )0 0

0 0

( ) ( ) ( ) ( ) ( )

( )

pt p t

pt p pt

g t s t h t s e h t s h e d

s e h e d s e I

τ

τ

τ τ

τ τ

+∞−

−∞

+∞−

−∞

= ∗ = ∗ =

= =

∫

∫

With ( ) 0 for 0h t t≡ <

0 00

( ) ( ) ( )pt p ptLg t s e h e d s e H pττ τ

∞−= =∫

Operating the excitation of the following input signal:

One gets:

The difference is I

For a causal system, it obtains:



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0

2. ( ) is the Laplace transform of the impulse response of the causal LTI system.

( ) ( )

L

ptL

H p

H p h t e dt∞

−= ∫

0

( ) ( ) ( )t

g t s h t dτ τ τ= −∫

( )( ) ( ) ( ) ( )( )

LL L L L

L

G pG p S p H p H pS p

= ⇔ =

1. ( ) is called system functionLH p

3. Because of the restriction to causality, it follows for the convolution integral:



N T S


( ) ( )LH H jω ω=

2Re 0, ( ) j -axis is covered by convergence area ensuring:Lp H pβ ω= <

1 2( ) ( ) ( )L L LH p H p H p= ⋅

4. When combining 2 causal, effectless LTI-system (filters), one obtains:

Non-reactive combination of two causal LTI-system

5. ( ) does not automatically equal the system function (p) on the _LH H j axisω ω

When moving from , the following cases c( ) an occ( ur) : LH p to H ω

Case 1:

( )s t( )( )

1

1L

h t

H p( )( )

2

2L

h t

H p( )g t



N T S


0Re 0p β= =

( ) ( ) function with suitable Dirac's delta functionsLH H jω ω= +

1Re 0 the axis is outside of the convergence are( ) does not s

aexi t

p jHωω

β= > ⇒ −→

Case 2:

Case 3:0

( )h t dt∞

< ∞∫

Note: for a causal system where the impulse response is integrable:

such case does not occur



N T S


Convergence areas of the Laplace transform

jωp - plane

σ0

2β 0β 1β



N T S


202

2 21 0

1( )( ) 1( )L

U p pCH pU p ppL

pC

ωω

= = =++

0 0( ) sin( ) ( )h t t tω ω ε= ⋅

[ ]

[ ] [ ]

00 0

00 0 0 0

1( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )2

H j jj

jj

ωω πδ ω ω πδ ω ω πδ ωπ ω

ω δ ω ω δ ω ω π δ ω ω δ ω ωω

⎧ ⎫⎡ ⎤= + − − ∗ +⎨ ⎬⎢ ⎥

⎣ ⎦⎩ ⎭⎧ ⎫

= + − − ∗ + + − −⎨ ⎬⎩ ⎭

Example: Given is the following network

The corresponding system can be derived as:

0iR =L

C

( )1u t( )1U p

( )2u t( )2U p



N T S


0 00

1 1 1 1( ) ( ) dvvj j v j

δ ω ω δ ωω ω ω ω

+∞

−∞

± ∗ = ± =− ±∫

[ ]0 00 0

0 0

1 1( ) ( ) ( )2 2

Hj j j j j j

ω ω πω δ ω ω δ ω ωω ω ω ω

⎡ ⎤= − + − − +⎢ ⎥− +⎣ ⎦

2 20 0 0

2 20 0 0 0 0

1 1( )( ) ( ) 2LH p

p p j p j j p j p jω ω ωω ω ω ω ω

⎡ ⎤= = = ⋅ −⎢ ⎥+ − ⋅ + − +⎣ ⎦

1( ) ( ) ( )

m

L n nn

H H j aω ω π δ ω ω=

= + −∑

where:

Simplification results:

The corresponding system function:

[ ]00 0( ) ( ) ( ) ( )

2LH H jj

ω πω ω δ ω ω δ ω ω= + − − +



N T S

3.3.6 The System function HL(p)When moving from , one can conclude from the existance ( ):( ) ( ) LH to HH p ωω

If ( ) exists for Re 0 and ( ) is represented as an analytical function of , ( ) can be written as:

( ) Re 0

L

L

L

H p p HH p

pH p H pj

ωω

=

⎛ ⎞= ≥⎜ ⎟

⎝ ⎠

i



N T S

3.3.7 Network Function as System Function

0 0

0

0

...( ) ( )...

mi

mii m

L n ni n

ii

a pa p aH p N pb p bb p

=

=

+ += = =

+ +

∑

∑

With networks consisting of LTI elements, the network function N(p) is a rational fraction function in p with constant, real-valued coefficients:



N T S

3.3.8 Pole Zero plots

00

1

1

( )( )

( )

i

i

rm i

iL

rn i

i

a p pH p

b p p

µ

ν∞

=

∞=

−=

−

∏

∏

01

Order of the numerator polynom ii

r mµ

=

= =∑

1

Order of the denominator polynom nii

rυ

∞=

= =∑m

n

aKb

=

0If the roots of the numerator polynomial and the denominator polynomial ,one gets:

p p∞

where:

poles

zeros

Real constant:

( ) ( ) ( )L L LG p H p S p= ( ) ( ) ( )g t h t s t= ∗

With poles and zeroes, the transmission properties are fully described:

(apart from K)



N T S


Example of Pole-Zero plot

0

j ωω

1j−

1j2p∞

02p

1p∞

3p∞

1− 1

01p

0

σω

p - plane



N T S


0

01

1

( )

i

i

ri

iL

ri

i

p pH p K

p p

µ

ν∞

=

∞=

−=

−

∏

∏

0 01 1

( ) arctan ( ) ( )i i i ii i

p K r p r pµ ν

ϕ ϕ ϕ∞ ∞= =

= + −∑ ∑

0 0arctan

0K

KKπ>⎧

= ⎨± <⎩

0 0( )0 0 0( ) i ij j p pi i ip p p p e p p eϕ −− = − = −

( ) iji ip p p p e ϕ∞

∞ ∞− = −

The representation of pole-zero diagram can be used to determine the magnitude and phase and shows the influence of any zero and pole clearly:

and

Thus:magnitude

phase

where:



N T S


Method to determine the influence of a Zero and a Pole from that Pole-zero plot

Imj p

Re p

0ip p−

0iϕ

p

0ip



N T S


( )( ) ( ) j pL LH p H p e ϕ= ⋅

With the representation for:

the influence of poles and zeros becomes clear

Method to determine the Influence of Poles and Zeros on the Magnitude and phase to a point on jw_axis

jω

( ) ( )point wanted L

p jH H

ωω ω

=

=

p - plane

0ij pω −

σ

0iϕ

iϕ∞

ip∞

ij pω ∞−

0ip



N T S


• Poles and zeros with the same distance to the regarded point cancel each other

• The system function is mainly influenced by the corresponding pole and zero with the closest distance to p.

•A quick estimation for the transfer function is important for technical applications:

( )( ) ( ) ( ) jLH j H A e ϕ ωω ω ω= =



N T S


Low-pass system: Number of poles > number of zeros

because

( ) 0A ωω

=→ ±∞

jω

p - plane

σ

low-pass (3rd order)



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3.3.8 Pole Zero plotsBand-pass:

( ) (0) 0

One or more zeros at the origin.No. of poles > No. of zeros

A and Aωω

=→ ±∞

i

ii

jω

σ

p - plane

band-pass (4th order)



N T S

3.3.8 Pole Zero plotslim ( )A K

ωω

→±∞=High-pass:

• At least one zero at origin

• No. poles = No. zeros

jω

σ

p - planehigh-pass (3rd order)



N T S

3.3.8 Pole Zero plotsBand-stop:

0

lim ( ) 0 lim ( ) 0A and Aω ω ω

ω ω→±∞ →

≠ =

• one zero or more at center frequency

• same number of poles and zeros

• No zero at origin



N T S


01 02

1 2

( ) ( ) ( )L

j p j pH j H A K K

j p j pω ω

ω ω ωω ω∞ ∞

− −= = = =

− −

All-pass:

• No. poles = No. zeros

• Symmetry of pole/zero locations concern to jω-axis

jω

10j pω −

10p

20j pω −

1j pω ∞−

1p∞

2j pω ∞−

2p∞ 20p

regarded frequencyp jω=

all-pass (2nd order)

σ



N T S


System with all-pass

jω

σ

Minimum-phase-system All-pass

jω

σ σ

jω

1



N T S


All-pass as Lattice type Filter for Phase correction.

Causal and stable LTI-systems:

• Poles only located in the left p-half-plane and on the imaginary axis.

• Zeros can be located in both half-planes.

σ

jω



N T S


( ) ( )( ) a jbLH e eω ωω − −=

N N N

pp j p j jσ ωσ ωω ω ω

= + → = = + = Σ + Ω

0 00 0

( )1 1

1 1

( ) ( )( )

( ) ( )

i i

i i

r rmN i i

m ni iLN N

r rnN i i

i i

P P P PH p K K

P P P P

µ µ

ν ν

ωω

ω ∞ ∞

−= =

∞ ∞= =

− −= =

− −

∏ ∏

∏ ∏

01 1

and i ii i

r m r nµ ν

∞= =

= =∑ ∑

The effect of the single poles and zeros on the transmission system can be overviewed:

Normalization of p can be carried out as:

where



N T S


ln ( ) ( ) ( )pLN NH j a jbω = − Ω − Ω

( ) ln( ( ))pN Na A ωΩ = − Ω

00 0 ,

1 1( ) ln ln ( ) ln ( )i i

p

r rm nN N i i i i

i ia K r P P r P P

µ ν

ω ∞−∞ ∞

= =

− Ω = + − − −∑ ∑

0 01 1

( ) arctan ( ) ( )m nN N i N i i N i

i i

K r rµ ν

ϕ ω ϕ ϕ−∞ ∞

= =

Ω = + Ω − Ω∑ ∑

A logarithmation gives:

with : frequency normalized damping ratio

One can obtain:

and for the frequency normalised phase function:



N T S


2U1U CR

2

1

11( ) 1 1L

U pCH pU pRCR

pC

= = =++

11p

RC∞ = −

1N RC

ω =

Example: Given is the following circuit:

One gets:with

A suitable normalization frequency is:

11 1( ) and 1

11L

N

H P Pp Pω

∞⇒ = = = −++



N T S


Pole-Zero diagram of the RC-Circuit

N

j j ωωω

=

1j pω ∞−

11−

N

σσω

=

1p∞

p - plane



N T S


1− 12− 2

1( )

( )2

1 11 1

Aj

ωω ω

= =′+ ′ +

n

ωωω

′ =

0 .707

Magnitude and Phase of the RC-Circuit

( ) ( )arctan arctanN

b ωω ωω

⎛ ⎞′ ′= = ⎜ ⎟

⎝ ⎠2π

0 11−

2π

−

n

ωω

ω′ =



N T S

3.3.8 Pole Zero plots2

1( )1GrN

dd

τ ϕ= − Ω =Ω +ΩAppropriate envelope delay:

Envelope delay of RC-Circuit

grΓ

11− 22− 0

N

ωωω

′=



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3.3.9 Minimum Phase systems

0 0 0 0Re Re and Im Imy x y xp p p p= − =

• An LTI system with zeros in the right half-plane is said to contain all-pass

• Such a system can be divided into: The all-pass which only effect on the phase

The all-pass-less or minimum phase system which exhibits the same magnitude frequency response but with a smaller phase than the original system.

Transformation of non-minimum phase system into minimum phase:

Theorem: LTI system without poles and zeros in the right half-plane are stable and of minimum phase.

For minimum-phase systems, the phase b( ) = ( )is fully prescrib

Theoremed by ( ).

:A

ω ϕ ωω

−



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3.3.9 Minimum Phase Systems

jω

σ

System with all-pass

jωjω

1σ σ

Minimum-phase-system All-pass



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3.3.10 StabilityAn LTI system ( ) ( ) is stable if:s t g t→

2( )g t M< < ∞1( )s t M< < ∞

1( ) ( ) ( ) ( ) where ( )s t g t s h t d s Mτ τ τ τ+∞

−∞

→ = − < < ∞∫

1( ) ( ) ( ) ( ) ( ) ( )g t s h t d s h t d M h t dτ τ τ τ τ τ τ τ+∞ +∞ +∞

−∞ −∞ −∞

= − ≤ − < −∫ ∫ ∫

( ) ( ) for all th t d h t dτ τ τ+∞ +∞

−∞ −∞

− = < ∞∫ ∫

From this, one can conclude:

2( )g t M< < ∞This results to if:



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3.3.10 Stability

00

0 1

0 1

( )( )

( )

i

i

mri

iii i

L nri

i ii i

p pa pH p K

b p p p

µ

ν∞

= =

∞= =

−= =

−

∑ ∏

∑ ∏

1 1 ,

( )( )

irik

L ki k i

AH p Kp p

ν ∞

= = ∞

= ⋅−∑∑

Reasoning: A system can be described by a fractional rational system function

or can be described by the following equation:

(for n > m)

or may have, in the simplest case, just single roots in the denominator polynomial:

1 ,

( )n

iL

i i

AH p Kp p= ∞

=−∑



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3.3.10 Stability

,, . 1

1 1

( ) ( )( 1)!

ii

rp t kik

i k

Ah t K e t tk

ν

ε∞

∞ −

= =

⎡ ⎤= ⋅ ⋅ ⋅⎢ ⎥−⎣ ⎦

∑ ∑

, .

1( ) ( )i

np t

ii

h t K A e tε∞

=

⎡ ⎤= ⋅ ⋅ ⋅⎢ ⎥⎣ ⎦∑

The transform into the time domain results in the general case to

or for simple poles to:

( . ) 11. In the general case, the impulse response consists of time function .ip t ke t∞ −

( ) 1i ip t re t∞ ∞ −where the term have to be respected

Re 0i ip σ∞ ∞= <

All these terms get smaller for a growing t, if:



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3.3.10 Stability

( ) 1lim 0 for Re 0i ip t rit

e t p∞ ∞ −∞→∞

= <

This means that the corresponding pole is located in the left p-half-plane:

Theorem: The poles of the system function of a stable causal network are all located in the left half-plane.

Theorem: The poles of the system function of a stable network can be located in the left half-plane and on the imaginary axis.

On the imaginary axis, they must be single. The appropriate denominator polynomial must have roots with negative real part or single roots with a real part of zero. This is the case of a special (conditional) stability!

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