Delayed Trandfer Functions

7/28/2019 Delayed Trandfer Functions

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Delayed Trandfer Functions.Docx Behzad 1

Systems with delayed responses:

Assume a dynamics in either continuous or discrete time domain

sys.LTI( )snT ( )sh nT

Figure 1: Impulse response for a LTI network operates in sampled-time regime

The z-transform of the system response, in a general form would resemble, { }( ) ( )sh nT H z= .

Let us assume a LTI system (e.g. in Figure 2), that provides the same response, however,

includes a certain delay in its output, as shown below:

( )

sys.

Delayed

LTI

( )snT ( )s sh nT qT

Figure 2: System response to an impulse excitation, which includes sq T delay

Time Shifting Theorem1: If x(t) = 0 for t < 0 and x(t) has z-transform X(z) for the delayed

function it is { }( ) ( )nx t nT z X z = .

By recalling the time shifting theorem, the TF for such dynamics, denoted as ( )H z would fall in

the form sown in (1)

{ }

1( ) ( ) ( ).

delayed

s s qH z h nT qT H zz= =

(1)

1 Also called real translation theorem


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A z-domain strictly proper rational transfer functions, ( )( )H z , which is analytic

within can be expressed in terms of the sum of the partial fraction bases, as follows

1 ( ) .

Pq

iii

rH z z z p

== (2)

Where complex values i

p and ir represent poles and corresponding residues respectively, and

P is the number of the poles.

The transfer function inz-domain (1), or its equivalent form (2) can be analytically converted to

the s-domain by applying Bilinear Transformation, shown in (3)

.ss zs

+=

(3)

An inverse transformation from z-domain to s-domain is also possible by the following form of

the Bilinear Transformation

1

1

zz s

z

=

+ (4)

Where z is complex discrete frequency and 2 2 ss

FT

= = , in which sT and sF represent the

sampling interval and sampling rate (frequency) respectively.

By applying Bilinear Transformation (BL), it would be

{ } { }1 1

( ) ( ) ( )q qP P

q i i

i ii i

z r z rH s BL H z BL z H z BL BL

z p z p

= =

= = = = =

1 1

( )

q q qP Pi i

i ii i

r rs s sH s

s ss s sp p

s s

= =

+ +

= = = + + +

(5)


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Therefore; ( ) ( ), 0.q

sH s H s q

s

=

+ (6)

The time domain equivalent for (6) can be consider as follows;

{ } { }1 1 1 1( ) ( ) ( ) ( ) .q q

s

s sh t qT H s H s H s

s s

= = = + +

(7)

It is summarized as a convolution between two time signal,

1( ) ( ).

q

s s

sh nT qT h t

s

= +


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Corollary: When the transformation between s andz domains is carried out through the bilinear

transformation the poles of discrete domain model are related to the poles of the equivalent

continues domain model by the bilinear transformation too.

Proof:

From (5) { }( ) ( ) ( )s

H s BL H z Hs

+= =

(8)

qi

ii

z rBL

z p

(9)

( )1 1

( )q qP P ii

i ii ii

r srs sH sss s s p p s

ps

= =

+ = = = + + + +

( )( ) ( )

( )

1 1

1

11 1

1

iq qP P

ii

ii ii i

i

rs

pb ss s

ps s p p ss

p

= =

+ = = + + + +

Then, it is

( )

1

1 ( )

1

1

iq P

i

ii

i

rs

psH z

pss

p

=

+ = + +

, (10)

for which the poles are in the form of

( )

1.1

iii

in s domain

pp p

= + (11)


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With comparing (11) and (4), the definition for bilinear transformation, it is concluded that the

poles for transfer function in z-domain can be mapped to the s-domain by applying bilinear

transformation to obtain the poles for the equivalent s-domain model.

However, the zeros cannot be related to each other by a similarly simple mapping.

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Delayed Trandfer Functions