Frequency Modulation KEEE343 Communication Theory Lecture #15, April 28, 2011 Prof. Young-Chai Ko koyc@korea.ac.kr
Summary Angle Modulation Properties of Angle Modulation Narrowband Frequency Modulation
Properties of Angle-Modulated Wave Property 1: Constancy of transmitted wave The amplitude of PM and FM waves is maintained at a constant value equal to the carrier amplitude for all time. The average transmitted power of angle-modulated wave is a constant P av = 1 2 A2 c Property 2: Nonlinearity of the modulated process m(t) =m 1 (t)+m 2 (t) s(t) =A c cos [2 f c t + k p (m 1 (t)+m 2 (t))] s 1 (t) =A c cos(2 f c t + k p m 1 (t)), s(t) 6= s 1 (t)+s 2 (t) s 2 (t) =A c cos(2 f c t + k p m 2 (t))
[Ref: Haykin & Moher, Textbook]
Property 3: Irregularity of zero-crossings Zero-crossings are defined as the instants of time at which a waveform changes its amplitude from a positive to negative value or the other way around The irregularity of zero-crossings in angle-modulation wave is attributed to the nonlinear character of the modulation process. The message signal m(t) increases or decreases linearly with time t, in which case the instantaneous frequency f i (t) of the PM wave changes form the unmodulated carrier frequency f c to a new constant value dependent on the constant value of m(t)
Property 4: Visualization difficulty of message waveform The difficulty in visualizing the message waveform in angle-modulated waves is also attributed to the nonlinear character of angle-modulated waves. Property 5: Tradeoff of increased transmission bandwidth for improved noise performance The transmission of a message signal by modulating the angle of a sinusoidal carrier wave is less sensitive to the presence of additive noise
Example of Zero-Crossing Consider a modulating wave m(t) given as m(t) = at, t 0 0, t < 0 where a is the slope parameter. In what follows we study the zero-crossing of PM and FM waves for the following set of parameters f c = 1 4 [Hz] a = 1 volt/s
[Ref: Haykin & Moher, Textbook]
Phase modulation: phase-sensitivity factor PM wave is k p = 2 radians/volt. Then, the s(t) = Ac cos(2 f c t + k p at), t 0 A c cos(2 f c t), t < 0 t n /2 Let denote the instance of time at which the PM wave experiences a zerocrossing; this occurs whenever the angle of the PM wave is an odd multiple of. Then we may set up 2 f c t n + k p at n = 2 + n, n =0, 1, 2,... t n as the linear equation for. Solving this equation for, we get the linear formula t n = 1 2 + n 2f c + k p a = 1 2 t n + n, n =0, 1, 2,... f c =1/4 [Hz] and a = 1 volt/s
Frequency modulation Let k f =1. Then the FM wave is s(t) = Ac cos(2 f c t + k f at 2 ), t 0 A c cos(2 f c t), t < 0 Invoking the definition of a zero-crossing, we may set up 2 f c t n + k f at 2 n = 2 + n, n =0, 1, 2,... t n = 1 ak f f c + s f 2 c + ak f 1 2 + n!, n =0, 1, 2,... t n = 1 4 1+ p 9 + 16n, n =0, 1, 2,... f c =1/4 [Hz] and a = 1 volt/s
Comparing the zero-crossing results derived for PM and FM waves, we may make the following observations once the linear modulating wave begins to act on the sinusoidal carrier wave: For PM, regularity of the zero-crossing is maintained; the instantaneous frequency changes from the unmodulated value of f c =1/4Hz to the new constant value of f c + k p (a/2 )= 1 2 Hz. For FM, the zero-crossings assume an irregular form; as expected, the instantaneous frequency increases linearly with time t
Relationship between PM and FM An FM wave can be generated by first integrating the message signal m(t) with respect to time t and thus using the resulting signal as the input to a phase modulation. A PM wave can be generated by first differentiating m(t) with respect to time t and then using the resulting signal as the input to a frequency modulator. We may deduce the properties of phase modulation from those frequency modulation and vice versa.
[Ref: Haykin & Moher, Textbook]
Narrow-Band Frequency Modulation Narrow-Band FM means that the message signal has narrow bandwidth. Consider the single-tone wave as a message signal, which is extremely narrow banded: m(t) =A m cos(2 f m t) FM signal Instantaneous frequency f i (t) = f c + k f A m cos(2 f m t) = f c + f cos(2 f m t) f = k f A m Phase i(t) = 2 Z t 0 f i ( ) d =2 = 2 f c t + f f m sin(2 f m t) apple f c t + f 2 f m sin(2 f m t)
Definitions Phase deviation of the FM wave Modulation index of the FM wave: = f f m f m Then, FM wave is s(t) =A c cos[2 f c t + sin(2 f m t)] s(t) =A c cos(2 f c t) cos( sin(2 f m t)) A c sin(2 f c t)sin( sin(2 f m t)) For small compared to 1 radian, we can rewrite cos[ (2 f m t)] 1, sin[ sin(2 f m t)] sin(2 f m t) s(t) A c cos(2 f c t) A c sin(2 f c t)sin(2 f m t)
[Ref: Haykin & Moher, Textbook]
Polar Representation from Cartesian Consider the modulated signal which can be rewritten as where s(t) =a(t) cos(2 f c t + (t)) s(t) =s I (t) cos(2 f c t) s Q (t)sin(2 f c t) s I (t) =a(t) cos( (t)), and, s Q (t) =a(t)sin( (t)) and a(t) = s 2 I(t)+s 2 Q(t) 1 2, and (t) = tan 1 apple sq (t) s I (t)
Approximated narrow-band FM signal can be written as s(t) A c cos(2 f c t) A c sin(2 f c t)sin(2 f m t) Envelope a(t) =A c 1+ 2 sin 2 (2 f m t) 1/2 Ac 1+ 1 2 1/2 2 sin 2 (2 f m t) Angle (t) =2 f c t + (t) =2 f c t + tan 1 ( sin(2 f m t)) for small Using the power series of the tangent function such as tan 1 (x) x 1 3 x3 +
Angle can be approximated as (t) 2 f c t + sin(2 f m t) 1 3 3 sin 3 (2 f m t) Ideally, we should have (t) 2 f c t + sin(2 f m t) The harmonic distortion value is D(t) = 3 3 sin3 (2 f m t) The maximum absolute value of D(t) is D max = 3 3
For example for D max = 0.33 3 =0.3, =0.009 1% which is small enough for it to be ignored in practice.
Amplitude Distortion of Narrow-band FM Ideally, FM wave has a constant envelope But, the modulated wave produced by the narrow-band FM differ from this ideal condition in two fundamental respects: The envelope contains a residual amplitude modulation that varies with time The angle i(t) contains harmonic distortion in the form of thirdand higher order harmonics of the modulation frequency f m