4: Amplitude Modulation - ArraytoolIntroductionDSB-SC ModulationAmplitude ModulationBandwidth...

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Introduction DSB-SC Modulation Amplitude Modulation Bandwidth Efficient Modulations FDM AM Receivers Noise Summary

4: Amplitude Modulation

Y. Yoganandam, Runa Kumari, and S. R. Zinka

Department of Electrical & Electronics EngineeringBITS Pilani, Hyderbad Campus

August 14, 17, 19, 21, 24 & 26, 2015

4: Amplitude Modulation Communication Systems, Dept. of EEE, BITS Hyderabad

Outline

1 Introduction

2 DSB-SC Modulation

3 Amplitude Modulation

4 Bandwidth Efficient Modulations

6 AM Receivers

7 Noise

8 Summary

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

Modulation

To alter or adapt (the voice) according to the circumstances, one’s listener, etc.

Baseband signals produced by various information sources are not alwayssuitable for direct transmission over a given channel. These signals are

usually further modified to facilitate transmission. This conversion process isknown as modulation.

Modulation

Carrier Signal:Ac cos (ωct + θ)

Amplitude Modulation:AAM = f (m (t))

Frequency Modulation:ωFM = g (m (t))

Phase Modulation:θPM = h (m (t))

Modulation

Amplitude Modulation

Amplitude

−1.5

−0.5

Message wave envelopeCarrier wave

Amplitude Modulation

Amplitude

−1.5

−0.5

Message wave envelopeCarrier wave

Why Modulation?

• Ease of radiation

• Simultaneous transmission of several signals

• Effecting the exchange of SNR with bandwidth

Why Modulation?

Basic Terminology

• Modulation

• Modulating Signal

• Carrier Signal

• Modulated Signal

• Sidebands

• Aliasing

• Demodulation

Basic Terminology

• Modulation

• Carrier Signal

• Sidebands

• Aliasing

• Demodulation

Basic Terminology

• Modulation

• Carrier Signal

• Sidebands

• Aliasing

• Demodulation

Basic Terminology

• Modulation

• Carrier Signal

• Sidebands

• Aliasing

• Demodulation

Basic Terminology

• Modulation

• Carrier Signal

• Sidebands

• Aliasing

• Demodulation

Basic Terminology

• Modulation

• Carrier Signal

• Sidebands

• Aliasing

• Demodulation

Basic Terminology

• Modulation

• Carrier Signal

• Sidebands

• Aliasing

• Demodulation

Basic Terminology

• Modulation

• Carrier Signal

• Sidebands

• Aliasing

• Demodulation

Baseband Communication

The term baseband is used to designate the band of frequencies of the signaldelivered by the source or the input transducer.

In telephony, the baseband is the audio band of (voice signals) of 0 to 3.5 kHz.

In television, the baseband is the video band occupying 0 to 4.3 MHz.

For digital data or PCM using bipolar signaling at a rate of Rb pulses per sec-ond, the baseband is 0 to Rb Hz.

Usually, baseband signals cannot be transmitted over a radio link but are suit-able for transmission over a pair of wires , coaxial cables, or optical fibers.

Carrier Communication

Communication that uses modulation to shift the frequency spectrum of asignal is known as carrier communication.

In this mode, one of the basic parameters (amplitude, frequency, or phase) of asinusoidal carrier of high frequency ωc is varied in proportion to the basebandsignal m (t). This results in amplitude modulation (AM), frequency modula-tion (FM), or phase modulation (PM), respectively.

Modulation is used to transmit both analog as well as digital baseband signals.

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

DSB-SC Modulation

m(t)(Modulating signal)

DSB-SC Modulation

cosωct(Carrier)

DSB-SC Modulation

m(t)(Modulating signal) (Modulated signal)

m(t) cosωct

cosωct(Carrier)

DSB-SC Modulation

m(t) cosωct

cosωct(Carrier)

DSB-SC Modulation

m(t) cosωct

cosωct(Carrier)

m(t) cosωctm(t)

DSB-SC Modulation

m(t) cosωct

cosωct(Carrier)

m(t) cosωctm(t)

0 2πB-2πB

DSB-SC Modulation

m(t) cosωct

cosωct(Carrier)

m(t) cosωctm(t)

-ωc +ωc

LSB USBLSBUSB

0 2πB-2πB

ωc≥2πB

DSB-SC Demodulation

m(t) cosωct

DSB-SC Demodulation

m(t) cosωct

cosωct(Carrier)

DSB-SC Demodulation

m(t) cosωct

cosωct(Carrier)

DSB-SC Demodulation

m(t) cosωct

cosωct(Carrier)

ω+2ωc-2ωc 0

DSB-SC Demodulation

m(t) cosωct

cosωct(Carrier)

Low-pass filtere(t)

m(t)12

ω+2ωc-2ωc 0

A/2A/2

Theoretical Explanation of DSB-SC

Modulation:

m (t)⇐⇒ M (ω)

m (t) cos ωct⇐⇒ 12[M (ω + ωc) + M (ω−ωc)] (1)

Demodulation:

e (t) = m (t) cos2 ωct =12[m (t) + m (t) cos 2ωct]

e (t)⇐⇒ 12

M (ω) +14[M (ω + 2ωc) + M (ω− 2ωc)] (2)

Modulation:

m (t)⇐⇒ M (ω)

Demodulation:

e (t)⇐⇒ 12

M (ω) +14[M (ω + 2ωc) + M (ω− 2ωc)] (2)

Modulation:

m (t)⇐⇒ M (ω)

Demodulation:

e (t)⇐⇒ 12

M (ω) +14[M (ω + 2ωc) + M (ω− 2ωc)] (2)

DSB-SC Modulators – 1: Nonlinear Modulator

cosωct

m(t) Σ

cosωct

m(t) Σ

cosωct

x1(t) NLy1(t)

cosωct

z(t)Σ

4bm(t) cosωct

cosωct

z(t) BPF±ωc

Let input-output characteristics of NL device be approximated by a powerseries:

y (t) = ax (t) + bx2 (t) (3)

Then the summer output z (t) is given by

z (t) = y1 (t)− y2 (t) =[ax1 (t) + bx2

1 (t)]−[ax2 (t) + bx2

2 (t)]

Substituting the two input signals x1 (t) = cos ωct + m (t) and x2 (t) =cos ωct−m (t) in the above equation gives

z (t) = 2am (t) + 4bm (t) cos ωct. (5)

From the above equation, it can be said that the circuit acts as a balancedbridge for carrier signal. Do you know what is meant by that?

y (t) = ax (t) + bx2 (t) (3)

z (t) = y1 (t)− y2 (t) =[ax1 (t) + bx2

1 (t)]−[ax2 (t) + bx2

2 (t)]

z (t) = 2am (t) + 4bm (t) cos ωct. (5)

y (t) = ax (t) + bx2 (t) (3)

z (t) = y1 (t)− y2 (t) =[ax1 (t) + bx2

1 (t)]−[ax2 (t) + bx2

2 (t)]

z (t) = 2am (t) + 4bm (t) cos ωct. (5)

y (t) = ax (t) + bx2 (t) (3)

z (t) = y1 (t)− y2 (t) =[ax1 (t) + bx2

1 (t)]−[ax2 (t) + bx2

2 (t)]

z (t) = 2am (t) + 4bm (t) cos ωct. (5)

y (t) = ax (t) + bx2 (t) (3)

z (t) = y1 (t)− y2 (t) =[ax1 (t) + bx2

1 (t)]−[ax2 (t) + bx2

2 (t)]

z (t) = 2am (t) + 4bm (t) cos ωct. (5)

DSB-SC Modulators – 2: Switching Modulators

m(t)w(t)

BPF±ωc

m(t)w(t)

BPF±ωc t

m(t) cosωct2π

DSB-SC Modulators – 3: Diode Bridge Switch

Acosωct

Duringpositive

Duringnegative

Bandpassfilterm(t) km(t) cosωct

DSB-SC Modulators – 4: Ring Modulator

Bandpassfilter km(t) cosωct

Acosωct

vi=m(t)w0(t)

Impact of Asynchronous Carrier

m(t) cosωct

cosωct(Carrier)

Low-pass filtere(t)

m(t)12

ω+2ωc-2ωc 0

A/2A/2

Demodulation process is similar to that of modulation except that we use alowpass filter instead of bandpass filter. Thus all the circuits used for modu-lation can be used for demodulation also.

But having a synchronous carrier at the receiver end is a must. This compli-cates the receiver design.

m(t) cosωct

cosωct(Carrier)

Low-pass filtere(t)

m(t)12

ω+2ωc-2ωc 0

A/2A/2

m(t) cosωct

cosωct(Carrier)

Low-pass filtere(t)

m(t)12

ω+2ωc-2ωc 0

A/2A/2

m(t) cosωct

cosωct(Carrier)

Low-pass filtere(t)

m(t)12

ω+2ωc-2ωc 0

A/2A/2

Let the local carrier with phase and frequency offset be 2 cos [(ωc + ∆ω) t + δ] .

Then the product at the receiver is given as

e (t) = 2m (t) cos ωct cos [(ωc + ∆ω) t + δ]

= m (t) {cos [(2ωc + ∆ω) t + δ] + cos [(∆ω) t + δ]} .

After lowpass filtering,

e0 (t) = m (t) cos [(∆ω) t + δ] . (6)

So, after lowpass filtering,

e0 (t) = m (t) cos [(∆ω) t + δ] .

If ∆ω = 0,e0 (t) = m (t) cos δ. (7)

If δ = 0 and ∆ω 6= 0,e0 (t) = m (t) cos (∆ωt) . (8)

Even if ∆f is 1 Hz, the effect is very annoying.

e0 (t) = m (t) cos [(∆ω) t + δ] .

If ∆ω = 0,e0 (t) = m (t) cos δ. (7)

e0 (t) = m (t) cos [(∆ω) t + δ] .

If ∆ω = 0,e0 (t) = m (t) cos δ. (7)

e0 (t) = m (t) cos [(∆ω) t + δ] .

If ∆ω = 0,e0 (t) = m (t) cos δ. (7)

e0 (t) = m (t) cos [(∆ω) t + δ] .

If ∆ω = 0,e0 (t) = m (t) cos δ. (7)

Ensuring Synchronous Carrier at the Receiver

To ensure identical carrier frequencies at the transmitter and the receiver, wecan use quartz crystal oscillators, which are very stable (at least at low carrierfrequencies).

At very high frequencies, a carrier, or pilot, is transmitted at a reduced level(usually about -20dB) along with the sidebands. After separating the pilotsignal using a very narrow-band filter, it is amplified and used to synchronize(with the help of a phase-locked loop or PLL) the local oscillator.

Phase-Locked Loop (PLL)

A sin (ωct+θi)

B cos (ωct+θo)

x(t)A sin (ωct+θi)

B cos (ωct+θo)

Loopfilter

eo(t)x(t)A sin (ωct+θi)

B cos (ωct+θo)

Loopfilter

B cos (ωct+θo)Phase Detector

Loopfilter

B cos (ωct+θo)ω = ωc + ceo(t)

Loopfilter

B cos (ωct+θo)

ω = ωc + ceo(t)

θe = θi-θo

eo = sinθe

Let the input to the PLL be A sin (ωct + θi), and let the VCO output beB cos (ωct + θo). Then

x (t) = AB sin (ωct + θi) cos (ωct + θo)

[sin (2ωct + θi + θo) + sin (θi − θo)] .

After filtering, error signal eo (t) is given as

eo (t)=AB2

sin (θi − θo) =AB2

sin θe, (9)

where θe = θi − θo.

eo (t)=AB2

sin θe, (9)

eo (t)=AB2

sin θe, (9)

Signal-Squaring Method

BPF±2ωc

x(t)( )2 PLL 2:1 Frequency

dividerk cosωctm(t) cosωct c cos 2ωct

Narrow-bandfilter

The squarer output x (t) is

x (t) = [m (t) cos ωct]2 =12

m2 (t) +12

m2 (t) cos 2ωct.

Since m2 (t) is a non-negative signal, it can be written as

m2 (t) = k + φ (t) .

Thenx (t) =

m2 (t) + k cos 2ωct + φ (t) cos 2ωct. (10)

BPF±2ωc

Narrow-bandfilter

x (t) = [m (t) cos ωct]2 =12

m2 (t) +12

m2 (t) cos 2ωct.

m2 (t) = k + φ (t) .

Thenx (t) =

BPF±2ωc

Narrow-bandfilter

x (t) = [m (t) cos ωct]2 =12

m2 (t) +12

m2 (t) cos 2ωct.

m2 (t) = k + φ (t) .

Thenx (t) =

BPF±2ωc

Narrow-bandfilter

x (t) = [m (t) cos ωct]2 =12

m2 (t) +12

m2 (t) cos 2ωct.

m2 (t) = k + φ (t) .

Thenx (t) =

BPF±2ωc

Narrow-bandfilter

x (t) = [m (t) cos ωct]2 =12

m2 (t) +12

m2 (t) cos 2ωct.

m2 (t) = k + φ (t) .

Thenx (t) =

Costas Loop

m(t) cos (ωct+θi)

The mechanism is very similar to that of PLL except that

eo (t) ∝ sin 2θe.

Costas Loop

VCOm(t) cos (ωct+θi)

2 cos (ωct+θo)

2 sin (ωct+θo)

−π/2

Costas Loop

VCOm(t) cos (ωct+θi)

2 cos (ωct+θo)

2 sin (ωct+θo)

−π/2

Costas Loop

Lowpassfilter

m(t) cos (ωct+θi)

2 cos (ωct+θo)

2 sin (ωct+θo)

m(t) cos θe

m(t) sin θe

−π/2

Costas Loop

Lowpassfilter

m(t) cos (ωct+θi)

2 cos (ωct+θo)

2 sin (ωct+θo)

m(t) cos θe

m(t) sin θe

m2(t) sin 2θe12

−π/2

Costas Loop

Lowpassfilter

(narrow band)VCO

Lowpassfilter

m(t) cos (ωct+θi)

2 cos (ωct+θo)

2 sin (ωct+θo)

m(t) cos θe

m(t) sin θe

K sin 2θe

m2(t) sin 2θe12

−π/2

Costas Loop

Lowpassfilter

(narrow band)VCO

Lowpassfilter

m(t) cos (ωct+θi)

2 cos (ωct+θo)

2 sin (ωct+θo)

m(t) cos θe

m(t) sin θe

K sin 2θe

m2(t) sin 2θe12

−π/2

Costas Loop

Lowpassfilter

(narrow band)

Output

Lowpassfilter

m(t) cos (ωct+θi)

2 cos (ωct+θo)

2 sin (ωct+θo)

m(t) cos θe

m(t) sin θe

K sin 2θe

m2(t) sin 2θe12

−π/2

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

Amplitude Modulation (with Carrier)

DSB-SC receivers are expensive as they have to maintain the synchronismwith that of the transmitter.

The solution is to send the carrier along with the signal. This implies that thetransmitter has to transmit extra power in the high power carrier.

So, amplitude modulated signal is given by

ϕAM (t) = m (t) cos ωct + A cos ωct = [A + m (t)] cos ωct. (11)

Under Modulation

Amplitude

−0.5

Under Modulation

Amplitude

−0.5

Under Modulation

Amplitude

−0.5

Under Modulation

Amplitude

−1.5

−0.5

Under Modulation

Amplitude

−1.5

−0.5

Over Modulation

Amplitude

0−0.5

Over Modulation

Amplitude

0−0.5

Over Modulation

Amplitude

−0.5

Over Modulation

Amplitude

−2.5

−0.5

Over Modulation

Amplitude

−2.5

−0.5

Modulation Index (µ)

From the previous slides, the condition for envelope detection of an AM signalis

A + m (t) ≥ 0. (12)

The above equation is equivalent to

A ≥ mnp, (13)

where mnp is the absolute negative peak amplitude.

From the above discussion, we define modulation index (µ) as

µ =mnp

A. (14)

When µ > 1 (over-modulation), envelop detection is not possible and onlyoption available is synchronous detection.

A + m (t) ≥ 0. (12)

A ≥ mnp, (13)

µ =mnp

A. (14)

A + m (t) ≥ 0. (12)

A ≥ mnp, (13)

µ =mnp

A. (14)

A + m (t) ≥ 0. (12)

A ≥ mnp, (13)

µ =mnp

A. (14)

A + m (t) ≥ 0. (12)

A ≥ mnp, (13)

µ =mnp

A. (14)

Power Efficiency (η)

ϕAM (t) = m (t) cos ωct︸︷︷︸side−bands

+A cos ωct︸︷︷︸carrier

Pc =A2

2and Ps =

m2 (t)

The sideband power is the useful power and the carrier power is the powerwasted for convenience. So, η, the power efficiency is defined as

η =useful powertotal power

Pc + Ps=

m2 (t)

m2 (t) + A2× 100%. (15)

For the special case of tone modulation, m (t) = µA cos ωmt. So, m2 (t) =

(µA)2 /2 and

η =µ2

2 + µ2 × 100%. (16)

Pc =A2

2and Ps =

m2 (t)

Pc + Ps=

m2 (t)

m2 (t) + A2× 100%. (15)

(µA)2 /2 and

η =µ2

2 + µ2 × 100%. (16)

Pc =A2

2and Ps =

m2 (t)

Pc + Ps=

m2 (t)

m2 (t) + A2× 100%. (15)

(µA)2 /2 and

η =µ2

2 + µ2 × 100%. (16)

Pc =A2

2and Ps =

m2 (t)

Pc + Ps=

m2 (t)

m2 (t) + A2× 100%. (15)

(µA)2 /2 and

η =µ2

2 + µ2 × 100%. (16)

AM Modulators

AM signals can be generated by any DSB-SC modulator if the modulatingsignal is A + m (t) instead of just m (t).

But because there is no need to suppress the carrier in the output, the modu-lating signal do not have to be balanced. This results in a considerably simplermodulator for AM.

AM Modulators

BPF±ωc

c cosωct

When c� m (t), the voltage across terminals bb′ is

vbb′ = [c cos ωct + m (t)]w (t)

= [c cos ωct + m (t)][

(cos ωct− 1

3cos 3ωct +

cos 5ωct− · · ·)]

cos ωct +2π

m (t) cos ωct︸︷︷︸AM

+ other terms.

AM Modulators

BPF±ωc

c cosωct

(cos ωct− 1

3cos 3ωct +

cos 5ωct− · · ·)]

cos ωct +2π

+ other terms.

AM Modulators

BPF±ωc

c cosωct

(cos ωct− 1

3cos 3ωct +

cos 5ωct− · · ·)]

cos ωct +2π

+ other terms.

AM Modulators

BPF±ωc

c cosωct

(cos ωct− 1

3cos 3ωct +

cos 5ωct− · · ·)]

cos ωct +2π

+ other terms.

AM Modulators

BPF±ωc

c cosωct

(cos ωct− 1

3cos 3ωct +

cos 5ωct− · · ·)]

cos ωct +2π

+ other terms.

AM Modulators

BPF±ωc

c cosωct

(cos ωct− 1

3cos 3ωct +

cos 5ωct− · · ·)]

cos ωct +2π

+ other terms.

AM Demodulation

The AM signal can be demodulated coherently by a locally generated carrier.However, this defeats the very purpose (i.e., using non-coherent demodula-tion) of AM.

So, we shall consider here the following non-coherent demodulation schemes:

1 Rectifier detection

2 Envelop detection

AM Demodulation

2 Envelop detection

AM Demodulation

2 Envelop detection

Rectifier Detector

+−[A+m(t)] cosωct

Rectifier Detector

[A+m(t)]

Rectifier Detector

[A+m(t)]/π

Rectifier Detector

Lowpassfilter[A+m(t)] cosωct

[A+m(t)]/π

Rectifier Detector

Lowpassfilter[A+m(t)] cosωct

[A+m(t)]/π

Rectifier Detector

Lowpassfilter[A+m(t)] cosωct vo(t)

Rectifier Detector

Lowpassfilter[A+m(t)] cosωct vo(t)

Envelop Detector

+−[A+m(t)] cosωct RC Capacitor

discharge vo(t)

Envelop Detector

discharge vo(t)

Envelop Detector

discharge vo(t)

Envelop Detector

discharge vo(t)

Envelop Detector

discharge vo(t)

Envelop Detector

discharge vo(t)

Envelop Detector

discharge vo(t)

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

Bandwidth Efficient Modulation Schemes

• Single Side-Band Modulation (SSB)

• Vestigial Side-Band Modulation (VSB)

• Quadrature Amplitude Modulation (QAM)

SSB-SC Modulation

ω-ωc +ωc

LSB USBLSBUSB

SSB-SC Modulation

ω-ωc +ωc

LSB USBLSBUSB

ω-ωc +ωc

SSB-SC Modulation

ω-ωc +ωc

LSB USBLSBUSB

ω-ωc +ωc

SSB-SC Modulation

ω-ωc +ωc

LSB USBLSBUSB

ω-ωc +ωc

ω0-2ωc +2ωc

Time-Domain Representation of SSB Signals

M+(ω)

M−(ω)

M+(ω)

M−(ω)

ω-ωc +ωc

M+(ω−ωc)M−(ω+ωc)

ω-ωc +ωc

M−(ω−ωc)M+(ω+ωc)

M+ (ω)⇐⇒ m+ (t)M− (ω)⇐⇒ m− (t)

Since M+ (ω) and M− (ω) are conjugates, m+ (t) and m− (t) are also conju-gates. Also, because m+ (t) + m− (t) = m (t), we can express

m+ (t) =12[m (t) + jmh (t)]

m− (t) =12[m (t)− jmh (t)]

where mh (t) is unknown.

From the above set of equations, it follows that

M+ (ω) =12[M (ω) + jMh (ω)] (17)

M− (ω) =12[M (ω)− jMh (ω)] .

M+ (ω)⇐⇒ m+ (t)M− (ω)⇐⇒ m− (t)

m+ (t) =12[m (t) + jmh (t)]

m− (t) =12[m (t)− jmh (t)]

M+ (ω) =12[M (ω) + jMh (ω)] (17)

M− (ω) =12[M (ω)− jMh (ω)] .

M+ (ω)⇐⇒ m+ (t)M− (ω)⇐⇒ m− (t)

m+ (t) =12[m (t) + jmh (t)]

m− (t) =12[m (t)− jmh (t)]

M+ (ω) =12[M (ω) + jMh (ω)] (17)

M− (ω) =12[M (ω)− jMh (ω)] .

M+ (ω)⇐⇒ m+ (t)M− (ω)⇐⇒ m− (t)

m+ (t) =12[m (t) + jmh (t)]

m− (t) =12[m (t)− jmh (t)]

M+ (ω) =12[M (ω) + jMh (ω)] (17)

M− (ω) =12[M (ω)− jMh (ω)] .

Since M+ (ω) = M (ω) u (ω),

M+ (ω) =12

M (ω) [1 + sgn (ω)]

M (ω) +12

M (ω) sgn (ω) . (18)

Comparing (18) and (17) gives

Mh (ω) = −jM (ω) sgn (ω) = [M (ω)] [−jsgn (ω)] .

Since, 1/πt⇐⇒ −jsgn (ω), taking inverse Fourier transform on both sides ofthe above equation gives

Mh (ω) =12

ˆ ∞

−∞

m (τ)

t− τdτ =

ˆ ∞

−∞

m (α)

t− αdα. (19)

The right-hand side of the above equation defines the Hilbert transform ofm (t).

M+ (ω) =12

M (ω) [1 + sgn (ω)]

M (ω) +12

M (ω) sgn (ω) . (18)

Mh (ω) =12

ˆ ∞

−∞

m (τ)

t− τdτ =

ˆ ∞

−∞

m (α)

t− αdα. (19)

M+ (ω) =12

M (ω) [1 + sgn (ω)]

M (ω) +12

M (ω) sgn (ω) . (18)

Mh (ω) =12

ˆ ∞

−∞

m (τ)

t− τdτ =

ˆ ∞

−∞

m (α)

t− αdα. (19)

M+ (ω) =12

M (ω) [1 + sgn (ω)]

M (ω) +12

M (ω) sgn (ω) . (18)

Since, 1/πt⇐⇒ −jsgn (ω),

taking inverse Fourier transform on both sides ofthe above equation gives

Mh (ω) =12

ˆ ∞

−∞

m (τ)

t− τdτ =

ˆ ∞

−∞

m (α)

t− αdα. (19)

M+ (ω) =12

M (ω) [1 + sgn (ω)]

M (ω) +12

M (ω) sgn (ω) . (18)

Mh (ω) =12

ˆ ∞

−∞

m (τ)

t− τdτ =

ˆ ∞

−∞

m (α)

t− αdα. (19)

SSB signal ΦUSB (ω) can be expressed as

ΦUSB (ω) = M+ (ω−ωc) + M− (ω + ωc) .

Taking inverse transform of the above equation gives

ϕUSB (t) = m+ (t) ejωct + m− (t) e−jωct.

Substituting m+ (t) and m− (t) values gives

ϕUSB (t)= m (t) cos ωct−mh (t) sin ωct. (20)

Similarly, it can be shown that

ϕLSB (t)= m (t) cos ωct + mh (t) sin ωct. (21)

ΦUSB (ω) = M+ (ω−ωc) + M− (ω + ωc) .

Hilbert Transform – Physical Interpretation

|H(ω)|

θh(ω)+π/2

−π/2

Mh (ω) = [M (ω)]× [−jsgn (ω)]

|H(ω)|

θh(ω)+π/2

−π/2

Mh (ω) = [M (ω)]× [−jsgn (ω)]

|H(ω)|

θh(ω)+π/2

−π/2

Mh (ω) = [M (ω)]× [−jsgn (ω)]

SSB-SC Modulators

• Selective-Filtering Method

• Phase-Shift Method

SSB-SC Modulators

• Selective-Filtering Method

• Phase-Shift Method

Selective-Filtering Method

0 1000 2000 3000 4000Frequency f, Hz

ω-ωc +ωc

0 1000 2000 3000 4000Frequency f, Hz

ω-ωc +ωc

Phase-Shift Method

cos ωct

ϕUSB (t) = m (t) cos ωct−mh (t) sin ωct

Phase-Shift Method

cos ωct

Phase-Shift Method

cos ωct

Phase-Shift Method

DSB-SCmodulator

cos ωct

m(t) cos ωct

Phase-Shift Method

−π/2

DSB-SCmodulator

−π/2

cos ωct

m(t) cos ωct

sin ωct

Phase-Shift Method

−π/2

DSB-SCmodulator

DSB-SCmodulator−π/2

cos ωct

mh(t) mh(t) sin ωct

m(t) cos ωct

sin ωct

Phase-Shift Method

−π/2 Σ

DSB-SCmodulator

DSB-SCmodulator−π/2

SSB signal

cos ωct

mh(t) mh(t) sin ωct

m(t) cos ωct

sin ωct

SSB-SC Demodulation

SSB signal is given by

ϕSSB (t) = m (t) cos ωct∓mh (t) sin ωct.

Assuming that we have a synchronized local carrier at the receiver end, wecan demodulate the signal as shown below:

ϕSSB (t) cos ωct =12

m (t) (1 + cos 2ωct)∓ 12

mh (t) sin 2ωct

m (t) +12

m (t) cos 2ωct∓ 12

mh (t) sin 2ωct︸︷︷︸can be filtered out using an LPF

. (22)

SSB-SC Demodulation

m (t) (1 + cos 2ωct)∓ 12

mh (t) sin 2ωct

m (t) +12

. (22)

SSB-SC Demodulation

m (t) (1 + cos 2ωct)∓ 12

mh (t) sin 2ωct

m (t) +12

. (22)

SSB-SC Demodulation

m (t) (1 + cos 2ωct)∓ 12

mh (t) sin 2ωct

m (t) +12

. (22)

SSB-SC Demodulation

m (t) (1 + cos 2ωct)∓ 12

mh (t) sin 2ωct

m (t) +12

. (22)

Envelope Detection of SSB+C Signal

SSB signal with a carrier (SSB+C) can be expressed as

ϕSSB+C (t) = A cos ωct + [m (t) cos ωct + mh (t) sin ωct]= [A + m (t)] cos ωct + mh (t) sin ωct

=√[A + m (t)]2 + m2

h (t)×[A + m (t)] cos ωct + mh (t) sin ωct√

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

So, the envelope of the SSB+C signal is given by

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

ϕSSB+C (t) = A cos ωct + [m (t) cos ωct + mh (t) sin ωct]

= [A + m (t)] cos ωct + mh (t) sin ωct

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

=√[A + m (t)]2 + m2

[A + m (t)]2 + m2h (t)

=√[A + m (t)]2 + m2

h (t) cos (ωct + θ) . (23)

E =√[A + m (t)]2 + m2

√1 +

2m (t)A

+m2 (t)

A2 +m2

h (t)A2

≈ A[

1 +m (t)

]= A + m (t). ( Assuming A� m (t)) (24)

Vestigial Sideband (VSB) Modulation

ω-ωc +ωc

LSB USBLSBUSB

ω-ωc +ωc

LSB USBLSBUSB

ω-ωc +ωc

LSB USBLSBUSB

ω-ωc +ωc

2 cosωct

BPFHi(ω)

VSB signal

2 cosωct

BPFHi(ω)

VSB signal

2 cosωct

VSB signal e(t)

2 cosωct

BPFHi(ω)

VSB signal

Lowpasfilter Ho(ω)

2 cosωct

VSB signal m(t)e(t)

ΦVSB = [M (ω + ωc) + M (ω−ωc)]Hi (ω)

e (t)⇐⇒ [ΦVSB (ω + ωc) + ΦVSB (ω−ωc)]

M (ω) = [ΦVSB (ω + ωc) + ΦVSB (ω−ωc)]Ho (ω)

= M (ω) [Hi (ω + ωc) + Hi (ω−ωc)]Ho (ω)

HenceHo (ω) =

1[Hi (ω + ωc) + Hi (ω−ωc)]

, |ω| ≤ 2πB (25)

HenceHo (ω) =

1[Hi (ω + ωc) + Hi (ω−ωc)]

, |ω| ≤ 2πB (25)

HenceHo (ω) =

1[Hi (ω + ωc) + Hi (ω−ωc)]

, |ω| ≤ 2πB (25)

HenceHo (ω) =

1[Hi (ω + ωc) + Hi (ω−ωc)]

, |ω| ≤ 2πB (25)

HenceHo (ω) =

1[Hi (ω + ωc) + Hi (ω−ωc)]

, |ω| ≤ 2πB (25)

Quadrature Amplitude Modulation (QAM)

cos ωct

−π/2

sin ωct

cos ωct

−π/2

sin ωct

ΣQAM signal

cos ωct

−π/2

sin ωct

m1(t) cos ωct + m2(t) sinωct

cos ωct

−π/2 −π/2

2cos ωct

sin ωct 2sin ωct

QAM signal

cos ωct

−π/2 −π/2

2cos ωct

sin ωct 2sin ωct

Lowpassfilter

Lowpassfilterx1(t)

QAM signal

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

A Simple FDM Example

Channel 3

12 kHz300-3400 Hz

Channel 3

12 kHz300-3400 Hz

Mixer 3Channel 3

12 kHz300-3400 Hz

Mixer 3Channel 3

12 kHz300-3400 Hz

8.6 − 11.7 kHz

Mixer 3Channel 3

12 kHz300-3400 Hz

8.6 − 11.7 kHz

Mixer 2

Mixer 3

Channel 2

Channel 3

12 kHz

16 kHz300-3400 Hz

300-3400 Hz

8.6 − 11.7 kHz

12.6 − 15.7 kHz

Mixer 1

Mixer 2

Mixer 3

Channel 1

Channel 2

Channel 3

12 kHz

16 kHz

20 kHz300-3400 Hz

300-3400 Hz

8.6 − 11.7 kHz

12.6 − 15.7 kHz

16.6 − 19.7 kHz

Mixer 1

Mixer 2

Mixer 3

Channel 1

Channel 2

Channel 3

12 kHz

16 kHz

20 kHz300-3400 Hz

300-3400 Hz

8.6 − 11.7 kHz

12.6 − 15.7 kHz

16.6 − 19.7 kHz

8.6 kHz 19.7 kHz

Output

Common bus

Mixer 1

Mixer 2

Mixer 3

Channel 1

Channel 2

Channel 3

12 kHz

16 kHz

20 kHz300-3400 Hz

300-3400 Hz

8.6 − 11.7 kHz

12.6 − 15.7 kHz

16.6 − 19.7 kHz

8.6 kHz 19.7 kHz

Output

Common bus

A More Complicated FDM Modulation Plan

Voice channels(0.3 - 3.4 kHz)

0.3 kHz

3.4 kHz

0.3 kHz

3.4 kHz

Channelcarriers

0.3 kHz

3.4 kHz

104 kHz

96 kHz

88 kHz

80 kHz

72 kHz

64 kHz

Chan 11

Chan 12

Channelcarriers

0.3 kHz

3.4 kHz

104 kHz

96 kHz

88 kHz

80 kHz

72 kHz

64 kHz

60 kHz

Chan 1

Chan 2

Chan 3

Chan 4

Chan 5

Chan 6

Chan 7

Chan 8

Chan 9

Chan 10

Chan 11

Chan 12

Channelcarriers

0.3 kHz

3.4 kHz

104 kHz

96 kHz

88 kHz

80 kHz

72 kHz

64 kHz

60 kHz

Chan 1

Chan 2

Chan 3

Chan 4

Chan 5

Chan 6

Chan 7

Chan 8

Chan 9

Chan 10

Chan 11

Chan 12

Channelcarriers

Basic12 - channel

group(60 - 108 kHz)

0.3 kHz

3.4 kHz

104 kHz

96 kHz

88 kHz

80 kHz

72 kHz

64 kHz

60 kHz

Chan 1

Chan 2

Chan 3

Chan 4

Chan 5

Chan 6

Chan 7

Chan 8

Chan 9

Chan 10

Chan 11

Chan 12

Channelcarriers

Basic12 - channel

group(60 - 108 kHz)

Groupsubcarriers

0.3 kHz

3.4 kHz

104 kHz

96 kHz

88 kHz

80 kHz

72 kHz

64 kHz

612 kHz

468 kHz

584 kHz

516 kHz

420 kHz

60 kHz

Chan 1

Chan 2

Chan 3

Chan 4

Chan 5

Chan 6

Chan 7

Chan 8

Chan 9

Chan 10

Chan 11

Chan 12

Channelcarriers

Basic12 - channel

group(60 - 108 kHz)

Groupsubcarriers

0.3 kHz

3.4 kHz

104 kHz

96 kHz

88 kHz

80 kHz

72 kHz

64 kHz

612 kHz

468 kHz

584 kHz

516 kHz

420 kHz

552 kHz

504 kHz

408 kHz

456 kHz

360 kHz

312 kHz

60 kHz

Chan 1

Chan 2

Chan 3

Chan 4

Chan 5

Chan 6

Chan 7

Chan 8

Chan 9

Chan 10

Chan 11

Chan 12

Channelcarriers

Basic12 - channel

group(60 - 108 kHz)

Groupsubcarriers

0.3 kHz

3.4 kHz

Basic60 - channelsupergroup

(312 - 552 kHz)

104 kHz

96 kHz

88 kHz

80 kHz

72 kHz

64 kHz

612 kHz

468 kHz

584 kHz

516 kHz

420 kHz

552 kHz

504 kHz

408 kHz

456 kHz

360 kHz

312 kHz

60 kHz

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

Tuned Radio Frequency (TRF) Receiver

Receiverantenna

Antennacouplingnetwork

Receiverantenna

RF stage

Receiverantenna

RFamp.

RF stage

Receiverantenna

RFamp.

RF stage

Receiverantenna

RFamp.

RF stage Detectorstage

Receiverantenna

RFamp.

Audiodetector

Receiverantenna

RFamp.

Audiodetector

Audiostage

Receiverantenna

RFamp.

Audiodetector

Audioamplifiers

Speaker

Audiostage

Receiverantenna

Multistage Amplifiers

0.7 1.0 1.3

n = 1n = 2

0.7 1.0 1.3

n = 1n = 2n = 3

0.7 1.0 1.3

n = 1n = 2n = 3n = 4

0.7 1.0 1.30

0.7 1.0 1.3

n = 1n = 2n = 3n = 4

0.7 1.0 1.30

0.7 1.0 1.3

n = 1n = 2n = 3n = 4

TRF Receiver – Disadvantages

• Bandwidth is inconsistent and varies with center frequency (due to skineffect)

• High-frequency, multi-stage amplifiers are susceptible to breaking intooscillations

• Gains are not uniform over a very wide frequency range

�With the development of the superheterodyne receiver, TRF receivers are

seldom used except for special-purpose, single-station receivers.

Superheterodyne AM Receiver

Edwin Howard Armstrong (1890 – 1954)

• Inventor of regenerative andsuper-regenerative circuits

• Invented superheterodynereceiver

• Armstrong was also the inventorof modern frequency modulation(FM) radio transmission

The most prolific and influential inventor in radio history

RF amplifierwith bandpassfilters tubanleto desired ωc

Localoscillator

ωc+ωIF

Localoscillator

ωc+ωIF

[A+m(t)] cos ωct

Frequencyconverter(mixer)

Localoscillator

ωc+ωIF

[A+m(t)] cos ωct

IFamplifier

Localoscillator

ωc+ωIF

[A+m(t)] cos ωct

IFamplifier

Localoscillator

ωc+ωIF

[A+m(t)] cos ωct [A+m(t)] cos ωIFt

IFamplifier Detector

Localoscillator

ωc+ωIF

IFamplifier Detector

Audioamplifier

Speaker

Localoscillator

ωc+ωIF

fIF = 455 KHz

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

Noise in Baseband Systems

Inputm(t)

TransmitterST

Channel

Channel noisen(t)

ReceiverSi Ni,

So No,Output

so(t) no(t),

Sn(ω)

ω2πB-2πB

Assuming that the filters are ideal and channel is distortion-less,

No = 2ˆ B

0Sn (ω) df = 2

df = NB. (26)

Inputm(t)

TransmitterST

Channel

Channel noisen(t)

ReceiverSi Ni,

So No,Output

so(t) no(t),

Sn(ω)

ω2πB-2πB

No = 2ˆ B

0Sn (ω) df = 2

df = NB. (26)

Inputm(t)

TransmitterST

Channel

Channel noisen(t)

ReceiverSi Ni,

So No,Output

so(t) no(t),

Sn(ω)

ω2πB-2πB

No = 2ˆ B

0Sn (ω) df = 2

df = NB. (26)

Inputm(t)

TransmitterST

Channel

Channel noisen(t)

ReceiverSi Ni,

So No,Output

so(t) no(t),

Sn(ω)

ω2πB-2πB

No = 2ˆ B

0Sn (ω) df = 2

df = NB. (26)

So, SNR for baseband systems is given as

SNR =So

NB= γ. (27)

• SNR of 5 – 10 dB is barely intelligible• Telephone Quality : 25 – 30 dB• TV Quality : 45 – 55 dB

SNR =So

NB= γ. (27)

SNR =So

NB= γ. (27)

QAM – Revisited

cos ωct

−π/2 −π/2

2cos ωct

sin ωct 2sin ωct

Lowpassfilter

Lowpassfilterx1(t)

QAM signal

QAM – Revisited

cos ωct

−π/2 −π/2

2cos ωct

sin ωct 2sin ωct

Lowpassfilter

Lowpassfilterx1(t)

QAM signal

Bandpass Noise Representation

ωωc−ωc

Sn(ω)

n (t) = nc (t) cos ωct + ns (t) sin ωct n2 = n2c = n2

ωωc−ωc

Sn(ω)

ωωc−ωc

Sn(ω)

ωωc−ωc

Sn(ω)

ω2πB

Sns(ω)Snc(ω) or

n (t) = nc (t) cos ωct + ns (t) sin ωct

n2 = n2c = n2

ωωc−ωc

Sn(ω)

ω2πB

Sns(ω)Snc(ω) or

DSB-SC Systems

√2 cos ωct

DSB-SC Systems

√2 cos ωct

DSB-SC Systems

n(t)√2 cos ωct

DSB-SC Systems

Si Ni,

Bandpassfilter

ωc + 2πB yi(t)

√2 cos ωct

DSB-SC Systems

Si Ni,

Bandpassfilter

ωc + 2πB yi(t)

√2 cos ωct √2 cos ωct

DSB-SC Systems

Si Ni, So No,

Bandpassfilter

ωc + 2πB

Basebandfilter

yi(t) yo(t)

DSB-SC Systems

Si Ni, So No,

Bandpassfilter

ωc + 2πB

Basebandfilter

yi(t) yo(t)

Demodulator

DSB-SC Systems

Si Ni, So No,

Bandpassfilter

ωc + 2πB

Basebandfilter

yi(t) yo(t)

Demodulator

Receiver

DSB-SC Systems

Si Ni, So No,

Bandpassfilter

ωc + 2πB

Basebandfilter

yi(t) yo(t)

Demodulator

Receiver

yi =√

2m (t) cos ωct + n (t)

ST = Si = m2 (t)

Ni = 2NB

yo = m (t) +1√2

nc (t)

So = m2 (t)

N0 = NB

DSB-SC Systems

Si Ni, So No,

Bandpassfilter

ωc + 2πB

Basebandfilter

yi(t) yo(t)

Demodulator

Receiver

yi =√

ST = Si = m2 (t)

Ni = 2NB

yo = m (t) +1√2

nc (t)

So = m2 (t)

N0 = NB

DSB-SC Systems

Si Ni, So No,

Bandpassfilter

ωc + 2πB

Basebandfilter

yi(t) yo(t)

Demodulator

Receiver

yi =√

ST = Si = m2 (t)

Ni = 2NB

yo = m (t) +1√2

nc (t)

So = m2 (t)

N0 = NB

DSB-SC Systems

Si Ni, So No,

Bandpassfilter

ωc + 2πB

Basebandfilter

yi(t) yo(t)

Demodulator

Receiver

yi =√

ST = Si = m2 (t)

Ni = 2NB

yo = m (t) +1√2

nc (t)

So = m2 (t)

N0 = NB

SSB-SC Systems

2 cos ωct

SSB-SC Systems

2 cos ωct

SSBfilter

SSB-SC Systems

2 cos ωct

SSBfilter

SSB-SC Systems

Si Ni,

BandpassfilterLSB yi(t)

2 cos ωct

SSBfilter

SSB-SC Systems

Si Ni,

BandpassfilterLSB yi(t)

2 cos ωct 2 cos ωct

SSBfilter

SSB-SC Systems

Si Ni, So No,

BandpassfilterLSB

Basebandfilter

yi(t) yo(t)m(t)

SSBfilter

SSB-SC Systems

Si Ni, So No,

BandpassfilterLSB

Basebandfilter

yi(t) yo(t)m(t)

Demodulator

SSBfilter

SSB-SC Systems

Si Ni, So No,

BandpassfilterLSB

Basebandfilter

yi(t) yo(t)m(t)

Demodulator

Receiver

SSBfilter

SSB-SC Systems

Si Ni, So No,

BandpassfilterLSB

Basebandfilter

yi(t) yo(t)m(t)

Demodulator

Receiver

SSBfilter

yi = [m (t) cos ωct∓mh (t) sin ωct]+n (t)

ST = Si = m2 (t)

Ni = NB

yo = m (t) + nc (t)

So = m2 (t)

N0 = NB

SSB-SC Systems

Si Ni, So No,

BandpassfilterLSB

Basebandfilter

yi(t) yo(t)m(t)

Demodulator

Receiver

SSBfilter

ST = Si = m2 (t)

Ni = NB

yo = m (t) + nc (t)

So = m2 (t)

N0 = NB

SSB-SC Systems

Si Ni, So No,

BandpassfilterLSB

Basebandfilter

yi(t) yo(t)m(t)

Demodulator

Receiver

SSBfilter

ST = Si = m2 (t)

Ni = NB

yo = m (t) + nc (t)

So = m2 (t)

N0 = NB

SSB-SC Systems

Si Ni, So No,

BandpassfilterLSB

Basebandfilter

yi(t) yo(t)m(t)

Demodulator

Receiver

SSBfilter

ST = Si = m2 (t)

Ni = NB

yo = m (t) + nc (t)

So = m2 (t)

N0 = NB

AM Systems – Synchronous Demodulation

Synchronous demodulation is identical to DSB-SC in every respect except forthe addition carrier. So the signal at the input of the receiver is

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Hence, the desired signal power is

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

The output noise will be exactly the same as that of DSB-SC:

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

AM Systems – Synchronous DemodulationSynchronous demodulation is identical to DSB-SC in every respect except forthe addition carrier. So the signal at the input of the receiver is

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t)

= A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)

)m2 (t)

A2 + m2 (t). (28)

yi (t) =√

2 [A + m (t)] cos ωct + n (t) .

Si =(√

2)2 [A + m (t)]2

2= A2 + m2 (t) + 2Am (t) = A2 + m2 (t).

So = m2 (t)No = NB

m2 (t)NB

=A2 + m2 (t)NB

m2 (t)

A2 + m2 (t)=

)m2 (t)

A2 + m2 (t). (28)

AM Systems – Envelop Detection

In case of envelop detection, signal at the input of the receiver is

yi (t) = [A + m (t)] cos ωct + n (t)= [A + m (t) + nc (t)] cos ωct + ns (t) sin ωct. (29)

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

Since we are using envelop detection, output of the demodulator is given as

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Depending upon whether A + m (t) � n (t) or A + m (t) � n (t), we can usedifferent approximations. We will discuss only small-noise noise case. Forfurther explanation, see B. P. Lathi’s book.

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

=[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Si =[A + m (t)]2

= A2 + m2 (t) + 2Am (t) =A2 + m2 (t)

2. (30)

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t)

=A2 + m2 (t)

2. (30)

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

= Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

yo = Ei (t)

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

Si =[A + m (t)]2

2= A2 + m2 (t) + 2Am (t) =

A2 + m2 (t)2

. (30)

yo = Ei (t) =

√[A + m (t) + nc (t)]2 + n2

s (t). (31)

AM Systems – Envelop Detection (Small Noise Case)

yi (t) = [A + m (t)] cos ωct + n (t)= [A + m (t) + nc (t)] cos ωct + ns (t) sin ωct.

For small noise case, envelope Ei can be approximated as

yo = Ei (t) =√[A + m (t) + nc (t)]2 + n2

s (t) ≈ A + m (t) + nc (t) . (32)

ST = Si =A2 + m2 (t)

Ni = n2 (t) = 2NB

So = m2 (t)

N0 = 2NB

yo = Ei (t) =√[A + m (t) + nc (t)]2 + n2

s (t) ≈ A + m (t) + nc (t) . (32)

ST = Si =A2 + m2 (t)

Ni = n2 (t) = 2NB

So = m2 (t)

N0 = 2NB

yo = Ei (t) =√[A + m (t) + nc (t)]2 + n2

s (t) ≈ A + m (t) + nc (t) . (32)

ST = Si =A2 + m2 (t)

Ni = n2 (t) = 2NB

So = m2 (t)

N0 = 2NB

yo = Ei (t) =√[A + m (t) + nc (t)]2 + n2

s (t) ≈ A + m (t) + nc (t) . (32)

ST = Si =A2 + m2 (t)

Ni = n2 (t) = 2NB

So = m2 (t)

N0 = 2NB

yo = Ei (t) =√[A + m (t) + nc (t)]2 + n2

s (t) ≈ A + m (t) + nc (t) . (32)

ST = Si =A2 + m2 (t)

Ni = n2 (t) = 2NB

So = m2 (t)

N0 = 2NB

So, signal to noise ratio is given as

A2 + m2 (t)

A2 + m2 (t)2NB

A2 + m2 (t). (33)

So, signal to noise ratio is given as

A2 + m2 (t)

A2 + m2 (t)2NB

A2 + m2 (t). (33)

Outline

1 Introduction

2 DSB-SC Modulation

6 AM Receivers

7 Noise

8 Summary

4: Amplitude Modulation - ArraytoolIntroductionDSB-SC ModulationAmplitude ModulationBandwidth...

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