Amplitude Modulation Ang Man Shun October 30, 01 Reference Hwei P. Hsu Analog and Digital Communication Summary Message Carrier Simple AM DSB-LC DSB-SC SSB / VSB Equation m(t) Large Carrier Unity A m cos ω m t A c cos ω c t cos ω c t (A c + m(t)) cos ω c t = A c cos ω c t + m(t) cos ω c t A c cos ω c t + ma c ma c + ma c m(t) cos ω c t + ma c ma c cos (ω c ± ω m ) t Bandwidth Advantage Disadvantage Simple AM DSB-LC ω m $ Low η DSB-SC ω m Highη $$$ SSB/VSB ω m High η Small BW $$$$$ 1
1 Introduction Message signal and Carrier Signal in time domain A m cos ω m t = A m (ejω mt + e jω mt ) m(t) Unity Carrier cos ω c t = 1 (ejωct + e jωct ) Larger Carrier A c cos ω c t = A c (ejω ct + e jω ct ) frequency component in same amplitude Negative frequency is not real, just mathematical ω c > ω m, if not,aliasing 1.1 Continuous Wave Modulation CW modulation is just a multiplication, for montone message and larger carrier : (A m cos ω m t) (A c cos ω c t) By identity cos A cos B = cos (A + B) + cos (A B) A m A c { + } = A ma c 4 { e j(ω c ω m )t + e j(ω c ω m )t + e j(ω c+ω m )t + e j(ω c ω m )t } One cosine term has frequency spectral line, so cosine terms have 4 spectral line Frequency spectra line translated Negative frequency is not real, it is only mathematical, so although it has 4 spectra, it actually has only spectra 1. The signal s Fourier Transfroms For general message m(t) F M (jω m ) CW Modulated signal of general message with large carrier m(t)a c cos ω c t = A c m(t) ( e jωct + e jωct) F A c [M (jω m + jω c ) + M (jω m jω c )]
Amplitude Modulation There are 3 to 4 basic types of AM modulation : Simple AM (DSB-LC), DSB-SC, SSB, and VSB.1 Simple AM signal in Time Domain AM : Message signal embed into the amplitude of carrier Simple AM is also called : Double Sideband Large Carrier (DSB-LC) Modultion For general message signal m(t) : x AM (t) = (A c + m(t)) cos ω c t For simple monotone message signal m(t) = A m cos ω m t : x AM (t) = (A c + A m cos ω m t) cos ω c t = A c cos ω c t + A m cos ω m t cos ω c t By cos A cos B = = A c cos ω c t + A c m cos ω m t cos ω c t where m = A m A c cos (A + B) + cos (A B) = A c cos ω c t + ma c [ + ] = A c cos ω c t + ma c Simple AM / DSB-LC modulated signal is thus x AM (t) = (A c + m(t)) cos ω c t = A c cos ω c t + m(t) cos ω c t x AM (t) = (A c + m(t)) cos ω c t = A c cos ω c t + ma c Envelope of modulated signal will follow the message signal Message m = A m A c 100% m = 100% A m = A c m > 100% overmodulation A m > A c Requirements for AM / DSB-LC : A c > A m, ω c ω m 3
. Simple AM signal in Frequency Domain For general message signal The simple AM signal m(t) F M(jω m ) Bandwidth = [0, ω m ] = ω m The AM signal s Fourier Transform x AM (t) = (A c + m(t)) cos ω c t = A c cos ω c t + m(t) cos ω c t F {x AM (t)} = F {A c cos ω c t} + F {m(t) cos ω c t} = A c F {cos ω c t} + 1 F { m(t) ( e jω ct + e jω ct )} = A c F {cos ω c t} + 1 F { m(t)e jω ct } + 1 F { m(t)e jω ct } = A c ˆ F As m(t) M(jω m ) thus m(t)e ±jat F M (jω ± ja) = A c cos ω c te jωt dt + 1 ˆ ˆ m(t)e j(ω ω c)t dt + 1 ˆ ( e j(ω ω c )t + e j(ω+ω c)t ) dt + 1 M (jω jω c) + 1 M (jω jω c) m(t)e j(ω+ω c)t dt X AM (jω) = A c δ (ω ± ω c) + 1 M (jω jω c) + 1 M (jω jω c) Bandwidth is twice of original bandwidth : BW AM = W = ω m AM wave contains sideband with bandwidth of each band as W LSB = W USB = W = ω m 4
.3 Double-Sideband Suppressed-Carrier Modulation DSBSC Recall, simple AM / DSB-LC for m(t) = A m cos ω m t x AM (t) = (A c + m(t)) cos ω c t = A c cos ω c t + m(t) cos ω c t x AM (t) = A c cos ω c t + ma c Message Carrier contains no info, so suppresse it to enhance power efficiency After dropping the carrier term, x DSB SC (t) = m(t) cos ω c t = m(t) cos ω c t x DSB SC (t) = ma c Message For general message signal m(t) F M(t), the Fourier Transform is F { } } x(t) DSB-SC = F {m(t) cos ωc t} = F {m(t) ejω ct + e jω ct = 1 [M (jω jω c) + M (jω + jω c )].4 Generation & Demodulation of DSB signal 5
For Simple AM / DSB-LC After mixing : x AM (t) cos ω c t = [A c + m(t)] cos ω c t After passing LPF : A c + m(t) After passing capacitor to block the DC : m(t) Thus, with suitable amplifier, the original signal can be recovered. * Then this demodulator works A c = [A c + m(t)] 1 + cos ω ct = A c + m(t) + A c + m(t) cos ω c t Since carrier is suppressed, so A m 0, and thus m = A m for DSB-SC signal. A c so modulation index is meaningless.5 Single-Sideband Suppressed Carrier To improve power efficiency, further dropping one sideband For monotone signal, the DSB-SC signal is x DSB SC (t) = ma c After dropping one sideband After LPF LSB x SSB LSB (t) = ma c After HPF HSB x SSB USB (t) = ma c Vestigial Sideband Signal Since SSB require a very sharp cut-off filter to remove one sideband, such filter is not easy to implement Thus, the requirement is relaxed by allowing a vestige part : Vestigial Sideband signal 6
3 Power of the AM Signal 3.1 Review of Root-Mean-Square value For a function A cos ωt, the RMS value is : ˆ 1 T RMS (A cos ωt) = (A cos ωt) 1 dt = A T T 0 ˆ T 0 1 cos ωtdt = A T ˆ T 0 1 + cos ωt dt Since cos θ is orthogonal to 1, so the second integral is zero ˆ 1 T 1 = A T 0 dt + 1 ˆ T T cos ωtdt = A T 0 T = A }{{} So the RMS value of a function of A cos ωt is A Then, recall that simple AM / DSB-LC signal of montone message has the form x AM (t) = (A c + m(t)) cos ω c t = A c cos ω c t + A cm Then the RMS value is Thus the total power, by P = A R, is x AM,RMS = A c + A cm + A cm 0 + A cm,am = A c R + A cm + A cm The power used to transmitte information for simple AM is thus : When m = 1 η AM = P Info = A cm + A cm A c R + A cm + A cm = m 4 + m 4 1 + m 4 + m 4 = m 4 + m = m + m η AM = 1 3 3 = 66.6%Power Lost Therefore, simple AM signal is not power-efficienct. η AM < 33.3% = sup η AM In summary, for simple AM / DSB-LC signal, The efficiency is limited to 33% The carrier signal is present even if nothing is being transmitted The circuitary is relatively simple (only envelop detector is required! ) Bandwidth is ω m 7
For Dobule Sideband Suppressed Carrier of montone message, the wave form is x DSB (t) = x AM (Without Carrier) = A cm Thus, the RMS value is + A cm Thus, the Total power is And hence, the power efficiency is x DSB,RMS = A cm + A cm,dsb = A cm + A cm η DSB = P Info = = 100% (Ideal) The power efficiency of DSB singal is very good, but the tradeoff is it require relatively expensive circuitry in the receiver In summary, for DSB signal It have must higher power efficiency ( 100% ) But it has same bandwidth as simple AM, ω m It require relatively expensive circuitry in the receiver For SSB singal, a sideband filter, either high pass or low pass, is concatenated to the receiver circuit. For Single Sideband Suppressed Carrier of montone message, the wave form is Thus, the RMS value is Thus, the Total power is x SSB (t) = A cm cos (ω c ± ω m ) t x DSB,RMS = A cm And hence, the power efficiency is,ssb = A cm In summary, for SSB signal, η SSB = P Info = = 100% (Ideal) It has high power efficiency ( 100% ) It has relatively most expensive circuitry ( An extra sideband filter ) It cut bandwidth in half, BW SSB = ω m END 8