INTRODUCnON
TO
PAR
IN
LINEAR NETWORKS
HUGO VIFIA~
1400 FOUNTAIN GROVE PARKWAY
SANTA ROSA, CALIFORNIA 95404
iI
HEWLETT
PACKARD
www.HPARCHIVE.com
OBJECTIVES
"EXPLAIN"
HOW ANETWORK MODIFIES A BlGNAI.
WHAT PARAMETEM GENERATE DISTORTION
HOW TO MINIMIZE THEIR EFFECT
FIGURE OF MERIT OF A SYSTEM
FIDELITY WITH WHICH A SIGNAL OR
INFORMATION IS TRANSMITTED OR
REPRODUCED.
DEVIATIONS FROM PERFECT TRANSMISSION:
HARMONIC DISTORTION
CROSSTALK, INTERFERENCE
ERROR RATE, f¥N- DEGRADATION
1
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DISTORTION IS DUE TO
ONE OR MORE OF THE BASIC
NETWORK CHARACTERISTICS
CD
0
0
LEVEL
TIME
DEPENDING
DEPENDING
FREQUENCY
DEPENDING
"'---....y
'--------..y--------/
NON LINEAR
1
LINEAR
TIME VARIANT
TIME IttJVARIANT
EFFECTS:
NEW SIGNAL COMPONENTS
ARE GENERATED
51GNAL COMPONENTS
ARE MODIFIED?
LINEAR NETWORKS
I
INPUT
f(t)
OUTPUT
0 > - - - - -..... e--_L1_N_EA_R_ _
_
IF
fi(t)
-.
f Z (t)
-.
NETWORK
r-------0 9 (t)
LINEAR
91(t)
}
9z (t)
THEN
al f l(t)+a Zf Z (t)'" a191(t)+aZ9z(t)
SUPERPOSITION APPLIES
jwt
FOR A SINUSOIDAL INPUT FUNCTION f(t) = eTHE STEADY STATE OUTPUT
FUNCTION get) IS GIVEN By THE TRANSFER FUNCTION H(jw) MULTIPLIED
BY THE INPUT FUNCTION e-Jwt:
- 11(" )
9 (t) -
M
-jwt -_ At)
-j0(w) -jwt _ A( ) -j[wt+0(W)]
~w· e
.e
OJ e
JW . e
~'---v--J
AMPLITUDE PHASE
CHARACTERISTIC CHARACTERISTIC
Gejw)= H(jw)· F(jw)
get) = h(t) f(t)
*
2
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LINEAR NETWORKS
"-----+------,-----1.- t
TIME
'-+----+-------,----1.-
9
H<jw)
p'"
TIME
t
f<¢i<>n)
G~w)
F{jw)
A(wn)
-----_:
,,
I --------l-------JL--
Wn
A(w n)
....,~
I
w:
FREQUENCY
wn
",,(w n)
I
•
W
wn FREQUENCY
DOES A LINEAR NETWORK IMPLY NO DISTORTION?
YES FOR CW SIGNALS, HOWEVER ...
MAGNITUDE VARIATIONS WITH FREQUENCY
f(t)= sinwt +~sin 3wt +ksinsUlt
F{jW)
H{jW)
.!.
3
1}
1
I
3"
-5
1
5
FREQUENCY
FREQUE~CY
3
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FREQUENCY
PHASE VARIATIONS WITH FREQUENCY
f(t)=sinwt + ~Sin 3wt + kSin Swt
A(w)
w
C
E
:z
f---.------
'-'
~
F(j(l,)
lZl(WL)~---;F~R::;:EQ~UC=EN:-:-:C::-y:-----1~
G(juJ)
o f---.-::;---,FR,+,-,E~QT"'IU:=E:..:.Nc=.:y'-------1~
0
I
I
-180
:
I
I
I
"3 5
I
5
I
I
I
o
FREQUENCY
3
FREQUENCY
_ISO°
DI6TORTIONLESS TRANSMISSION
get) =Af(t-to )
f(t)
H ideaI
0
A+-----,~-----,
to
4
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I
t
IDEAL TRANSFER CHARACTERISTIC
A(w)
h
r----I
I
I
I
0
I
A(w) = CONST=A
- - - - - - - - - - - - . - ,- - - - - - ,
I
r
I
I
r
I
I
I
wIc
I
!
-liw
¢(w)
+liw
.-w
r-_
I
I
--_
:
-- : -.. -....
~--_
-liw
We, +Aw
---+----='""+-<::::-----i-'---=---..:-----,---+----.----- W
l
~-..
I
-0
----_ ¢o=± n7r -- -- __(t())
--- . . -J -..
--""""'.........:
---
I
--~ __
-
-..: ~ ~4J.,.__0
I
---_
I
I
- - -----N
I
---. __
I
-_
I
-I...
I
I
--J
CRITERIA FOR DISTORTJONLESS TRANSMISSION
OVER LIMITED BANDWIDTH
A(w) NETWORK
TRANSFER FUNCTIONS
1.
A
MAGNITUDE
BANDPA5S
FILTER
CONSTANT AMPLITUDE
OVER BANDWIDTH OF INTEREST
A(w) = A = CO~STANT
'--------'----'------__ w
</J(w)
-
BW --
FREQUENCY
-.
BW --
FREQUENCY
2. LINEAR PHASE OVER
BANDWIDTH OF INTEREST
'/J(w) = -wto = L1~EAR WITH w
PHASE
to =
PROPAGATION DELAY
OF AN IDEAL NETWORK
5
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NON IDEAL TRANSFER CHARACTERISTIC
A ideal
A(w)
j
.~
,"
"
"I
I
,,'
I
I t
\
'
,,
..
-AW We tAw
M(w)
-AW
, We., tAw
I'
,
,
:,,
,
I
I
II
,
=canst;
' ...
,
w
,,
,
+Aw
¢(w)
-AW
I<:-',---t-,--+,- I - ,-
+Aw
":~
I
I
We.
1.1)
~P'.:"
,\Q
................
t:.¢(w)
-110-\1
<Ds:
Al(
w
~-___
Cause: l::!.A, A(3-calculation ofeffec.ts on Signals
IDEAL AND NON-IDEAL
PART OF A NETWORK
f(t)o
H (jw)
=
=
Hi
~r
I
I
•
JI---T-
6H
0.
± A· e- jwto•
'-y--1
g(f)
,
Hi (j w) • Hn'j (jw)
t - jwt
~
!
H(iw)
I•
I
)LHid~/IHncn-idealll
= ±Aoe
=
Hejw)
fp(t) =Aof(t-to?
(1+L\A(w))e
[1 +LIA (w)
- jl::!.¢(w)
~<)l(w) + J: l> ¢(w) + -- -]
[1 +AA(w) - jA0Cw) +
0
\
••
°
higher order terms]
Y
Hni (jw)or~H(jw)
Hi (jw)
6
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J
APPROXIMAnONS FOR 4AC/I1) & A~(C&I):
CD
EXAMPLE:
® 6.A = an COS n~w for -b..wL w< + 6.w
FOURIER EXPANSION:
n~
~~(w) = ~
b..A(w) =
,
an'COS
bn·sin
6.A
n~w
n~w
~ (t) = ~-1Fo(jW{1 + aln e jn't'lil+a;
\~
t- 2'11
of ripple number
~l/.)
periods
overAw
?(llJ
:.Shift Theorem:~n~I-.±n~1
w
Jt
fo(t+nl.')
g (t)= fo(t)+a2n fo(t-n~)+an
2
Result: Echoes for discrete signals
:
1-
l
an
o
f--~---+-I--~--+-----lT
iii.t~~
iI\'
to
l
._,
\
~
~¢ = -b n sin n?!w:.
@
v
l
•
~
'-~
2
Discrete Signals
CALCULATE EFFECT OF 6.A, b..¢ ON SIGNA.L f(t) .... g(t)
SIGNAL DISPERSION DUE TO ANON IDEAL
AMPLITUDE AND PHASE CHARACTERISTIC
f(t)
Hideal
i
r--
111 ~II~~}~
~(i:¢i":" -" i
~
Ht-to)
-Ito 1-
H
_
°
t::.rl.
+.E.'P
-=1'=
+&
get)
t V I \lllllI=r~
9b _t!lIIii
b
~:su't 2:
nr nT
+
~igher order terms
7
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0
EFFECTS OF ECHOS
ON DISCREET SIGNALS
CROSSTALK
ERROR, CROSSTALK
TI-1RESHOLD - ~ - - ---.!. ~
1
1
I
I
I
TDM - CHANNELS I
I
I
GUARD
~~~
_1_0~1~\_1.-1.I II I~' \ I
I ~I
,\ I
I
I
BANDS
I
--I
1
1..1/ "
--l""'-------O>+---Lt-----f'"'""L~-~~---_t__'''''4_---->O'__._
2
1 1\,3/"
I 1
,_/
+
4
@: 1 dB ripple generates echoes only ~ ZO dB down
RISE TIME DEGRA.Di\TION
-0---'
I
/1
1
1
1
I
1
I
1
1
I
....-!-:"---+---"'--I------,1--....
+
~tr~
:
~ --2.n?;'~
;
HOW CAN DISTORTION BE MINIMIZED?
1. MEASURE I:1A,
/:).0, (r*)
2. EQUALIZE/MINIMIZE I:1H
EXAMPLE: GAIN (AMPLITUDE) AND PHASE EQUALIZING
A(w)
¢lW)
... w
INCOt-iVEt-iIENT
10 ADJUST
TO ASLOPED
REFERENCE
... '"
'----------
DELAY
EQUALIZER
/--- -
--...,
* MISMl\lQ.1 AND MULll PATH TRANSMISSION
/
MUCH EA51ER
TO ADJUST
TO A CONSTt\NT
(GROUP DELA'()
"
'--------~l.U
8
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REFERENCE.
TWO ASPECTS OF GROUP DELAY
®
CD SIGNAL PROPAGATION
t9
=-
SIGNAL DISTORTION
d0(w) = t + d[a0(w)]
dw
~ , dw ,
constant' deviation' .....AQI
d0
dw
ideal
THE GROUP DELAY -
~~
non ideal
TIlE DERIVATIVE OF THE
DETERMINES THE PROPAGATION
PHASE CHARACTERISTIC CAN
DELAY OF SIGNAL ENERGY OR
BE USED TO SPECIFY AND
MEASURE THE DEVIATiON
INFORMATION (ENVELOPE DELAY).
FROM THE IDEAL (LINEAR)
THE PHASE DELAY -
~
PHASE CHARACTERISTIC.
IT IS A PHASE LINEARITY
DETERMINES THE STEADY STATE
PHASE RELATIONSHIP
MEASUREMENT. ONLY THE
B~TWEEN
INPUT AND OUTPUT (CARRIER
DEVIATiON FROM THE CONSTANT
DELAY) FOR AN IDEAL NETWORK
(GROUP DELAY) IS IMPORTANT
t9 = tp.
IN THIS CASE.
PROPAGATION DELAY AND ELECTRICAL LENGTH
FOR AN IDEAL PHASE CNARACTEIlISTIC
f(t,n, g(t, [)
t =ELECTRlCAl LEII6TH
.--+~---J'----'o.--....-
t
~>~-t
to=i= -~
in vacuum
r
0
C
w
dw
'-y---l
'--y-/
d0
t o = -=--=-'--y-/
PROPAGATION
DELAY
PHASE
DELAY
GROUP
DELAY
9
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PHAgE DELAY AND GROUP DELAY
ENVELOPE
get) = e(t-tg )
CA.RRIER
COS
[We (t-tp)]
g(t)
¢(w)
CARRIER: tp(Ul) =_ 0~(JJ)
r-------,- w
t
CARRIER
1WIIII1fI-- t
,
-1
ENVELOPE: t (w)
g
= _ d dw
(2\ (w)
PROPAGATION DELAY FOR A
NON IDEAL PHASE CHARACTERISTIC
_(1M) ~
1M
eI,C""
CIiU
f(t)
~---.-='+---_W
tc. :; tg (0) LP
=t (We) BP
W--
10
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DISTORTION EFFECTS ON MODULATED SlGNALS~
CD TIME
DOMAIt-J MODULATION:
(PULSE MODULATION)
INFORMATIOt-J It-J SIGt-JAL
AMPLITUDE AND TIME:
.
} {DISCRETE SIGNALS:
f(t) = 1· -M <t<M
... FOURIERAPPROACHFORAH
{ 0:
t> IMI
ECHO DISTORTION
~t
-Lh.
® FREQUEt-JCY DOMAIN
MODULATION:
(CARRIER MODULATiON)
INFORMATION IN CARRIER AMPLITUDE/ANGLE
(ENVELOPE)
w
-"t
ENVELOPE
f(t) =
CARRIER
[1+o«t)].e-
~t
AM} CONTINUOUS
SIGt-IALS:
jwct
POWER SERIES
APPROACH
"'t
. r.
f(t) = e - JLLlJct~t)
CARRIER
]
Ir:::.
PM/.M
ENVELOPE
DISTORTION
/
z:.... ID1DDrzl:1[]'jLt
11
ENVELOPE
~
SJGNAL DISTORTION DUE TO LINEAR BUT
FREQUENCY DEPENDING TRANSFER CHARACTERISTIC
t·_·,.
- '.,
AM DI5TORTION
T
....
-,
? €.
"
....
_L..--..L----'----_ _
°9
LPF
"'--~ -----------------
.... ....
---..t.. .
USB ATTENUATED:
.....
W
,.:').
~!1.
-- -- -1-------------1..1
/-'
,.,.
I
'\.~~~l
.
"
\
EFFECT:
, ,
:~
CHANGE OF INDEX MODULATION (m)
ODD HARMONIC DISTORTION OF
ENVELOPE FOR SMALL INDEX m.
.;:,':
"
AM TO PM CONVERSION
f
11
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APPROXIMATIONS FOR AA(cu) & 11~(JAJ)
® POWER
EXAMPLE:
SERIES EXPANSION
=a 1 w+a 2 w2.
M(w) = a1 W+ a2. <.02.+ a~ <.0 3 + .
~A
/::"(/)(w) = b o + b2. <.02. + b 3 <.0 3 + .
g(t) = 3'-1 Fo(jw)· [1+a l W+ a2.w2.J
:. DIFFERENTIATION THEOREM
a 1 ·w -a'f(t)
a2. £.02. -
a1 f (t)
get) = fO(t)+alfo(t)+a2.fJt)
RESULT: TABLE FOR BASIC SIGNALS:
AM-f(t) = Re
[1+'" (t)] e -jwctJ
'fM-f (t) = Re [e -j [Wet t f(t>] ]
T
MODULATED SIGNALS
-6w
We
I 8(t) = 3'-Fo (jw)' [lMA-
tAW
jMT:JJ
CA1.CULA1E EFFECT OF l:>.A, 6.~ ON
get)
DISTORTION ON AM-SIGNALS
.r
[Q(t)]]
f(t)=R [l+d(t)]e-jWct o-@-og(t)= Re
e
~to«t)+P(t~2.+Q(tl.e -JLWcttarctg 1 to«t) + P(t)
t:.A(w) = a1wtd1.w2ta3w3
l:>A(w) = b 2 w 2 +b 3 w3
/ . "1(t)
<1>1
/
/
/
,,
r1
TABLE 1
0(
PlJREAM
'f= CO~T.
81
82
P",
Q",
a1~
-a 20(
a3
b2
f(t)
1
(t)
b3
al
b2
83«
b3 «
-b 2 0(
8 1b2.«
EXAMPLE:
o«t)= m cos W m t
0<
= -m,wm sin Wmt gUadrature Phase
0< = -m wm2. cos wm t In Phase
Quadrature Phase
0(
= mw 3m sin wm t
12
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AM·SIGNAL9~
DISTORTION ON
CD EXAMPLE:
USB ATTENUATED BY
E
J
DISTORTION ON AM-SIGNALS:
Dl~O
P(t)
r
ENVELOPE
p (t) -- 0
• t
i
"<
b,L~--~"--------T--~
r(t)] r
~
Q(t)
Q(t)=at
0
Q(t)
(t) ~ i+offiV
..........
~~~=.y.-----""""-J"'--'_.
13
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t
""'---/
t
EXAMPLE FOR AM TO PM CONVERSION
COLOR TV SIGNAL:
BRIGHT
---tUNE
~/
, 'f-
~OC-
DARK
..
COLOR
SATURATION
PROBLEM: COLOR CHANGE AS A FUNCTION OF BRIGHTNESS
DISTORTION ON ANGULAR MODULATION
e[e-j E-uct+r(t>J] ~ get) = ReV[l+PCt~\Q(tre -jrct+rCt>tarct9 ~~J]
f(t) =R
D.A(w)
= a 1 w ta zw Z+a 3 w.3
!::o.t/J(W)= b 2 w Z + b 3 w 3
'f (t)
PURE 'fM
o<=COIlST.= 1
~---I
a1
az
I
I
\
I
\.... --
83
I
bz
I
CARRIER '.
ba
\
f(t)
a 1 b Z.
P
Q
r
a1r
a2'f'Z
.! ...
8 3(T -T)
bzt .,.
3b3 Tr
3a1bz
~"
I
r
~,\
\
\
\ i
-azi
- 3a3
·z
bz'r
tr
"
~--r-
rr
'3 '"
~(i
-'3'r>
81 bz(r-T
;l;Q'f:
','; I---i.
" ' , P(t)
I
/ ~
\. i
I
I
,~fCt)
CARRIER\ \
,\ ~f>.Y'
"I
.\ ,
': 1
)
9(t)
EXAMPLE:
ret): mcoswmt
r
r
i
=-mw m Sinwmt 'f2=m1.~mlCOS2wmt+l]
= -m wh, COS wm t
= m wtr, sin tum t
14
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r3=m:lIlf[sin~wmt-35inwmt]
DISTORTION 011_ MODULATION
CD EXAMPLE:
PM - SIGNAL
b.C/J = b 2 w 2
f~liIIaI
!::J.A=o
A(/J
~W
-AW
We.
iittl~ ----
A,W
Fr
i
P(t)
bz
~
r- elminate by limiting""'=
HARMON Ie DISTORTION:
@:W m =2'1f'10K"z" b2.= O.5deg .. ~2! 1 %
+-------"
,
m»z
:,
,
,
,,
\
15
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DISTORT. . . . . MODULATION
A H <jU))
MODULATOR
~~OD.
f: <t)
~@--O
.
Pv
ORIGINAL
DISTORTED
FM-ENVELOPE
FM-ENVELoPE
fe~(t)
e
, : ;ru
~
ru ll::, ru ~
t"
We
Feejw)
DEMODULATOR
:\
. \
-AU)
~en
ge<t;-r;eW
tg ~
. '----------1:----...'"
. to
l4r
t
GeejUl)
--i--'--
~w ~~
ge'it)
Wuru
EXAMPLE: HARMONIC DISTORnON
=3oonsl
@:/::,t
9
W m ~ 2 7r. 10 kHz
d
Z
== 1% - . SECOND HARMONIC DIGroRTlQN
WJ.lAT HAPPENS WWEN
MORE THAN ONE SIGNAL IS PRESENT:
f(t) = e -j [Wet + T1 (t) + i 2 (t)J-+ g (t) =
TABLE
a3
P
r
1
I
I a3[t~-fJ I
-
r
t:. ('f2.):::: Q 1('('2) = DIFFERENTIAL PHASE
/t:.T/(T2.)::::P1(Iz) = DIFFERENTiAl GAIN
16
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V(1 +p)2+ Q2. e- .... arct91~p
Qr
-3a 3 rr
SUMMARY AND CONCLUSIONS
DISTORTION EFFECTS ON MODULATED SIGNALS:
• CHANGE OF INDEX MODULATION
-- CHANGE IN BRIGHTNESS
(TV) OR LOUDNESS VS. FREQUEt.JCY
• NON-LINEAR ENVELOPE DISTORTION
-- HARMONIC DISTORTION
• MODULATION CONVERSION (AM TO PM)
-- COLOR CHAt.JGE VS. It.JTENSITY, CROSSTALK) ETC.
• INTERMODULATION: DIFFERENTIAL GAIN At.JD PHASE
-- MORE THAN ONE SIGNAL COMPONENT PRESENT
RESPONSIBLE PARAMETERS:
• L\A(W), L\lZ>(W) OR ~tg(W)
MEASUREMENT PRINCIPLES FOR ~
4A,4~ & ~t9
® SEPARATION BY SUBSTITUTION
@
SEPARATION BY DIFFERENTIATION
"~(w) =I~O + ~~o~)}~~~
t:.A(w)
= A(w)-~o
dA
dw
SUBSTITUTED
CONSTANT
+
d [bACw)]
~
AMPLITUDE
LINEARITY
SUBSTITUTED
LINEAR TERM
t:.(/J(W)
=0
-d0 = d [L\0(W)] + t
0
dw
dw
+
= ~(W)- £..Oto
~
PHASE LINEARITY
SUBSTITUTED
CONSTANT
~tg(W)
=
~
-d[Ml>(w)] = t (W)-t =b.t
dw
9
0
9
tg(W)- to
17
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@ 414. A(/J MEASUREMENT BY SUBSTITUTION
~A(w)
CD
= A(w) - A o
t.A(w)
A(w)
liA =DEVIATlON FI«lM CONgfAHT AMPLITUDE
EXPANDED
f----'---+--f----+-+---'I..--+w
LL-------------'+w
~(/)(W)
¢(w)
= ¢(W)-wto
Ml(w)
®
1<::-----------1.0
.-----
IE-,---.....-----I------\--+---+--W
MEMUIIEMBIf EMMPLE • DEVIATION
FROM LINEAR PHASE
MAGNITUDE AND PHASE CHARACTERISTIC OF BAND PASS FILTER
A = 5dB/div (top)
(/) = 9O o/div (bottom)
/).A
= 1 dB/div (top)
=
10o /div (bottom)
J., = 4.2 m
M/J
J.=O
CW = 1.1 GHz
CW = 1.1 GHz
±l\F = 50 MHz.
±l\F = 50 MHz
18
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(j) 'A, A0 MEASUREMENT BY DJFFERENTIATJON
dA(w)
dW
A(w)
CD
L..-L------------'-+w
_ d0(w)
Q!(w)
dw
®
""=:----------w
•
(,of
f--------------'+w
o
DIFFERENTIATION PROCESS: Analog, Numerical,Modulation
AA, 4(/J MEA9UREMENT BY DJFFERENTIATION
dA(w)
A(w)
dw
AMPliTUDE LINEARITY
dA = 0+ d[AA(w)]
dw
dw
-.---+l-----L....L.---.L...L
u.>
d0(w)
-de;)
PHMoE LINEARITY OR GROUP DELAY
1---+---1------W
df)
_ dOl _ t lw) _ t· + d [A o (w)]
dw- g~ - 0
'---------'----w
19
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GROUP DELAY AS
DISTORTION PARAMETER
CRITERIA FOR DISTORTION LESS TRANSMISSION:
H
ideal =
<A(W) = A = constant
<;Zl(w)
=wto = linear with w
1
NECESSARY CONDITION:
(SUFFICIENT IF ~w «1 ~ NARROW BAND)
d~(w)
dw
b.H
_ <:A(W) non ideal-
= to = canst .
Ao
=AA(w)
(Zl(w) - wt = 6.C/J (w)
~
0
- At ( )_ -d[~0(W)]
t 9 ()
W - t0 - u 9 W dw
GROUP DELAY MEASUREMENT
PRINCIPLES:
t g (w)= - d0(w) BY DEFINITiON
dw
¢(W)
FREQUENCY
1.------------- W
,
tg(W)
DEVIATION FROM
CONSTA.NT GROUP DELAY
GROUP
DELAY
I
A0
•
L--
- - ' -_ _
W
FREQUENCY
®
DIFFERENTIATION
DEVIATION FROM CONSTANT
GROUP DELAY INDICATES DISTORTiON
~ ENVELOPE DELAY
TOTAL DELAY INDICATES
TRANSIT TIME
20
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DIFFERENTIATION TECHNIQUES
FOR GROUP DELAY MEASUREMENTS
® NUMERICAL DIFFERENTIATiON
tg(w)=_d(2l(W)
dw
¢(w)
= -lim ~0 =-lim 01-0 1
... W
AW.O
W2-Wl
AW.o
W
APERTURE
21
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GROUP DELAY RESOLUTION
AS A FUNCTION OF APERTURE AND PHASE RESOLUTION
GROUP DELAY
RESOLUTION
100 n5
10 ns
1 ns
-
PHASE DETECTOR
RESOLUTION
0.1 ns
2.78
kHz
27.8
278
kHz
kHz
2.78
MHz
27.8
MHz
FREQUENCY
APERTURE
(Af)
EFFECTS OF INCREASING APERTURE
b.f= 200kHz
IH=ZO kHz.
~
--a
s:
Q
--~
~
~
<Il
~
9
~
~
~
~
::>
0
~
\!)
I>l
~
FREQUENCY-
FREQUENCY
f'I.IASE
¢
FREQUENCY-
PHASE.
III
-.lAf f+I
i ____
li.rI
I
REDUCE. NOISE
I
I
I
I
(LARGER SIN)
1----t
- -I
A
9 - 360° li-f
22
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LIMITATION DUE TO
MEASUREMENT APERTURES
l:::.tg(w)
W
1-----'\----F--+--~--_7I_____+--
--I~w ...
I.~--Wp~--~I
l:::.w p
Wo
e = 1 - COS oc: = 1 - cos _6._00
2wp
V
II1I
GROllP DELAY ERROR DUE /
TO MEASUI1:EMENT APEImlRE
10
I
14.3==
I
EXAMPlE:
WI' = 27l'· 1 MHz.
min I:J,w =211'·100 kHz
ma)l. e =5%
IJ
I
1
IJ
4.5
I
I
I
VAWI' [%1 =
0.1
7
1 1.42
Aperture .100RIpple Period
W
'1
I I I IIII
10
I
20
23
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~w [%]
p
NOISE LIMITATIONS
GROUP DELAY MEASUREMENTS
NOISE
~;;<_ VECTOR
,\
/
/
I
SIGNAL/NOISE
(FOR A GIVEN
\
J
A¢
BI>.NDWIDTH)
I
/
Atg
*
/
80dB
±O.OOGo ±0.06ns
SIGNAL
*
CAUSES OF NOISE
SOURCE RESIDUAL FM
D.UT NOISE FIGURE
PHASE DETECTOR NOISE
60dB
±O.OGo
±O.6ns
40dB
±O.Go
±G.ons
20dB
±G.Oo ±GO.Ons
FOR l:i.f = 278 KHz.
-I
A¢
WI-lERE t g = 3GOo' .6. f
NOTE: AS l1f - 0, THE
SIGNAL/NOISE - 0 dB
SIGNAL TO NOISE DEGRADAnON
DUE TO DIFFERENTIATION
f(t)
1'---\.-----1---\------,t------t
N(t)
Sew) set)
r~~bf----'W
2
Ws
BW
2
Sill
f(t)
=set) t
T
N(t) @: f(t) = lsi sin wst + n~ I Nnl sin
sw
-7
df
dt
~
wn t
= Wsisl cos wst + n± Wn INn! cos wnt
1'\11
T
24
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LIMITATIONS DUE TO THE
DIFFERENTIATION PROCESS
SiN for~t9
SiN for~0
GROUP DELAY AND DEVIATION
FROM UNEAR PHASE FOR THE
SAME SIGNAL TO NOISE RAno
[:€VIATION FROM W.lEAR PHASE:
~¢: 10 deS/div
GROUP DELAY
Atg: SOn6/div
Aperture: 200 kHz.
I~""'"
CWo 70MHz
5MHL
±~F:
25
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NOISE REDUCTlON USING
SI6NAL AVERAGING
r- -,
1
I
I
I
I
+
[JZEJ
(I/N)SN
NEW OO~
(NOT DISPlAYED)
I
I(H/N)I
{H/N)I\N-\
LASf SfORED AVERAGE
(DISPLAYED)
AN
NEW STORED A'IERAGE
(MEW OISPLA'O
_ : _MElT LIMITATJONe
IN GROUP DlLAY
S
APERTURE:
MEASUREMENT APERTURE DUE TO t:.w
NOISE:
6/N RATIO IN SOURCE, RECEIVER, DEVICE, ETC.
HARDWARE
LIMITATIONS: DETECTOR RESOLUTION, SOURCE RESOLUTION, ETC.
PRINCIPLE
RELATED
LIMITATIONS: DIFFERENTIATION, NON LINEAR DISTORTION
HOW DOES ~ t 9 COMPARE TO ~¢
t::.¢: WHEN EVER POSSIBLE SINCE IT HAS MUCH LESS AMBI6UETY.
~t9: 5/N CAN BE IMPROVED BY AVERAGING.
26
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RELATIONSHIP BETWEEN DISPERSION
AND GROUP DELAY DEYIATIONS
A¢
.-.t
=:::.wT lb
-w~
(
~2b
n=1
~lb AAAAfj4b
11(/J(w) =-b·sin(nTlO)
I1tg (w) =nTb·cos(nTw)
CONVERSION TABLE :t
DEVIATION FROM
to
CONST~N1 GROUP DEL~Y
t>.
tg (W)
..
t>.lll(W):: -
-III
0
Jt.tg(w)dw
-tW
f-----+--------1
~
-loll
DEVI~TION
fROM
LINEAR PHASE
t>.~(W)
0
~
}Co
}-
27
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2C
o
CONVERSION TABLE:It
DEVIATION FROM
DEVIATION FROM
TO
LINEAR PHASE
-
CONSTANT GROUP DELAY
Mg (w) = ~~ [Ml(w)]
-00
.. 6
-w
A¢(W)=-b,W tW
Atg(W)=b.
{(J
1-
b,
1-
b2
M/J (w) =-b2w2
A¢(w) = -b3W~
!-b
3
A¢(W) = -b4 W4
a) LINEAR DELAY
GIVEN SPEC:
Atg ..:::: ins
OVER 1MHz
~}2ns
~c-~
-w
~w
1 MHz.
WHAT DEVIATION
FROM LINEAR
PHASE CAN WE
SPECIFY?
SOLUTION:
NEW SPEC:
STEP 1: WHAT IS THE NORMALIZED
GROUP DELAY FUNCTION?
Atg(tu) =C1W
STEP 2: WHAT IS THE CORRESPONDIN6
PHASE CHARACTERISTIC?
li¢(w) =~C1W2.
STEP 3: CALCULATE C1 FROM SPEC.
AND SUBSTITUTE IN Ml>(w)
11 t
C1=~
[sl
=[r~dJ
STEP 4: SUBSTlTUOE NUMERiCAl VALUES
AND SOLVE FOR u-adJor[degJ
2- -Mg·oo
li""w)-:!'
!!.s·w
¥'\'
- 2.
W
Z.
28
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f:,.¢ (rad)
=
-3
-lns·21T·1MHz==21T·.1O [rad]
A¢ (deg) =
~. ~!&(rad) == o.3G@egloverlMHz
CONT: CONVERSIONS
6 tg (10) --+ 6~(w)
b) PARABOLIC DELAY
SPEC:
Atg(W)
"
/1}10nS
~+w
-w
'----r--'
1 MHz.
SAME STEPS MIN EXAMPLE a):
1. ,6.tg(w)= czw z
2. A(/J(w) =
-1 C
ZW
-1
3. A(/J(W) = "3.
~ti"l
1.20
3
~.(.IJ
{h/).¢(W)
+w
3 -1
=:3Mg·w
z
4. A¢[rdd]= 2;·10- [rad]=U[deg]
29
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we
-w
'-v-------'
1 MHz.
HEWLETT'ft PACKARD
PRINTED IN U.S.A.
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