ece 4710: lecture #22 1 frequency modulation fm spectrum is very difficult to calculate in general ...
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ECE 4710: Lecture #22 1
Frequency Modulation
FM spectrum is very difficult to calculate in general Useful to develop simple approximations when the
modulation index, , is small or large
Recall that
“small” so F B / 5 Narrowband FM “large” so F > 5 B Wideband FM
FM complex envelope
where
BF
)()( tjceAtg
dmDtt
f )()(
rad/s are units )(
abledummy vari
mD f
ECE 4710: Lecture #22 2
Narrowband FM
Narrowband FM = NBFM Restrict phase angle to small value For small values the exponential function in complex
envelope can be approximated using first two terms of Taylor series expansion
So NBFM signal at Tx output
2.0|)(| t
1for 1 xxex
1 |)(|for )](1[)( )( tjtjAeAtg ctj
c
]2sin[]2cos[)(Re
]2sin[]2cos[)](1[Re
)(Re)( 2
tfjtftjAA
tfjtftjA
etgts
cccc
ccc
tfj c
ECE 4710: Lecture #22 3
Narrowband FM
NBFM signal at Tx output (continued)
Carrier Term does NOT depend on (t) modulation» Just like large carrier (LC) in AM
Sideband Term depends on (t) modulation and is 90° out of phase with respect to carrier» Note that (t) is modulating amplitude of sine !!
» Different than AM where sidebands are in phase with LC
]2sin[)(]2cos[
]2sin[]2cos[)(Re)(
tftAtfA
tfjtftjAAts
cccc
cccc
Carrier Term Sideband Term
ECE 4710: Lecture #22 4
NBFM Spectrum
2.0BF
f
)2sin()( tftm m
Carrier
Sidebands
]2sin[)(]2cos[)( tftAtfAts cccc
)()([2
)]()([2
)( ccf
ccc ffMffM
f
Dffff
AfS
LC DSB
ECE 4710: Lecture #22 5
NBFM Tx
LC
DSB
]2sin[)(]2cos[)( :NBFM tftAtfAts cccc
ECE 4710: Lecture #22 6
Wideband FM
FM instantaneous frequency of s(t) varies in direct proportion to modulating signal voltage m(t) If m(t) stays at one particular voltage more than another then s(t) will
spend more time at one particular frequency
PSD of s(t) should have more power at frequencies associated with m(t) voltages that are more likely
Probability Density Function (PDF) of m(t) can be used to approximate PSD for s(t) for wideband FM (WBFM)
)(
2)(
2
2)(
2
c
f
mc
f
m
f
c ffD
fffD
fD
Af
P
)( of PDF where tmfm
ECE 4710: Lecture #22 7
Wideband FM
Equal probability for any voltage between Vp
and +Vp
Uniform PDF for m(t) and rectangular spectrum for
WBFM
Triangle Modulating m(t) PDF of m(t)
Approximate WBFM spectrum for triangle modulation
ECE 4710: Lecture #22 8
Wideband FM Tx
VCO = Voltage Controlled Oscillator
Phase Lock Loop (PLL) configuration produces DC output at LPF to
ensure stable carrier when center frequency of VCO drifts away from fc
ECE 4710: Lecture #22 9
Digital FM Signals
What happens if a digital waveform is used for m(t) in an FM system?? Instantaneous frequency of s(t) varies in time about
assigned carrier frequency in direct proportion to modulating signal voltage m(t)
Two voltage states (± Ac) two discrete frequency values
(fH, fL) @ output of FM Tx
)(21
)( tmDftf fci
)(tm+Ac
VCOL
H
f
f
Ac
ECE 4710: Lecture #22 10
Digital FM Signals
)(ts
VCOL
H
f
f
Hf Lf Hf Lf Hf Lf
Binary Frequency Shift Keying (BFSK)
)(tm+Ac
Ac
1 0 1 0 1 0
1 0 1 0 1 0
special case of WBFM with digital m(t)
ECE 4710: Lecture #22 11
BFSK Signal Spectrum
Binary Digital m(t) PDF of m(t)
Approximate WBFM spectrum for BFSK
ECE 4710: Lecture #22 12
Digital AM & FM
BPSK Signal
PolarModulation
BFSK Signal
Special Case of DSB-SC
Special Case of
FM
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
ECE 4710: Lecture #22 13
FDM
Frequency Division Multiplexing = FDM Technique to transmit multiple messages simultaneously
over a wideband channel Modulate multiple messages onto subcarriers
» Composite baseband signal formed from multiple signals
Composite signal spectrum most not have overlapping spectrum message “crosstalk” (interference)
Composite signal modulated onto main RF carrier Any type of modulation can be used for carrier and sub-
carriers (AM, FM, DSB, etc.)» Different modulation types can be used within same FDM system» Example: DSB-SC for subcarriers and FM for carrier FM Stereo
ECE 4710: Lecture #22 14
FDM Tx
Transmitter
Composite Baseband Signal Spectrum
ECE 4710: Lecture #22 15
FDM Rx
Receiver
1. Received FDM signal demodulated to recover composite baseband signal
2. Passed thru multiple bandpass filters to separate individual subcarriers
3. Subcarriers demodulated to recover multiple message signals
ECE 4710: Lecture #22 16
FM Stereo Broadcast
FM Stereo Broadcast in U.S. is FDM system FM carrier modulation DSB-SC sub-carrier modulation Left & Right (L/R) audio channels for stereo signal L+R signal used to allow mono Rx (outdated)
» 15 kHz baseband audio signal BW
L-R signal modulated on 38 kHz DSB-SC sub-carrier» 38 kHz sub-carrier signal BW (2 x 15 kHz)» 38 / 2 = 19 kHz pilot tone for coherent demodulation of sub-carrier
Rx uses sum & difference of L+R and L-R to recover L and R channels for stereo speakers
ECE 4710: Lecture #22 17
FM Stereo Tx
Transmitter
Composite Baseband Signal Spectrum
23kHz
53kHz
fL + R
Composite Baseband Signal Spectrum
23kHz
53kHz
L + R
DSB-SC SpectrumLR
ECE 4710: Lecture #22 18
FM Stereo Rx
L + R
L R
fp = 19 kHz
fVCO = 38 kHz
ECE 4710: Lecture #22 19
FM Stereo Broadcast
FCC can authorize FM station for subsidiary communication authorization (SCA) Up to 4 SCA channels authorized First SCA subcarrier is usually 67 kHz
» Second analog audio program» Background “elevator” music Musac service
Secondary service of FM station for revenue generation
Other subcarriers used for data or analog audio» Radio Broadcast Data System (RBDS)
57 kHz subcarrier with 1200 bps dataAuxillary text information (music titles, artist, traffic reports, etc.)
ECE 4710: Lecture #22 20
FM Stereo Spectrum
fComposite Baseband Signal Spectrum with SCA Channels
23kHz
53kHz
L + R
DSB-SC SpectrumLR
67kHz
57kHz
2nd Audio“Musac”Data
75kHz
59kHz
1. Non-Commercial Stations 88.1 to 91.9 MHz
2. Commercial Stations 92.1 to 107.9 MHz
3. RF Channel BW = 200 kHz
4. F = 75 kHz from RF channel carrier frequency
5. 2F = 150 kHz < 200 kHz channel BW guard BW to minimize interference between stations