chapter 6 - analog modulation and demodulationsilage/chapter6svu.pdf · ee4512 analog and digital...

EE4512 Analog and Digital Communications Chapter 6

Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation


Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation•• Amplitude ModulationAmplitude Modulation

•• Pages 306Pages 306--309309


•• The analytical signal for double sideband, large carrier The analytical signal for double sideband, large carrier amplitude modulation (DSBamplitude modulation (DSB--LC AM) is:LC AM) is:

ssDSBDSB--LC LC AMAM(t(t) =) = AACC (c + (c + s(ts(t)) )) coscos (2(2ππ ffCC t)t)

where where c c is the is the DC bias DC bias or or offset offset and and AACC is the carrier is the carrier amplitude. The continuous analog signal amplitude. The continuous analog signal s(t)s(t) is a baseband is a baseband signal with the information content (voice or music) to be signal with the information content (voice or music) to be transmitted.transmitted.


•• The baseband power spectral density (PSD) spectrum of The baseband power spectral density (PSD) spectrum of the information signal the information signal s(ts(t) or ) or S(fS(f) for voice has significant ) for voice has significant components below 500 Hz and a bandwidth of < 8 kHz:components below 500 Hz and a bandwidth of < 8 kHz:

S(fS(f) =) = FF(s(t(s(t))))The The singlesingle--sidedsided spectrum of the modulated signal is:spectrum of the modulated signal is:

FF(A(ACC (c + (c + s(ts(t)) )) coscos (2(2ππ ffCC tt)) = )) = S(fS(f –– ffCC))

Power Spectral Density of s(t)Power Spectral Density of s(t)500 Hz500 Hz 8 kHz8 kHz

dBdB


•• TheThe singlesingle--sidedsided (positive frequency axis) spectrum of the (positive frequency axis) spectrum of the modulated signalmodulated signal replicates the baseband spectrum as a replicates the baseband spectrum as a doubledouble--sidedsided spectrum about the carrier frequency.spectrum about the carrier frequency.

Carrier 25 kHzCarrier 25 kHzDoubleDouble--sided spectrumsided spectrum

Baseband spectrumBaseband spectrum


•• TheThe doubledouble--sided modulated spectrum about the carrier sided modulated spectrum about the carrier frequency has an frequency has an lowerlower ((LSBLSB) and ) and upperupper ((USBUSB) sideband.) sideband.


•• The modulated DSBThe modulated DSB--LC AM signal shows an LC AM signal shows an outer envelope outer envelope that follows the polar baseband signal s(t)that follows the polar baseband signal s(t)..


•• The analytical signal for double sideband, suppressed The analytical signal for double sideband, suppressed carrier amplitude modulation (DSBcarrier amplitude modulation (DSB--SC AM) is:SC AM) is:

ssDSBDSB--SC SC AMAM(t(t) =) = AACC s(t) s(t) coscos (2(2ππ ffCC t)t)

where where AACC is the carrier amplitude. The is the carrier amplitude. The singlesingle--sided sided spectrum of the modulated signal replicates the baseband spectrum of the modulated signal replicates the baseband spectrum as a doublespectrum as a double--sided spectrum about the carrier sided spectrum about the carrier frequencyfrequency but but without without a carrier component.a carrier component.


•• The analytical signal for double sideband, suppressed The analytical signal for double sideband, suppressed carrier amplitude modulation (DSBcarrier amplitude modulation (DSB--SC AM) is:SC AM) is:


where where AACC is the carrier amplitude. The modulated signal is the carrier amplitude. The modulated signal ssDSBDSB--SC SC AMAM(t(t) looks similar to s(t) but has a temporal but not ) looks similar to s(t) but has a temporal but not spectral carrier component.spectral carrier component.


•• The DSBThe DSB--LC AM and the DSBLC AM and the DSB--SC AM modulated signals SC AM modulated signals have the same have the same sidebandssidebands..

Carrier 25 kHzDSB-LC AM

DSB-SC AM No carrier


•• The modulated DSBThe modulated DSB--LC AM and the DSCLC AM and the DSC--SC AM signals SC AM signals are different.are different.


•• The modulated DSBThe modulated DSB--SC AM signal has an SC AM signal has an envelope envelope that that follows the polar baseband signal s(t) but not an outer follows the polar baseband signal s(t) but not an outer envelopeenvelope..


Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation•• Coherent DemodulationCoherent Demodulationof AM Signalsof AM Signals

•• Pages 309Pages 309--315315


•• The DSBThe DSB--SC AM signal can be simulated in SC AM signal can be simulated in SystemVueSystemVue..

SVU Figure 1.64 modifiedSVU Figure 1.64 modified


•• The DSBThe DSB--SC AM SC AM coherent receivercoherent receiver has a has a bandpass filterbandpass filtercentered at fcentered at fCC and with a bandwidth of and with a bandwidth of twice twice the bandwidth the bandwidth of s(t) because of theof s(t) because of theLSB and USBLSB and USB. The. Theoutput of the multiplieroutput of the multiplieris is lowpass filteredlowpass filtered withwitha bandwidth equal toa bandwidth equal tothe bandwidth of s(t).the bandwidth of s(t).

r(t) = r(t) = γγ ssDSBDSB--SCSC(t(t) + ) + n(tn(t))


Bandpass filter

Lowpass filter

S&M Figure 6S&M Figure 6--44

z(t)


•• The DSBThe DSB--SC AM received signal is r(t) = SC AM received signal is r(t) = γγ ssDSBDSB--SCSC(t(t) + ) + n(tn(t).).The bandpass filter passes the modulated signal but filters The bandpass filter passes the modulated signal but filters the noise:the noise:

z(tz(t) = ) = γγ ssDSBDSB--SCSC(t(t) + n) + noo(t) S&M Eq. 6.3(t) S&M Eq. 6.3

nnoo(t) has a Gaussian distribution.(t) has a Gaussian distribution.The bandpass filter has a centerThe bandpass filter has a centerfrequency of ffrequency of fCC = 25 kHz and a= 25 kHz and a--3 dB bandwidth of 8 kHz3 dB bandwidth of 8 kHz(25 (25 ±± 4 kHz).4 kHz).

Gaussian noise

Bandpass filter

no(t)


•• The filter noise nThe filter noise noo(t) has a (t) has a flat power spectral density flat power spectral density within the bandwidth of the bandpass filter:within the bandwidth of the bandpass filter:

PSD

no(t)

fC = 25 kHz

21 kHz 29 kHz


•• The filter noise nThe filter noise noo(t) can be described as a (t) can be described as a quadraturequadraturerepresentation:representation:

nnoo(t) = (t) = WW(t) (t) coscos (2(2ππ ffCCtt) + ) + ZZ(t) sin ((t) sin (22ππ ffCCtt) S&M Eq. 5.62R) S&M Eq. 5.62R

In the coherent receiver the noise is processed:In the coherent receiver the noise is processed:

nnoo(t) (t) coscos (2(2ππ ffCCtt)) = = WW(t) cos(t) cos22 (2(2ππ ffCCtt) + S&M Eq. 6.5 ) + S&M Eq. 6.5 ZZ(t) (t) coscos (2(2ππ ffCCtt)) sin (2sin (2ππ ffCCtt))

PSD

fC = 25 kHz

21 kHz 29 kHz


•• Applying the Applying the trignometrictrignometric identityidentity the filter noise nthe filter noise noo(t) is:(t) is:

nnoo(t) (t) coscos (2(2ππ ffCCtt)) = = ½½ WW(t)(t) + ½½ WW(t) (t) coscos (4(4ππ ffCCtt) +) +½½ ZZ(t) sin (4(t) sin (4ππ ffCCtt) S&M Eq. 6.5) S&M Eq. 6.5

After the lowpass filter in the receiver the demodulated After the lowpass filter in the receiver the demodulated signal is:signal is:

ssdemoddemod(t) = (t) = ½½ γγ AAC C s(t) + s(t) + ½½ WW(t)(t) S&M Eq. 6.7S&M Eq. 6.7

PSD

fC = 25 kHz

21 kHz 29 kHz


•• The transmitted DSBThe transmitted DSB--SC AM signal is:SC AM signal is:


The average normalized biThe average normalized bi--sided power of sided power of ssDSBDSB--SCSC(t(t) is ) is found in the spectral domain with found in the spectral domain with S(fS(f) =) = FF (s(t)):(s(t)):

[ ]−∫2

2trans C C

1P = A S(f f ) + S(f + f ) df2

S&M Eq. 6.8S&M Eq. 6.8


•• The dualThe dual--sided spectral do not sided spectral do not overlap overlap (at zero frequency) (at zero frequency) and the and the cross termscross terms are zero so that:are zero so that:

where Pwhere Pss is the average normalized power of s(t).is the average normalized power of s(t).

[ ]−∫2

2trans DSB-SC C DSB-SC C

2

trans s

1P = A S (f f ) + S (f + f ) df2

AP = P2

S&M Eq. 6.9S&M Eq. 6.9


•• The The average normalized power of s(t) is found in the average normalized power of s(t) is found in the spectral domain:spectral domain:

In a In a noiseless channelnoiseless channel the power in the demodulatedthe power in the demodulatedDSBDSB--SC AM signal is:SC AM signal is:

∫ ∫2 2

s CP = S(f) df = S(f + f ) df S&M Eq. 6.10S&M Eq. 6.10

= =2

2 2demod, noiseless s trans

1 γP γ A P P4 2 S&M Eq. 6.11S&M Eq. 6.11


•• The The average normalized power of the processed noise is:average normalized power of the processed noise is:

The signalThe signal--toto--noise power ratio then is:noise power ratio then is:

=processed noise o1P N (2 B)4

= =

2

2transtrans

coherent DSB-SCo

o

γ P γ P2SNR 1 N B N (2 B)4

S&M Eq. 6.12S&M Eq. 6.12

2 B2 B


•• The DSBThe DSB--SC AM coherent receiverSC AM coherent receiver requires a phase requires a phase and frequency synchronous reference signal. If the and frequency synchronous reference signal. If the reference signal has areference signal has aphase error phase error φφ then:then:

S&M Eq. 6.17S&M Eq. 6.17

coscos (2(2ππ ffCCtt + + φφ))



ϕ

=coherent DSB-SC phase error

2 2 trans

o

SNR

γ cos PN B


•• The DSBThe DSB--SC AM coherent receiverSC AM coherent receiver requires a phase requires a phase and frequency synchronous reference signal. If the and frequency synchronous reference signal. If the reference signal has areference signal has afrequency error frequency error ∆∆f f then:then:

SSdemoddemod frequency frequency errorerror(t(t) =) =½½ γγ AAC C s(t) s(t) coscos (2(2ππ ∆∆f t)f t)+ + ½½ XX(t(t) ) coscos (2(2ππ ∆∆f t)f t)+ + ½½ YY(t(t) sin (2) sin (2ππ ∆∆f t)f t)

S&M Eq. 6.18S&M Eq. 6.18





•• Although the noise component remains the same, the Although the noise component remains the same, the amplitude amplitude of the demodulated signal varies with of the demodulated signal varies with ∆∆ff::

SSdemoddemod frequency frequency errorerror(t(t) =) =½½ γγ AAC C s(t) s(t) coscos (2(2ππ ∆∆f t)f t)+ + ½½ XX(t(t) ) coscos (2(2ππ ∆∆f t)f t)+ + ½½ YY(t(t) sin (2) sin (2ππ ∆∆f t)f t)

S&M Eq. 6.18S&M Eq. 6.18





•• The The frequency error DSBfrequency error DSB--SC AMSC AM signal can be simulated in signal can be simulated in SystemVueSystemVue..


Frequencysweep


Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation•• NonNon--coherent Demodulationcoherent Demodulationof AM Signalsof AM Signals

•• Pages 315Pages 315--326326


•• The The nonnon--coherent AMcoherent AM (DSB(DSB--LC) receiver uses an LC) receiver uses an envelope envelope detector detector implemented as a implemented as a semiconductor diodesemiconductor diode and a and a lowlow--pass filterpass filter::

The DSBThe DSB--LC AM analytical signal is:LC AM analytical signal is:

ssDSBDSB--LC LC AMAM(t(t) = ) = AACC (c + (c + s(ts(t)) )) coscos (2(2ππ ffCC t)t)

where c is the DC bias (offset).where c is the DC bias (offset).


•• The envelope detector is a The envelope detector is a halfhalf--wave rectifierwave rectifier and and provides a provides a DC bias DC bias ((cc) to the processed DSB) to the processed DSB--LC AM LC AM signal :signal :

c = DC bias


•• The output of the halfThe output of the half--wave diode rectifier is lowwave diode rectifier is low--pass pass filtered to remove the carrier frequency and outputs the filtered to remove the carrier frequency and outputs the envelope which is the information:envelope which is the information:


•• The DSBThe DSB--LC AM signal can be decomposed as:LC AM signal can be decomposed as:

ssDSBDSB--LC LC AMAM(t(t) = ) = s(ts(t) ) coscos (2(2ππ ffCC t) + At) + AC C c c coscos (2(2ππ ffCC t)t)S&M Eq. 6.20RS&M Eq. 6.20R

The average normalized power of the information term: The average normalized power of the information term:

=2C

info term SAP P2 S&M Eq. 6.23S&M Eq. 6.23


•• The average normalized transmitted power is:The average normalized transmitted power is:

Since Since s(ts(t) + c must be >= 0 to avoid distortion in the) + c must be >= 0 to avoid distortion in theDSBDSB--LC AM signal: c LC AM signal: c ≥≥ | min [| min [s(ts(t)] | or c)] | or c22 ≥≥ ss22(t) for all t.(t) for all t.

[ ]=

=

∫T

2carrier term C C

02 2C

carrier term

1P A c cos(2πf t) dtT

A cP2

S&M Eq. 6.24S&M Eq. 6.24


•• Therefore cTherefore c22 ≥≥ PPs s and for DSBand for DSB--LC AMLC AM::

The power efficiency The power efficiency ηη of a DSBof a DSB--LC AM signal is:LC AM signal is:

≥carrier term info termP P S&M Eq. 6.28S&M Eq. 6.28

η ≤ info term info term

carrier term info term trans DSB-LC AM term

P P = = 0.5P + P P

S&M Eq. 6.29S&M Eq. 6.29


•• The DSBThe DSB--LC AM signal LC AM signal wastes wastes at least half the at least half the transmitted power because the power in the carrier term transmitted power because the power in the carrier term has no information:has no information:

The The modulation index modulation index m is defined as:m is defined as:

≥carrier term info termP P η ≤ 0.5

[ ] [ ][ ] [ ]

−max s(t) + c min s(t) + cm =

max s(t) + c + min s(t) + cS&M Eq. 6.30S&M Eq. 6.30


•• The The modulation index modulation index m defines the power efficiency but m defines the power efficiency but m m must be less than 1. If m > 1 then min [must be less than 1. If m > 1 then min [s(ts(t) + c] < 0 and ) + c] < 0 and distortion occurs.distortion occurs.

[ ] [ ][ ] [ ]

−max s(t) + c min s(t) + cm =

max s(t) + c + min s(t) + cS&M Eq. 6.30S&M Eq. 6.30


•• The The average normalized power of the demodulation average normalized power of the demodulation noiseless DSBnoiseless DSB--LC AM signal is:LC AM signal is:

Then the signalThen the signal--toto--noise power ratio for the DSBnoise power ratio for the DSB--LC AM LC AM signal is:signal is:

= 2demod, noiseless info termP 2 γ P

= =2 2

info term trans DSB-LCnoncoherent DSB-LC

o o

2 γ P γ PSNR N (2 B) N B

S&M Eq. 6.40S&M Eq. 6.40

2 B2 B

S&M Eq. 6.39S&M Eq. 6.39


Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation•• Coherent and NonCoherent and Non--CoherentCoherentAM DemodulationAM Demodulation

•• Pages 51Pages 51--5555


•• The The coherent coherent AM (DSBAM (DSB--LC) analog communication system LC) analog communication system can be simulated in can be simulated in SystemVuSystemVuee..

SVU Figure 1.64SVU Figure 1.64


•• The The nonnon--coherentcoherent AM (DSBAM (DSB--LC) analog communication LC) analog communication system can also be simulated in system can also be simulated in SystemVuSystemVuee..



•• The The nonnon--coherent AMcoherent AM (DSB(DSB--LC) receiver is the LC) receiver is the crystal crystal radio radio which needs no batteries! Power for the highwhich needs no batteries! Power for the high--impedance ceramic earphone is obtained directly from the impedance ceramic earphone is obtained directly from the transmitted signaltransmitted signal. For simplicity, the RF BPF is omitted . For simplicity, the RF BPF is omitted and the audio frequency filter is a simple and the audio frequency filter is a simple RCRC network.network.



Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation•• Frequency Modulation andFrequency Modulation andPhase ModulationPhase Modulation

•• Pages 334Pages 334--343343


•• The analytical signal for an The analytical signal for an analog phase modulatedanalog phase modulated (PM) (PM) signal is:signal is:

ssPMPM(t(t) =) = AACC coscos [2[2ππ ffCC t + t + αα s(ts(t)] S&M Eq. 6.53)] S&M Eq. 6.53

where where αα is the is the phase modulation constantphase modulation constant radrad/V and /V and AACC is is the carrier amplitude. The continuous analog signal the carrier amplitude. The continuous analog signal s(t)s(t) is a is a baseband signal with the information content (voice or baseband signal with the information content (voice or music) to be transmitted.music) to be transmitted.


•• The analytical signal for an The analytical signal for an analog frequency modulatedanalog frequency modulated(FM) signal is:(FM) signal is:

ssFMFM(t(t) =) = AACC coscos{ 2{ 2ππ [f[fCC + k + k s(ts(t)] t + )] t + φφ] S&M Eq. 6.53] S&M Eq. 6.53

where where k k is the is the frequency modulation constantfrequency modulation constant Hz / V, Hz / V, AACCis the carrier amplitude and is the carrier amplitude and φφ is the initial phase angle atis the initial phase angle att = 0. t = 0. The continuous analog signal The continuous analog signal s(ts(t)) is a baseband is a baseband signal with the information content.signal with the information content.


•• The The instantaneous phaseinstantaneous phase of the PM signal is:of the PM signal is:

ΨΨPMPM(t(t) = 2) = 2ππ ffC C t + t + αα s(ts(t) S&M Eq. 6.56) S&M Eq. 6.56

The The instantaneous phaseinstantaneous phase of the FM signal is:of the FM signal is:

ΨΨFMFM(t(t) = ) = 22ππ [f[fCC + k + k s(ts(t)] t + )] t + φφ] S&M Eq. 6.57] S&M Eq. 6.57

The instantaneous phase is also call the The instantaneous phase is also call the angleangle of the signalof the signal..The The instantaneous frequencyinstantaneous frequency is the time rate of change of is the time rate of change of the angle:the angle:

f(tf(t) = (1/2) = (1/2ππ)) ddΨΨ(t) / (t) / dtdt S&M Eq. 6.58S&M Eq. 6.58


•• The instantaneous frequency of the unmodulated carrier The instantaneous frequency of the unmodulated carrier signal is:signal is:

ffcarriercarrier(t(t) = d) = dΨΨcarriercarrier(t(t) / ) / dtdt = = d/dtd/dt {2{2ππ ffCCtt + + φφ} S&M Eq. 6.59} S&M Eq. 6.59

The instantaneous phase is also:The instantaneous phase is also:t t tt

ΨΨ(t) = (t) = ∫∫ f (f (λλ) d) dλλ = = ∫∫ f (f (λλ) d) dλλ + φ S&M Eq. 6.60S&M Eq. 6.60--∞∞ 00

There are There are practical limitspractical limits on instantaneous frequency and on instantaneous frequency and instantaneous phase. To avoid ambiguity and distortion in instantaneous phase. To avoid ambiguity and distortion in FM signals due to FM signals due to phase wrappingphase wrapping::

k k s(ts(t) ) ≤≤ ffCC for all t S&M Eq. 6.61for all t S&M Eq. 6.61


•• To avoid ambiguity and distortion in PM signals due to To avoid ambiguity and distortion in PM signals due to phase wrapping:phase wrapping:

--ππ < < αα s(ts(t) ) ≤≤ ππ radians for all t S&M Eq. 6.61radians for all t S&M Eq. 6.61

Since FM and PM are both change the angle of the carrier Since FM and PM are both change the angle of the carrier signal as a function of the analog information signal signal as a function of the analog information signal s(ts(t), FM ), FM and PM are called and PM are called angle modulationangle modulation..

For example, is this signal FM, PM or neither:For example, is this signal FM, PM or neither:t t

x(tx(t) = A) = ACC coscos { 2{ 2ππ ffCCtt ++ ∫∫ k s(k s(λλ) d) dλλ + + φφ}} S&M Eq. 6.60S&M Eq. 6.60--∞∞


•• The instantaneous phase of the signal is:The instantaneous phase of the signal is:tt

ΨΨxx(t(t) = 2) = 2ππ ffCCtt ++ ∫∫ k s(k s(λλ) d) dλλ + + φφ S&M Eq. p. 336S&M Eq. p. 336--∞∞

which is which is notnot a linear function of a linear function of s(ts(t) so the signal is ) so the signal is not PMnot PM. . The instantaneous frequency of the signal is:The instantaneous frequency of the signal is:

ffxx(t(t) = (1/2) = (1/2ππ) ) ddΨΨxx(t(t) / ) / dtdt = = ffCC + k + k s(ts(t) / 2) / 2ππ

and the frequency difference and the frequency difference ffxx –– ffC C is a linear function of is a linear function of s(ts(t) ) so the signal is FM.so the signal is FM.

The The maximum phase deviationmaximum phase deviation of a PM signal isof a PM signal ismax | max | ααs(ts(t) |. The ) |. The maximum frequency deviationmaximum frequency deviation of a FM of a FM signal is signal is ∆∆f = max | k f = max | k s(ts(t) |. ) |.


•• The spectrum of a PM or FM signal can be developed as The spectrum of a PM or FM signal can be developed as follows: S&M follows: S&M EqsEqs. 6.64 through 6.71. 6.64 through 6.71

C C m

C m

C m

C m C m

C

v(t) = A sin(2π f t + β sin 2π f t)v(t) = Re { exp(j 2π f t + j β sin 2π f t) }

exp(j 2π f t + j β sin 2π f t) = cos (2π f t + β sin 2π f t) + j sin (2π f t + β sin 2π f t)v(t) = Im { A exp(2π

now

( )

∞

∞

∞

∞

∑

∑

C m

m n mn = -

m n mn = -

f t + jβ sin 2π f t) }

exp(j β sin 2π f t) = c exp(j 2π n f t)

exp(j β sin 2π f t) = J β exp(j 2π n f t)

now

after further development Bessel function of the first kindBessel function of the first kind


•• Bessel functions of the first kind Bessel functions of the first kind JJnn((ββ)) are tabulated for FM are tabulated for FM with single tone fwith single tone fmm angle modulation (S&M Table 6.1):angle modulation (S&M Table 6.1):

n

β


•• For single tone fFor single tone fmm angle modulation the spectrum is periodic angle modulation the spectrum is periodic and infinite in extentand infinite in extent::

n

β

( )∞

∞∑C n m C

n = -v(t) = A J β sin[2π (n f + f ) t] S&M Eq. 6.72S&M Eq. 6.72


•• The The complexity complexity of the Bessel function solution for the of the Bessel function solution for the spectrum of a single tone angle modulation can be spectrum of a single tone angle modulation can be simplified by the simplified by the CarsonCarson’’s Rule approximations Rule approximation for the for the bandwidth bandwidth BB. Since . Since ββ = = ∆∆f / ff / fmm::

B = 2 (B = 2 (ββ + 1) f+ 1) fmm = 2 (= 2 (∆∆f + ff + fmm) Hz) Hz

n

β

S&M Eq. 6.74S&M Eq. 6.74


•• CarsonCarson’’s Rules Rule for the approximate bandwidth of an angle for the approximate bandwidth of an angle modulated signal was developed by John R. Carson in modulated signal was developed by John R. Carson in 1922 while he worked at AT&T. Prior to this in 1915 he 1922 while he worked at AT&T. Prior to this in 1915 he presaged the concept of presaged the concept of bandwidthbandwidthefficiencyefficiency in AM by proposingin AM by proposingthe suppression of a sidebandthe suppression of a sideband(see S&M p. 326(see S&M p. 326--333)333)::

B = 2 (B = 2 (ββ + 1) f+ 1) fmm = 2 (= 2 (∆∆f + ff + fmm) Hz ) Hz

18861886--19401940


•• The The normalized powernormalized power within the Carsonwithin the Carson’’s Rule bandwidth s Rule bandwidth for a single tone angle modulated signals is:for a single tone angle modulated signals is:

Note that JNote that J--nn((ββ) = ) = ±± JJnn((ββ) so that ) so that JJ--nn22((ββ) = J) = Jnn

22((ββ) and for the ) and for the normalized power calculation the sign of J(normalized power calculation the sign of J(ββ) ) is not used.is not used.

( )= ∑2 β+1

2Cin-band, sinusoid n

n = -(β+1)

AP J β 2 S&M Eq. 6.75S&M Eq. 6.75

Spectrum of single tone FM modulation


Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation•• Frequency ModulationFrequency Modulation

•• Pages 55Pages 55--5757


•• The analog FM transmitter and receiver can be simulated in The analog FM transmitter and receiver can be simulated in SystemVueSystemVue. A bandpass audio filter removes the low . A bandpass audio filter removes the low frequency components in the voice signal for clarity. frequency components in the voice signal for clarity.



•• A phaseA phase--locked loop (PLL) token has a frequency output locked loop (PLL) token has a frequency output which tracks the frequency deviation which tracks the frequency deviation ∆∆f which is f which is proportional to the voice signal. proportional to the voice signal.



•• The PLLThe PLLtoken istoken issomewhatsomewhatcomplex.complex.



•• The analog FM power spectral density PSD of the voice The analog FM power spectral density PSD of the voice signal has a bandwidth predicted only by Carsonsignal has a bandwidth predicted only by Carson’’s Rule s Rule since it is not a single tone.since it is not a single tone.

PSDPSD

VoiceVoice


•• Here fHere fmaxmax = 4 kHz, k = 25 Hz/V and = 4 kHz, k = 25 Hz/V and ∆∆ffmaxmax = 40(25) = 1 kHz. = 40(25) = 1 kHz. The CarsonThe Carson’’s Rule approximate maximum bandwidths Rule approximate maximum bandwidthB = 2 (B = 2 (∆∆f + ff + fmm) = 10 kHz or ) = 10 kHz or ±± 5 kHz (but seems wrong!)5 kHz (but seems wrong!)

PSDPSD

VoiceVoice40

Bandwidth

fC


•• A 200 Hz A 200 Hz single tonesingle tone FM signal has a PSD with periodic FM signal has a PSD with periodic terms at fterms at fCC ±± n fn fmm = 25 = 25 ±± 0.2 n kHz.0.2 n kHz.

PSDPSD

fC200 Hz200 Hz


•• Here fHere fmm = 200 Hz, k = 25 Hz/V and = 200 Hz, k = 25 Hz/V and ∆∆ffmax max = 40(25) = 1 = 40(25) = 1 kHz. The CarsonkHz. The Carson’’s Rule approximate maximum bandwidths Rule approximate maximum bandwidthB = 2 (B = 2 (∆∆f + ff + fmm) = 2.4 kHz or ) = 2.4 kHz or ±± 1.2 kHz:1.2 kHz:

PSDPSD

fC200 Hz200 Hz

BandwidthBandwidth


•• Since Since ββ = = ∆∆f / ff / fmm = 1 kHz / 0.2 kHz = 5 and the Bessel = 1 kHz / 0.2 kHz = 5 and the Bessel function predicts a bandwidth of 2 n ffunction predicts a bandwidth of 2 n fmm = 2(12)(200) = = 2(12)(200) = 4.8 kHz (since n = 12 for 4.8 kHz (since n = 12 for ββ = 5 from Table 6.1):= 5 from Table 6.1):

PSDPSD

fC200 Hz200 Hz

BandwidthBandwidth


Chapter 6Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation•• Noise in FM and PM SystemsNoise in FM and PM Systems

•• Pages 347Pages 347--355355


•• A general angle modulated transmitted signal, where A general angle modulated transmitted signal, where ΨΨ(t) (t) is the instantaneous phase, is the instantaneous phase, is:is:

ssangleangle--modulatedmodulated(t(t) = A) = ACC coscos [[ΨΨ(t(t)] )] S&M Eq. 6.86S&M Eq. 6.86

The received signals is:The received signals is:

rrangleangle--modulatedmodulated(t(t) = ) = γγ AACC coscos [[ΨΨ(t)] + (t)] + n(tn(t) S&M Eq. 6.87) S&M Eq. 6.87


•• The analytical signal for PM is:The analytical signal for PM is:

ssPMPM(t(t) = A) = ACC coscos [[ΨΨ(t)](t)] = AACC coscos [2[2ππ ffC C t t + + αα s(ts(t)])]S&M Eq. 6.53S&M Eq. 6.53

After development the SNR for demodulated PM is:After development the SNR for demodulated PM is:

SNRSNRPMPM = (= (αγαγ AACC))22 PPS S / (2 N/ (2 Noo ffmaxmax) ) S&M Eq. 6.98S&M Eq. 6.98

where where −−ππ < < αα s(ts(t) ) ≤≤ ππ for all t.for all t.


•• The analytical signal for FM is:The analytical signal for FM is:

ssFMFM(t(t) = A) = ACC coscos [[ΨΨ(t)](t)] = AACC coscos [2[2ππ ffC C t t + + ∫∫ k s(k s(λλ) d) dλλ]]S&M Eq. 6.53S&M Eq. 6.53

After development the SNR for demodulated FM is:After development the SNR for demodulated FM is:

SNRSNRFMFM = 1.5 (k = 1.5 (k γγ AAC C /(2/(2ππ) ) ))22 PPS S / (N/ (Noo ffmaxmax33) ) S&M Eq. 6.98S&M Eq. 6.98

where k where k s(ts(t) ) ≤≤ ffCC for all t.for all t.


End of Chapter 6End of Chapter 6

Analog ModulationAnalog Modulationand Demodulationand Demodulation

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