Angle Modulation: Phase Modulation or Frequency Modulation Basic Form of FM signal: Constant Amplitude Information is contained in  (t) Define Phase Modulation Index, m p, to restrict  (t) such that Instantaneous Phase:  (t)  (t) = m p a(t) Total Instantaneous Phase Reference Phase Phase Deviation Normalized Information Signal

Visualizing the FM Phasor... …down at the “Complex Plane” Reference Phasor cc cc FM Phasor  (t) m p = |  MAX | Reference Phase Constant Amplitude cc  (t) = m p a(t)

Instantaneous Frequency Deviation Time Waveform: Phasor Notation: “Carrier Frequency” “Frequency Deviation” Can be very large and still have |a(t)| < 1 High rates of change implies Wide Bandwidth Instantaneous “Frequency Deviation”

Voltage Controlled Oscillator (VCO) f c (hz or R/s ) k 0 (hz/v or R/s/v) VCO fcfc k0k0 v in f out “Free running” frequency Frequency deviation:  f(t) “Reference” phase Phase Deviation:  (t) Frequency of VCO Output Deviation Sensitivity

Simple Case For Analysis

FM Spectrum Spectral Analysis: High Math. Spectrum is characterized by spectral components spaced at + n  m from  c, with amplitudes determined by Bessel Functions J n (m p ). Now all we have to do is take the Fourier Transform of this thing... cc mm J0J0 J3J3 J2J2 J1J1 J4J4

J 1 (m) J 0 (m) J 2 (m) J 3 (m) J 4 (m) J 5 (m) J 6 (m) J 7 (m) J 8 (m) J 9 (m) J 10 (m)

Bessel Function Tabulation The Bessel Function values determine the relative voltage amplitudes of their respective sidebands. The squares of the Bessel Function values determine the relative power amplitudes of their respective sidebands. For any value of m: If our Bandwidth includes N sidebands, then :