02 - signals and modulation

7/25/2019 02 - Signals and Modulation

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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)

02 - signals and modulation

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