cs study notes

8/18/2019 CS Study Notes

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he transmission medium can 'e a metallic ca'le, optical fi'er

ca'le, Earth/s atmosphere, or a com'ination of to or more

tpes of transmission sstems.

$n the receiver, the incoming signals are filtered, amplified,

and then applied to the demodulator and decoder circuits,

hich e0tracts the original source information from the

modulated carrier.

he cloc- and carrier recover circuits recover the analogcarrier and digital timing (cloc-) signals from the incoming

modulated ave since the are necessar to perform the de+

modulation process.

%$RE 2+1 "implified 'loc- diagram of a digital radio

sstem.

INFORMATION CAPACITY, BITS, BIT RATE, BAUD,

AND MARY ENCODING

Information Capacity, Bits, an Bit Rat!

$α

B 0 t (2.2)

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here $3 information capacit ('its per second)

B 3 'andidth (hert4)

t = transmission time (seconds)

%rom Equation 2+2, it can 'e seen that information capacit

is a linear function of 'andidth and transmission time and is

directl proportional to 'oth.

$f either the 'andidth or the transmission time changes, a

directl proportional change occurs in the information

capacit.

he higher the signal+to+noise ratio, the 'etter the

performance and the higher the information capacit.

athematicall stated, the Shannon limit_for information

capacity is

(2.5)

or

(2.6)

here $ 3 information capacit ('ps)

B 3 'andidth (hert4)

N

S 3 signal+to+noise poer ratio (unitless)

%or a standard telephone circuit ith a signal+to+noise poer

ratio of 1&&& (5& dB) and a 'andidth of 2.7 -84, the

"hannon limit for information capacit is

$ 3 (5.52)(27&&) log1& (1 9 1&&&) 3 2.; -'ps

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"hannon/s formula is often misunderstood. he results of the

preceding e0ample indicate that 2.; -'ps can 'e propagated

through a 2.7+-84 communications channel. his ma 'e true,

'ut it cannot 'e done ith a 'inar sstem. o achieve an

information transmission rate of 2.; -'ps through a 2.7+-84

channel, each sm'ol transmitted must contain more than one

'it.

"#"#" M#ary Encoin$

M-ary is a term derived from the ord inary!

M simpl represents a digit that corresponds to the num'er of

conditions, levels, or com'inations possi'le for a given

num'er of 'inar varia'les.

%or e0ample, a digital signal ith four possi'le conditions

(voltage levels, frequencies, phases, and so on) is an +ar

sstem here " 6. $f there are eight possi'le conditions, = < and so forth.

he num'er of 'its necessar to produce a given num'er of

conditions is e0pressed mathematicall as

M N 2log= (2.=)

here N 3 num'er of 'its necessar M 3 num'er of conditions, levels, or com'inations

possi'le ith N 'its

Equation 2+= can 'e simplified and rearranged to e0press the

num'er of conditions possi'le ith N 'its as

#

N

"M (2.)

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Baud 3

N

f (2.11)

B comparing Equation #!$% ith Equation #!$$ the 'aud and

the ideal minimum ?quist 'andidth have the same value

and are equal to the 'it rate divided ' the num'er of 'its

encoded.

"#% AMP)ITUDE#S*IFT +EYING

he simplest digital modulation technique is amplitude-shift

keying (!"#), here a 'inar information signal directl

modulates the amplitude of an analog carrier.

!"# is similar to standard amplitude modulation e0cept

there are onl to output amplitudes possi'le. !mplitude+

shift -eing is sometimes called digital amplitudemodulation (@!).

athematicall, amplitude+shift -eing is

(2.12)

here

&ask (t) " amplitude+shift -eing ave

vm(t) 3 digital information (modulating) signal (volts)

!A2 3 unmodulated carrier amplitude (volts)

c 3 analog carrier radian frequenc (radians per second, 2Cf ct )

$n Equation #!$#' the modulating signal Dvm(t) is a

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normali4ed 'inar aveform, here 9 1 V 3 logic 1 and +1 V

3 logic &. herefore, for a logic 1 input, vm(t) 3 9 1 V, Equation

#!$# reduces to

and for a logic & input, vm(t) 3 +1 V, Equation #!$# reduces to

hus, the modulated ave &ask (t)' is either ! cos(c t) or &.

8ence, the carrier is either on or off' hich is h

amplitude+shift -eing is sometimes referred to as on-off keying (FF#).

%igure 2-# shos the input and output aveforms from an !"#

modulator.

%rom the figure, it can 'e seen that for ever change in theinput 'inar data stream, there is one change in the !"#

aveform, and the time of one 'it (t ) equals the time of one

analog signaling element (t,).

B 3 f ' A1 3 f ' 'aud 3 f ' A1 3 f '


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%$RE 2+2 @igital amplitude modulation: (a) input 'inarG

(') output @! aveform

he entire time the 'inar input is high, the output is a constant+

amplitude, constant+frequenc signal, and for the entire time the

'inar input is lo, the carrier is off.

he rate of change of the !"# aveform ('aud) is the same as

the rate of change of the 'inar input ('ps).

Eamp-! "#.

@etermine the 'aud and minimum 'andidth necessar to pass a

1& -'ps 'inar signal using amplitude shift -eing.

"olution

%or !"#, ? 3 1, and the 'aud and minimum 'andidth are

determined from Equations 2.11 and 2.1&, respectivel:

B 3 1&,&&& A 1 3 1&,&&&

'aud 3 1&, &&& A1 3 1&,&&&

;


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he modulating signal is a normali4ed 'inar aveform here a

logic 1 3 9 1 V and a logic & 3 +1 V. hus, for a logic l input,

vm(t) 3 9 1, Equation 2.15 can 'e reritten as

%or a logic & input, vm(t) 3 +1, Equation 2.15 'ecomes

>ith 'inar %"#, the carrier center frequenc (f c) is shifted

(deviated) up and don in the frequenc domain ' the 'inar

input signal as shon in %igure 2+5.

%$RE 2+5 %"# in the frequenc domain!s the 'inar input signal changes from a logic & to a logic 1

and vice versa, the output frequenc shifts 'eteen to

frequencies: a mar-, or logic 1 frequenc (f m), and a space, or

logic & frequenc (f s). he mar- and space frequencies are

separated from the carrier frequenc ' the pea- frequenc

deviation (If) and from each other ' 2 If.

%requenc deviation is illustrated in %igure 2+5 and e0pressedmathematicall as

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If 3 Jf m K f sJ A 2 (2.16)

here If " frequenc deviation (hert4)

Jf m K f sJ 3 a'solute difference 'eteen the mar- and

space frequencies (hert4)

%igure 2+6a shos in the time domain the 'inar input to an %"#

modulator and the corresponding %"# output.

>hen the 'inar input (f ) changes from a logic 1 to a logic &

and vice versa, the %"# output frequenc shifts from a mar- ( f m)

to a space (f s) frequenc and vice versa.

$n %igure 2+6a, the mar- frequenc is the higher frequenc (f c* If) and the space frequenc is the loer frequenc (f c + If),

although this relationship could 'e Lust the opposite.

%igure 2+6' shos the truth ta'le for a 'inar %"# modulator.

he truth ta'le shos the input and output possi'ilities for a

given digital modulation scheme.

%$RE 2+6 %"# in the time domain: (a) aveform: (') truth

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ta'le

"#/#. FS+ Bit Rat!, Ba&, an Ban'it(

$n %igure 2+6a, it can 'e seen that the time of one 'it (t ) is the

same as the time the %"# output is a mar- of space frequenc

(t s )! hus, the 'it time equals the time of an %"# signaling

element, and the 'it rate equals the 'aud.

he 'aud for 'inar %"# can also 'e determined '

su'stituting ? " 1 in Equation 2.11:

'aud 3 f ' A 1 3 f '

he minimum 'andidth for %"# is given as

B 3 J(f s K f ') K (f m K f ')J

3 J(f s K f m)J 9 2f '

and since J(f s K f m)J equals 2If ' the minimum 'andidth can 'e

appro0imated as

B 3 2(If 9 f ') (2.1=)

here

B3 minimum ?quist 'andidth (hert4)

If " frequenc deviation J(f m K f s)J (hert4)

f " input 'it rate ('ps)

Eamp-! "#"

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%$RE ;+= %"# modulator, t ', time of one 'it 3 1Af 'G f m mar-

frequencG f s, space frequencG ., period of shortest ccleG

1A1, fundamental frequenc of 'inar square aveG f 1, input 'it

rate ('ps)

"ince it ta-es a high and a lo to produce a ccle, the highest

fundamental frequenc present in a square ave equals the

repetition rate of the square ave, hich ith a 'inar signal is

equal to half the 'it rate. herefore,

f a 3 f ' A 2 (2.1)

here

f a 3 highest fundamental frequenc of the 'inar input signal

(hert4)

f b = input 'it rate ('ps)

he formula used for modulation inde0 in % is also valid for

%"#G thus,

h 3 If A f a (unitless) (2.17)

here

h 3 % modulation inde0 called the h+factor in %"#

f o 3 fundamental frequenc of the 'inar modulating

signal (hert4)

If " pea- frequenc deviation (hert4)

he pea- frequenc deviation in %"# is constant and alas at

its ma0imum value, and the highest fundamental frequenc is

equal to half the incoming 'it rate. hus,

2

2

JJ

sm

f

f f

h

−

=

or

1=


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sm

f

f f h

JJ −= (2.1


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%$RE 2+7 ?oncoherent %"# demodulator

he %"# input signal is simultaneousl applied to the inputs of

'oth 'andpass filters (B*%s) through a poer splitter.

he respective filter passes onl the mar- or onl the space

frequenc on to its respective envelope detector.

he envelope detectors, in turn, indicate the total poer in each

pass'and, and the comparator responds to the largest of the to

poers.

his tpe of %"# detection is referred to as noncoherent detection.

he incoming %"# signal is multiplied ' a recovered carrier

signal that has the e0act same frequenc and phase as the

transmitter reference.

8oever, the to transmitted frequencies (the mar- and space

frequencies) are not generall continuousG it is not practical to

reproduce a local reference that is coherent ith 'oth of them.Honsequentl, coherent %"# detection is seldom used.

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%$RE 2+< Hoherent %"# demodulator

he most common circuit used for demodulating 'inar %"#

signals is the phaselocked loop (*NN), hich is shon in 'loc-

diagram form in %igure 2+;.

%$RE 2+; *NN+%"# demodulator

!s the input to the *NN shifts 'eteen the mar- and space

frequencies, the dc error &oltage at the output of the phase

comparator follos the frequenc shift.

Because there are onl to input frequencies (mar- and space),

there are also onl to output error voltages. Fne represents a

logic 1 and the other a logic &.

Binar %"# has a poorer error performance than *"# or ! and,

consequentl, is seldom used for high+performance digital radio

sstems.

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$ts use is restricted to lo+performance, lo+cost, asnchronous

data modems that are used for data communications over analog,

voice+'and telephone lines.

"#/#/ Contin&o&s#P(as! Fr!3&!ncy#S(ift +!yin$

Hontinuous+phase frequenc+shift -eing (H*+%"#) is 'inar

%"# e0cept the mar- and space frequencies are snchroni4ed

ith the input 'inar 'it rate.

>ith H*+%"#, the mar- and space frequencies are selected such

that the are separated from the center frequenc ' an e0actmultiple of one+half the 'it rate (f m and f s " n,f A #)' here n 3

an integer).

his ensures a smooth phase transition in the analog output signal

hen it changes from a mar- to a space frequenc or vice versa.

%igure 2+1& shos a noncontinuous %"# aveform. $t can 'e

seen that hen the input changes from a logic 1 to a logic & and

vice versa, there is an a'rupt phase discontinuit in the analog

signal. >hen this occurs, the demodulator has trou'le folloing

the frequenc shiftG consequentl, an error ma occur.

%$RE 2+1& ?oncontinuous %"# aveform

%igure 2+11 shos a continuous phase %"# aveform.

2&


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%$RE 2+11 Hontinuous+phase "# aveform

?otice that hen the output frequenc changes, it is a smooth,

continuous transition. Honsequentl, there are no phase

discontinuities.

H*+%"# has a 'etter 'it+error performance than conventional

'inar %"# for a given signal+to+noise ratio.

he disadvantage of H*+%"# is that it requires snchro+

ni4ation circuits and is, therefore, more e0pensive to implement.

"#4 P*ASE#S*IFT +EYING

Phase-shift keying (*"#) is another form of angle-modulated'

constant-amplitude digital modulation.

"#4#. Binary P(as!#S(ift +!yin$

he simplest form of *"# is inary phase-shift keying

(B*"#), here ? " 1 and M " #!

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he 'alanced modulator has to inputs: a carrier that is in

phase ith the reference oscillator and the 'inar digital data.

%or the 'alanced modulator to operate properl, the digital

input voltage must 'e much greater than the pea- carrier

voltage.

his ensures that the digital input controls the onAoff state of

diodes @1 to @6. $f the 'inar input is a logic 1(positive

voltage), diodes @ 1 and @2 are forard 'iased and on, hile

diodes @5 and @6 are reverse 'iased and off (%igure 2+15').

>ith the polarities shon, the carrier voltage is developed

across transformer 2 in phase ith the carrier voltage across

1. Honsequentl, the output signal is in phase ith the reference

oscillator.

$f the 'inar input is a logic & (negative voltage), diodes @l and

@2 are reverse 'iased and off, hile diodes @5 and @6 are

forard 'iased and on (%igure ;+15c). !s a result, the carrier

voltage is developed across transformer 2 1


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%$RE ;+15 (a) Balanced ring modulatorG (') logic 1 inputG(c) logic & input

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%$RE 2+16 B*"# modulator: (a) truth ta'leG (') phasor

diagramG (c) constellation diagram

"#4#.#" Ban'it( consi!rations of BPS+5

$n a B*"# modulator. the carrier input signal is multiplied '

the 'inar data.

$f 9 1 V is assigned to a logic 1 and +1 V is assigned to a logic

&, the input carrier (sin ct) is multiplied ' either a 9 or + 1 .

he output signal is either 9 1 sin ct or +1 sin ct the first

represents a signal that is in phase ith the reference oscillator,

the latter a signal that is 1


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Each time the input logic condition changes, the output phase

changes.

athematicall, the output of a B*"# modulator is

proportional to

B*"# output 3 Dsin (2Cf at) 0 Dsin (2Cf ct) (2.2&)

here

f a 3 ma0imum fundamental frequenc of 'inar

input (hert4)

f c 3 reference carrier frequenc (hert4)

"olving for the trig identit for the product of to sine

functions,

&.=cosD2C(f c K f a)t K &.=cosD2C(f c 9 f a)t

hus, the minimum dou'le+sided ?quist 'andidth (B) is

f c 9 f a f c 9 f a +(f c 9 f a) or +fc 9 fa

2f a

and 'ecause f a 3 f ' A 2 ' here f b = input 'it rate,here B is the minimum dou'le+sided ?quist 'andidth.

%igure 2+1= shos the output phase+versus+time relationship

for a B*"# aveform.

Nogic 1 input produces an analog output signal ith a &O

phase angle, and a logic & input produces an analog output

signal ith a 1


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!s the 'inar input shifts 'eteen a logic 1 and a logic &

condition and vice versa, the phase of the B*"# aveform

shifts 'eteen &O and 1


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athematicall, the demodulation process is as follos.

%or a B*"# input signal of 9 sin ct (logic 1), the output of the

'alanced modulator is

output 3 (sin ct )(sin ct) 3 sin2ct (2.21)

or

sin2ct 3 &.=(1 K cos 2ct) 3 &.= + &.=cos 2ct

filtered out

leaving

output 3 9 &.= V 3 logic 1

$t can 'e seen that the output of the 'alanced modulator contains

a positive voltage (9D1A2V) and a cosine ave at tice the

carrier frequenc (2 ct ).

he N*% has a cutoff frequenc much loer than 2 ct, and,

thus, 'loc-s the second harmonic of the carrier and passes onl

the positive constant component. ! positive voltage represents a

demodulated logic 1.

%or a B*"# input signal of +sin ct (logic &), the output of the

'alanced modulator is

output 3 (+sin ct )(sin ct) 3 sin2ct

or

sin2ct 3 +&.=(1 K cos 2ct) 3 &.= 9 &.=cos 2ct

filtered out

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leaving

output 3 + &.= V 3 logic &

he output of the 'alanced modulator contains a negative

voltage (+DlA2V) and a cosine ave at tice the carrier

frequenc (2ct).

!gain, the N*% 'loc-s the second harmonic of the carrier and

passes onl the negative constant component. ! negative

voltage represents a demodulated logic &.

0PS+ transmitt!r.

! 'loc- diagram of a *"# modulator is shon in %igure 2+

17. o 'its (a di'it) are cloc-ed into the 'it splitter. !fter 'oth

'its have 'een seriall inputted, the are simultaneousl parallel

outputted.

he $ 'it modulates a carrier that is in phase ith the reference

oscillator (hence the name P$P for Pin phaseP channel), and the

'it modulate, a carrier that is ;&O out of phase.

%or a logic 1 3 9 1 V and a logic &3 + 1 V, to phases are

possi'le at the output of the $ 'alanced modulator (9sin ct

and + sin ct), and to phases are possi'le at the output of the

'alanced modulator (9cos ct), and (+cos ct).

>hen the linear summer com'ines the to quadrature (;&O

out of phase) signals, there are four possi'le resultant

phasors given ' these e0pressions: 9 sin ct 9 cos ct, 9 sin

ct + cos ct, +sin ct 9 cos ct, and +sin ct + cos ct.

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%$RE 2+17 *"# modulator

Eamp-! "#4

%or the *"# modulator shon in %igure 2+17, construct the truth

ta'le, phasor diagram, and constellation diagram.

"olution

%or a 'inar data input of 3 F and $3 &, the to inputs to the $

'alanced modulator are +1 and sin ct, and the to inputs to the

'alanced modulator are +1 and cos ct.

Honsequentl, the outputs are

$ 'alanced modulator 3(+1)(sin ct) 3 +1 sin ct

'alanced modulator 3(+1)(cos ct) " +1 cos ct and the output

of the linear summer is

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%igure 2+1; shos the output phase+versus+time relationship

for a *"# modulator.

%$RE 2+1; Futput phase+versus+time relationship for a *"# modulator.

"#4#"#" Ban'it( consi!rations of 0PS+

>ith *"#, 'ecause the input data are divided into to

channels, the 'it rate in either the $ or the channel is equal to

one+half of the input data rate (f 'A2) (one+half of f 'A2 " f 'A6 )!

his relationship is shon in %igure 2+2&.

%$RE 2+2& Bandidth considerations of a *"# modulator

$n %igure 2+2&, it can 'e seen that the orse+case input

condition to the $ or 'alanced modulator is an alternative

1A& pattern, hich occurs hen the 'inar input data have a

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11&& repetitive pattern. Fne ccle of the fastest 'inar

transition (a 1A& sequence in the $ or channel ta-es the

same time as four input data 'its.

Honsequentl, the highest fundamental frequenc at the input

and fastest rate of change at the output of the 'alance.:

modulators is equal to one+fourth of the 'inar input 'it

rate.

he output of the 'alanced modulators can 'e e0pressed

mathematicall as

(2.22)

here

he output frequenc spectrum e0tends from f/c 9 f ' A 6 to f/c +f ' A 6 and the minimum 'andidth (f ?) is

Eamp-! "#6

5=


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%or a *"# modulator ith an input data rate (f ') equal to 1&

'ps and a carrier frequenc 7& 84, determine the minimum

dou'le+sided ?quist 'andidth (f N ) and the 'aud. !lso, compare

the results ith those achieved ith the B*"# modulator in

E0ample 2+6. se the *"# 'loc- diagram shon in %igure 2+17

as the modulator model.

"olution

he 'it rate in 'oth the $ and channels is equal to one+half of

the transmission 'it rate, or

f ' 3 f '1 3 f ' A 2 3 1& 'ps A 2 3 = 'ps

he highest fundamental frequenc presented to either 'alanced

modulator is

f a3 f ' A 2 3 = 'ps A 2 3 2.= 84

he output ave from each 'alanced modulator is

(sin 2Cf at)(sin 2Cf ct)

&.= cos 2C(f c K f a)t K &.= cos 2C(f c 9 f a)t

&.= cos 2CD(7& K 2.=)84t K &.= cos 2CD(7& K

2.=)84t

&.= cos 2C(7.=84)t + &.= cos 2C(72.=84)t

he minimum ?quist 'andidth is

B3(72.=+7.=)84 3 =84

he sm'ol rate equals the 'andidth: thus,

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sm'ol rate 3 = mega'aud

he output spectrum is as follos:

$t can 'e seen that for the same input 'it rate the minimum

'andidth required to pass the output of the *"# modulator is

equal to one+half of that required for the B*"# modulator in

E0ample 2+6. !lso, the 'aud rate for the *"# modulator is one+

half that of the B*"# modulator.

he minimum 'andidth for the *"# sstem descri'ed in

E0ample 2+ can also 'e determined ' simpl su'stituting

into Equation 2+1&:

B 3 1& 'ps A 2 3 = 84

"#4#"#% 70PS+ r!c!i2!r85

he 'loc- diagram of a *"# receiver is shon in %igure 2+

21. he poer splitter directs the input *"# signal to the $

and product detectors and the carrier recover circuit. he

carrier recover circuit reproduces the original transmit carrier

oscillator signal. he recovered carrier must 'e frequenc and

phase coherent ith the transmit reference carrier. he *"#

signal is demodulated in the $ and product detectors, hich

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2PSK)!

"#9 BAND:IDT* EFFICIENCY

Band.idth efficiency (sometimes called information density or

spectral efficiency, often used to compare the performance of

one digital modulation technique to another.

athematical 'andidth efficienc is

3ert4

sits

34 and.idthimum

psrateit ontransmissi B

A

)(min

)(==η (2.27)

>here Bη 3 'andidth efficienc

DIFFERENTIA) P*ASE#S*IFT +EYING

5ifferential phase-shift keying (@*"#) is an alternative form of

digital modulation here the 'inar input information is

contained in the difference 'eteen to successive signaling

elements rather than the a'solute phase.

"#;#. Diff!r!ntia- BPS+

"#;#.#I DBPS+ transmitt!r.

%igure 2+57a shos a simplified 'loc- diagram of a

differential inary phase-shift keying (@B*"#) transmitter. !n

incoming information 'it is Q?FRed ith the preceding 'it

prior to entering the B*"# modulator ('alanced modulator).

%or the first data 'it, there is no preceding 'it ith hich to

compare it. herefore, an initial reference 'it is assumed. %igure 2+

57' shos the relationship 'eteen the input data, the Q?FR

output data, and the phase at the output of the 'alanced modulator.

$f the initial reference 'it is assumed a logic 1, the output from the

Q?FR circuit is simpl the complement of that shon.

$n %igure 2+57', the first data 'it is Q?FRed ith the

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here

H 3 carrier poer (atts)

N 3 noise poer (atts)

"tated in dB, H A ? (dB) 3 1& log DH A ?

3 H(dBm) K ?(dBm) (2.52)

Energ per 'it is simpl the energ of a single 'it of information.

athematicall, energ per 'it is

E ' 3 H ' (A'it) (2.55)

here

8 3 energ of a single 'it (Loules per

'it)

' 3 time of a single 'it (seconds)H 3 carrier poer (atts)

"tated in dB, E '(dB) 3 1& log E ' (2.56)

and 'ecause " $9f ' here f is the 'it rate in 'its per second,

8 can 'e reritten as

E ' 3 H A f ' (A'it) (2.5=)

"tated in dB, E '(dB) 3 1& log H A f ' (2.5)

3 1& log H K 1& log f ' (2.57)

6=


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?oise poer densit is the thermal noise poer normali4ed to a 1+

84 'andidth (i.e., the noise poer present in a 1+84 'andidth).

athematicall, noise poer densit is

?o 3 ? A B (>A84) (2.5A 84 ) (2.61)

"tated in dBm, ?o(dBm) 3 1& log (#A&.&&1) 9 1& log (2.62)

Energ per 'it+to+noise poer densit ratio is used to compare

to or more digital modulation sstems that use different

transmission rates ('it rates), modulation schemes (%"#, *"#,

!), or encoding techniques (+ar).

athematicall, 8 ' 9N o is

8 ' 9N o " (/9f ) 9 ( N9B) (2.65)

here 8 ' 9N o is the energ per 'it+to+noise poer densit ratio.

Rearranging Equation 2.65 ields the folloing e0pression:

8 ' 9N o " (/9N ) : ( B9f ) (2.66)

here

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