1. Frequency Modulation ECE 4710: Lecture #21 Overview:
Mathematical analysis of FM signal & spectrum is very complicated
FM is normally a wideband modulation method where BFM >> baseband signal B
Spectrally inefficient
FM is a non-linear modulation method
AM & DSB-SC are linear methods
Non-linear method allows one to trade RF signal BW for non-linear increase in Rx S/N
Power for bandwidth tradeoff
FM is the most widely used analog modulation method for mobile radio  used extensively from 1940-present day
Spectrally efficient digital methods have replaced FM in many mobile radio applications (e.g. cellular)
Congested spectrum requires spectral efficiency for large user populations
ECE 4710: Lecture #21

2. Frequency Modulation
Complex envelope for phase modulation (PM) and frequency modulation (FM) is:
The real envelope R(t) = | g(t) | = Ac  constant value
PM and FM are constant envelope modulation methods
AM & DSB-SC are linear since R(t) is linearly  to m(t)
Bandpass signal is:
Relationship between q (t) & m(t) determines type of modulation  PM or FM
For PM we have
Linear relationship between phase and modulating signal
Still nonlinear modulation since:
ECE 4710: Lecture #21

3. Frequency Modulation
In the constant Dp is the phase sensitivity of the phase modulator with units of rad/V
Larger Dp means greater phase change in s(t) for every volt of m(t)  assuming m(t) is voltage signal
FM is a special case of PM
Frequency is the time derivative of phase: So
Let Df = frequency deviation constant with units rad/V•s then
ECE 4710: Lecture #21

4. PM & FM Circuits
PM circuit  pass unmodulated RF signal through circuit that introduces time variation in phase
FM circuit  vary frequency of tuned RF oscillator circuit by varying resonant frequency of the circuit
Varactor diodes have variable capacitance controlled by amount of voltage applied across diode
Time varying capacitance used to generate time-varying phase for PM and time-varying frequency for FM
ECE 4710: Lecture #21

5. PM & FM Circuits
PM Circuit
FM Circuit
ECE 4710: Lecture #21

6. PM & FM
PM: and FM:
So and
Integrator and differentiator circuits (op-amp) can be used on baseband m(t) to generate PM from FM and vice versa
Integrator + PM circuit = FM output
Differentiator + FM circuit = PM output
ECE 4710: Lecture #21

7. Frequency Modulation ECE 4710: Lecture #21
For this class we are mainly interested in FM since it is the most widely used analog modulation method in the world  analog mobile radio
Bandpass signal &
Instantaneous frequency of s(t) is
and for FM we have
Instantaneous frequency varies in time about assigned carrier frequency by an amount determined by Df and modulating information signal m(t)
ECE 4710: Lecture #21

8. Frequency Modulation ECE 4710: Lecture #21
For illustration purposes assume m(t) is sinusoidal:
DF = peak frequency deviation
ECE 4710: Lecture #21

9. Note Constant FM Signal Envelope!!
Frequency Modulation
Note Constant FM Signal Envelope!!
fc + DF fc
fc - DF
Non-linear Class C or D power amplifiers with high DC to RF efficiency can be used for FM since amplitude of RF signal does not contain any information
ECE 4710: Lecture #21

10. Frequency Modulation Instantaneous frequency
Frequency of s (t) at specific point in time
Fourier Transform frequency
FT spectrum for s (t) is evaluated for interval - < t < +
FT spectrum represents frequency content of s (t) over all time  **average** frequency content
Peak frequency deviation is given by
ECE 4710: Lecture #21

11. Frequency Modulation ECE 4710: Lecture #21
Peak frequency deviation related to RF signal BW
Increasing amplitude of modulating signal (Vp) increases DF and RF signal BW
Spectral components appear farther an farther away from fc
Note that average power of FM signal is constant value =
Independent of FM signal BW
As BW  the spectral components near fc must decrease in strength since the average power remains constant
AM or DSB-SC  increase in m(t) affects Tx output power but not RF signal BW
FM  increase in m(t) affects RF signal BW but not Tx output power
ECE 4710: Lecture #21

12. Frequency Modulation Frequency modulation index: FM signal spectrum
B is absolute bandwidth of m(t)
For sinusoidal m(t)  B = fm  sinusoid frequency
Sinusoidal modulation is often used for demonstration purposes and simplified calculations
Actual m(t) is usually non-deterministic (e.g. voice)
FM signal spectrum
and
g(t) is nonlinear function of m(t)  no general relationship relating G(f ) to M(f ) !!  evaluated on case by case basis for deterministic waveforms only
ECE 4710: Lecture #21

13. FM Signal Spectrum ECE 4710: Lecture #21
Assume sinusoidal modulation signal:
where
Complex envelope is:
Baseband signal spectrum can be shown to be:
Infinite series line spectrum (e.g. d) spaced by fm with amplitudes determined by first order Bessel functions Jn (b)
Jn (b) can only be evaluated numerically
ECE 4710: Lecture #21

14. Bessel Functions ECE 4710: Lecture #21 1. Note that for f = fc  n = 0
2. Carrier spectral amplitude is  | J0 (b ) |
3. Modulation index b determines carrier
strength
4. b can be selected such that | J0 (b ) | = 0
 e.g. b = 2.4, 5.5, etc.
ECE 4710: Lecture #21

15. FM Signal Spectrum
ECE 4710: Lecture #21

16. FM Signal Spectrum
ECE 4710: Lecture #21

17. FM Signal Spectrum RF signal BW depends on bf and B
Computation of RF signal BW using standard definitions (3-dB BW) is very difficult for anything but simplified m(t)  sinusoidal
Computer computations for non-deterministic waveforms (data, voice, etc.)
Carson’s rule  approximate BW for 98% total power 
Simple and therefore very useful
Widely used rather than computational approach for estimating signal BW
ECE 4710: Lecture #21