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

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

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

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

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

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

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

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

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

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

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

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

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

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

FM Signal Spectrum ECE 4710: Lecture #21

FM Signal Spectrum ECE 4710: Lecture #21

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