Echivalarea sistemelor analogice cu sisteme digitale

Prof.dr.ing. Ioan NAFORNITA

An analogic system is composed from circuits

A digital system is a computer algorithm

An analogic

(continuous time) signal

Digital signal, a numerical sequence, obtained by sampling and AD conversion

Constant signal with digital frequency 0 cycles/sample

The maximum digital frequency is 0.5 cycles/sample

There are some “problems” with sampling. The signal

having a 0 digital frequency can be obtained from many

analogic signals

The signal with 0.5 digital frequency can be also

obtained from many analogic signals

Sampling

Sampling – time-domain discretization

The Sampling Theorem

Ideal Sampling.

A sample of x(t) is obtained

multiplying the continuous-time signal x(t) with the rectangular impulse uΔ(t ):

Another sample can be obtained using the same rectangular impulse shifted by kTs.

0x t u t x u t

1

2 2u t t

s s sx t u t kT x kT u t kT

sampling process :

small values of Δ:

ideal sampling of the signal x(t) :

k k

s s sx t u t kT x kT u t kT

0

0

lim

limsT

k ks s s

u t t

u t kT t kT x kT t

ˆsT

ks sx t x t t x kT t kT

sT s s

k

x t x t t x kT t kT

Ideal Sampling

Mathematical model

System model

Spectrum of Ideal Sampled Signal

ideal sampled signal:

the spectrum of the ideal sampled signal :

sT s s

k

x t x t t x kT t kT

ˆsTX x t t F

The Fourier transform of a product is convolution of the Fourier transforms. The Fourier transform of the periodic Dirac’s distribution is also a periodic Dirac’s distribution.

The effect of the convolution of a specified function with the periodic Dirac’s distribution is the periodic repetition of the considered function.

1 2 2ˆ2sT

ks sX x t t X k

T T

F

1 2 1 2ˆk ks s s s

X X k X kT T T T

The spectrum of an ideal sampled signal is the periodic repetition of the spectrum of the original signal. The period is inverse proportional with the sampling step Ts.

s sk

x t x kT t kT

1 2ˆks s

X X kT T

Spectrum of original signal

Spectrum of periodic Dirac distribution

Spectrum of ideal sampled signal

Aliasing error 1 2ˆ

ks sX X k

T T

Sampling Band-limited Signalsx(t)-band-limited

MX if 0

• successive replica of the original signal spectrum are not superimposed and the spectrum of the original signal can be recovered by low-pass ideal filtering for

2s M

Ideal Low-pass Filter

t

tsinthpH c

c

the aliasing error can be avoided.

For perfect reconstruction :

c sM M

2s M Sampling freq.

Cutoff freq for the low pass filter

0r sH T

The frequency response of the reconstruction filter is:

Its response :

with the spectrum:

,

0, c

rc

cs

sT

H T p

c sM M

ˆr rx t x t h t

ˆ

1

r r

kcs s

s

X X H

X k T p XT

, . . .rx t x t a e w

When the condition: is not verified, the aliasing error appears.

M Ms

2 Me

Reconstruction

ˆ

sin

sin

sin

sin2

r r

c

k

c

k

c

k

cc

k s c

s s s

s s s

ss s

s

ss

s

x t h t x t

tT x kT t kT

t

tx kT T t kT

t

t kTx kT T

t kT

t kTx kT

t kT

sinc

cr s r s

tH T p h t T

t

If the finite energy signal x(t) is band limited at ωM , ( X(ω)=0 for |ω | > ωM), it is uniquely determined by its samples if the sampling frequency is higher or equal than twice the maximum frequency of the signal:

the original signal can be reconstructed from its samples a.e.w:

if the cut-off frequency of ideal low-pass reconstruction filter :

sx nT n

2s M

sin2 ccr

k s c

ss

s

t kTx t x kT

t kT

c sM M

WKS (Whittaker, Kotelnikov, Shannon) Sampling Theorem

The link between the two frequency axes of spectra of the discrete signal and of the corresponding continuous signal is:

spectrum periodic

The maximum frequencies are also related:

sT dX

; s sM MM

T T

Ideal Low-pass Filtering Reconstruction

The signal reconstructed from curves of type sin x / x.

Interpolation

,

,

sin

sin

2

1,

0,

Ms

k M

s sMk k

ss

s

ss s n k

n k

T n kx nT x kT

T n k

n kx nT x kT x kT x nT

n k

for n k

for n k

Linear interpolation• approximates the signal using straight lines

that unify points determined by the samples

• Reconstruction filter is triangular.

• errors

22sinsin

2

2

r s s

ss

s

s

T

H T TT

Reconstruction by Zero Order Extrapolation

2

2 2

2

2sin sin2 2

22

sinsin2

2

Ts

T Tj jr s

T jj sr s

s

s s

ss

s ss

s

s

s

T TT

h t p t e e TT

T

H e T eT

The frequency response of the reconstruction filter is not perfectly flat in the pass band.

The Fourier Transform of Distributions

1) The spectrum of the Dirac’s distribution

for any test function (t):

the Dirac’s distribution is even.

Hence, we have obtained:

0or 0

dtttdttt

1j t j tt e t e dt t

1t

2) The spectrum of the constant 1(t)

duality 221 t

cc 2

Approximation of continuous-time systems with discrete-

time systems

The continuous-time systems are replaced by discrete-time systems even for the processing of continuous-time signals.

1. Impulse invariance method

2. Step invariance method

3. Finite Difference Approximation (FDA)

4. Bilinear Transform

General Method for Quality Evaluation of Approximation

for Band-limited Systems

AD converter – Analog to Digital Converter

DA converter –Digital to Analog Converter

- transfer function of the analog

system to be approximated

- transfer function of equivalent

digital system.

The digital system - minimizes error .

Suppose:

a

d

a

H s

H z

e

H j

.

0, for

0, for

Signal : not defined

M

a M

a T

X

x t t t t

Identification of the impulse response of the system h(t)

-identifies a system,

.

sin -identifies a

with M

t

t

t

T

not

necessarilly band limited

band - limited

system

Identification of the impulse response of a band-limited system using the cardinal sine.

Frequency response of the band-limited system that must be identified

Spectrum of the input signal (cardinal sine)

Spectrum of the output signal

1. Impulse Invariance Method

1 ; .

1 1

1Minimum error :

a a d

a a

d d d

a d

y t h t x n nT y nT h nT

y n n h n h nT T

e nT h nT h nTT

d ah n Th nT

1

1

, band-limited / / 2 (half of sampling freq.)

Response of the c.t. system, at moments :

.

Response of the d.t. system:

Z .

Trans

a a M s

a a a

d d d d

-

x t X s T

nT

y nT X s H s nT

y nT y n X z H z

L

1 1

formation without error , 0 :

, or:

Z

d d a

d d a a

e nT

y n y nT y nT

X z H z X s H s nT

L

The digital system transfer function depends on the input signal due

to the presence of functions and .a dX s X z

-11Zd a a

d

H z X s H s nTX z L

• Change the input signal : changes the digital system.

• No perfect approximation.

• Approximation using “standard” signals:– Unit step (t)– Ramp signal t (t).

-11Zd a a

d

H z X s H s nTX z L

Approximation by sum of unit step signals, shifted and weighted

k kk

x t a t t

Approximation by sum of ramp signals, shifted and weighted

k k kk

x t a t t t t

Impulse invariance method for digital systems equivalent with band-limited c.t.

systems

1 1For sinc ; ;

The Laplace transform of the sampled signal is

ˆ

a d d a

a

a a T an

snTa a

n n

tx t x n n h n Th nT

T T T

y nt

Y s y t t h nT t nT

h nT t nT h nT e

L L

L

The Z transform of the output signal from the digital system is

1ˆ ; ;

1 2ˆFor any sampled signal we have

Replacing :

sT

n nd d a a d z e

n n

a ak

H z h n z Th nT z Y s H zT

kY j H j j

T T

j s

1 2ˆ .a ak

kY s H s j

T T

2 ;sTd az e

k

kH z H s j

T

.kT

ereereejy ; zxrezjωσs

TTjTjsTjΩ

2 ;

Relation between s and z planes for the Impulse Invariance Method

Left half plane <0 interior of unit disc

|z|=r<1

segment [-/T, /T) on imaginary axis one wrapping on the unit disc

Right half plane >0 exterior of unit disc |z|=r>1

Imaginary unit disc |z|=1

axis =0

• Avoid aliasing errors from the frequency response of the digital system obtained:

• frequency response of the analog system: completely included in freq. band -π/T , π/T.

• band-limited analog system

• Sampling freq.

TM /

2 / 2s MT

Impulse invariance method: link between frequency responses of

equivalent systemsTransfer functions

Freq. responses

2 ; ; ,

2 2.

sT

jd az e

k

jd a a

k kT

kH z H s j z e s j

T

kH e H k H

T T

The frequency response of the digital system is the same with the frequency response of the analog system of limited band for frequency less than half of sampling frequency

; and a d T MH HT T

• frequency response of the digital system ~ comb

Example: RC circuit

0

0

0

0

0

0

0

1 1 1 ; .

1

1Freq. response .

1

1Analog system: .

Digital system: .

aa a a

a

tt

a

nT

d a

dy tRC y t x t H s

sdt RC

Hjj

h t e t e t

Th n Th nT e n

non band limited system aliasing errors

The impulse response of the discrete system is

.

- small enough compared to (time constant of the circuit)

non band limited system aliasing errors (freq. response of

the digital

nT

d a

Th n Th nT e n

T

system affected at higher frequencies)

0

0

1 1

1

1 1

1 1

1

t

a a a

d d d T

T nT

T

Y s H s X s e ts s

T

Y z H z X z

e z z

Te e n

e

Unit step response

11 ; 1 .

1Error: 0

1 , 0 ; 1 1 , 0 and 1.

, 1 , 0.

1For , 0,905 , 0,

10

t T nT

a d T

a d

nT nT

a d

nT

T

Ty t e t y n e e n

ee nT y nT y nT

T Ty nT e n y nT e n

T Te nT e n

Te e nT

101 .n

e

10

0 0

0

0

0

1 0,1 (Higher T/ , error even smaller)

102

T/ 0.1

Sampling frequency 20

At 10 , the magnitude response of the analog system2

1 110 0.0318 or

1 10 10

n

s

s

s

Te nT e

T

Hj

020log 10 30

aliasing error negligeable

H dB

Frequency response

1 ;

1

1

1

a

d Tj

Hj

H

e e

2. Step Invariance Method

Input signal: unit step

Step response of the analog signal

Digital input signal:

1a ax t t X s

s

1a a a ay t h t t Y s H s

s

Z1

1

1d dx n n X zz

d as n s nT

Step responses:

and a a d ds t h t t s n h n n

d as n s nT

1 111

adH z H sn nT

sz

Z L

1 11 ad

H sH z z nT

s

Z L

Digital system:

Example: RC circuit

Step response

1

Step response of the equivalent digital system

1

with impulse response

1

t

a

nT

d

T nT

d a

s t e t

s n e n

h n e e n n h n

.

For 1: . nT

d

T

T T Th n e n n

aa a

dy tRC y t x t

dt

n

Error

, 1.

a

d a

a d

t h t

h n h nT

T Te nT h nT h n n

3. Finite difference approximation

0

1, time constant

1

1

aa a

a

dy tRC y t x t RC

dt

H ss

1d da a at nT

y n y ndy t y nT y nT T

dt T T

First derivative approximation

1 1T

y n y n x nT

11

11

1 1

11 1d a zs

T

H z H szz

T T T

Transfer function of the digital system:

0

k 0 0

0

0

11

Mk

k k kN Mk

k k a Nk kkk

kk

k kp p

t nT kk kp

b sd y t d x t

a b H sdt dt a s

d u tC u n p

dt T

General case

11

0

k 0 0 0 0

1 11

1 11 1

kk kp p

kkp

N k M kp pp p

k k k kk kp k p

zC z U z U z

T T

a C y n p b C x n pT T

1

1 1

0 0

1

0

11

0

Apply transform:

1 1

1

1

transfer functions, original c.t. system equivalent d.t. system,

finite differ

k kN M

k kk k

kM

kk

d ak zN sT

kk

z

z za Y z b X z

T T

zb

TH z H s

za

T

ence approximation.

...

1

1

1

1; d a z

sT

zH z H s s

T

Relation Between s and z planes Finite Difference Approximation

1

2 2

and .

1 1

11

; 0 11

jz x jy re s j

zs z

T sT

r z rT T

2 2

2

2 2 2

2

2 2

1 and

1 1

1notation: or

1

or 0, because generally 0

Tx y

T T

y xT x x

yx x yx

x y x x

2 22

The real and imaginary parts of describe a circle:

1 1

2 2

plane: imag. axis plane: contour of circle C= 1/2,0 , radius 1/2

z

x y

s z

Relation c.t. frequency & d.t. frequency :

= tg arctg y

T Tx

2 221 1

2 2x y

0 1/ 2,center 1/ 2,0

0 of 1/ 2,center 1/ 2,0

Imaginary axis , 0 1/ 2,center 1/ 2,0

z

z

s j z

z

Left half plane disc

Right half plane exterior disc

disc

21 1

1 1

j Tx jy

j T T

s-plane | z-plane

• Stable analog system => stable digital system

• Analog system and digital system: identical freq. response if imaginary axis on s-plane = unit circle on the z-plane.

• Not true for this method!!!

2 22

or 36

1 1and =1 very close

2 2

limited at : tg 36 36

36 2 very high value!!!!!

Sampling theorem 2 at limit 36

Very high sampling frequ

a M M M M

s M

s M s s

x y z

H ω T

ency appropriate for

modeling low frequency systems (electromechanical systems)

0

1

1time constant: .

1

1

1

a

d

RC

H ss

TTH z

zT

Example: first order LPF

Finite difference approximation

Finite difference approximation

higher accuracy !

Finite difference approximation

4. Bilinear Transform

1t

a

dyx t y t x d H s

dt s

Area An (ABCD)~ integral In: numerical methods (trapezoidal rule)

1

nT

n

n T

I x d

Input signal

• Bilinear transform:

1

1

1

2 2

1 12

nT

n

n T

I y nT y n T x d

x nT x n T TAB CD AD

Ty n y n x n x n

1

1

1

1 2 1

1

1

2 1d a zs

T z

T zH z H s

z

1

1

1

1 2 1

1

2 1;

1 d a zs

T z

zs H z H s

T z

Relation of the s and z planes for the bilinear transformation method

1

1

2 2

2 2

12 1 2 1 1

2

12 2

12 2

0 1.

0 1

0 1.

Tsz

s zTT z s

T T

zT T

z

z

z

0 1, left half plane (s plane) unit disc (z plane)

So: 0 1, imaginary axis (s plane) unit circle (z plane)

0 1, right half plane (s plane) outside the unit disc

z

z

z

Relation between frequency responses for the bilinear transform

Analog and digital frequencies connection:

2 or

2

2arctg2

tgT

T

2arctg2

12 1

12

1 and 2arctg 2

Imaginary axis ( plane) unit circle (z plane).

Analog and digital frequencies:

2 or: 2arctg .

2 2

Tj j

Tj

s j z e reT

j

Tr z

s

Ttg

T

Distorted digital freq.response due to non-linear relation between frequencies!!!