peter pazmany catholic university · 2011.10.05.. tÁmop – 4.1.2-08/2/a/kmr-2009-0006 . 1....

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 1

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

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITYConsortium 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.

PETER PAZMANY

CATHOLIC UNIVERSITY

SEMMELWEIS

UNIVERSITY

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 2

Peter Pazmany Catholic University

Faculty of Information Technology

ELECTRICAL MEASUREMENTS

Fundamentals of signal processing

www.itk.ppke.hu

(Elektronikai alapmérések)

A jelfeldolgozás alapjai

Dr. Oláh András

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 3

Electrical measurements: Fundamentals of signal processing

Lecture 3 review• Deprez instrument, hand instruments• Measuring alternating current or voltage• RMS (Root Mean Square)• Measurement error• Measuring very high and very low voltage• Digital voltmeter• Level measurement• Waveform measurement• Measuring time – philosophical considerations• Measuring frequency• Measuring time• The ELVIS system

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Outline• About the decibel• Description of signals in transform domain (Fourier and Laplace

transformation)• The bandwidth of signal• Analog-to-Digital Conversion• The noise

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 5


About the decibel: definition• The decibel is the ratio of two power quantities:

• When referring the measurements of field amplitude (voltage quantity) can beconsider the ratio of the squares of the quantities (the two resistors are the samevalue, ie. R1= R2) :

• The decibel can depict high range of values on expressive scale. For examplethe range between 1kV and 1μV means 109:1 ratio, which is only 180dB value.

dB PP

= ⋅ ⎛⎝⎜

⎞⎠⎟

10 2

1log P

P

dB2

1

1010⎛⎝⎜

⎞⎠⎟=

dBU RU R

UU

RR

= ⋅⎛

⎝⎜

⎞

⎠⎟ = ⋅ ⎛

⎝⎜⎞⎠⎟+ ⋅ ⎛

⎝⎜⎞⎠⎟

10 20 1022

2

12

1

2

1

1

2log

//

log log

dB UU

= ⋅ ⎛⎝⎜

⎞⎠⎟

20 2

1log

UU

dB2

1

2010⎛⎝⎜

⎞⎠⎟=

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About the decibel: resolution• The resolution is a fundamental parameter in measurements

(roughly it means the capability of measurement device todifferentation of two close values).

• It can characterizes the relative sensitivity of the measurement:– For example, 4000 digits range DVM (Digitális Voltage Meter) has 4000:1

nominal resolution, in decibel scale this resolution is 72 dB.– An other example: n bit ADC has 2n different quantization levels, 10lg(2n/1)

= 6n, ie. The increasing of the dynamic is 6 dB per bit.– Comment: the resolution is often measured in percentage (% = 10-2), and

the “excellent” resolution is expressed in ppm (parts per million = 10-6).

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About the decibel: definition• We can convert an absolute power or voltage measure x into dB scale:

P[dB]=10 lg( P/ Pref ) or U[dB]=20 lg( U/ Uref ) where xref is reference value.

• The used reference can be recognized by the notation:– dBV (feszültség “egység”): the common voltage reference is UREF = 1V

effective value (Root Mean Square)– dBFS : FS: Full Scale– dBc : c: carrier– dBr : r: relative, the application determines the reference value

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About the decibel: some tricks1:1 → 0 dB10:1 → 20 dB (obvious conversions: log(1) = 0, log(10) = 1)2:1 → 6 dB (Note: log(2) ≅ 0,3)4 = 2⋅2 → 6 + 6 = 12 dB (log(x⋅y) = log(x) + log(y))8 = 2⋅4 → 6 + 12 = 18 dB9 (“ between 8 → 18 dB and 10 → 20 dB” by linear interpolation) → 19 dB3 ( 9=3⋅3) → 9.5 dB6 = 2⋅3 → 6 + 9.5 = 15.5 dB5 (“between 4 and 6”, by interpolation) → 14 dB7 (by interpolation) → 17 dB

arány 1:1 2:1 3 4(=2⋅2) 5 6(=2⋅3) 7 8(=2⋅4) 9 10:1

dB 0 6.02 9.54 12.04 13.98 15.56 16.90 18.06 19.08 20

dB = 20log (rate)

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Signal decomposition• In exanimation and description of informatics systems the signals

should be treated as the sum of harmonic signals (Fourieranalyses).

• Question: What conditions must be satisfied to compose a signalas the sum of harmonic components?

• We give the engineering approach to define the Fourier (signalspectrum) and Laplace transformations.

• According to the signal spectrum we can define the signal (and thesystem) bandwidth: it is the difference between the upper andlower frequencies in a contiguous set of frequencies.

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Categories of analog time signals• Limited energy:

• Limited support:

• Entrant:

• Periodicity:

∫−

∞→∞<

2/

2/

2 )(1limT

TT

dttxT

( ) 0 if or a bx t t T t T= ≤ ≥ aT bT

,...2,1,0,1,2... )()( −−=+= kkTtxtx

( ) 0 if 0x t t= <

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Signal decomposition – basic idea

x(t)

t

What are the signal characteristics ?What frequencies contained in the signal?What kind of amplifier bandwidth should be used…etc. ?

From this representation

can not be answered

sk(t)

t

Basic signal: ( ) ,...2,1,0 2sin)( 0 == ktkfAts kk π

Amplitude Frequency

( ) ,...2,1,0 2sin)()( 0 ==≈ ∑∑ ktkfAtstxk

kk

k π

We get answers for all technical questions!!!

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 12


Signal decomposition

Signal Decomposition

(transformation)

Meaningful representation for the given engineering task

Technical specification

Design of signal processing

etc.

What are the basic signals ???

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Advantages

Signal

Physically difficult to interpret

Basic signal 1

Basic signal 2

Basic signal n

Physically easy to interpret

Linear System

Const 1 · basic signal1

Const 2 · basic signal2

Const n · basic signaln

The effect of linear system can be easily

interpreted

Characteristics of linear system: const 1, const 2, …., const n

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Choice of base signals

( )tkfjAk 02exp π ( )( )kk tkfjB ϕπ +02exp

( )tkfjAH kk 02exp π

( )tkfjAk 02exp π Eigenfunction of a linear system

System( )tkfjAk 02exp π ( )tkfjAk 02exp πConst·

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Mathematical discussion

Lin. inv. system h(t)x(t) y(t)

( ) ( ) ( ) ( )∫∫∞

∞−

∞

∞−

−=−= ττττττ dtxhdxthty )(

( ) ( ) ( )k kh s t d const s tτ τ τ∞

−∞

− = ⋅∫02( ) j kf tk ks t A e π= ⋅

( ) ( ) ( )0 0 0 0 0 02 ( ) 2 2 2 2 2j kf t j kf t j kf j kf t j kf j kf tk k k kh A e d h A e e d A e h e d const A eπ τ π π τ π π τ πτ τ τ τ τ τ

∞ ∞ ∞− − −

−∞ −∞ −∞

= = = ⋅∫ ∫ ∫

( )∫∞

∞−

−== ττ τπ dehkfHconst kfj 020 :)(

t

δ(t) Dirac-delta impulsesignal

h(t)=Φ(δ(t))Impulse response

function

t

Convolution

??

!!

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 16


Signals in the spectral domainCan we composite x(t) as the sum of ?tkfj

k ets 02)( π=

If x(t) is periodic signal, then 02( ) jk f tk

kx t c e π

∞

=−∞

= ∑

Tf 1:0 = ( ) 02

0

1:T

jk f tkc x t e dt

Tπ−= ∫

x(t)

t

)(tx kc FOURIER SERIES

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Consequence

Lin. inv. system h(t)

x(t) y(t) Lin. inv. system H

kx ky

( ) ( )∫∞

∞−

−= τττ dxthty )(

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

)3(0000)2(0000)(0000)0(

0

0

0

fHfH

fHH

H

( )∫∞

∞−

−== ττ τπ dehkfHconst kfj 020 :)(

Hxy =

( ) kk xkfHy 0=

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 18


Problem: not all signal is periodic∫∞

∞−

−= dtetxfX ftj π2)(:)( FOURIER TRANSFORMATION

Time domain Frequency domain

dttdx )(

)(2 ffXj π

∫t

duux0

)( )(

21 fX

fj π

∫∞

∞−

dttx )(2 ∫∞

∞−

dffX )(2

FT basic properties: Linearity, Translation, Modulation, Convolution, Scaling, Parseval's theorem

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Signal’s spectrumExample: x(t)=u(t)e-αt → X(ω)=1/(α+jω)

Problems:1. The Dirac delta function has not FT2. Contstans signal has not FT3. FT of periodic signals

( )2 2

1X ωα ω

=+ ( )arc arctanX ωω

α= −

( ) F sin =?tω

( ) F =?tδ F const. =?

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Spectrum of rectangular signal

ω

X(ω)

t

x(t)

ε/2- ε/2

( ) ( )sin / 2/ 2

X fωε

ωε=

( )1 if 2=0 otherwise

tx tε

εε

⎧ <⎪⎨⎪⎩

( ) ( )

( ) ( )

-j2 2-j -j

22-j 2 j 2 -j 2 j 2 j 2 -j 2

1 ee e-j

1 e e 2 e e 2 e e-j -j2 j2

sin 22 sin 22

tt tF x t x t dt dt

ε εω

ω ωε

εε

ωε ωε ωε ωε ωε ωε

ε ω

ε ω εω εωωε

ωεεω ωε

∞

−∞ −−

⎡ ⎤= = = =⎢ ⎥

⎣ ⎦

− − −= = = =

= =

∫ ∫

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Signal’s spectrum: the Dirac delta Solution:

– Rectangular approximation

t

x(t)

ε/2- ε/2

ω

X(ω)

( ) ( )sin / 2/ 2

X fωε

ωε=

( )1 if 2=0 otherwise

tx tε

εε

⎧ <⎪⎨⎪⎩

( ) ( )0

limt x tεεδ

→= ( ) ( )

0lim 1F t F x tεε

δ→

= =

( )const. const.F fδ= ⋅

1.

2.

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 22


Signal’s spectrum: sine wave Solution:

3. Sine wave in the frequency domain

|X(ω)|

ωω0-ω0

( ) ( ) ( ) ( ) ( ) ( )0 0j -j

0 0 0e e 1sin2j 2 j 2 j

t t

F x t t F x t x t X Xω ω

ω ω ω ω ω⎧ ⎫

⎡ ⎤= − = − − +⎨ ⎬ ⎣ ⎦⎩ ⎭

Fourier TransformationModulation

( ) ( ) ( )0 0 011 sin2j

F tω δ ω ω δ ω ω⎡ ⎤⋅ = − − +⎣ ⎦


Bandwidth of a signal: the concept• It is desirable to classify signals according to their frequency-domain

characteristics (their frequency content):– Low-frequency signal: if a signal has its spectrum concentrated about zero

frequency– High-frequency signal: if the signal spectrum concentrated at high

frequencies.– Bandpass-signal: a signal having spectrum concentrated somewhere in the

broad frequency range between low frequencies and high frequencies.


Bandwidth of a signal: the concept (cont’)

• The quantative measure of the range over which the spectrumis concentrated is called the bandwidth of signal.

• We shall say that a signal is bandlimited if its spectrum iszero outside the frequency range | f | ≥ B, where B is theabsolute bandwith. The absolute bandwidth dilemma:– Bandlimited signals are not realizable! – Realizable signals have infinite bandwidth!– (No signal can be time-limited and bandlimited simultaneosuly.)


Bandwidth of a signal: the concept (cont’)• In the case of a bandpass signal (fmin ≤ f ≤ fmax), the term

narrowband is used to describe the signal if its bandwidthB= fmax − fmin,

is much smaller than the median frequency(fmax + fmin)/2.

Otherwise, the signal is called wideband.• There are many bandwidth definitions depending on

application:– noise equivalent bandwidth– 3 dB bandwidth– η% energy bandwidth


The noise equivalent bandwidthIt is definied as the bandwidfth of a system with a rectangulartransfer funtiuon that receives as much noise as the system underconsideration

f

White noise PSD

B

( )S f


The 3 dB bandwidthIs the bandwidth at which the absolute value of the spectrum(energy spectrum or PSD) has decreased to a value that is 3 dBbelow its maximum value.

fBε

( ) ( ) ( )2, ,X f X f S f( )max max

fX X f=

maxXε ⋅

0.5ε =


The η% energy bandwidthIs the bandwidth that contains η % of total emitted.

fB90%

( ) ( )2 ,X f S f

90%


Frequency ranges of some natural signals

Biological Signals

Type of Signal Frequency Range [Hz]Electroretinogram 0 - 20

Pneumogram 0 - 40

Electrocardiogram (ECG) 0 -100

Electroenchephalogram (EEG) 0 - 100

Electromyogram 10 - 200

Sphygmomanogram 0 - 200

Speech 100 - 4000

Seismic signalsSeismic exploration signals 10 - 100

Eartquake and nuclear explosion signals 0.01-10

Electromagnetic signals

Radio bradcast 3x104 - 3x106

Common-carrier comm. 3x108 - 3x1010

Infrared 3x1011 - 3x1014

Visible light 3.7x1014 - 7.7x1014

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The convolution

( ) ( )∫∞

∞−

−= τττ dxthty )(Time domain

Frequency domain ( )∫ ∫∫∞

∞−

−∞

∞−

∞

∞−

− =−== dtedtxhdtetyfY ftjftj ππ τττ 22 )()()(

( ) ( ) ( )∫ ∫∫ ∫∞

∞−

∞

∞−

−−−∞

∞−

∞

∞−

− =−=−= ττττττ τπτππ ddteetxhddtetxh fjtfjftj 222 )()(

( ) ( ) )()()()( 2222 fXfHdueuxdehddueeuxh fujfjfjfuj ∫ ∫∫ ∫∞

∞−

∞

∞−

−−∞

∞−

∞

∞−

−− === πτπτππ ττττ

)()()( fXfHfY =

The Fourier transform translates between convolution and multiplication of functions.

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Consequence

( ) ( )∫∞

∞−


( )∫∞

∞−

−= ττ τπ dehfH fj2:)(

)()()( fXfHfY =

Frequency response Impulse response function


x(t) y(t) Lin. inv. systemH(f)

)( fX )( fY

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 32


Problem: not all signal is absolutely integrable

∫∞

∞<o

dttx )( not satisfied, but ∫∞

− ∞<0

)( dtetx tαIf x(t) entrance and

then tetx α−)( has Fourier Transform

( ) ∫∫∞

+−∞

−−− ==ℑ0

)2(

0

2 )()(:)( dtetxdteetxetx tfjftjtt παπαα

∫∞

∞−

−= dtetxsX st)()( LAPLACE TRANSFORM

fjs πα 2: += „complex frequency”

∫=G

stdsesXj

tx )(21)(π

There are a lot of algebraic methods available for inverse transform

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Advantage of Laplace transformation

⎭⎬⎫

⎩⎨⎧

∞<= ∫∞

0

)(:)(: dttxtxX F

⎭⎬⎫

⎩⎨⎧

∞<= ∫∞

−

0

)(:)(: dtetxtxX tL α

LF XX ⊂

We extend algebraic apparatus to broader function class.

The (complex) frequency lost the direct physical content

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 34


Consequence

( ) ( )∫∞

∞−


( )∫∞

∞−

−= ττ τdehsH js:)(

)()()( sXsHsY =

Transfer function Impulse response function


x(t) y(t) Lin. inv. system H(s)

)(sX )(sY

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 35


SummaryRepresentation Computation of output

signal Properties

Time domain – impulse response function

( )∫∞

∞−

−= τττ dxthty )()(

Not intuitive, complicate mathematicalapparatus (convolution integral)

Frequency domain ( ) )()( ωωω jXjHjY =

∫∞

∞−

−= dtethjH ftj πω 2)(:)(

Intuitive, simple mathematical apparatus

Complex frequency domain

( ) )()( sXsHsY =

∫∞

−=0

)(:)( dtethsH st

Not intuitive, but simple mathematical apparatus

Comment: Calculation of Fourier Transform for discrete signal is DTFT, in practice DFT (FFT) [→see Signal Processing course].

Comment: In math see integral transformation

( ) ( ) ( )2

1

,t

t

y p K p t x t dt= ∫Fourier, Laplac, Hilbert, Poisson, etc.In 2 dimension: walsch, wavelet, etc.

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Characterization of linear invariant systems

Linear Invariant system (eg.: filter)

Input signalOutput signal

( )∫∞

∞−

−= ττ τπ dehfH fj2:)( ( )∫∞

∞−

−= ττ τdehsH js:)(

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Signal manipulation in frequency domain

x(t)

t f

( )X f

f

( )H f

f

( ) ( ) ( )Y f H f X f=

FT

t

( ) ( )∫∞

∞−


IFT

Lowpass filter

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 38


Signal manipulation in frequency domain

x(t)

t f

( )X f

f

( )H fHighpass filter

f

( ) ( ) ( )Y f H f X f=

FT

t

( ) ( )∫∞

∞−


IFT


Analog-to-Digital ConversionSignal analysis and processing is engaged with studying the differentphenomena of nature and draw conclusions about how the observed quantitiesare changing in time. All applications have one thing in common, signals arestudied as a function of time and the analysis is carried out by a computer.However, computers can only process digital sequences, thus the analog signalmust first be converted into a binary sequence.

Analog to Digital Conversion

analog signal, x(t) binary sequence, cn

00100111101001110111


NotationsThe underlying notation is summarized by the following table:

ˆkx

Signal Time VoltageAnalog signal x(t) Continuous Continuous

Sampled signal x(n) or x(nT) Discrete Continuous

Quantized signal Discrete Discrete

Coded signal cn Discrete Binary


x(t) x(nT) ≡ x(n) ( )x n

Sampling Quantization

T ΔT Optimal representation

cn

Coding

Compressing

Analog-to-Digital Conversion • ADC has three main steps:

– sampling when sample the value of the signal x(t) at certain discrete timeinstants obtaining a sequence xk;

– quantization when the values of the samples xk are rounded to someallowed discrete levels (referred to as quantization levels) and having afinite set of these levels they can then easily be represented by binarycodewords.

– coding when quantization symbols are mapped into binary codewords


The challenge of ADC• Question:

– Is there any loss of information in the course of the conversion?– What is the optimal representation of signals by binary

sequences (in terms of length …etc.) ?• Fundamental challenges of sampling and of quantization:

choosing proper sampling frequency and quantization levels.ADC is fully characterized by

– the sampling frequency (denoted by fs);– the number of quantization levels (N),– and the rule of quantization.

• Optimizing ADC means that we seek the optimal values of theseparameters in order to obtain efficient binary representation ofsignals with minimum loss of information.


SamplingSampling is carried out by a switch and temporary we assumethat the switch is ideal (i.e. the holding period is zero).

x(t) xs(t)

Sampling

T Δt

Analog signal Real sampled signal


Sampling (cont’)

? x(t)

Reconstructed analog signal

x(t) x (nT)

Sampling

T ΔT

Analog signal Sampled signal

Sampling switch

Can analog signal be reconstructed from their samples without any loss?


The sampling theorem(Shannon – Kotelnikov 1949)

If a bandlimited signal x(t) (the band is limited to B) is sampledwith sampling frequency fs ≥ 2B then x(t) can be uniquelyreconstructed form its samples as follows:

where

( ) ( ) ( )n

x t x nT h t nT∞

=−∞

= −∑

( ) ( )sin 22

2Bt

h t TBtπ

π=


Phenomena of aliasingIf the sample frequency is not chosen to be high enough (i.e. frequency fs ≥ 2B),then Xs(f) then there is an overlap in the spectrum, which implies that X (f) cannotbe regained from Xs(f) .

Aliasing


Summarizing of samplingIn the case of practical sampling first we obtain xs(t) from x(t) and then fromxs(t) the original signal x(t) can be regained by letting xs(t) pass through alowpass filter.

Filtering

x(t)

Reconstructed analog signal

H(f)

f

Lowpass filter

x(t) xs(t)

Sampling

T ΔT

Analog signal Real sampled signal

Sampling switch


QuantizationWe assume that the signal is already sampled and we deal with samples x(n).Since each sample has continuous amplitude, quantization is concerned tomapping x(n) into which may have only a finite number of values.

( ) 1 2ˆ , ,..., ,Nx n Q α α α∈ =

Quantization

( )x n R∈

Sampled signal Quantified signal

( )x n

ˆ( )x n


Quantization (cont’)• Quantization always entails loss of information due to the

rounding process.• The design of a quantizer is concerned with two parameters:

– number of quantization levels;– location of quantization levels (uniform or non-uniform);

• The quality of quantization is described by a Signal-to-Quantization Noise Ratio (SQNR) where the average signalpower is compared to the noise power resulting from thequantization error:

average signal power:average noise power due to quantization

SQNR =

[ ]( : 10log )dBSQNR SQNR=


• Signal value is rounded off to predefinedthresholds called as quantization valueswhich are equidistantly placed.

• Notations:– the sample range is [-C,C]– the distance between the thresholds is ∆,– the number of quantization level is N = 2C/ ∆ = 2n,

where n represents the number of bits by which thequantized signal can be represented.

– the error signal is and -∆/2≤ ε ≤ -∆/2.ˆ: x xε = −

Uniform quantization

The quantization characterictics and the quantization error function


Uniform quantization (cont’)


Modeling the quantization noise Since the nature of errors are random the specific value of ε depends on the valueof the current sample, thus ε is regarded as a random variable subject to uniformprobability density function, and the average noise power is

( )/ 2 / 2 2

2 2 2

/ 2 / 2

1( )12

E u p u du u duεεΔ Δ

−Δ −Δ

Δ= = =

Δ∫ ∫


SQNR of the uniform quantization• In the case of full-scale sine wave (with amplitude C ):

• In the case of random input variable subject to uniform probability densityfunction over the interval [-C,C]:

• In the case of sine wave with amplitude A (in normal operation i.e. A<C)

2 22 2

2 2

/ 2 3 4 3 3: 2/12 2 2 2

nC CSQNR N= = = =Δ Δ

[ ]( : 6.02 1.78)dBSQNR n= +

( )2 22 2

2 2

2 /12 4: 2/12

nC CSQNR N= = = =Δ Δ

[ ]( : 6.02 )dBSQNR n=

[ ] ( ): 6.02 1.78 20log /dBSQNR n C A= + −


Non-uniform quantization• Uniform quantization suffer from one bottleneck: if the sample

to be quantized does not exploit the full range of quantization(i.e. [-C,C] the interval) then SNR can deteriorate severly. Asresult a user having smaller dynamic range suffers a drop inQuality of Service (QoS).

• Non-uniform quantization is way to compensate this effect:smaller dynamic range there are plenty of quantization levels(to help the users with smaller dynamics) whereas in the case oflarge dynamic signal there are less quantization levels


Non-uniform quantization (cont’)

Probability density function of samples in the case of small and large dynamics


Non-uniform quantization (cont’)

The implementation of nonlinear quantization can be reduced to applying anequidistant quantizer preceded by a proper nonlinear distortion function l(x).


The optimal non-uniform quantization• The optimal characteristics l(x) can be found by solving the following

problem:

• This optimization is a hard problem itself ( solved in the domain of functionalanalysis), but it is made more difficult by the fact that real life processes arenonstacionary (the sample p.d.f. p(x) is changing in time) and as result thisproblem must be solved again and again in order to adopt to the changingnature of the process.

( )

( )( )

2

opt ( )

2

( ) : max1

C

xC

Cl x

xC

u p u dul x

p x dxl x

−

− ′

∫

∫


The logarithmic quantization• To circumvent the difficulties of optimization, we are satisfied by choosing

an lopt(x) subject to a modified objective function which guarantees uniformSQRN:

• One can easily see that if x2 ~ 1 / l´(x)2, then indeed the SNR is constant andindependent of px(u). Thus l´(x) ~ 1 / x, from which l(x) ~ log(x), whichentails logarithmic quantization.

( )

( )( )

2

opt ( )

2

( ) : max .1

C

xC

Cl x

xC

u p u dul x const

p x dxl x

−

−

=

′

∫

∫


The logarithmic quantization (cont’)

Characteristics of logarithmic guantizer


The logarithmic quantization (cont’)

Non-Uniform Quantization( )y n

CompressionUniform

quantization Expansion

( )l x

x

( )y n ( )x n( )x n ( )1l x−

x

The real compressor l(x) is chosen differently in Europe (“A-law”) or in the US and Far East (“μ-law”).


Quantization errors: zero drift


Quantization errors: gain error


Quantization errors: integral nonlinearity


Quantization errors: differential nonlinearity


AD converters and main performancesMany various AD converters have been designed and developed.However, currently on the market there are only a few main typesof them: successive approximations register SAR, pipeline, delta-sigma, flash and integrating converters.


AD converters and main performances (cont’)• We can see that there is no one universal AD converter – the

converters of high speed are of the poor resolution and vice versa– accurate (large number of bits) converters are rather slow.

• The most commonly used are the SAR (SuccessiveApproximation Register) and Delta-Sigma converters. SARconverters are very accurate, operate with relatively highaccuracy (16-bit) and wide range of speed – up to 1 MSPS.

• The Delta-Sigma converters (16-bit and 24-bit) are used whenhigh accuracy and resolution are required. Recently, theseconverters are still in significant progress.


Successive Approximation Register (SAR)The principle of operation of the SAR device resembles the weighting on thebeam scale. Successively the standard voltages in sequence: Uref/2, Uref/4,Uref/8... Uref/2n are connected to the comparator. These voltages are comparedwith converted Ux voltage.

-

+SHUx

analoguesignal

Controlled voltage source

Controlled voltage source

registerUref

digitalsignalUcom

p


SAR (cont’)If the connected standard voltage is smaller than the converted voltage in theregister this increment is accepted and the register sends to the output 1 signal. Ifthe connected standard voltage exceeds the converted voltage the increment isnot accepted and register sends to the output 0 signal.

time

Ux

1 1 1 10

Uref/2

Uref/4Uref/8

Uref/16Uref/32

Ucomp


Performance trade-offs of ADC

In the realization of the ADC converters improving the sample rate and theresolution at the same time are conflicting requirements.


Available ADC on the market

Part Type Bits Sampling rate Manufacturer Price, $

ADC180 Integration 26 2048ms Thaler 210

ADS1256 Delta-sigma 24 300kHz Texas 9AD7714 Delta-sigma 24 1kHz AD 9AD1556 Delta-sigma 24 16kHz AD 27MAX132 Integration 18 63ms Maxim 8AD7678 SAR 18 100kHz AD 27ADS8412 SAR 16 2MHz AD 23MAX1200 Pipeline 15 1MHz Maxim 20AD9480 pipeline 8 500MHz AD 200MAX105 Flash 6 800MHz Maxim 36


Characteristics of ADC per application

Application Architecture Resolution Sampling rate

AudioSARDelta-sigma

10-16 bits14-18 bits

85-500 kHz48-50kHz

MedicalSARDelta-sigma

8-16 bits16 bits

50-500 kHz192 kHz

Automatic controlSARDelta-sigma

8-16 bits16 bits

40-500 kHz250Hz

Wireless comm.SARDelta-sigma

8 bits13 bits 270kHz

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The noise• In signal processing the noise can be considered unwanted

data without meaning, in other words the noise is an error orundesired random disturbance of a useful information signal.

• The measurement signals are usually accompanied by somenoises and interferences, sometimes of the level comparableto the level of the measured.

• The typical interference signals are generated by the electricpower lines, electrical machines, lighting equipment,commutating devices, radio communication transmitters,atmospheric discharges or cosmic noises. There are alsointernal sources of noises – resistors and semiconductordevices (thermal Noise, shot noise, etc.).

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Classification according to power spectral density• The noise is by definition derived from a random signal, we

can describe it by its statistical properties (mean, variation,correlation, etc.)

• What does spectral analysis mean for a random signal? (Weknow for deterministic signal: Fourier Transformation)

• Correlation function:

• For stationary stochastic signal R(τ) is constant, the powerspectral density is by definition its Fourier Transformation:

( )( )[ ] ( )[ ]

2

E x t x tR

τ μ μτ

σ+ − −

=

( ) ( ) j

-

S R e dωτω τ τ∞

−

∞

= ∫

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Colors of noise

Blue noise f Grey noise Violet noise f 2

White noise Pink noise, 1/f Brownian noise 1/f 2

http://upload.wikimedia.org/wikipedia/commons/0/02/White_noise_spectrum.png

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Thermal noise (Johnson – Nyquist noise)• Phenomenon: it is an electronic noise inside an electrical

conductor at equilibrium regardless of any applied voltage.• Cause: the thermal agitation of the charge carriers .• Description : white Gaussian distribution with variance per

hertz of bandwidth:

where kB is the Boltzmann’s constan, T is the resistor'sabsolute temperature in kelvins, and R is the resistor value inohms. For a given ∆f bandwidth (eg.: R=1kΩ, T=300K, ¯vn=4.07 nV/√Hz)

– (eg.:Δf=10kHz, vn= 400 nV)

2 4n Bv k T R= ⋅ ⋅ ⋅

4n n Bv v f k T R f= Δ = ⋅ ⋅ ⋅ ⋅Δ

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Thermal noise (Johnson – Nyquist noise) (cont’)

3 dB

10 dBm

-133 dBm

Power Spectral density:

( )B

2

1h fk T

R h fS fe

⋅⋅

⋅ ⋅ ⋅=

−

( ) B2S f R k T≈ ⋅ ⋅ ⋅

4n Bv k T R f= ⋅ ⋅ ⋅ ⋅Δ

(R=50Ω)

Bk Tfh

<< some THz

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Shot noise• Phenomenon: detectable statistical fluctuations in current

measurement.• Cause: the current being carried by discrete charges

(electrons) whose number per unit time fluctuates• Description: individual Poisson processes, together (law of

large numbers!) normal distribution with standard deviation:

where q is the elementary charge, Δf is a the bandwidth inHertz over which the noise is measured, and I is the averagecurrent through the device (eg.: I=100mA, Δf=1Hz,σi=0.18nA)

2i q I fσ = ⋅ ⋅ ⋅Δ

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The spectrum of thermal and shot noise

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Other noises• Flicker noise• Burst noise (or popcorn noise)• Interference noise• Quantization noise

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Summary• The decibel is logarithmic function of the ratio of two power

(or voltage) quantities.• The Fourier transform is the operation that decomposes a

signal into its constituent frequencies.• Computer based measurements can only process digital

sequences, thus the analog signal must first be converted intoa binary sequence.

• The noise is by definition derived from a random signal, wecan describe it its statistical properties

• Next lecture: Positioning systems

peter pazmany catholic university · 2011.10.05.. tÁmop – 4.1.2-08/2/a/kmr-2009-0006 . 1....

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