Biological Signals

(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 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.

PETER PAZMANY CATHOLIC UNIVERSITY

SEMMELWEIS UNIVERSITY

(2)

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

(3)

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

(4)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 4

Outline

• About the decibel

• Description of signals in transform domain (Fourier and Laplace transformation)

• The bandwidth of signal

• Analog-to-Digital Conversion

• The noise

(5)

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 be consider the ratio of the squares of the quantities (the two resistors are the same value, ie. R₁= R₂) :

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

dB P

= ⋅ ⎛ P

⎝⎜ ⎞ 10 ²⎠⎟

1

log ^P

P

2 dB

1

1010

⎛⎝⎜ ⎞

⎠⎟ =

dB U R

U R

U U

R

= ⋅ ⎛ R

⎝⎜ ⎞

⎠⎟ = ⋅ ⎛

⎝⎜ ⎞

⎠⎟ + ⋅ ⎛

⎝⎜ ⎞

10 ² 20 10 ⎠⎟

2 2

1

2 1

1 2

log /

/ log log

dB U

= ⋅ ⎛U

⎝⎜ ⎞ 20 ²⎠⎟

1

log U

U

2 dB

1

1020

⎛⎝⎜ ⎞

⎠⎟ =

(6)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 6

About the decibel: resolution

• The resolution is a fundamental parameter in measurements (roughly it means the capability of measurement device to differentation 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).

(7)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 7

About the decibel: definition

• We can convert an absolute power or voltage measure x into dB scale:

P^[dB]=10 lg( P/ P_ref) or U^[dB]=20 lg( U/ U_ref) where x_ref is reference value.

• The used reference can be recognized by the notation:

– dBV (feszültség “egység”): the common voltage reference is U_REF = 1V effective value (Root Mean Square)

– dBFS : FS: Full Scale – dBc : c: carrier

– dBr : r: relative, the application determines the reference value

(8)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 8

About the decibel: some tricks

1:1 → 0 dB

10:1 → 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 dB

9 (“ between 8 → 18 dB and 10 → 20 dB” by linear interpolation) → 19 dB 3 ( 9=3⋅3) → 9.5 dB

6 = 2⋅3 → 6 + 9.5 = 15.5 dB

5 (“between 4 and 6”, by interpolation) → 14 dB 7 (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)

(9)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 9

Signal decomposition

• In exanimation and description of informatics systems the signals should be treated as the sum of harmonic signals (Fourier analyses).

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

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

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

(10)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 10

Categories of analog time signals

• Limited energy:

• Limited support:

• Entrant:

• Periodicity:

−

∫

∞

→ < ∞

2 /

2( ) lim 1

T

T T x t dt

T

( ) 0 if _a or _b

x t = t T≤ t T≥ ^T^a ^T^b

,...

2 , 1 , 0 , 1 , 2 ...

) (

)

(t = x t + kT k = − − x

( ) 0 if 0 x t = t <

(11)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 11

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

s_k(t)

t

Basic signal: s_k⁽t⁾ = A_k ^sin(²πkf₀t)k = ⁰^,¹^,²^,...

Amplitude Frequency

(2 ) 0,1,2,...

sin )

( )

(^t ≈

∑

^s ^t =

∑

^A ^kf₀^t ^k = x

k k k

k π

We get answers for all technical questions!!!

(12)

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 ???

(13)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 13

Advantages

Signal

Physically difficult to

interpret

Basic signal₁ Basic signal ₂

Basic signal _n

Physically easy to interpret

Linear System

Const₁ · basic signal₁ Const₂ · basic signal₂

Const _n · basic signal_n

The effect of linear system can be easily

interpreted Characteristics of linear system: const ₁, const ₂, …., const _n

(14)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 14

Choice of base signals (

^j ^kf ^t

)

A_k exp 2π ₀ ^B_k ^exp

(

^j

(

²^π^kf₀^t ⁺^ϕ_k

) ) (

^j ^kf ^t

)

A

H_k _k exp 2π ₀

(

^j ^kf ^t

)

A_k exp 2π ₀ Eigenfunction of a linear system

System

(

^j ^kf ^t

)

A_k exp 2π ₀ ^Const· ^A_k ^exp

(

^j²^π^kf₀^t

)

(15)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 15

Mathematical discussion

Lin. inv. system

x(t) h(t) y(t)

∫

( ) ( )

∫

^∞ ( ) ( )

∞

−

∞

−

=

−

= h t τ x τ dτ h τ x t τ dτ t

y( )

( ) (k ) k^{( )}

h τ s t τ τd const s t

∞

−∞

− = ⋅

∫

2 0

( ) ^j ^{kf t}

k k

s t = A e⋅ ^π

( ) k ^j² ^{kf t}⁰⁽ ⁾ ( ) k ^j² ^{kf t}⁰ ^j² ^kf⁰ k ^j² ^{kf t}⁰ ( ) ^j² ^kf⁰ k ^j² ^{kf t}⁰

h τ A e ^π ^τ dτ h τ A e ^π e ^{π τ}dτ A e ^π h τ e ^{π τ}dτ const A e ^π

∞ ∞ ∞

− − −

−∞ −∞ −∞

= = = ⋅

∫ ∫ ∫

∫

( )

∞

−

= −

= H kf h τ e ^π ^τdτ const ( ₀): ^j² ^kf⁰

t

δ(t) Dirac-delta impulse signal

h(t)=Φ(δ(t)) Impulse response

function

t

Convolution

??

!!

(16)

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

Signals in the spectral domain

Can we composite x(t) as the sum of ?s_k (t) = e ^j²^π^kf⁰^t If x(t) is periodic signal, then ( ) _k ^jk² ^{f t}⁰

k

x t ^∞ c e ^π

=−∞

=

∑

f T1

0 := ( ) ² ⁰

0

: 1

T

jk f t

ck x t e dt

T

π

=

∫

−

x(t)

t

) (t

x c^k ^FOURIER_SERIES

(17)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 17

Consequence

Lin. inv. system h(t)

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

H

xk y^k

( ) ( )

∞

∫

∞

−

= h t τ x τ dτ t

y( )

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

=

) 3 ( 0

0 0

0 )

2 ( 0

0

0 0

) ( 0

0 0

0 )

0 (

0 0

0

f H f

H f

H H

H

∫

( )

∞

−

= −

= H kf h τ e ^π ^τdτ const ( ₀): ^j² ^kf⁰

Hx y =

( ) _k

k H kf x

y = ₀

(18)

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

Problem: not all signal is periodic

∞

∫

∞

−

= x t e− dt f

X ( ) : ( ) ^j²^π^ft FOURIER TRANSFORMATION

Time domain Frequency domain dt

t

dx( ) j2πfX( f )

∫

^t ^x ^u ^du

0

)

( _j₂¹_π_f ^X⁽^f ⁾

∫∞

∞

−

dt t

x²( ) ∫^∞

∞

−

df f X²( )

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

(19)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 19

Signal’s spectrum

Example: x(t)=u(t)e^-αt→ X(ω)=1/(α+jω)

Problems:

1. The Dirac delta function has not FT 2. Contstans signal has not FT

3. FT of periodic signals

( )

₂¹ ₂

X ω

α ω

= + ^arc^X

( )

^ω ^{= −}^arctan _α^ω

{ ( ) }

F sin ωt =?

{ } ( )

F δ t =?

{ }

F const. =?

(20)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 20

Spectrum of rectangular signal

ω X(ω)

t x(t)

ε/2 - ε/2

( )

^sin

(

^{/ 2}

)

X f ωε/ 2

= ωε

( )⁼ ¹ ^if ²

0 otherwise

x t_ε t ε

⎧ ε <

⎪⎨

⎪⎩

{

( )

}

^{( )}

( ) ( )

-j

2 2

-j -j

2 2

-j 2 j 2 -j 2 j 2 j 2 -j 2

e e 1 e

-j

1 e e 2 e e 2 e e

-j -j2 j2

sin 2

2 sin 2

2

t t t

F x t x t dt dt

ε ε

ω ω ω

ε ε ε

ωε ωε ωε ωε ωε ωε

ε ω

ε ω εω εω

ωε ωε

εω ωε

∞

−∞ − −

⎡ ⎤

= = = ⎢ ⎥ =

⎣ ⎦

− − −

= = = =

= =

∫ ∫

(21)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 21

Signal’s spectrum: the Dirac delta

Solution:

– Rectangular approximation

t x(t)

ε/2 - ε/2

ω X(ω)

( )

^sin

(

^{/ 2}

)

X f ωε/ 2

= ωε

( )⁼ ¹ ^if ²

0 otherwise

x t_ε t ε

⎧ ε <

⎪⎨

⎪⎩

( ) ( )

lim0

t x t_ε δ ε

= → ^F

{ }

^δ

( )

^t ⁼ ^lim_ε_→₀ ^{F x t}

{

^ε

^{( )} }

⁼¹

{

^const.

}

^const.

( )

F = ⋅δ f

1.

2.

(22)

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)|

ω₀ ω -ω₀

( ) ( )

{

0

} ^{( )}

^j ⁰

^{( )}

^-j ⁰

⁽

0

⁾ ⁽

0

⁾

e e 1

sin 2j 2j 2j

t t

F x t t F x t x t X X

ω ω

ω = ^⎧⎨ − ^⎫⎬ = ⎡⎣ ω ω− − ω ω+ ⎤⎦

⎩ ⎭

Fourier Transformation Modulation

{

^{1 sin}

( )

⁰

}

_2j¹

⁽

⁰

^{) (}

⁰

⁾

F ⋅ ω t = ⎡⎣δ ω ω− −δ ω ω+ ⎤⎦

(23)

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.

(24)

Bandwidth of a signal: the concept (cont’)

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

• We shall say that a signal is bandlimited if its spectrum is zero outside the frequency range | f | ≥ B, where B is the absolute bandwith. The absolute bandwidth dilemma:

– Bandlimited signals are not realizable!

– Realizable signals have infinite bandwidth!

– (No signal can be time-limited and bandlimited simultaneosuly.)

(25)

Bandwidth of a signal: the concept (cont’)

• In the case of a bandpass signal (f_min ≤ f ≤ f_max), the term narrowband is used to describe the signal if its bandwidth

B= f_max− f_min,

is much smaller than the median frequency (f_max+ f_min)/2.

Otherwise, the signal is called wideband.

• There are many bandwidth definitions depending on application:

– noise equivalent bandwidth – 3 dB bandwidth

– η% energy bandwidth

(26)

The noise equivalent bandwidth

It is definied as the bandwidfth of a system with a rectangular transfer funtiuon that receives as much noise as the system under consideration

f White noise PSD

B

( )

S f

(27)

The 3 dB bandwidth

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

f B_ε

( ) ^, ( ) ²^, ( )

X f X f S f

max max ( )

X = f X f

Xmax

ε ⋅

ε =0.5

(28)

The η% energy bandwidth

Is the bandwidth that contains η % of total emitted.

f B_90%

( ) ²^, ( )

X f S f

90%

(29)

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 signals Seismic 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

(30)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 30

The convolution

( ) ( )

∫

∞

−

= h t τ x τ dτ t

y( ) Time domain

Frequency domain

∫

^∞

∫ ∫ ( )

∞

−

∞ −

∞

−

∞

−

− = − =

= y t e dt h x t d e dt f

Y( ) ( ) ^j²^π^ft (τ) τ τ ^j²^π^ft

( ) ∫ ∫ ( )

^{( )}

∫ ∫

^∞

∞

−

∞

−

∞ −

∞

−

∞

−

− = − =

−

= h(τ)x t τ e ^j²^π^ftdtdτ h(τ)x t τ e ^j²^π^f ^t ^τ e ^j²^π^f^τdtdτ

( )

⁽ ⁾

( )

⁽ ⁾ ⁽ ⁾

)

( x u e ² e ² dud h e ² d x u e ² du H f X f

h ^j ^fu ^j ^f

∫

^j ^f

∫

^j ^fu

∫ ∫

^∞

∞

−

∞

−

∞ −

∞

−

∞

−

− = =

= τ ^π ^π^τ τ τ ^π^τ τ ^π

) (

)

( f H f X f

Y =

The Fourier transform translates between convolution and multiplication of functions.

(31)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 31

Consequence

( ) ( )

∫

∞

−

= h t τ x τ dτ t

y( )

∫ ( )

∞

−

= h τ e− ^π ^τdτ f

H( ): ^j² ^f

) ( ) ( )

( f H f X f

Y =

Frequency response Impulse response function

H(f)

) ( f

X Y( f )

(32)

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

Problem: not all signal is absolutely integrable

∫

∞

<

o

dt t

x( ) not satisfied, but

∫

^∞ ⁻ ^< ^∞

0

)

(t e dt x ^α^t

If x(t) entrance and

then x(t)e⁻^α^t has Fourier Transform

(

⁻

)

⁼

_∫

^∞ ⁻ ⁻ ⁼

_∫

^∞ ⁻ ⁺

ℑ

0

) 2 (

0

2 ( )

) ( :

)

(t e x t e e dt x t e dt

x ^α^t ^α^t ^j ^π^ft ^α ^j ^π^f ^t

∫

∞

−

= x t e− dt s

X( ) ( ) ^st LAPLACE TRANSFORM

f j

s :=α + 2π „complex frequency”

∫

=

G

stds e

s j X

t

x ( )

2 ) 1

(

π

There are a lot of algebraic methods available for inverse transform

(33)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 33

Advantage of Laplace transformation

⎭⎬

⎫

⎩⎨

⎧ < ∞

= ^∞

∫

0

) ( :

) (

: x t x t dt

X ^F

⎭⎬

⎫

⎩⎨

⎧ < ∞

=

∫

^∞ ⁻

0

) ( :

) (

: x t x t e dt

X ^L ^α^t

L

F

X

X ⊂

We extend algebraic apparatus to broader function class.

The (complex) frequency lost the direct physical content

(34)

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

Consequence

( ) ( )

∞

∫

∞

−

= h t τ x τ dτ t

y( )

∫ ( )

∞

−

= h τ e− ^τdτ s

H( ): ^js

) ( ) ( )

(s H s X s

Y =

Transfer function Impulse response function

H(s)

) (s

X Y(s)

(35)

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

Summary

Representation Computation of output signal

Properties

Time domain – impulse response

function ^∞∫ ( )

∞

−

= h t τ x τ dτ t

y( ) ( ) Not intuitive,

complicate mathematical apparatus

(convolution integral) Frequency domain Y(jω)= H( )jω X(jω)

∞∫

∞

−

= h t e− dt j

H( ω): ( ) ^j²^π^ft

Intuitive, simple mathematical apparatus

Complex frequency

domain Y(s)= H( )s X(s)

∫∞

= − 0

) ( : )

(s h t e dt

H ^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

y p =

∫

K p t x t dt Fourier, Laplac, Hilbert, Poisson, etc.

In 2 dimension: walsch, wavelet, etc.

(36)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 36

Characterization of linear invariant systems

Linear Invariant system (eg.: filter)

Input signal Output

signal

∫ ( )

∞

−

= h τ e− ^π ^τdτ f

H( ): ^j² ^f ^∞

∫ ( )

∞

−

= h τ e− ^τdτ s

H( ): ^js

(37)

2011.10.05.. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 37

Signal manipulation in frequency domain

x(t)

t f

( )

X f

f

( )

H f

f

( ) ( ) ( )

Y f = H f X f

FT

t

( ) ( )

∞

∫

∞

−

= h t τ x τ dτ t

y( )

IFT Lowpass filter

(38)

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 f

Highpass filter

f

( ) ( ) ( )

Y f = H f X f

FT

t

( ) ( )

∞

∫

∞

−

= h t τ x τ dτ t

y( )

IFT

(39)

Analog-to-Digital Conversion

Signal analysis and processing is engaged with studying the different phenomena of nature and draw conclusions about how the observed quantities are changing in time. All applications have one thing in common, signals are studied as a function of time and the analysis is carried out by a computer.

However, computers can only process digital sequences, thus the analog signal must first be converted into a binary sequence.

Analog to Digital Conversion

analog signal, x(t) binary sequence, c_n

00100111101001110111

(40)

Notations

The underlying notation is summarized by the following table:

ˆ_k x

Signal Time Voltage

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

(41)

x(t) x(nT) ≡ x(n) ^{x n}^ˆ( )

Sampling Quantization

T ΔT Optimal

representation

c_n Coding

Compressing

Analog-to-Digital Conversion

• ADC has three main steps:

– sampling when sample the value of the signal x(t) at certain discrete time instants obtaining a sequence x_k;

– quantization when the values of the samples x_k are rounded to some allowed discrete levels (referred to as quantization levels) and having a finite set of these levels they can then easily be represented by binary codewords.

– coding when quantization symbols are mapped into binary codewords

(42)

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 f_s);

– the number of quantization levels (N), – and the rule of quantization.

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

(43)

Sampling

Sampling is carried out by a switch and temporary we assume that the switch is ideal (i.e. the holding period is zero).

x(t) x_s(t)

Sampling T Δt

Analog signal Real sampled signal

(44)

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?

(45)

The sampling theorem

(Shannon – Kotelnikov 1949)

If a bandlimited signal x(t) (the band is limited to B) is sampled with sampling frequency f_s ≥ 2B then x(t) can be uniquely reconstructed form its samples as follows:

where

( ) ( ) ( )

n

x t ^∞ x nT h t nT

=−∞

=

∑

−

( )

^{sin 2}

( )

2 2

h t T Bt

Bt π

= π

(46)

Phenomena of aliasing

If the sample frequency is not chosen to be high enough (i.e. frequency f_s ≥ 2B), then X_s(f) then there is an overlap in the spectrum, which implies that X (f) cannot be regained from X_s(f) .

Aliasing

(47)

Summarizing of sampling

In the case of practical sampling first we obtain x_s(t) from x(t) and then from x_s(t) the original signal x(t) can be regained by letting x_s(t) pass through a lowpass filter.

Filtering

x(t)

Reconstructed analog signal H(f)

f Lowpass filter

x(t) x_s(t)

Sampling

T ΔT

Analog signal Real sampled signal

Sampling switch