02 - signals and modulation
TRANSCRIPT
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Signals and
Modulation
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The Nature of Electronic Signals
Signals are generally classified in terms of theirtime or frequency domain behaviour Time domain classifications include:
Static:Unchanging (DC)
Quasi static:Sloly changing (am!lifier drift) "eriodic:Sine ave #e!etitive:"eriodic but varying ($lectrocardiogra!h) Transient:%ccur once only (im!ulse) Quasi transient:#e!etitive but seldom (radar !ulses)
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Sinusoidal Signals
Most acoustic and electromagnetic sensors e&!loit the!ro!erties of sinusoidal signals 'n the time domain such signals are constructed of
sinusoidally varying voltages or currents constrainedithin ires
here vc(t) Signal
Ac Signal am!litude (*)
c +requency (rad,s)
fc +requency (h-)
t Time (s)
tfAtAtv ccccc 2coscos)( ==
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Generating Sinusoidal Signals
+requency
selective netor./ain
+eedbac.
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Sinusoidal Signals in the Frequency
Domain
0o uncertainty in
the measurement
of the frequency
1,Measurement
Uncertainty
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The Fourier Series
2ll continuous !eriodic signals can be re!resented by afundamental frequency sine ave and a collection of
sine and,or cosine harmonics of that fundamental sine
ave
The +ourier series for any aveform can be e&!ressedin the folloing form:
here: anbn 2m!litudes of the harmonics (can be -ero)
n 'nteger
)sin()cos(2
)(
1 1
0 tnbtnaa
tv
n n
nn
=
=
++=
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Calculating the Fourier Coefficients
The am!litude coefficients can be calculated as follos
3ecause only certain frequencies determined by integer n are
alloed the s!ectrum is discrete The term a4,1 is the average value of v(t) over a com!lete cycle
Though the harmonic series is infinite the coefficients become so
small that their contribution is considered to be negligible5
=
=
t
n
t
n
dttntvT
b
dttntvT
a
0
0
)sin()(2
)cos()(2
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Number of Harmonics Required
2n $C/ trace ith a fundamental frequency of about6517- can be re!roduced ith 84 to 94 harmonics (a
bandidth of about 6447-)
2 square ave on the other hand may require u! to 6444
harmonics because e&tremely high frequency terms are
contained ithin the transitions
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Sampled Signals and Digitisation
To !rocess signals ithin a com!uter requires that theybe sam!led !eriodically and then converted to a digital
re!resentation
To ensure accurate reconstruction: the signal must be sam!led at a rate hich is at least double the
highest significant frequency com!onent of the signal5 This is
.non as the 0yquist rate5
The number of discrete levels to hich the signal is quantised
must also be sufficient5
Signal reconstruction involves holding the signalconstant (-ero order hold) during the !eriod beteen
sam!les then !assing the signal through lo!ass filter
to remove high frequency com!onents generated by the
sam!ling !rocess
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Generating Signals in !T"!#
M2T;23 has a com!rehensive number of builtinfunctions to facilitate the generation of both !eriodic and
a!eriodic functions
Square /enerates a square ave
Satooth /enerates a triangular ave Sine /enerates a sine ave
$&! /enerates an e&!onential
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$sing the FFT in !T"!#
M2T;23 has a good +ast +ourier Transform (++T) function but besure to remember toe folloing
Select a signal of length 1n
+or a real in!ut the out!ut s!ectrum is folded
The s!ectrum is com!le& so use the 654e4?@ A sam!le !eriod (s)
t>(6:14B9)dt@ A total time (n>66)
f>644@ A signal frequency (7-)
sig>cos(1!if5t)@ A generate time domain sig
sub!lot(166) !lot(tsig)@ A !lot time domain signal
s!ect > abs(fft(sig))@ A ++T to obtain s!ectrum
freq>(4:14B8)5,(dt14B9)@ A generate the freq a&essub!lot(161) !lot(freqs!ect)@ A !lot the s!ectrum
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!liasing% Time Domain
&llustration
% r i g i n a l S i n e a v e
S a m ! l e d D 5 D D & ! e r c y c le
S a m ! l e d 6 5 ? ? & ! e r c y c le
S a m ! l e d 6 5 6 6 & ! e r c y c l e
Data sam!led at less than
the 0yquist rate a!!ears to
be shifted in frequency
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!liasing% Frequency Domain
&llustration 'n the frequency domain an analog
signal may be re!resented in terms ofits am!litude and total bandidth asshon in the figure5
2 sam!led version of the same signal
can be re!resented by a re!eatedsequence s!aced at the sam!lefrequency (generally denoted fs)
'f the sam!le rate is not sufficientlyhigh then the sequences ill overla!
and high frequency com!onents illa!!ear at a loer frequency (albeit!ossibly ith reduced am!litude)
2 n a l o gS i g n a l S ! e c t r u m
S a m ! l e dS i g n a l S ! e c t r u m
S a m ! l e dS i g n a l S ! e c t r u m
f s
f s
: f s
: f s
2 l i a s e dS i g n a l
S i g n a l n o t2 l i a s e d
Chir! to E447-
sam!led at 6.7-
Sam!led
at E447-
Sam!led
at 1E47-
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!mplitude odulation '!(
2 modulation technique in
hich the am!litude of the
carrier is varied in accordance
ith some characteristic of the
baseband modulating signal5
't is the most common form of
modulation because of the
ease ith hich the baseband
signal can be recovered fromthe transmitted signal
[ ] tftvAtv cbcam 2cos)(1)( +=
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)ercentage odulation
'n general 2amF6 otherise a !hase reversal occurs and
demodulation becomes more difficult5
The e&tent to hich the carrier has been am!litude
modulated is e&!ressed in terms of apercentage
modulationhich is Gust calculated by multi!lying 2amby
6445
[ ] tftfAAtv caamcam 2cos2cos1)( +=
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! in the Frequency Domain
To determine the characteristics of the signal in thefrequency domain it can be reritten in the folloing
form (using a trig identity)
't can be seen that this is made u! from three
inde!endent frequencies: The original carrier
2 frequency at the difference beteen the carrier and thebaseband signal
2 frequency at the sum of the carrier and the baseband signal
[ ]tfftffAA
tfAtv acacamc
ccam )(2cos)(2cos
2
2cos)( +++=
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! Demodulation% Simulation
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! Demodulation% The Crystal Set
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Frequency odulation 'F(
2 modulation technique in hich the frequency of the
carrier is varied in accordance ith some characteristicof the baseband signal5
The modulating signal is integrated because variations in
the modulating term equate to variations in the carrier
!hase5
The instantaneous angular frequency can be obtained by
differentiating the instantaneous !hase as shon
+=
t
bccfm dttvktAtv )(cos)(
)()( tkvdttvkt
dt
dbc
t
bc +=
+=
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Sinusoidal F odulation +or sinusoidal modulation the formula for
+M is as follos:
'n this case hich is the ma&imum!hase deviation is usually referred to as
the modulation index The instantaneous frequency in this case
is
So the ma&imum frequency deviationdefined as f
[ ]ttAtv accfm sincos)( +=
tfff
tf
aac
aac
cos
cos22
+=
+=
aff =
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F Spectrum
$ven though the instantaneousfrequency lies ithin the range fcH,fthe s!ectral com!onents of the signaldonIt lie ithin this range
The s!ectrum com!rises a carrier
ith am!litude Jo() ith sidebandson either side of the carrier at offsetsof a 1a ?a K5
The bandidth is infinite hoever forany most of the !oer is confinedithin finite limits
CarsonIs #ule states that thebandidth is about tice the sum ofthe ma&imum frequency deviation!lus the modulating frequency
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F Demodulation
Demodulation of +M is
commonly achieved by
converting to 2M folloed by
envelo!e detection
Sim!lest conversion is to !assthe signal through a frequency
sensitive circuit li.e a lo!ass
or band!ass filter called a
discriminator Modern methods include ";;
and quadrature detection
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"inear Frequency odulation
'n most active sensors thefrequency is modulated in alinear manner ith time
Substituting into the standardequation for +M e obtain thefolloing result
tAbb=
+=
+= 2
2cos)(
cos)(
tA
tAtv
tdtAtAtv
bccfm
t
bccfm
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Frequency odulated Continuous *a+e
)rocessing
'n +requency Modulated Continuous Lave (+MCL)
systems a !ortion of the transmitted signal is mi&ed ith
(multi!lied by) the returned echo5
The transmit signal ill be shifted from that of the
received signal because of the round tri! time
Calculating the !roduct
+= 2)(
2)(cos)( t
AtAtv
bccfm
+
+= 222 )(
2)(cos
2cos)()( t
Att
AtAtvtv bc
bccfmfm
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FC* continued,--
$quating using the trig identity for the !roduct of to sines
cosA5cosB>45Ecos(A+B)Hcos(A-B)N
The first cos term describes a linearly increasing +M signal (chir!) at
about tice the carrier frequency5 This term is generally filtered out5
The second cos term describes a beat signal at a fi&ed frequency
The signal frequency is directly !ro!ortional to the delay time andhence is directly !ro!ortional to the round tri! time to the target
( )
++
++
=
2
22
2.cos
22cos
2
1)(
bcb
cb
bbc
outA
tA
AtAtA
tv
2b
beat
Af =
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)ulse !mplitude odulation
Modulation in hich the am!litude ofindividual regularly s!aced !ulses in a!ulse train is varied in accordance ithsome characteristic of the modulatingsignal
+or timeofflight sensors the am!litude isgenerally constant
The ability of a !ulsed sensor to resolveto closely s!aced targets is determinedby the !ulse idth
The ?d3 bandidth of a !ulse is
here S!ectral Lidth (7-)
"ulseidth (sec)
1
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Frequency Shift .eying 'FS.(
This modulation technique is thedigital equivalent of linear +Mhere only to differentfrequencies are utilised
2 single bit can be re!resentedby a single cycle of the carrierbut if the data rate is not criticalthen multi!le cycles can be used
Demodulation can be achieved
by detecting the out!uts of a !airof filters centred at the tomodulation frequencies
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Effect of the Number of Cycles per #it on
the Signal Spectrum
E Cycles !er 3it %ne Cycle !er 3it
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)hase Shift .eying ')S.(
Usually binary !hase coding5 Thecarrier is sitched beteen H,694according to a digital basebandsequence5
This modulation technique can be
im!lemented quite easily using abalanced mi&er or ith a dedicated3"SO modulator
Demodulation is achieved bymulti!lying the modulated signal by acoherent carrier (a carrier that is
identical in frequency and !hase to thecarrier that originally modulated the3"SO signal)5
This !roduces the original 3"SOsignal !lus a signal at tice the carrierhich can be filtered out5
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Spectrum of a #)S. Signal
/ne Cycle per #it Random odulation
Unmodulated Carrier
Modulated Carrier
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Stepped Frequency odulation
2 sequence of !ulses aretransmitted each at a slightlydifferent frequency
The !ulse idth (in the radarcase) is made sufficiently ideto s!an the region of interest
but because it is so ide itcannot resolve individualtargets ithin that region
'f all of the !ulses are!rocessed together theeffective resolution is im!rovedbecause the total bandidth isidened by the total frequencydeviation of the sequence of!ulses5
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+iltering
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Filter Types
2 filter is a frequency selective netor. that !asses certain
frequencies of an in!ut signal and attenuates others
The three common ty!es of filter are:
7igh "ass
;o "ass
3and "ass 7igh !ass filter bloc.s signals belo its cutoff frequency and !asses
those above
;o !ass filter !asses signals belo its cutoff frequency and
attenuates those above
3and !ass filter !asses a range of frequencies hile attenuatingthose both above and belo that range
2 fourth less common configuration is a bandsto! or notch filter that
attenuates signals at a s!ecific frequency or over a narro range of
frequencies and !asses all other frequencies5
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Filter &mplementations
The maGor filter categories are as follos3utterorth (ma&imally flat)
Chebyshev (equi ri!!le)
3essel (linear !hase)$lli!tical
0ote that the !ass band is s!ecified by its half !oer
!oints (45848 of the !ea. voltage gain)
'f the gain is !lotted in d3 then it ould be calculatedusing 14log64(gain) to convert to !oer5
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#utter0orth
This a!!ro&imation to an ideal lo !ass filter is basedon the assum!tion that a flat res!onse at -ero
frequency is most im!ortant5
The transfer function is an all!ole ty!e ith roots that
fall on the unit circle
+airly good am!litude and transient characteristics
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Chebyshe+
The transfer function is also all!ole but ith rootsthat fall on an elli!se5 This results in a series of equi
am!litude ri!!les in the !ass band and a shar!er
cutoff than 3utterorth5
/ood selectivity but !oor transient behaviour
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#essel
%!timised to obtain a linear !hase res!onse hichresults in a ste! res!onse ith no overshoot or ringing
and an im!ulse res!onse ith no oscillatory
behaviour
"oor frequency selectivity com!ared to the other
res!onse ty!es
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Elliptic
7as -eros as ell as !oles hich create equiri!!lebehaviour in the !ass band similar to Chebyshev
filters
Peros in the sto! band reduce the transition region so
that e&tremely shar! rolloff characteristics can be
achieved
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Filter Rolloff and &nsertion "oss
The rate at hich a signalis attenuated as a functionof frequency is .non asrolloff
The total attenuation at as!ecific frequency is .noas the insertion loss
'nsertion loss at a !articularfrequency are determined
by the filter order #olloff for lo!ass filters
asym!totic to nd3,octave #olloff for band!ass filters
asym!totic to ?nd3,octave 0ote 6 octave > doubling in frequency 6 decade > 64 & frequency
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Filter Frequency Responses
0ote that the cutoff frequency (1447- in this case) s!ecified in M2T;23
is equal to the ?d3 !oint for the 3utterorth filter and to the !assband
ri!!le for the Chebyshev and $lli!tic filters
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Specifying a #utter0orth #andpass Filter
A 3ans!ass filter
fs > 144eH4?@
ts > 6,fs@
fmat > B454eH4?@ A centre frequency
bmat > 6454eH4?@ A bandidth
l>1ts(fmatbmat,1)@ A loer band limit
h>1ts(fmatHbmat,1)@ A u!!er band limit
n>lhN@baN>butter(?n)@ A th order
hN>freq-(ba641B)@
freq>(4:641?),(1444ts641B)@
Asemilog&(freq14log64(abs(h)))@
!lot(freqabs(h))@
grid
title(3and!ass +ilter Transfer +unction)
&label(+requency (.7-))@
ylabel(/ain)
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Convolution
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Con+olution% Definition
;inear Time 'nvariant (;T') systems arecom!letely characterised by their im!ulseres!onse
2n arbitrary in!ut can be e&!ressed as a
eighted su!er!osition of time shifted im!ulses Therefore the system out!ut is Gust the eighted
su!er!osition of the time shifted im!ulseres!onses
This eighted su!er!osition is termed theconvolution sum(discrete time) or convolutionintegral(continuous time)
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Con+olution% Time of Flight
# a d a r
+or timeofflight sensors the im!ulse
res!onse of the roundtri! !ro!agationto the !oint target is a signal that is
delayed in time and attenuated in
am!litude h(t)= a(-) here are!resents the attenuation and the
round tri! time delayThis is an ;T' system hich allos us
to calculate the system res!onse by
ta.ing the convolution of the transmit
!ulse ith the target (made of many
!oint reflectors)