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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)
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. $f there are eight possi'le conditions, = < and so forth.
he num'er of 'its necessar to produce a given num'er of
conditions is e0pressed mathematicall as
M N 2log= (2.=)
here N 3 num'er of 'its necessar M 3 num'er of conditions, levels, or com'inations
possi'le ith N 'its
Equation 2+= can 'e simplified and rearranged to e0press the
num'er of conditions possi'le ith N 'its as
#
N
"M (2.)
6
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Baud 3
N
f (2.11)
B comparing Equation #!$% 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
7
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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+;.
%$RE 2+; *NN+%"# demodulator
!s the input to the *NN shifts 'eteen the mar- and space
frequencies, the dc error &oltage at the output of the phase
comparator follos the frequenc shift.
Because there are onl to input frequencies (mar- and space),
there are also onl to output error voltages. Fne represents a
logic 1 and the other a logic &.
Binar %"# has a poorer error performance than *"# or ! and,
consequentl, is seldom used for high+performance digital radio
sstems.
1;
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$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
56
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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
6
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