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