Angle Modulation, III Lecture topics FM Modulation (review) FM Demodulation Spectral pre-emphasis/de-emphasis to improve SNR
NBFM Modulation For narrowband signals, k f a(t) 1 and k p m(t) 1, ˆϕ NBFM A(cosω c k f a(t)sinω c t) ˆϕ NBPM A(cosω c k f m(t)sinω c t) We can use a DSB-SC modulator with a phase shifter. Phase modulation Frequency modulation
NBFM: Bandpass Limiter In practice, this modulation will not be perfect, and there will be some amplitude modulation remaining. To fix this, follow with a limiter and a bandpass filter. For FM, m(t) a(t) Asin(ω c t) + ˆφ FM (t) φ FM (t) π/2 Acos(ω c t) Narrowband FM Modulator Limiter Bandpass Filter For PM, the message m(t) does not need to be integrated.
Armstrong s indirect method Armstrong wanted to generate WBFM using NBFM and frequency multipliers. Suppose we have a nonlinear device with the response y(t) = a 0 +a 1 x(t)+a 2 x 2 (t)+ +a n x n (t) Let a(t) = t m(u)du and x(t) = Acos(ω ct+k f a(t)). Then y(t) = a 0 +a 1 cos(ω c t+k f a(t))+ +a n (cos(ω c t+k f a(t))) n A bandpass filter isolate the FM signal with carrier frequency nω c.
FM Demodulation Frequency-selective filter RC high-pass filter: H(f) = j2πrc 1+j2πRC j2πrc (2πRC 1) RLC circuit with carrier frequency ω c < ω 0 = 1/ LC Differentiator Slope detection Zero-crossing detectors Phase-locked loop (not discussed today)
Derivative Theorem for Fourier Transform If G(f) is the Fourier transform of g(t), then and Proof : dg(t) dt = d dt = t dg(t) dt j2πf G(f) g(τ)dτ G(f) j2πf + 1 2 G(0)δ(f) G(f)e j2πft dt d dt G(f)ej2πft dt = j2πf G(f)e j2πft dt By the Fourier inversion theorem, j2πf G(f) is transform of g (t).
Slope-Detecting Filter Information in an FM signal is contained in the instantaneous frequency ω i (t) = ω c +k f m(t) We can extract ω i using a slope-detecting filter, where H(f) = a2πf +b
FM Demodulator and Differentiator
Envelope Detection using Ideal Differentiator If ω = k f m p < ω c we can use envelope detection. ϕ FM (t) = d ( t )) Acos (ω c t+k f m(u) du dt t ) = A sin (ω c t+k f m(u) du (ω c +k f m(t)) t ) = A(ω c +k f m(t)) sin (ω c t+k f m(u)du π Envelope of ϕ FM (t) is A(ω c +k f m(t)). Important that A is constant.
FM Detection Circuits RC high-pass filter. The transfer function is The impulse response is H(f) = j2πrcf 1+j2πRCf j2πrcf (2πRC 1) h(t) = δ(t) e t/rc u(t)
Advantages of FM FM is less susceptible to amplifier nonlinearities. If input is and the output is x(t) = Acos(ω c t+ψ(t)) y(t) = a 0 +a 1 x(t)+a 2 x 2 (t)+ = c 0 +c 1 cos(ω c t+ψ(t))+c 2 cos(2ω c t+2ψ(t))+ The extra terms have spectrum outside the carrier signal band. They will be blocked by bandpass filter. Nonlinearities in AM cause signal distortion. For y(t) = ax(t)+bx 3 (t), y(t) = am(t)cosω c t+bm 3 (t)cos 3 ω c t = (am(t)+ 3 4 bm3 (t))cosω c t+ 1 4 bcos3ω ct FM is preferred for high-power applications, such as microwave relay towers.
Advantages of FM (cont.) FM can adjust to rapid fading (change of amplitude) using automatic gain control (AGC). FM is less vulnerable to signal interference from adjacent channels. Suppose interference is I cos((ω c +ω)t). Then received signal is where r(t) = Acos(ω c t)+icos((ω c +ω)t) = (A+I cosωt)cosω c t Isinωtsinω c t = E r (t)cos(ω c t+ψ(t)) ( ) Isinωt ψ(t) = tan 1 I sinωt (I A) A+Icosωt A The output of an ideal frequency modulator is ψ(t) for FM is y d (t) = Iω A cosωt, which is inversely proportional to amplitude A.
Noise and FM Suppose that the power spectrum of noise is flat over an FM channel. E.g., white noise has constant power spectrum H Z (f) = N 0 2 The power of the noise in a frequency band of width 2B is 2 fc+b f c B N 0 2 df = 2BN 0 The transfer function for FM demodulator satisfies H(f) = af +b. This filter increases noise at higher frequencies. We can reduce high frequency noise by using pre-emphasis/de-emphasis.
FM Pre-emphasis and De-emphasis Pre-emphasis: RLC high pass filter. De-emphasis: RC low pass filter.
FM Pre-emphasis and De-emphasis (cont.) The linear pre-emphasis range is f 1 = 2.1 khz to f 2 = 30 khz. Pre-emphasis filter has transfer function If f f 1 then H p (f) 1. H p (f) = f 2 f 1 +j2πf f 1 f 2 +j2πf If f 1 f f 2 then which is a differentiator! H p (f) j2πf f 1 The corresponding de-emphasis filter has transfer function. H d (f) = f 1 j2πf +f 1 1 H p (f)
FM Pre-emphasis and De-emphasis Filters 5 4 3 2 1 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x 10 4 2 1.5 1 0.5 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x 10 4 2.5 2 1.5 1 0.5 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x 10 4
FM Pre-emphasis and De-emphasis Filters (cont.) 10 2 10 1 10 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10 0 10 1 10 2 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10 0 10 2 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000