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()

 Carrier wave

 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 )= expj( 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 cos2πf1t + 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 cos2πf1t + 2πk1 m(τ )dτ  + a2 A1 cos 2πf1t + 2πk1 m(τ )dτ   ∫0   ∫0  2 2 t t   a2 A1 a2 A1   = a1 A1 cos2πf1t + 2πk1 m(τ )dτ  + + cos4πf1t + 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 cos2πf1t + 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 cos2πfct + 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= A1 + 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

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

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 coswc 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