Amplitude Modulation “AM” with Envelope Detector Large S/N limit m1< s(t) ≅ sin ωm t ≤ 1

Recei ved = y(t) = Ac [ 1 + m s ( t)] cos ωc t + n c ( t)cos ω c t + n s ( t)s in ω c t j ω t = Re { Y (t)e c } slowly varying

y, low-pass filtered (envelope)

“overmodulation” = distortion (“undermodulation = wasted power)

Lec 16b.6-1 2/2/01 T1 Amplitude Modulation “AM” with Envelope Detector

Recei ved = y(t) = Ac [ 1 + m s ( t)] cos ωc t + n c ( t)cos ω c t + n s ( t)s in ω c t j ω t = Re { Y (t)e c } slowly varying

Im { Y(t)} Y(f ) 2W Y(t)

ns(t) n(t)

R e { Y(t)} f 0 0 fc A [1 + m s(t)] nc(t) N = kT c o 2

Lec 16b.6-2 2/2/01 T2 Amplitude Modulation “AM” with Envelope Detector

Im { Y(t)} Y(f ) 2W Y(t)

ns(t) n(t)

R e { Y(t)} f 0 0 fc A [1 + m s(t)] nc(t) N = kT c o 2 NY(t) ≅ Ac [ 1 + m s (t)] + nc (t) envelope = detected signal + noise

2 2 2 2 2 Note: 4WNo = n c cos ω c t + n s sin ω c t = n c

222 Sout 222 2 Ac ms (t) ≅ Ac m s ( t) nc (t) = Nout 4WNo Lec 16b.6-3 2/2/01 T3 Amplitude Modulation “AM” with Envelope Detector

222 Sout 222 2 Ac ms (t) ≅ Ac m s ( t) nc (t) = Nout 4WNo

A 2 2 ( 1 + m s(t)) 2 Sin ( c ) 2 ≅ where Sin = y signal (t) Nin 4WNo

22 ∆ SNi i 1m+ s (t) 11+ 2 Noi se figure FAM = = ≥ = 3 2 ⇒ FAM ≥ 3 2 SNo o 2m 22 s 1

provided that Ac >> nc (large S/N limit)

Lec 16b.6-4 2/2/01 T4 AM Performance (small S/N limit)

Im { Y}

φ (t) n A1c [ + m s (t)]

n(t) ≅ Ac [ 1 + m s ( t)] cos φn (t)

φn (t) R e {} Y

Y(t) ≅ n(t) + A cos φ (t) + A m s ( t)cos φ (t) c n c  n multiplicative noise!

Want Sin N in ≥ 10 for fully i ntel ligibl e AM ⇒ "AM threshold"

(i.e. A c (1 + m s (t) ) > 3 n (t))

Lec 16b.6-5 2/2/01 T5 Frequency and Phase Modulation (FM, PM)

Transmitted: x(t ) = A c cos [ω+φ c t (t) ]

Phase Modulation ("PM"): φ(t) =∆ K's (t) Frequency Modulation ("FM"):

d φ ∆ t =∆ω(t) = 2 π K s (t) r s-1 for s(t) < 1 , or φ (t) = s(τ )d τ dt ( ) ∫

vo x(t) frequency-sensitive +Ac discriminator output t function 0 ωc -A ω c FM signal

v(t) vo filter, low pass filter, receiver: y(t) discriminator output limiter DC block Lec 16b.6-6 2/2/01 envelope detector V1 FM Bandwidth Expansion Factor β*

d φ S(f) =∆ω(t) = 2 π K s (t) ⎡ r s-1⎤ for s(t) < 1 dt ⎣ ⎦

β=∗ ∆ KW f -W 0 W

Y(f ) intrinsic bandwidth P ∗ B ≅ 2W ( 1 +β), so 2W < B ≤ 2K + W

f 0 B ≅ 2K

∗ For PM: B ≅ 2 (K ' + 1) W ( not proven here) (if β >> 1)

Lec 16b.6-7 2/2/01 V2 Vector Signal Analysis of PM/FM

j ωc t Recei ved s i gna l y(t) = x(t) + n(t) = r(t)cos (ω c t +ψ (t)) = Re { Y(t)e }

Im {Y (t)} Y(t) rn(t) 2 2 x(t) >> n (t) ns(t) φn - φ(t)

r(t) φn η(t) Ac φ(t) ψ(t) Re { Y(t)} 0 ∆ ψ (t) ≅φ (t) + rn (t)sin (φ n −φ ) A c = φ (t) + ns (t ) A c

Discriminator output:

PM: v(t) =ψ (t) =φ (t) +η (t) =K ' s(t ) + n s (t ) A c  ∆ Lec 16b.6-8 FM: v(t) =ψ (t) 2π= Ks(t ) + n s (t ) 2π A c where ψ = dψ dt 2/2/01 V3 Calculation of PM Sout/Nout

Recall n(t) = n c ( t) cos ω c t − n s ( t) s in ω c t output N(f) slowly slowly kTR 2 varying varying

n(t2 ) = n(t 2 ) = n(t 2 ) = 2NB c s o f 0 B kTR 2

PM: v(t) =ψ (t) =φ (t) +η (t) =K ' s(t ) + ns (t ) A c

22 2 K' s (t) 22 ⎡ A c 2 ⎤ Therefore PM: So N out = = K ' s (t) • 2 ⎢ ⎥ ⎡n(t 2 ) A 2 ⎤ ⎢ 2N o W ⎥ ⎣ s c ⎦ ⎣ ⎦ 22 CNR (want large K ' s (t), 2N o B ≅ 4N o W but approaches FM) for case B = 2W(K´<<1)

Lec 16b.6-9 2/2/01 V4 Calculation of FM Sout/Nout ∆ FM: v(t) =ψ (t) 2π= Ks(t ) + n  s (t ) 2π A c where ψ = dψ dt

n(ts ) ↔ Ns (f ) n(t s ) ↔ j ωN(f s ) ↓ ⇒ ↓ ↓ ∆ 2 2 2 2 N = Ns R(τ) ↔ ω N(fs ) =ω N o n s o

2 FM : S N = K 22 s (t) ⎡ n 2 (t) ( 2π A ) ⎤ out out ⎣ c ⎦

Sout 2W 2 W 4N 3 1 4(2πf ) 2N df = o • W (2A π ) ∫ o 2 3 c 0 A f c 0 W 4 segments

Lec 16b.6-10 2/2/01 V5 Calculation of FM Sout/Nout 2 22 ⎡ Ac 3⎤ Therefore S out N out = K s • ⎢ • 3 2No W ⎥ FM ⎣ 2 ⎦ Pc 3P = c β∗ 22s = 6[CNR ]β ∗ 3 s 2 2N o W (K/W)2 ∗ where CNR( Carr i er-to-No i se Ratio) = Pc 2N o B = P c 2N o 2W β

S "Wide-band FM" (WBFM) S N = ⎛ o ⎞ *2 out out WBFM ⎜ ⎟•3β ⎝ No ⎠DSBSC 2 Psc 2N oW

FM advantage for β≅∗ 5 ( FM radio, 2(β+ ∗ 1)W ≅200 kHz );

Lec 16b.6-11 ∗2 2/2/01 3 β ≅19 dB! V6 Calculation of FM Sout/Nout FM pre-emphasis & de-emphasis filters

2 N(ts ) Since FM noise → ∝ f2

f 0

Pre-emphasis signal ∝ f2 pre-transmission and de-emphasize signal + noise at receiver; this can yield ~10 dB improvement (depending…)

Lec 16b.6-12 2/2/01 V7 “FM Threshold” – (low SNR limit) Im { Y(t)} φ (t) n Ac -φ(t) ψ (t) ≅φn + sin( φ−φn ) ⇒ signal obliteration rn rn(t) A c ranges over 2π Y(t) ψ(t) Therefore must have Ac >> rn Re { Y(t)} to avoid multiplicative noise φn(t) (analogous FM 60 to coding SNROUT for automobile 50 gain via 5 40 40 bandwidths) 2 30 1 20 baseband t 10 0 S/N in threshold dropouts in nulls, (baseline) spaced ~λ/2 10 20 30 (dB)

Lec 16b.6-13 2/2/01 typical FM thresholds V8 Issues In Choosing Modulation Type

1) Desired output SNR 2) Cost of bandwidth ($, availability) (for communications or storage) 3) Standards imposed on channel, inexpensive equipment 4) Potential for source coding 5) Characteristics (noise, fading), potential for channel coding 6) Cost, power, weight, size, thermal constraints on system

Lec 16b.6-14 2/2/01 X1 Output SNR Requirements CD-quality audio: say 40 dB dynamic range (loudest power/“quiet” power)

+55 dB SNR ⇒ 95 dB so 20 LOG10L ≅ 95 Therefore L = 56,000 l eve l s of σ⇒ < 32,000 digital l eve l s, ⇒ 15-bits

FM-quality audio: say β* = 5 ⇒ ~50 dB (~35 dB available above ~15-dB threshold)

Intelligible speech: > 10dB

Video: studio-quality ~40 + dB

Video: home-quality ~20 - 35 + dB

Lec 16b.6-15 2/2/01 X2 Nominal Bandwidth Requirements

1) Voice: ~3 kHz (6kHz excellent) 2) Music ~15 + kHz 3) Video ~6 MHz (NTSC), 20 MHz (HDTV) 4) Data ~10 – 109→7 bits/sec; 104 OK often

Lec 16b.6-16 2/2/01 X3 State-Of-The-Art Source Coding 1) Voice ~1.2, 2.4, 4.8, 9.6 kbps; OK→good 32 – 64 kbps ~ uncompressed ↑ e.g. 8 kHz at 8 bits ~ 58 dB SNR 2) Music 128 – 256 kbps for ~ CD quality, stereo 3) Video 10 kbps jerky, blurred, or little change 56 – 128 kbps ~10 fps, 2562 pixels (lip-read threshold) and artifacts (moving details) 384 kbps good video conference quality 1.5 Mbps ⇒ “VCR” NTSC TV 6 Mbps ⇒ good NTSC TV 20 Mbps ⇒ HDTV 4) Data divide by 2 – 4 for typical miscellaneous data,

Lec 16b.6-17 lossless coding 2/2/01 X4 FM Hybrid Analog Communication System Laser Example 1 FM Modulation x(t) t Laser 0 ∼ ω (t) x(t) s(t) s(t) t 0 Here we use the standard definition of CNR for a superheterodyne receiver.

2 ∗ 3 ∗ S OUT / NOUT = [CNR ] 6 sβ (where β = B / 2W) ↑ Baseband Optical superheterodynes are limited by photon noise that fluctuates with S(t), so the expression here is approximate. Since optical links have great bandwidth, β* can be very large. Lec 16b.6-18 2/2/01 Y1 AM Hybrid Analog Communication System Laser Example 2 AM Modulation x(t) Laser x(t) t 0

S(t1 ) The CNR applies to the unmodulated laser and its detector within the passband of the detector output corresponding to the spectrum of the signal s(t). This CNR must be above the AM threshold of ∼10 dB in order for 22 S OUT / N OUT = [CNR ] ms to apply.

Lec 16b.6-19 2/2/01 Y2 FM/AM Analog Communication System Laser Example 3 FM/AM Modulation s 2 (t) ⇒ AM near f1 Hz Laser Sm t 0 s1(t) s2(t) [FM] f1 (t) var ies with s1 ( t) [FM ] f1 ⎡ 22⎤ ⎡ 2 ∗ 3 ⎤ ∗ S OUT / NOUT ≅ [CNR ] m s1 6s2 β where B = 2Wβ for s 2 (t) ⎣ ⎦AM ⎣ ⎦ FM Assume avalanch photo diode: ⎛ g2 ⎞ ∗ ⎛ PD ⎞ 2kThf [CNR] APD =η( P s / hf ⎡ 2W β⎤ ) ⎜ 1+ + ⎟ ⎣ ⎦ ⎜ G 2 ⎜ P ⎟ 2 ⎟ ⎝ ⎝ s ⎠ RPL η s (eG ) ⎠ ⎡ 22⎤ PD = dark current + background power (W) . Want [CNR] m s1 ⎣ ⎦ AM to be over FM threshold ≅ 15 dB; then chooseβ ∗ to yi eld des i red

S 0 / N O (say 50 dB total) Lec 16b.6-20 2/2/01 Y3