Description digital signals and systems in time domain

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Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

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Description digital signals and systems in time domain

Digitális jelek és rendszerek időbeli leírása és analízise

András Olás, Gergely Treplán, Dávid Tisza

Digitális- neurális-, és kiloprocesszoros architektúrákon alapuló jelfeldolgozás

Digital- and Neural Based Signal Processing &

Kiloprocessor Arrays

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Introduction – review of signals

• A signal is a physical quantity in space, time and other dimensions in the physical reality (e.g. voltage, current)

• In mathematical sense a signal is a model of the physical quantity, a function of one or more independent variables (usually time or frequency) e.g.:

• A discrete time (DT) signal is a function of time where the domain of time consist of a discrete set.

It is a sequence in a mathematical sense e.g.:

( )

¹ ^sⁱⁿ

( )

^·exp

( )

^t ^,^a¹ ²^,

f t = a t − = t ∈ℝ⁺

( )

^siⁿ

( )

^·exp

( )

^k ^,^a ^2,^k ^{{0,1, 2,} ^,^30}

f k = a k − = ∈ …

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Introduction – review of signals

• A DT signal has values only where the domain has elements, it is undefined at other places.

( )

¹ ^siⁿ

( )

^·exp

( )

^k ^,^a¹ ^2,^k ^{{0,1, 2,} ^,^30}

f k = a k − = ∈ …

(8)

Introduction – review of sampling, quantization

Signal Time Value

Analog signal ^x(t) ^Continuous ^Continuous Sampled signal x(n) or x(nT) Discrete Continuous

Quantized signal ^Discrete ^Discrete Coded signal ^c_n ^Discrete ^Binary

ˆ_k x

x(t) x(nT) ≡ x(n) ^{ˆx n}^{( )}

Sampling Quantization

T ∆T Optimal

representation

c_n Coding

Compressing

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Introduction – review of sampling, quantization

• We assume that we have a sufficiently precise quantizer so we neglect the effect of quantization when we are analyzing discrete time systems and signals (we are working only with the sampled signal)

• The consequence of the sampling theorem is that the

signals are to be analyzed are reconstructable without

loss only from the sampled versions .

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Categories of discrete time signals

The most important properties of the DT signals are

• Time behavior, amplitude behavior, periodicity:

support (finite, infinite, entrant), energy, power, even-oddness

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Categories of discrete time signals

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Categories of discrete time signals

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Categories of discrete time signals

• If the energy of a signal is finite, the average power is null.

• There exist several signal of which it’s energy is infinite but the average power is finite. E.g.

( )

²

sin( )

lim 1

2 1 2

N

n N

N

x n n

x n N

=−

→∞

=

 

 

  =

+

 

 

∑

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Elementary DT signals

• Kronecker delta or unit sample

• Unit step function

• Unit ramp function

( ) ( )

^1, ^if ⁰

0, otherwise

x n

n

n =δ =  =



( ) ( )

^1, ^if ⁰

0, otherwise

x n u n

n =

=  ≥



( ) ( )

^, ^if ⁰

0, otherwise

r

x n u n n n ≥

=

= 



(15)

Elementary DT signals

• Exponential function

( )

ⁿ^, ^if ^,

( )

^, ^if ^,

( ) ( )

^· ^j ⁿ ⁿ^· ^{j n}

x n = a a∈ →ℝ x n ∈ℝ a∈ℂ → x n = r e ^ϕ = r e ^ϕ

(16)

Operations on DT signals – basic operations

• Addition:

• Multiplication with constant:

• Multiplication:

• Time shift:

• Accumulation:

• Discrete time convolution:

( )

^{( )} ^{( )}

y n = g n + h n

( )

^{· ( )}

y n = a g n

( )

^{( )·}

( )

y n = g n h n

( )

^k

{ } ( )

⁽ ⁾

y n = S g n = g n − k

( )

^{( )}

( )

^: ^{( )}^{· (} ⁾ ^{( )}^{· (} ⁾

k k

h

y n g n h n g k n k h k g n k

∞ ∞

∞

=− =−∞

− ≡

= ∗ =

∑ ∑

−

( )

0

( )

n

k

y n g k

=

∑

(17)

Operations on DT signals – basic operations

• Addition:

• Multiplication with constant

:

( ) ( ) ( ) ( ) ( )

( ) ( ) sin

y n g n h n

g n n

h n u n

= +

=

( ) ( ) ( )

( ) si

· n 2

y n a g n

g n n

a

=

(18)

Operations on DT signals – basic operations

• Multiplication:

• Time shift:

( ) ( ) ( ) ( ) ( )

·

( ) ( ) sin

r

y n g n h n

g n n

h n u n

=

( ) { } ( ) ⁽ ⁾ ( ) ( )

3

y n Sk g n g n k g n u n

k

= = −

=

= −

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Operations on DT signals – basic operations

• Accumulation:

( ) ( ) ( )

0

( ), 0.9

n

n k

y n g k g n u n

=

∑

=

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Operations on DT signals – basic operations

• Discrete Time convolution:

• Upper figure:

• Blue dots:

• Purple dots:

• Red bars:

• Lower figure:

• Blue dots:

• Red dot (sum of red bars):

value of convolution at time instant n

( )

^{( )}

( )

^: ^{( )}^{· (} ⁾

k

y n g n h n g k h n k

=

∞

−

= ∗ =

∑

−

( ) ^{sin 0.1·2}( ) ⁵ ⁵ ( ) ^0.2 ⁵ ⁵

0 otherwise

n n

h

n n

g  π − − ≤ ≤

≤ ≤ =

=  

 

( )

g k

( )

h n− k

( ) (^· )

g k h n−k

· ( ) ( )

k

h n k g k

∞

= ∞−

∑ −

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Operations on DT signals – basic operations

• Discrete Time convolution: ^{( )}

^{( )}

^: ^{( )}^{· (} ⁾

k

y n g n h n g k h n k

=

∞

−

= ∗ =

∑

−

( ) ^{sin 0.1·2}( ) ⁵ ⁵ ( ) ^0.2 ⁵ ⁵

0 otherwise

n n

h

n n

g ^ π − ^{− ≤ ≤}

≤ ≤ =

=  

 

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Operations on DT signals – basic operations

• Discrete Time convolution: ^{( )}

^{( )}

^: ^{( )}^{· (} ⁾

k

y n g n h n g k h n k

=

∞

−

= ∗ =

∑

−

( ) ^{sin 0.1·2}( ) ⁵ ⁵ ( ) ^0.2 ⁵ ⁵

0 otherwise

n n

h

n n

g ^ π − ^{− ≤ ≤}

≤ ≤ =

=  

 

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Operations on DT signals –algebraic properties of the discrete time convolution:

• Linear operator

• Commutativity

• Associativity

• Distributivity

• Associativity with scalar multiplication

• Multiplicative identity

• Complex conjugation

f ∗ = ∗g g f

( ) ( )

f ∗ ∗ =g h f ∗ ∗g h

( ) ( ) ( )

f ∗ g + h = f ∗ g + f ∗h

(

^f ^g

) ( )

^f ^g ^f

( )

^g ^, ^r^o

λ ∗ = λ ∗ = ∗ λ λ∈ℝ λ∈ℂ

, f f

δ δ

∃ ∗ =

f ∗ = ∗g f g

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Operations on DT signals –algebraic properties of the discrete time convolution:

• Integration:

• Differentiation:

• Time invariance:

( )( ) ( ) ( )

( ) ( )

( ) ^{( )} ^{( )}

·

d d d

R R R

n n n

f g x dx f x dx g x dx

f n g n f n g n

∞ ∞ ∞

∞ ∞ =

=− =− −∞

   

   

∗ =    

   

   

∗ =    

   

∫ ∫ ∫

∑ ∑ ∑

( )

( ) ( ) ( )

for discrete case, if the operator : 1

d df dg

f g g f

dx dx dx

Df n f n f n

D f g Df g f Dg

∗ = ∗ = ∗

= + −

∗ = ∗ = ∗

( ) ( ) ( )

k k k

S f ∗ g = S f ∗ = ∗g f S g

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Operations on DT signals – Signal representation by convolution (multiplicative identity):

Similarly as the definition of the Dirac delta we can use the same structure to define an arbitrary DT signal with convolution.

• Dirac delta (continuous time):

• Kronecker delta alternate definition (the property comes from definition):

• Signal representation by convolution:

every signal can be represented as a series of weighted Kronecker deltas

( )

^{( )}

( )

⁽ ⁾^{· (} ⁾ ^{( )}^· ⁽ ⁾

k k

y n

y n δ n y n δ k k y k δ n k

=− =−

∞ ∞

= ∗ =

∑

− =

∑

−

( )

⁰^{, and}

( )

has the property of

( )

1

0 0

t t t t

δ t δ ^+∞δ

−∞

+∞ =

=  =

 ≠

∫

( )

¹ ⁰^{, and}

( )

has the property of

( )

1

0 0 _n

n n n n

δ n δ ^∞ δ

=−∞

=

=  =

 ≠

∑

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Operations on DT signals – even-odd decomposition

An important property of a DT signal for it’s analysis is that it can be decomposed to even and odd parts:

( ) ( ( ) ( ) )

( ) ( ) ( )

even

odd

even odd

1 2 1 2

x n x n x n

x n x n x n

= + − 



= − −



= +

( )

^{sin 2}

(

²

)

x n = n +

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Discrete Time system

A DT system is an object which operates in a discrete time

fashion. We can describe a DT system with the input-output model:

• A DT system has an input (inputs)

• A DT system has an output (outputs)

• Single input single output – SISO

• Single input multiple output – SIMO

• Multiple input single output – MISO

• Multiple input multiple output – MIMO

• In this course we are dealing with SISO systems only

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Discrete Time system

A DT system’s input-output model is described by a mapping, a rule between the input and the output.

A DT system is a function which operates on the input.

We can describe the simplest DT systems with the basic signal operations mentioned earlier.

Discrete Time system

( )

x n ^{y n}

( )

( ) ( ( ) )

y n = Ψ x n

Input, stimulus

Output,

system response

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Operations of DT systems – basic operations

• Addition:

• Multiplication with constant:

• Multiplication:

• Time shift:

Shift register:

• Accumulation:

• Discrete time convolution:

( )

^{( )} ^{( )}

y n = g n + h n

( )

^{· ( )}

y n = a g n

( )

^{( )·}

( )

y n = g n h n

( )

^k

{ } ( )

⁽ ⁾

y n = S g n = g n − k

( )

^{( )}

( )

^: ^{( )}^{· (} ⁾

k

y n g n h n g k h n k

=

∞

−

= ∗ =

∑

−

( )

0

( )

n

k

y n g k

=

∑

( )

g n

( )

h n

( )

y n

( )

g n ^{y n}( ) ( ) a

g n

( )

h n

( )

y n

( )

g n ^{g n}( ⁻¹)

T

( )

g n ^{g n}( ⁻⁴)

T T T T

( )

g n ^{y n}( )

( ¹) T

y n−

( )

g n ^{y n}( )

( )

h n

(30)

Discrete Time system - Linearity

• A DT system is linear if:

• A DT system is non linear if that does not hold.

• Example of a linear DT system:

( ) (

^· ¹

( )

^· ²

( ) )

^·

(

¹

^{( )} )

^·

(

²

^{( )} )

y n = Ψ α x n + β x n = Ψα x n + Ψβ x n

(

( )

)

^{( )} ⁽ ⁾

( ) ( )

( )

^{( )} ^{( )}

( )

( ) ( )

( )

(

( )

) (

^{( )}

)

( )

( ) ( )

( )

1 2 1 2 1 2

2 1

1 2 1 1 2 2

2 1 2 1

2 1

LHS: · · · · 2 · 1 · 1

RHS: · · ( ) 2 1 ( ) 2 1

LHS=RHS linear system

u n u n

u n u n u n u n

u n u n u n

x n x n x n x n x n x n

x n x n x n x n x n x n

α β α β α β

α β α β

−

− −

Ψ = + −

Ψ + = + + − + −

   

   

Ψ + Ψ =  + − +  + − 

   

⇒

(31)

Discrete Time system - Linearity

• Example of a non linear DT system:

(

( )

)

^{( ) (} ⁾

( ) ( )

( ) (

^{( )} ^{( )}

)

( )

( ) ( )

( )

(

( )

) (

^{( )}

)

⁽ ⁾

( )

2

2 2

2

1 2 1 2 1 2

1

2 2

1 2 1 1 2 2

1 1

1

LHS: · · · · · 1 · 1

RHS: · · ( ) 1 ( ) 1

LHS RHS non li

u n u n

u n u n u n u n

u n u n u n

x n x n x n x n x n x n

x n x n x n x n x n x n

α β α β α β

α β α β

−

− −

Ψ = + −

Ψ + = + + − + −

   

   

Ψ + Ψ =  + − +  + − 

   

   

≠ ⇒ near system

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Discrete Time system – Time invariance

• A DT system is time invariant if the system operator is invariant to time shifts (the system does the “same”

on Monday, Tuesday, and on every holidays as well)

• A DT system is time variant if the system is dependent of the time when is it evaluated.

( ) (

^k

{ } ( ) )

^k

{ ⁽ ^{( )} ⁾ }

y n = Ψ S u n = S Ψ u n

( ) ( { } ( ) ) { ⁽ ^{( )} ⁾ } ^{( )}

1 2

k k

y n = Ψ S u n ≠ S Ψ u n = y n

(33)

Discrete Time system – Time invariance

• Example of a time invariant DT system:

( ( ) ) ^{( ) (} ⁾ { } ( )

( ) ⁽ ⁽ ⁾ ⁾ ⁽ ^{) (} ⁾

( ( ) )

{ } { ^{( ) (} ⁾ } ⁽ ^{) (} ⁾

2

2 2

1

LHS: 1

RHS: 1 1

LHS=RHS time invariant system

k

k k

u n u n u n

S u n u n k u n k u n k

S u n S u n u n u n k u n k

Ψ = + −

Ψ = Ψ − = − + − −

Ψ = + − = − + − −

⇒

(34)

Discrete Time system – Time invariance

• Example of a time variant DT system:

( ( ) ) ( ) ⁽ ⁾

{ } ( )

( ) ⁽ ⁽ ⁾ ⁾ ( ⁽ ⁾ ) ⁽ ⁽ ⁾ ⁾

( ( ) )

{ } { ^{( )} ⁽ ⁾ } ⁽ ⁾ ⁽ ⁾

2

2 2

2 1

LHS: 2 1

RHS: 2 1 2 1

LHS RHS time variant system

k

k k

u n u n u n

S u n u n k u n k u n k

S u n S u n u n u n k u n k

Ψ = + −

Ψ = Ψ − = − + − −

Ψ = + − = − + − −

≠ ⇒

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Discrete Time system - Causality

• A DT system is causal if the system’s next state can be fully determined by the combination of the input at that time instant and the input’s previous values. In other words the system is fully determined by the past and the present input.

• A typical causal system could be a real time audio processing codec or physical phenomenon

( ) (

^{x n}

( ) (

^, ^{x n} ^{1 ,}

) (

ⁿ

) )

y n = Ψ − …x − k

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Discrete Time system - Causality

• A DT system is anticausal if it only depends on the future input values

• A DT system is acausal or non-causal if the system’s response needs both future and past input values

• A typical acausal system could be an offline compression algorithm (e.g. zip), where we can seek for future samples to determine the “best” value at the present

( ) (

^{x n}

(

^k

)

^, ^, ^x

(

^{2 ,}

) (

¹

) )

y n = Ψ + … n + x n +

( ) (

^{x n}

(

^k

)

^, ^, ^{x n}

(

^{2 ,}

) (

^{x n} ^{1 ,}

) ( ) (

^{x n} ^, ^{x n} ^{1 ,}

)

^,

( ) )

y n = Ψ + … + + − … x n − m

(37)

Dynamical system – stability

• A dynamical system is said to be stabile if after a time in an abstract wide sense it stays somewhere and does not move elsewhere.

• There are different types of stability defined for a dynamical system. E.g.:

• Lyapunov stability – if all solutions of a DS start near an equilibrium point and stays close to it, the system is said to be Lyapunov stabile

• Marginal stability, asymptotic stability

• Orbital stability, structural stability,

• BIBO stability, etc.

(38)

Discrete Time system – BIBO stability

• A DT system is said to be Bounded Input – Bounded Output (BIBO) stabile if for every finite amplitude excitation it produces a finite amplitude response.

( ) ( )

( ) ( ( ) )

system . is BIBO stable iff

the system response: y ,

i

i x

i i y

M

n x M

x n

n n

Ψ ∀ ≤ < ∞

= Ψ ≤ < ∞ ∀

(39)

LTI systems

A discrete time system is an LTI (linear time invariant system) if the linearity and the time invariance property holds.

An LTI system is fully characterized by its h(n) impulse response function.

LTI system

( )

x n ^{y n}

( )

ⁿ ^LTI

(

^{x n}

( ) )

y = Ψ

Input, stimulus

Output,

system response

( ( ) ) ^{( ) ( )}

( ) ( ) ( )

LTI u n h n u

h x n

n y n = n

=

∗

Ψ ∗

(40)

LTI systems

Because an LTI system can be viewed as a convolution by it’s impulse response function (h(n)) every property which the convolution holds is true for an LTI system as well.

LTI system

( )

x n ^{y n}

( )

h n

Input, stimulus

Output,

system response

(41)

LTI systems – impulse response (constructive definition)

The impulse response of a system can be defined with the multiplicative identity property of the convolution:

The impulse response of a system is the system’s response to the Kronecker delta.

LTI system

( )

ⁿ

δ ^{h n}

( )

h n

Input, stimulus

Output,

system response

( ) ( ) ( )

^,

( ) ( ) ( ) ( )

y n = h n ∗x n x n = δ n ^⇒ y n = h n

(42)

LTI systems – consequences of the convolution properties

• Associativity– serial combination of LTI systems

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

( )

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

( )

( ) ( ) ( )

2 1

1 2

2 1

h n

g n

y n h n x n

y n h n h n x n

h n h n h n y n h n h n x n

= ∗

= ∗ ∗ 



= ∗

= ∗ ∗ 



LTI system

( )

x n

_{( )}

^{h n}

( )

^{y n}

( )

LTI system 1 g n

( )

h n1

LTI system 2

( )

h n2

(43)

LTI systems – consequences of the convolution properties

• Commutativity – switch ability of LTI systems

( ) ( ( ) ( ) ) ^{( )}

( ) ( ( ) ( ) ) ^{( )} ^{( )} ^{( ) ( )} ^{( ) ( )}

1 2

1 2 2 1

2 1

y n h n h n x n

h n h n h n h n h n y n h n h n x n

= ∗ ∗ 

= ∗ = ∗

= ∗ ∗ 

LTI system

( )

x n

_{( )}

^{h n}

( )

^{y n}

( )

LTI system 1 g n

( )

h n1

LTI system 2

( )

h n2

LTI system

( )

x n

_{( )}

^{h n}

( )

^{y n}

( )

LTI system 2 f n

( )

h n2

LTI system 1

( )

h n1

(44)

LTI systems – consequences of the convolution properties

• Commutativity – switch ability of impulse response and excitation

( ) ( ) ( )

y n h n x n y n x n h n

= ∗

LTI system 1

( )

x n ^{y n}

( )

h n

LTI system 2

( )

h n ^{y n}

( )

x n

(45)

LTI systems – consequences of the convolution properties

• Distributivity – parallel combination of LTI systems

LTI system

( )

x n ^{y n}

( )

( ) h n

g n

( )

h n

LTI system 1

( )

h n1

LTI system 2

( )

h n2

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

(

¹¹( ) ²( )

)

² ^{( )} ^{( )} ¹^{( )} ²^{( )}

y n h n x n

y n h n x n h n x n

h n h n h n y n h n h n x n

= ∗

= ∗ + ∗ 

= +

= + ∗ 

(46)

LTI systems – causality

An LTI system is causal iff it’s impulse response is an entrant DT signal.

( ) ( )

1

0

future s past and present s

· ( )

· ( ) · ( )

is an entrant signa

( ) ( )

"future" term must be 0 due to causality l

0, 1, 2,

k

k k

x x

y n h n x n h k x n k

x n k x n k

h n

h k h k

h n n

=−

−

=−∞

−

∞

=

∞

−

− =

= − + −

…

= ∗ =

⇒ ≡ = − −

∑

∑ ∑

(47)

LTI systems – BIBO stability

An LTI system is BIBO stable iff the system’s impulse response is absolute summable.

( ) ( )

( )

If a · ( ) system is BIBO stabile,

, < and < , must hold.

· ( ) · ( )

< , to hold, S must hol

( ) ( )

( )

f r o ( ) d

x y

x

y

k

k k

k

h k

y n h n x n h k y n

y n h k h k

y n h k

y

x n k

n x n M M n

x n k x n k

M

n n h k

∞

∞ ∞

∞

=−

=− =−

=−

=

<∞

−

∞

−

∀ ∀ ≤ ∞ ≤ ∞ ∀

= − ≤ −

≤

≤ ∞

=

∞

=

∗

∀ <

∑

∑ ∑

∑

(48)

LTI systems – BIBO stability

Consequently if an LTI system has an impulse response with finite number elements (limited support), the system is always BIBO stabile.

( ) ( ) { }

( ) ( )

\ if A

0 otherwi e

,

s

B h

k k A

h n n

h n

S h h n

B

n

=

∞

−∞ =

=  ∈ −∞ ∞ ≤ ≤ ⇒



=

∑

=

∑

≤ ∞

ℝ

(49)

LTI systems – FIR, IIR systems

An LTI system (or equivalently an LTI filter) is said to be of FIR type (finite impulse response) if it’s impulse response has limited support (finite length).

An system/filter is said to be of IIR type (infinite impulse response) if it’s impulse response has infinite support.

( )

{ }

( ) ( )

, 0, 0,1,1.2, 0.5, -0.4, 0.3, -0.2, 0.1, 0, 0,

0.9 0

0 otherwise

n II

F R

R

h I n h n n

… ↑ …

 − ≥

=





(50)

LTI systems – flow graph representation

• Every LTI system can be derived by the combination of the following basic operations: addition, multiplication, time shift

• Every FIR system is an IIR system

• There exist IIR systems where the impulse response function can be represented

by a closed form formula so they are implementable by a finite number of basic operations. They are called

recursive IIR systems.

• We deal with FIR and Recursive IIR. ^{FIR type}_systems

Recursive IIR systems IIR type systems

(51)

LTI systems – flow graph representation

A FIR type LTI system can be viewed as a purely feed forward type flow graph:

( ) ( )

0

· ( )

( ) ( )

M

k

y n h n x n h k x n k

=

= ∗ =

∑

−

FIR LTI

Output,

system response Input,

stimulus

(52)

LTI systems – flow graph representation

Most practically useful IIR filter can be represented by a finite element feedback and feed forward type flow graph:

( ) ( )

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

( ) ( )

0 0

0

1 0 1

0 1

· ( ), if the system can be represented by:

· ·

without the loss of generality we usually assume 1

· 1 ·

( ) ( )

· · 1 ·

· ·

N M

k k

N M

k

y n h n x n k

a y n k b x n k

a

y n a y n a y n N b x n b x n b x n M

x n

y n b x x

k

n b

h

=

∞

= =

−

− = −

=

+ − + + − = + − + +

= +

= =

−

∗

∑

∑ ∑

⋯ ⋯

(

ⁿ ^{− + +}¹

)

^⋯ ^b^M^·^{x n}

(

⁻ ^M

)

⁻ ^{a y n}¹^·

(

^{− − −}¹

)

^⋯ ^a^N^·^{y n}

(

⁻ ^N

)

(53)

LTI systems – flow graph representation

( ) ( ) ( ) ( )

( )

( ) ( )

( )

0 1 1

~ feed forward term ~ feedback term

· · 1 _M · · 1 _N·

u n g n

y n = b x n +b x n − + +⋯ b x n − M −a y n − − −⋯ a y n− N

IIR LTI

(54)

LTI systems – system equation

Implementable systems can be represented by finite element addition, multiplication and time shift operations. We are dealing with such systems.

These LTI systems can be described by a linear difference equation (system equation):

The order of this system/filter is the maximum number of time shift used either in the feed forward or the feedback path for the system:

( ) ( )

0

( ) ( ) ( )

0 0 0 1

· · , 1 · ·

N M M N

k k k k

a y n k b x n k a y n b x n k a y n k

= = = =

− = − = = − − −

∑ ∑ ∑ ∑

( )

filter order= max N M ,

(55)

Basic flow graph types – Direct form I

Direct form I implementation of a filter is the direct readout implementation of the system equation:

( )

⁰^·

( )

¹^·

(

¹

)

^M^·

( )

¹^·

(

¹

)

^N^·

( )

y n = b x n +b x n − + +⋯ b x n − M − a y n − − −⋯ a y n− N

IIR LTI

(56)

Basic flow graph types – Direct form II

Direct form II implementation of a filter is the direct readout of the modified system equation:

DF-II is a canonical representation respect to time delays.

( ) ( ) ( ) ( )

( )

⁰

( )

¹

( )

· 1 ·

· ·

N M

u n u n

u n x n N

y n b u n

a

u M

a

b n

− − −

= + +

=

−

− ⋯−

⋯

(57)

Basic flow graph types – Transposed forms

Transposed SISO filters can be constructed by exchanging the signal flow directions.

( ) { ( )

¹^{( ) ,}

}

¹ ^, ^M ¹ ^{: 0}

i i i

q n = S b u n +q ₊ n i = …M q ₊ =

IIR LTI – DF-I Transposed

(58)

Other flow graph types

There exist several other type of block diagrams which we don’t deal with within this course. E.g.

• Lattice – ladder type construction

• SOS (second order sections)

• Can be connected in serial or parallel fashion

• Special type serial and parallel constructed filters

Every structure type can implement the same LTI system, but for the actual implementation (let it be by hardware or software) all have consequences operating properties. E.g. internal numerical stability, scalability, numerical output precision, number of multiplication, adder, time shift elements used.

(59)

Canonical representations

A canonical representation respect to an element type (e.g. time shift) is the implementation which has the least number of element from that type.

Direct form II type is the canonical implementation respect to the time shift operation.

DF-I DF-II

# of time shifts=N + M # of time shifts=max(N M, )

(60)

System complexity – SISO, MIMO

SISO – single input single output SIMO – single input multiple output MISO – multiple input single output MIMO – multiple input multiple output

It is often useful or necessary to break more complex systems apart to SISO systems and join them together via known operations to get the original behavior and be analyzable.

Description digital signals and systems in time domain

Description digital signals and systems in time domain

Digital- and Neural Based Signal Processing &

Kiloprocessor Arrays

Contents

Contents

Contents

Introduction – review of signals

( )

( )

( )

( )

( )

( )

Introduction – review of signals

( )

( )

( )

Introduction – review of sampling, quantization

Introduction – review of sampling, quantization

• We assume that we have a sufficiently precise quantizer so we neglect the effect of quantization when we are analyzing discrete time systems and signals (we are working only with the sampled signal)

• The consequence of the sampling theorem is that the

signals are to be analyzed are reconstructable without

loss only from the sampled versions .

Categories of discrete time signals

Categories of discrete time signals

Categories of discrete time signals

Categories of discrete time signals

( )

( )

∑

Elementary DT signals

( ) ( )

( ) ( )

( ) ( )

Elementary DT signals

( )

( )

( ) ( )

Operations on DT signals – basic operations

( )

( )

( )

( )

( )

{ } ( )

( )

( )

∑ ∑

( )

∑

Operations on DT signals – basic operations

:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Operations on DT signals – basic operations

( ) ( ) ( ) ( ) ( )

( ) { } ( ) ( ) ( ) ( )

Operations on DT signals – basic operations

( ) ( ) ( )

∑

Operations on DT signals – basic operations

( )

( )

∑

Operations on DT signals – basic operations

• Discrete Time convolution: ( )

( )

∑

Operations on DT signals – basic operations

• Discrete Time convolution: ( )

( )

∑

Operations on DT signals –algebraic properties of the discrete time convolution:

( ) ( )

( ) ( ) ( )

(

) ( )

( )

Operations on DT signals –algebraic properties of the discrete time convolution:

( ) { } ( ) ⁽ ⁾ ( ) ( )

• Discrete Time convolution: ^{( )}

^{( )}

• Discrete Time convolution: ^{( )}

^{( )}

( ) ^{( )} ^{( )}

^{( )} )

^{( )} )