Principles of Communications

Meixia Tao

Dept. of Electronic Engineering Shanghai Jiao Tong University Chapter 3: Analog Modulation Selected from Ch 3, Ch 4.1-4.4, Ch 6.1-6.2 of of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi

Meixia Tao @ SJTU Topics to be Covered

AM/FM radio FM radio TV broadcast

Source Modulator Channel Demodulator Output

Amplitude modulation Angle modulation (phase/frequency) Effect of noise on amplitude modulation Effect of noise on frequency modulation

Meixia Tao @ SJTU 2 Modulation

What is modulation? Transform a message into another signal to facilitate transmission over a communication channel Generate a carrier signal at the transmitter Modify some characteristics of the carrier with the information to be transmitted Detect the modifications at the receiver

Why modulation? Frequency translation Frequency-division multiplexing Noise performance improvement

Meixia Tao @ SJTU 3 Analog Modulation

Characteristics that be modified in sin carrier

Amplitude → Amplitude modulation

Frequency → Angle modulation Phase

Meixia Tao @ SJTU 4 Amplitude Modulation Double-sideband suppressed-carrier AM (DSB-SC)

Baseband signal (modulating wave): mt()

Modulated wave

st()= ctmt () () = Amtcc ()cos(ωθ t+ 0 )

Meixia Tao @ SJTU 5 Spectrum of DSB-SC Signals

Spectrum of M(f) message M(0)

-W 0 W f

Spectrum of DSB-SC S(f) (1/2)AcM(0) USB LSB LSB USB

-f -W -f -f +W f -W f f c c c 0 c c fc+W 1 S( f ) = A [M ( f − f ) + M ( f + f )] 2 c c c

Translation of the original message spectrum to

Meixia Tao @ SJTU 6 Bandwidth and Power Efficiency

S(f) (1/2)AcM(0)

-f -W -f -f +W f -W f f c c c 0 c c fc+W

Required channel bandwidth BWc = 2 Required transmit power

TT/2 /2 112 22 2 Ps= lims ( t ) dt = lim Acc m( t ) cos (ωθ t+ 0 ) dt TT→∞ TT∫∫−−TT/2 →∞ /2 22 T /2 AAcc1 2 =limm ( t )[ 1+ cos(2ωθcmt += 20 )] dt P 22T →∞ T ∫−T /2

Meixia Tao @ SJTU 7 Demodulation of DSB-SC Signals

Phase-coherent demodulation

s(t) Product v(t) Low-pass vo(t) modulator filter

cos( 2πfct) Local PLL oscillator (phase-locked loop)

If there is a phase error φ, then

Scaled version of message signal Unwanted

Meixia Tao @ SJTU 8 Demodulation of DSB-SC Signals

Pilot-tone assisted demodulation

Add a pilot-tone into the transmitted signal

Filter out the pilot using a narrowband filter

Meixia Tao @ SJTU 9 Conventional AM Modulating wave Carrier wave: Baseband signal (normalized):

Modulated wave a ≤1 Modulation index: a Modulated wave

a >1

overmodulated

Meixia Tao @ SJTU 10 Spectrum of Conventional AM

Spectrum of M(f) M(0) message signal

-W 0 W f

(Ac/2)δ(f+fc) (Ac/2)δ(f-fc) S(f) (1/2)aAcM(0)

0 -fc-W -fc -fc+W fc-W fc fc+W f

AccAa Sf()=δδ( f −+ fcc) ( f + f) + Mf( −+ fc) Mf( + f c) 22

Meixia Tao @ SJTU 11 Bandwidth and Power Efficiency

Required channel bandwidth BWc = 2 Required transmit power

carrier power message power

Modulation efficiency 2 a 2 AP 2 power in sideband cm aP E = = 2 = m total power A2 a2 1+ aP2 c + AP2 m 22cm

Meixia Tao @ SJTU 12 Example

Message signal: mt ( ) = 3cos(200 ππ t ) + sin(600 t ) −5 Carrier: ct( )= cos(2 × 10t ) Modulation index: a=0.85 Determine the power in the carrier component and in the sideband components of the modulated signal

Meixia Tao @ SJTU 13 Demodulation of AM signals Envelop Detector

The simplicity of envelop detector has made Conventional AM a practical choice for AM-radio broadcasting

Meixia Tao @ SJTU 14 Single Sideband (SSB) AM

Common problem in AM and DSBSC: Bandwidth wastage

SSB is very bandwidth efficient H(ω)

ω

Meixia Tao @ SJTU 15 Expression of SSB signals

The baseband signal can be written as the sum of finite sinusoid signals n mt( )=∑ xi cos(2πθ ft i +≤ i ) , f ic f i=1 Then its USB component is n Ac mtc( )=∑ xi cos[ 2πθ ( f ci++ ft ) i )] 2 i=1 After manipulation nn Ac mc() t=∑∑ xicos(2πθ ft ii +− ) cos 2 πft c x isin(2 πθ ft ii + ) sin 2 πft c 2 ii=11 =

AAcc = mt()cos2ππ ftcc− mt ()sin2 ft 22Hilbert transform of m(t)

Meixia Tao @ SJTU 16 About Hilbert Transform

1∞ x ()τ 1 xt()⇔ xt () xt()= dτ = xt() ∗ πτ∫−∞ tt− π

−>jf,0 Hf()= j , f < 0 0,f = 0 H (ω) 相移 1 90° ω ω -90°

Meixia Tao @ SJTU 17 Generation of SSB-AM Signal

• The spectral efficiency of SSB makes it suitable for voice communication over telephone lines (0.3~3.4 kHz) • Not suitable for signals with significant low frequency components

Meixia Tao @ SJTU 18 Vestigial Sideband: VSB

VSB is a compromise between SSB and DSBSC

M(f) . VSB signal bandwidth is

B = W+fv . VSB is used in TV

-W 0 W f broadcasting and similar signals where low frequency components are significant

S(f)

-f c 0 fc f

W fv fv W

Meixia Tao @ SJTU 19 Comparison of AM Techniques

DSB-SC: more power efficient. Seldom used Conventional AM: simple envelop detector. AM radio broadcast SSB: requires minimum transmitter power and bandwidth. Suitable for point-to-point and over long distances VSB: bandwidth requirement between SSB and DSBSC. TV transmission

Meixia Tao @ SJTU 20 Signal Multiplexing

Multiplexing is a technique where a number of independent signals are combined and transmitted in a common channel These signal are de-multiplexed at the receiver Two common methods for signal multiplexing TDM (time-division multiplexing) FDM (frequency-division multiplexing)

Meixia Tao @ SJTU 21 FDM

LPF: ensure signal bandwidth limited to W

MOD (modulator): shift message frequency range to mutually exclusive high frequency bands

BPF: restrict the band of each modulated wave to its prescribed range

Meixia Tao @ SJTU 22 FDM application in telephone comm.

Voice signal: 300～3400Hz Message is SSB modulated. In 1st-level FDM, 12 signal are stacked in frequency, with a freq. separation of 4 kHz between adjacent carriers A composite 48 kHz channel, called a group channel, transmits 12 voice-band signals simultaneously In the next level of FDM, a number of group channel (typically 5 or 6) are stacked to form a supergroup channel Higher-order FDM is obtained by combining several supergroup channels => FDM hierarchy in telephone comm. systems

Meixia Tao @ SJTU 23 Quadrature-Carrier Multiplexing

Transmit two messages on the same carrier as

st()= Amc12 ()cos2 t( ππ ftcc) + Am ()sin2 t( ftc) cos() and sin() are two quadrature carriers Each message signal is modulated by DSB-SC Bandwidth-efficiency comparable to SSB-AM Synchronous demodulation of m1(t):

2 st( )cos( 2π ftc) = Am c12( t )cos( 2 π ftcc) + Am( t )sin( 2 ππ ftc) cos( 2 ft c) AA A =++ccm() t m ()cos4 t( ππ ft) c m()sin4 t( ft) 2211 cc 22

LPF

Meixia Tao @ SJTU 24 Application: AM Radio Broadcasting

Commercial AM radio uses conventional AM Superheterodyne receiver: from variable carrier freq of the incoming RF to fixed IF

Meixia Tao @ SJTU 25 Topics to be Covered

Source Modulator Channel Demodulator Output

Amplitude modulation Angle modulation (phase/frequency) Effect of noise on amplitude modulation Effect of noise on frequency modulation

Meixia Tao @ SJTU 26 Angle Modulation

. Either phase or frequency of the carrier is varied according to the message signal . The general form:

θ(t): the time-varying phase instantaneous frequency of s(t):

1dtθ () ft()= f + ic2π dt

Meixia Tao @ SJTU 27 Representation of FM and PM signals

Phase modulation (PM)

θ ()t= kmtp () where kp = phase deviation constant

Frequency modulation (FM)

1 d where kf = frequency fi() t−= f cf k mt () = θ () t deviation constant/frequency 2π dt sensitivity

t The phase of FM is θ()t= 2 π k md ( ττ ) f ∫0

Meixia Tao @ SJTU 28 Distinguishing Features of PM and FM

No perfect regularity in spacing of zero crossing Constant envelop, i.e. amplitude of s(t) is constant Relationship between PM and FM ∫ m(t)dt FM wave m(t) Phase integrator modulator π + AC cos[2 f ct k p ∫ m(t)dt]

AC cos(2πf ct)

d m(t) m(t) dt Frequency PM wave differentiator modulator AC cos[2πf ct + 2πk f m(t)]

AC cos(2πf ct) Will discuss the properties of FM only

Meixia Tao @ SJTU 29 Example: Sinusoidal Modulation

Sinusoid modulating wave m(t)

FM wave

d m(t) dt

PM wave

Meixia Tao @ SJTU 30 Example: Square Modulation

Square modulating wave m(t)

FM wave

PM wave

Meixia Tao @ SJTU 31 FM by a Sinusoidal Signal

Message

Instantaneous frequency of resulting FM wave

Frequency deviation: Carrier phase t ∆f θ(t )= 2 πτ( f ( ) −= f) d τsin(2 πft ) ∫0 ic m fm

= βπsin(2ftm ) Modulation index:

Meixia Tao @ SJTU 32 Example

Problem: a sinusoidal modulating wave of amplitude 5V and frequency 1kHz is applied to a frequency modulator. The frequency sensitivity is 40Hz/V. The carrier frequency is 100kHz. Calculate (a) the frequency deviation (b) the modulation index

Solution:

Frequency deviation ∆f = k f Am = 40×5 = 200Hz

∆f 200 Modulation index β = = = 0.2 fm 1000

Meixia Tao @ SJTU 33 Spectrum Analysis of Sinusoidal FM Wave

Rewrite the FM wave as

In-phase component Quadrature-phase component

Define the complex envelop of FM wave

~ jβ sin(2πfmt) s (t) = sI (t) + jsQ (t) = Ace ~ s (t) retains complete information about s(t)

j[2πfct+β sin(2πfmt)] ~ j2πfct s(t) = Re{Ace }= Re[s (t)e ]

Meixia Tao @ SJTU 34 ~ jβ sin(2πfmt) s (t) = Ace is periodic, expanded in Fourier series as ∞ ~ j2πnfmt s (t) = ∑cne n=−∞ where

n-th order Bessel function of the first kind 1 π Jn (ββ )= expj( sin x− nx) dx 2π ∫−π Hence,

cn = Ac J n (β )

Meixia Tao @ SJTU 35 ~ Substituting into s (t) ∞ ~ s (t) = Ac ∑ J n (β )exp( j2πnfmt) n=−∞

FM wave in time domain

FM wave in frequency-domain ∞ Ac S( f ) = ∑ J n (β )[δ ( f − fc − nfm ) +δ ( f + fc + nfm )] 2 c n=−∞

Meixia Tao @ SJTU 36 Properties of Bessel Function

Property 1: for small β ≤0.3 (Narrowband FM) Approximations

J 0 (β ) ≈1

J1(β ) ≈ β 2

J n (β ) ≈ 0, n >1

Substituting above into βA s(t) ≈ A cos(2πf t) + c cos[2π ( f + f )t] c c 2 c m βA J (β ) → 0 as β → ∞ − c cos[2π ( f − f )t] n 2 c m J (β ), n even β = n Approximate bandwidth = J −n ( ) − J n (β ), n odd Discuss the similarity between the conventional AM wave and a narrow band FM wave Meixia Tao @ SJTU 37 General Case

Goal: to investigate how and affect the spectrum Fix and vary and are varied

1.0 1.0

β = 5 β =1

fc fc 2∆f 2∆f

Meixia Tao @ SJTU 38 General Case

Fix and vary is fixed, but is varied

1.0 1.0

β = 5 β =1

fc fc 2∆f 2∆f

Meixia Tao @ SJTU 39 Effective Bandwidth of FW Waves

For large B is only slightly greater than For small The spectrum is limited to

Carson’s Rule:

B≈∆+2 ff 2mm = 2(1 +β ) f

Meixia Tao @ SJTU 40 Effective Bandwidth of FW Waves

99% bandwidth approximation The separation between the two frequencies beyond which none of the side-frequencies is greater than 1% of the unmodulated carrier amplitude i.e where is the max that satisfies

J n (β ) > 0.01

β 0.1 0.3 0.5 1.0 2.0 5.0 10 20 30

2nmax 2 4 4 6 8 16 28 50 70

Meixia Tao @ SJTU 41 Effective Bandwidth of FW Waves

A universal curve for evaluating the 99% bandwidth As increases, the bandwidth occupied by the significant side- frequencies drops toward that over which the carrier frequency actually deviates, i.e. B become less affected by

20

2

0.2 2

Meixia Tao @ SJTU 42 FM by an Arbitrary Message

Consider an arbitrary with highest freq. component W

Frequency deviation: ∆=f kf max mt ( ) Modulation index:

Carson’s rule applies as

Carson’s rule underestimates the FM bandwidth requirement Universal curve yields a conservative result

Meixia Tao @ SJTU 43 Example

In north America, the maximum value of frequency deviation is fixed at for commercial FM broadcasting by radio. Take , typically the maximum audio frequency of interest in FM transmission, the modulation index is

Using Carson’s rule,

Using universal curve,

Meixia Tao @ SJTU 44 Exercise

Assuming that mt ( ) = 10sinc ( 10 4 t ) , determine the transmission bandwidth of an FM modulated signal

with k f = 4000

Solution By Carson’s rule:

Meixia Tao @ SJTU 45 Application: FM Radio broadcasting

As with standard AM radio, most FM radio receivers are of super-heterodyne type . Typical freq parameters – RF carrier range = 88~108 MHz Limiter – Midband of IF = 10.7MHz – IF bandwidth = 200kHz discriminator – Peak freq. deviation = 75KHz

Baseband low-pass filter loudspeaker Audio amplifier with de-emphasis

Meixia Tao @ SJTU 46 Summary

Spectrum of sinusoidal FM Wave

∞ Ac S( f ) = ∑ J n (β )[δ ( f − fc − nfm ) +δ ( f + fc + nfm )] 2 c n=−∞

Carson rule approximation Universal curve approximation

Meixia Tao @ SJTU 47 Generation of FM waves

Direct approach Design an oscillator whose frequency changes with the input voltage => voltage-controlled oscillator (VCO) Indirect approach First generate a narrowband FM signal and then change it to a wideband signal Due to the similarity of conventional AM signals, the generation of a narrowband FM signal is straightforward.

Meixia Tao @ SJTU 48 Generation of Narrow-band FM

Consider a narrow band FM wave s1(t) = A1 cos[2πf1t +φ1(t)]

t where f1 = carrier frequency φ1(t) = 2πk1 m(τ )dτ ∫0 k1 = frequency sensitivity

Given φ1(t) <<1 with β ≤ 0.3, we may use cos[φ1(t)]≈1 sin[φ1(t)]≈ φ1(t)

Correspondingly, we may approximate s1(t) as

s1(t) = A1 cos(2πf1t)− A1 sin(2πf1t)φ1(t) t = A1 cos(2πf1t)− 2πk1 A1 sin(2πf1t) m(τ )dτ ∫0 Narrow-band FW wave

Meixia Tao @ SJTU 49 . Narrow-band frequency modulator

m(t) Product - integrator Modulator + Narrow-band + FM wave s1(t) A1 sin(2πf1t) Carrier wave -900 phase

shifter A1 cos(2πf1t)

. Next, pass s1(t) through a frequency multiplier

Narrow-band Wideband FM FM wave Wave Memoryless Band-pass nonlinear device filter

– The input-output relationship of the non-linear device is:

2 n s2 (t) = a1s1(t) + a2s1 (t) ++ ans1 (t)

– The BPF is used to Pass the FM wave centred at nf1 and with deviation n∆f1 and suppress all other FM spectra

Meixia Tao @ SJTU 50 Example: frequency multiplier with n = 2

. Problem: Consider a square-law device based frequency multiplier 2 s2 (t) = a1s1(t) + a2s1 (t) with t s1(t) = A1 cos2πf1t + 2πk1 m(τ )dτ ∫0 . Specify the midband freq. and bandwidth of BPF used in the freq. multiplier for the resulting freq. deviation to be twice that at the input of the nonlinear device . Solution: t t 2 2 s2 (t) = a1 A1 cos2πf1t + 2πk1 m(τ )dτ + a2 A1 cos 2πf1t + 2πk1 m(τ )dτ ∫0 ∫0 2 2 t t a2 A1 a2 A1 = a1 A1 cos2πf1t + 2πk1 m(τ )dτ + + cos4πf1t + 4πk1 m(τ )dτ ∫0 2 2 ∫0

fc=2f1 Removed by BPF with BW > 2∆f = 4∆f1

Meixia Tao @ SJTU 51 Generation of Wideband FM Signal

t s1(t) = A1 cos2πf1t + 2πk1 m(τ )dτ ∫0

Message Wideband signal Narrow-band Frequency FM signal Output Integrator phase multiplier BPF modulator

Ac cos(2π ft1 ) Crystal-controlled cos(2πflt) oscillator Mixer: perform up/down t conversion to shift the signal s(t) = Ac cos 2πfct + 2πk f m(τ )dτ ∫0 to the desired center freq.

fc = nf1 may not be the desired carrier frequency

k f = nk1

∆f = n∆f1

Meixia Tao @ SJTU 52 Exercise: A typical FM transmitter

Problem: Given the simplified block diagram of a typical FM transmitter used to transmit audio signals containing frequencies in the range 100Hz to 15kHz.

Desired FM wave: fc = 100MHz, ∆f = 75kHz.

Set β1 = 0.2 in the narrowband phase modulation to limit harmonic distortion.

Specify the two-stage frequency multiplier factors n1 and n2

Message signal Narrow-band Frequency Frequency FM signal Integrator phase multiplier Mixer multiplier

modulator n1 n2

Crystal-controlled Crystal-controlled oscillator oscillator 0.1MHz ?

Meixia Tao @ SJTU 53 Demodulation of FM Balanced Frequency Discriminator

t Given FM waves(t) = Ac cos2πfct + 2πk f m(τ )dτ ∫0 d t =−+π π π+ π ττ st() Acc 2 f 2 k f mt () sin2 ftc 2 k f m ( ) d dt ∫0 Hybrid-modulated wave with AM and FM Differentiator + envelop detector = FM demodulator Frequency discriminator: a “freq to amplitude” transform device Slope circuit Envelop Baseband H (f) detector + FM 1 signal wave ∑ - Slope circuit Envelop

H2(f) detector

j2πa( f − fc + B / 2), fc − B / 2 ≤ f ≤ fc + B / 2 = π + − − − ≤ ≤ − + H1( f ) j2 a( f fc B / 2), fc B / 2 f fc B / 2 H 2 ( f ) = H1(− f ) 0, elsewhere

Meixia Tao @ SJTU 54 Circuit diagram and frequency response

Meixia Tao @ SJTU 55 Application: FM Radio broadcasting

As with standard AM radio, most FM radio receivers are of super-heterodyne type . Typical freq parameters – RF carrier range = 88~108 Limiter MHz – Midband of IF = 10.7MHz discriminator – IF bandwidth = 200kHz – Peak freq. deviation = 75KHz Baseband low-pass filter loudspeaker Audio amplifier with de-emphasis

Meixia Tao @ SJTU 56 FM Radio Stereo Multiplexing

ml(t) + Stereo multiplexing is a form of FDM ∑ designed to transmit two separate + m (t) + + m(t) signals via the same carrier. r ∑ + ∑ - Widely used in FM broadcasting to K send two different elements of a program (e.g. vocalist and Frequency accompanist in an orchestra) so as to doubler cos(2πf t) give a spatial dimension to its c perception by a listener at the m(t) = [m (t) + m (t)] receiving end l r + [ml (t) − mr (t)]cos(4πfct) . The sum signal is left unprocessed in its + π baseband form K cos(2 fct) . The difference signal and a 38-kHz f = 19kHz subcarrier produce a DSBSC wave c . The 19-kHz pilot is included as a reference for coherent detection

10/4/2004 Meixia Tao @ SJTU 57 FM-Stereo Receiver

Baseband ml(t)+mr(t) + 2ml(t) LPF ∑ + To two loudspeakers m(t) + BPF centered Baseband ∑ at 2fc=38kHz LPF - 2mr(t) ml(t)-mr(t)

Frequency doubler

Narrow-band filter tuned to fc=19kHz

10/4/2004 Meixia Tao @ SJTU 58 Think …

AM vs. FM

Compared with AM, FM requires a higher implementation complexity and a higher bandwidth occupancy. What is the advantage of FM then?

Why AM radio is mostly for news broadcasting while FM radio is mostly for music program

Meixia Tao @ SJTU 59 Suggested Reading

Chapter 3 and Chapter 4.1-4.4 of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi

Meixia Tao @ SJTU 60 Topics to be Covered

Source Modulator Channel Demodulator Output

Amplitude modulation Angle modulation (phase/frequency) Effect of noise on amplitude modulation Effect of noise on frequency modulation

Meixia Tao @ SJTU 61 A Benchmark System

Baseband system: n(t) s (t) m + LPF

No carrier demodulation The receiver is an ideal LPF with bandwidth W Noise power at the output of the receiver W N P=0 df = N W n0 ∫−W 2 0 S P Baseband SNR is given by = R Nb NW0

Meixia Tao @ SJTU 62 Example:

Find the SNR in a baseband system with a bandwidth −14 of 5 kHz and with N 0 = 10 W/Hz. The transmitter power is 1kW and the channel attenuation is 10−12

Solution: −−12 3 9 PR =10 ×= 10 10 Watts

S P 10−9 =R = = −14 20 Nb NW0 10× 5000

=10log10 20 = 13dB

Meixia Tao @ SJTU 63 Effect of Noise on DSBSC

n(t) st() st() m0 (t) + BPF demod n (t) i n0 (t)

Modulated signal st()= Amtcc ()cosω t Input to the demodulator

rt()= st () + ni () t

=Amtc()cos wt cc +− n ()cos t wt cs n ()sin t wt c

Here ni(t) is a Gaussian narrow-band noise

N0 /2 ff −≤c W Sfn ()= i 0 otherwise Meixia Tao @ SJTU 64 In the demodulator, the received signal is first multiplied by a locally generated sinusoid signal r()cos( t wtc+=φφ ) Amt c ()cos wt cc cos( wt+ )

+nc()cos t wt cc cos( wt+−φφ ) n s ()sin t wt cc cos( wt+ ) 11 =++Amt()cosφφ Amt ()cos(2 wt ) 22c cc 1 ++[nt()cosφφ nt ()sin ] 2 cs 1 +[n( t )cos(2 wt+−φφ ) n ( t )sin(2 wt + )] 2 c cs c Assume coherent detector, we have φ = 0

Meixia Tao @ SJTU 65 Then the signal is passed through a LPF with bandwidth W 1 Sf()= Sf ( −+ f ) Sf ( + f ) nci n cn i c y() t=[ Amtcc () + n () t ] where 2 for f≤ W The output SNR can thus be defined as 1 2 APcm 2 S Po 4 APcm = = = NP 1 2WN o no P 0 4 nc 2 APcm Since the received power of DSBSC in baseband is PR = 2 The output SNR can be rewritten as DBSSC does not provide any SSP =R = SNR improvement over a NoDSB WN0 N b simple baseband systems

Meixia Tao @ SJTU 66 Effect of Noise on SSB Modulated signal: st()= Amtc ()cos wt cc± Amt ()sin wt c Input to the demodulator N0 /2 ff−≤c W/2 rt()= st () + n () t where Sfn ()= i i 0 otherwise =[ Amtcc() + n ()cos t] wt ccs +±[ Amt () − n ()sin t] wt c

Output of LPF: A 1 yt()=c mt () + n () t 22c Therefore, the output SNR is 1 2 APcm 2 S Po 4 APcm = = = NP 1 WN o no P 0 4 nc Meixia Tao @ SJTU 67 But in this case, 2 PR= AP cm Thus,

SSP SNR in an SSB system is =R = N WN N equivalent to that of a oSSB 0 b DSBSC system

Meixia Tao @ SJTU 68 Effect of Noise on Conventional AM

Modulated signal

s() t= Acc[ 1 + am ()cos t] w t Input to the demodulator

rt()= st () + ni () t

=++[ Acc Aamt() n c ()cos t] wt c − n s ()sin t wt c With coherent detector, after mixing and LPF: 1 y() t=++[ A A am() t n () t ] 1 2 cc c Removing DC component 1 y() t=[ A am () t + n () t ] 2 cc Meixia Tao @ SJTU 69 Exercise: please show that the output SNR is

SNR in conventional AM is always smaller than that in baseband.

Hint: 1 22 the received signal power is P= A1 + aP Rc2 m Modulation efficiency is

Meixia Tao @ SJTU 70 Performance of Envelope-Detector

Input to the envelope-detector

r() t=++[ Acc Aamt() n c ()cos t] wt c − n s ()sin t wt c Envelope of r(t)

2 2 Vr() t=+++[ A cc A am() t n c () t] n s () t If signal component is much stronger than noise

Vr() t≈+ A cc A am () t + n c () t After removing DC component, we obtain

y() t= Acc am () t + n () t At high SNR, performance of coherent detector and envelop detector is the same Meixia Tao @ SJTU 71 Performance of Envelope-Detector

If noise power is much stronger than the signal power

2 2 22 Vr() t= Ac( 1 + am () t) +++ nc () t n s () t 2 A cc n ()1 t( + am () t )

Ignore 1st term 2An () t ≈+22 +cc + (ntcs() nt ()) 1 22(1amt ()) ntcs()+ nt () 22 Vt()nc= nt () + nt s () An() t ε ≈+cc + 11+ε ≈+ Vn () t 12 ( 1am () t ) 2 Vtn () The system is operating below Ancc() t the threshold, no meaningful =++Vn () t ( 1am () t ) Vtn () SNR can be defined.

Meixia Tao @ SJTU 72 Exercise

Consider that the message is a WSS r.p M(t) with 2 autocorrelation function R M ( ττ ) = 16sinc (10000 ) . It is given = that mt () max 6 . We want to transmit this message to a destination via a channel with a 50dB attenuation and additive white noise −12 with PSD .Sf n ( ) = N 0 / 2 = 10 W/Hz . We also want to achieve an SNR at the modulator output of at least 50dB. What is the required transmitted power and the channel bandwidth if we employ the following modulation schemes? DSB-SC SSB AM with modulation index = 0.8

Meixia Tao @ SJTU 73 Topics to be Covered

Source Modulator Channel Demodulator Output

Meixia Tao @ SJTU 74 Effect of Noise in FM

The effect of additive noise is described by the changes of frequency, or the changes in the zero- crossings of the modulated FM single.

Meixia Tao @ SJTU 75 Effect of Noise on Angle Modulation

Block diagram of an angle demodulator S rt()= st () + nt () yt() st()+ nw () t N BPF LPF o Demod BW=Bc BW=W

Input to the demodulator is rt()=

=Accos[ωφ c t ++ () t] nt c ()cos ω cs t − nt ()sin ω c t

= Rn()cos( t wt cn+θ ()) t

Meixia Tao @ SJTU 76 Assume that the signal is much larger than the noise

rt()≈+ Acn R ()cos t(θφ n () t − () t) ⋅ Rt()sin(θφ () t− () t) ++φ −1 nn coswc t ( t ) tg Acn+− Rt()cos(θφ n () t () t) Rt() The phase term can be further n approximated as () Rty Rtn () Ac nts () θφrn()tt=+− () sin( θφ () tt ()) ()tt− () θ ()t θφn Ac e

Rtn () θ θ ()t φ()t y ()t n

ntc () 22 Ac En() t Meixia Tao @ SJTU 77 Therefore, the output of the demodulator is

ddRtn () yt()==θrf () t k mt ()+−sin(θφn ()t () t) 22ππdt dt Ac d =k mt() + Y () t fn2π dt

Desired signal Noise The noise component is inversely proportional to the signal amplitude Ac. (This is not the case for AM system)

Rtn () 1 Ytn()= sin(θφ n ()t −= () t) [ nt sc ()cos()φ t − nt ()sin() φ t] AAcc 1 = [ntsc()cosφφ− nt ()sin ] (Since φ () t is slowly varying) Ac

Meixia Tao @ SJTU 78 1 d The power spectral density of Yt n () is 2π dt 22 4π22f cos φφ sin Sf()= f2 Sf()+ Sf () 4π 2 YnAA nn sc cc 2 2 f f N0 fB≤ C 2 = Sf()= A2 2 nc c Ac 0 otherwise

Sfno ()Sfnf ()

At the output of LPF, the noise is limited to the freq. range [-W, W]

−B 2 − f f B 2 T -Wx Wx T f

Meixia Tao @ SJTU 79 Output SNR in FM

Now we can determine the output SNR in FM First, the output signal power is P= kP2 so fm The output noise power is 3 W N2 NW P= 00 f2 df = no ∫−W 22 AAcc3 Then, the output SNR is 22 2 S Ps 3kAfc P 3β fmP S =0 = m = N P2 W2 NW 2 N o no 0 (maxmt ( ) ) b

Meixia Tao @ SJTU 80 Key Observations

Rewrite

Power content of the normalized message Increasing β increases the output SNR, in contrast to AM Increasing the bandwidth (by Carson rule ) increases the output SNR. Thus, FM provides a way to trade off bandwidth for transmitted power Having a large B means having a large noise power. Thus, Increasing β cannot continue improving the performance indefinitely due to the threshold effect Increasing the transmitted power increases output SNR in both FM and AM systems, but the mechanisms are totally different.

Meixia Tao @ SJTU 81 Threshold Effect

There exists a specific SNR at the input of the demodulator below which the signal is not distinguishable from the noise

FM S0

N0 DSB

Si

Ni 0 a Meixia Tao @ SJTU 82 Comparison of Analog-Modulation

Bandwidth efficiency SSB is the most bandwidth efficient, but cannot effectively transmit DC VSB is a good compromise PM/FM are the least favorable systems Power efficiency (reflected in performance with noise) FM provides high noise immunity Conventional AM is the least power efficient Implementation complexity (transmitter and receiver) The simplest receiver structure is conventional AM FM receivers are also easy to implement DSB-SC and SSB-SC requires coherent detector and hence is much more complicated.

Meixia Tao @ SJTU 83 Think …

AM vs. FM

Compared with AM, FM requires a higher implementation complexity and a higher bandwidth occupancy. What is the advantage of FM then?

FM provides high noise immunity. This is why AM radio is mostly for news broadcasting while FM radio is mostly for music program.

Meixia Tao @ SJTU 84 Suggested Reading

Chapter 6.1-6.3 of of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi

Meixia Tao @ SJTU 85