Signals and their parameters

Signal

physical quantity, conveying somehow an information about the state of a physical system or mathematical model used for this purpose

signals

deterministic random

periodic

harmonic

non-periodic ergodic

poly-harmonic(distorted)

a set of possible meaningful parameters depends on the kind of a signalit is generally wise to know what we are going to measure and what we

may come across

finite duration(pulses)

infinite duration

analog digital

semi-periodic non-ergodic

Basic signals and their parametersconstant signal (DC)

harmonic signal (sinusoidal, AC)

t

( ) tfAty −=

1

)2sin( 00 ϕπ

0

0 period

TO

amplitud

e

A0

peak-

to-pe

ak v

alue

P-P

phase ϕϕϕϕ

time t

( ) ( )tynTty

fT

O

O

O

=+

=1

frequency f

f0

A0

frequency spectrum representation

Poly-harmonic signal

f1

A1

spectrum

f2 f3 f4

A2

A3

A4( ) π φ= ⋅ ⋅ ⋅ +∑ sin(2 )n o nn

y t A n f t

time representation

0.7spectrum

0.7spectrum

0 100 200 300 400 500 600

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

frequency [kHz]

amplitu

de [

j.u.

]

0 20 40 60 80 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

frequency [kHz]time [µs]

valu

e [a

.u]

valu

e [a

.u]

spectrum

Poly-harmonic signal – cont.

-0.5

0

0.5

1

1.5time representation

valu

e [a

.u.]

0.4

0.6

0.8spectrum

ampl

itud

e [j

.u.]

0

0.5

1

1.5time representation

valu

e [a

.u.]

0.5

1

1.5time representation

valu

e [a

.u.]

1

1.5time representation

1

1.5time representation

0 200 400 600-1.5

-1

time [us]

0 20 40 60 800

0.2

0.4

frequency [kHz]

ampl

itud

e [j

.u.]

0 200 400 600-1.5

-1

-0.5

time [us]

valu

e [a

.u.]

0 200 400 600-1.5

-1

-0.5

0

time [us]

valu

e [a

.u.]

0 200 400 600-1.5

-1

-0.5

0

0.5

time [us]

valu

e [a

.u.]

0 200 400 600-1.5

-1

-0.5

0

0.5

1

time [us]

valu

e [a

.u.]

Poly-harmonic signal – cont.

-1

-0.5

0

0.5

1

1.5time representation

valu

e [a

.u.]

-1

0

1

2

3phase spectrum

phas

e [r

ad]

-0.5

0

0.5

1

1.5time representation

valu

e [a

.u.]

0

2

4phase spectrum

phas

e [r

ad]

0

0.5

1

1.5time representation

valu

e [a

.u.]

1.5

2

2.5phase spectrum

phas

e [r

ad]

0.5

1

1.5time representation

valu

e [a

.u.]

1

1.5phase spectrum

phas

e [r

ad]1

1.5time representation

4

5phase spectrum

1.5time representation

0.5phase spectrum

0 200 400 600-1.5

-1

time [us]0 20 40 60 80

-2

frequency [kHz]

0 200 400 600-1.5

-1

-0.5

time [us]0 20 40 60 80

-4

-2

frequency [kHz]

0 200 400 600-1.5

-1

-0.5

0

time [us]

valu

e [a

.u.]

0 20 40 60 800

0.5

1

frequency [kHz]

phas

e [r

ad]

0 200 400 600-1.5

-1

-0.5

0

time [us]

valu

e [a

.u.]

0 20 40 60 80-0.5

0

0.5

frequency [kHz]ph

ase

[rad

]

0 200 400 600-1.5

-1

-0.5

0

0.5

time [us]

valu

e [a

.u.]

0 20 40 60 80-1

0

1

2

3

frequency [kHz]

phas

e [r

ad]

0 200 400 600-1.5

-1

-0.5

0

0.5

1

time [us]

valu

e [a

.u.]

0 20 40 60 80-2

-1.5

-1

-0.5

0

frequency [kHz]

phas

e [r

ad]

spectrum has two components spectrum has two components –– amplitude and phaseamplitude and phase

Basic paramitersaverage – AVG (mean value)

( )∫+

=

Tt

tAV dtty

Ty

0

0

1( )∫

+

∞→=

τ

τ τ

0

0

1lim

t

tAV dttyy

periodic signalnon-periodic signal

random signal

root-mean-square - RMS

( )∫+

=

τ021

limt

dttyy( )∫+

=

Tt

dttyy0

21( )∫∞→

=τ τ

0

21lim

tRMS dttyy( )∫=

tRMS dtty

Ty

0

21

What is the meaning of a RMS value?

the value of constant voltage (or current) dissipating on a resistance R the same amount of electrical power as given AC signal

( ) ( ) ( )( )

( )∫∫∫+++

=⋅=⋅=

Tt

t

Tt

t

Tt

tAV dttu

TRdt

Rtu

tuT

dttituT

P0

0

20

0

0

0

1111

= = ⋅ = ⋅

22 ;RMS

AV RMS RMS AV

UP R I U R P

RR

wat-meter

PM

substitution measurement of the RMS value

DC-component, AC-component, RMS-AC

Electronic frequency measurement

measured signal

reference

comparator(former) gate

gating oscillator

counterdisplay

1359.078 Hz

0 200 400 600-1.5

-1

-0.5

0

0.5

1

1.5time representation

time [us]

valu

e [a

.u.]

0 200 400 600

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time [us]

valu

e [a

.u.]

Basic parameters – cont.harmonic distotrion

RMS

n

nRMS

n

nRMS

n

nRMS

y

A

A

A

h

∑

∑

∑∞

=

∞

=

∞

= == 2

2

1

2

2

2

2

1

2

2

1

2

2

RMS

n

nRMS

RMS

n

nRMS

A

A

A

A

THD

∑∑∞

=

∞

= ==

f1

A1

f2 f3 f4

A2A3

A4

( ) ∑ +⋅⋅=n

nnn tfAty )2sin( ϕπ desired signal

spurs

f1 f2 f3 f4

extention for non-harmonic signals:Total Harmonic Distortion + Noise THD+NSpurious-Free Dynamic Range SFRD

f1

A1

f2 f3

A2 A3

SF

DR

dBm

NRMS

2

1

2

22

RMS

n

RMSnRMS

A

NA

NTHD

∑∞

=

+

=+

[ ][ ] [ ]dBmdBm

dBc

31 AA

SFDR

−

=

Non-linear distortion

1

1.5

input signal(sinusoid)

distorted transfer

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

464210

26251

13130.5

330.2

THD[%]

h[%]

k

-1.5

-1

-0.5

0

0.5

1

1.5

outp

ut

input

-1.5

-1

-0.5

0

0.5 distorted output signal

transfer characteristic

input/output relation

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

45412

24241

13120.5

THD[%]

h[%]

k

0

2

4

6

8

outp

ut

input

Pulsed waveforms

„charge”

( )dttiQ ∫=

„energy”

( )dttyE ∫= 2

ymin y max

t1 t2

0.5·ymax

FWHM

peak value( )tyy maxmax =

peak-peak value( ) ( )tytyy pp minmax −=−

FWHMFull Width at Half Maximum

( ) ( ) max5.02112 ytytyFWHM ttt==

−=

Pulsed waveforms (ANSI/IEEE 194-1977)

yTOP

y90

ymax

over

shoo

t

peak

val

ue

y10

yBASE

y50

tR tF

rise time fall time

yMIN

pulse duration

under

rshoo

t

pulse stoppulse start

puls

e am

plit

ude

offs

et

peak

-to-

peak

val

ue

Digital telecommunication signalbinary synchronous signal

0 5 10 15 20 25 30 35 40 45 50

1

bit

bit number

0

war

tość

logi

czna

1 2 3 4 5

t [ns]

0.5

volt

age

[V]

data rate 10 Gb/s

frameframe

preamble

~constant bits

data

~random bits

idle bits

constant

continous stream of data

binary asynchronous signal

preamble

~constant bits

data

~random bits

data packet

Random signalsnoise

-5

0

5

-5

0

5

-5

0

5

10000

12000

12000

14000

10000

12000

tim

e pl

ot

-5 0 50

2000

4000

6000

8000

10000

-5 0 50

2000

4000

6000

8000

10000

-5 0 50

2000

4000

6000

8000

10000

his

togr

amsp

ectr

um

-60

-50

-40

-30

-20

-10

-60

-50

-40

-30

-20

-10

-60

-50

-40

-30

-20

-10

Measurement primer – voltage

ideal voltmeter

V

IV

IV = 0 ⇒ RV = ∞

ZV = ∞

real voltmeter

V

CVRV

RV

~ 10 MΩ - 10G ΩCV

~ 100 pF – 1 nF

impedance of a voltmetervoltage divider effect

0.01 1 100 10k 1M

10k

10M

10G10G || 100 pF

10M || 100 pF

10M || 1 nF

f [Hz]

|Z|

[Ω]

impedance of a voltmetervoltage divider effect

V

RZ

RVUX UV

VZ

VXV RR

RUU

+=

VZ

Z

X

X

RRR

U

U

+=

∆

ZV

X

X

RR

U

U

⋅>

⇒<∆

100

%1

Measurement primer – current

ideal ammeter

AUA

UA = 0 ⇒ RA = 0

ZV = 0

current divider effect RA

real ammeter

A

CA

RAµV

shunt resistor

current divider effect

A

RA

IX

IA

AZ

ZXA RR

RII

+=

VZ

A

X

X

RRR

I

I

+=

∆

ZA

X

X

RR

I

I

⋅<

⇒<∆

01.0

%1

RZ

RA

0-400 µA: 1 mV/µA

0-400 mA: 1 mV/mA

0-20 A: 10 mV/A

0-5 mA: 100 Ω

0-500 mA: 1 Ω

0-10 A: 0.01Ω

CA

~ ???

1.251

1.251457

Measurement primer – AC voltages/currentsacuuracy af DC measurements is much greater than accuracy of AC measurements

⇒ AC → DC conversion

amplitude A0

(peak value)

VDC 0AUV ≈

signal after full-wave rectifier

mean valueU2AV

signal after wave rectifier

mean valueU1AV

0

0

318.0 A

AUV

≈

≈≈π

0

0

637.0

2

A

AUV

≈

≈≈π

LPF

LPF

VDC

VDC

Measurement primer – AC voltages/currentsI would like to measure the RMS value ???

waveform factor

AV

RMSW y

yk

1

=

crest factor

RMSC y

yk max=

kW kC

sinusoid 1.111 1.414sinusoid 1.111 1.414

rectified 1.571 2

triangle 1.155 1.732

sawtooth 1.155 1.732

square wave 1 1

= ⋅ ≈ ⋅sin1 11.11RMS W AV AVU k U U

sinusoid

( )= ⋅sin sind dRMS RMS W WU U k k

distorted signal

RMS value measurement

( )∫+

=

Tt

tRMS dtty

Ty

0

0

21

„exact” approach –True RMS Detector

multiplier

ux(t)

uy(t) ∫dtT1

⋅

square root

URMSuy(t) ∫T

averaging

⋅

(⋅⋅⋅⋅)2 + UOUT

UIN AD8361

(⋅⋅⋅⋅)2

-

(⋅⋅⋅⋅)2 UREF

UWY

UWE VGA

AD8362

exp(-⋅⋅⋅⋅/UX)

(⋅⋅⋅⋅)2

Primer – multimeter (digital – DMM)

AC/DC

A/C

AC

DC

R

AC

DC

R

display

3.1415

interfaceRS232GPIB

attenuator

I

R/U Urefshunt

C/U

T/U

Primer – time parameters

oscilloscope (scope)vertical (Y) amplifier

timebase generator s/div

V/div

cathode ray tubeCRT

t [s]

u(t) [V] Zwe

10 MΩ15 pF

s

V

vertical (Y) amplifier

s/div

V/dz displayADCsample memory GPU

clock generator

analog oscilloscope

digital oscilloscope

Zwe

10 MΩ15 pF

Primer – spectrum measurements

inputattenuator/amplifier

Hz/div

dB/dz displayBPFRMS

sweeping generator

spectrum analyser

Zwe

50 Ω

Hz

dBm

measuring amplitude characteristics

spectrum analyser

in

out

spectrum analyser

tracking generator

circuit under test