Lec 9 DC_ Spring 2012_Bandpass Modulation

8/2/2019 Lec 9 DC_ Spring 2012_Bandpass Modulation

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Bandpass Modulation

&

Demodulation

Engr. Ghulam Shabbir


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Modulation and Demodulation (MODEM)


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Digital Communication System


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Modulation and Demodulation (MODEM)


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Baseband vs. Bandpass Communications


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Bandpass transmission involves some translation ofthe baseband signal to some band of frequencycentered around fc

Bandpass Transmitter:

Carrier (high frequency pure sinusoidal generated by thelocal oscillator) is altered in response to a given low

frequency signal (message signal) generated by the source


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8

Bandpass Modulation and Demodulation

Bandpass Modulation is the process by which somecharacteristics of a sinusoidal waveform is variedaccording to the message signal.

Modulationshifts the spectrum of a baseband signalto some high frequency.

Demodulator/Decoder baseband waveform recovery


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Modulation Modulation - process (or result of the process)

of translation of the baseband message signalto bandpass (modulated carrier) signal at

frequencies that are very high compared to thebaseband frequencies.

Demodulation is the process of extracting thebaseband message back the modulated carrier.

An information-bearing signal is non-deterministic, i.e. it changes in an unpredictablemanner.


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Modulation & Demodulation

Baseband

Modulation

CarrierRadio

Channel

Synchronization/Detection/

Decision

Carrier

Data in Data out


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Modulation Modulation is a process of mixing a signal with a

sinusoid to produce a new signal. This new signal, conceivably, will have certain

benefits of an un-modulated signal, especially duringtransmission.

If we look at a general function for a sinusoid:f(t) =A sin (t+)

This sinusoid has the following 3 parameters that canbe altered, to affect the shape of the graph.

1. A, is called the magnitude, or amplitude of thesinusoid.

2. The term , is known as the frequency3. The term, is known as the phase angle.

All 3 parameters can be altered to transmit data.


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Modulation The sinusoidal signal that is used in the

modulation is known as the carrier signal, orsimply "the carrier".

The signal that is used in modulating the carriersignal (or sinusoidal signal) is known as the "datasignal" or the "message signal".

It is important to notice that a simple sinusoidalcarrier contains no information of its own.

In other words we can say that modulation isused because some data signals are not alwayssuitable for direct transmission, but themodulated signal may be more suitable.


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Why Modulate? Digital modulation is the process by which digital

symbols are transformed into waveforms that arecompatible with characteristics of the channel.

In the case of baseband modulation, thesewaveforms usually take the form of shaped pulses.

But in the case of bandpass modulation the shapedpulses modulate a sinusoid called a carrier wave, orsimply a carrier; for radio transmission the carrieris converted to an electromagnetic (EM) field ofpropagation to the desired destination.

The transmission of EM fields through space isaccomplished with the use of antennas.

Why it is necessary to use a carrier for the radiotransmission of baseband signals


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Why Modulate? The size of the antenna depends on the wavelength and

the application.

For cellular telephones, antennas are typically /4 in size,where wavelength is equal to c/f, and c, the speed of light, is

3 x 108 m/s.

Most channels require that the baseband signal be shifted toa higher frequency.

For example in case of a wireless channel antenna size isinversely proportional to the center frequency, this isdifficult to realize for baseband signals.

For speech signal f = 3kHz = c/f = (3x108) / (3x103) Antenna size without modulation /4=105 /4 meters = 15

miles - practically unrealizable

Same speech signal if amplitude modulated using fc= 900MHz will require an antenna size of about 8cm.

For this reason, carrier-wave or bandpass modulation is anessential step for all systems involving radio transmission.


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It means that we need height of antenna equal to 105m or 100 km! This is practically impossible!

However we can reduce its height by l/2, l/4, l/8 or up

to l/16. But even if we reduce it to l/16, it becomes 6.2km, which is still impossible! Therefore, we cannottransmit low frequency signals directly.

As per above equation, if we increase frequency ofelectrical signal the height will reduce.

However, this creates one more problem! We want totransmit electrical signals in audio range. But, ourhighest audio frequency is 20 kHz. For this, height ofantenna will be around 15 km. This height is also

impossible!

Why Modulate?


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This height is practically easily possible!

Also, our voice frequency constantly changes. So we willhave to change height of antenna constantly. So all theseproblems are absurd.

The only solution to this problem----- is the process ofmodulation.

In modulation very high frequency, carrier wave is taken.It is modulated (in either AM or FM style) by modulatingsignal, which we want to transmit actually.

After modulation, low frequency RIDESover carrier wave. This modulated carrier wave is connected to antenna for

transmission.

Now suppose we want to transmit 3 kHz signal, with 300MHz carrier wave. Then actually, 300 MHz signal is

transmitted. For this height of antenna will be

Why Modulate?


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Why Modulate? Now carrier signal will be transmitted with modulating

signal and at some distance, we shall receive it. In receiver, we want 3 kHz signal actually. So by another

process of DEMODULATION, we shall filter outunwanted 300 MHz signal and remaining PURE 3 kHzsignal will be used.

Bandpass modulation can provide other importantbenefits in signal transmission. If more than one signalutilizes a single channel, modulation may be used toseparate the different signals.

Modulation can be used to minimize the effects of

interference. A class of such modulation schemes, known as spread-

spectrum modulation, requires a system bandwidthmuch larger than the minimum bandwidth that would berequired by the message.


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Why Modulate?

Modulation can also be used to place a signal in afrequency band where design requirements, such afiltering and amplification, can be easily met. This is thecase when radio-frequency (RF) signals are converted toan intermediate frequency (IF) in a receiver.

This is evident that efficient antenna of realistic physicalsize is needed for radio communication system

For this reason, carrier-wave or bandpass modulation isan essential step for all systems involving radiotransmission.

Modulation also required if channel has to be shared byseveral transmitters (Frequency division multiplexing).


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WhyCarrier ?

Effective radiation of EM waves requires antennadimensions comparable with the wavelength:

Antenna for 3 kHz would be ~100 km long

Antenna for 3 GHz carrier is 10 cm long

Sharing the access to the telecommunication channelresources

Amplitude, frequency and phase of a carrier can bevaried according to the message signal.

Three ways of representing a modulated signal (M&P,I&Q and complex envelope).

ASK generation and demodulation/detection

FM (or FSK) equivalent to modulating the phase of the

carrier by the integral of the message signal.


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Bandpass modulation (either analog or digital) is the processby which an information signal is converted to a sinusoidal

waveform; for digital modulation, such a sinusoid of duration Tis referred to as a digital symbol.

The sinusoid has just three features that can be used todistinguish it from other sinusoids:

Amplitude

Frequency

Phase

Thus bandpass modulation can be defined as the processwhereby the amplitude, frequency or phase of an RF carrier, or

a combination of them, is varied in accordance with theinformation to be transmitted.

The general form of the carrier wave is

s(t)= A(t) cos(t)

where A(t)is the time-varying amplitude and (t) is the time-

varying angle.

DigitalBandpassModulation


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It is convenient to write (t) = (t) + (t), so that

s(t) = A(t) cos ( (t) +(t))

=A(t) cos (2fct +(t))

where 0 is the radian frequency of the carrier,(t) is the phase ; and = 2fc The three parameters (amplitude, frequency and

phase) can be varied in analog or digital form.

When varied in digital form, it is referred to asShifting & Keying.



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The basic bandpass modulation / demodulation

types are listed as under:


When the receiver exploits knowledge of thecarriers phase to detect the signals, the process iscalled coherent detection

When the receiver does not utilize such phasereference information, the process is called non-

coherent detection.


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Modulation Process

Modulation implies varying one or morecharacteristics (modulation parameters a1, a2, an) of a carrier f in accordance with the

information-bearing (modulating) basebandsignal.

Sinusoidal waves, pulse train, square wave,etc. can be used as carriers

1 2 31 2 3

, , , ... , (= carrier), , ,... (= modulation parameters)

(= time)

n

n

f f a a a a ta a a a

t


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Digital Bandpass Modulation TechniquesThree ways of representing bandpass signal:

(1) Magnitude and Phase (M & P)

Any bandpass signal can be represented as:

A(t) 0is real valued signal representing the magnitude

(t) is the genarlized angle (t)is the phase

The representation is easy to interpret physically, but often isnot mathematically convenient

In this form, the modulated signal can represent informationthrough changing three parameters of the signal namely:

Amplitude A(t) : as inAmplitude Shift Keying (ASK)

Phase (t) : as inPhase Shift Keying (PSK)

Frequency d(t)/dt : as in Frequency Shift Keying (FSK)

)](cos[)(cos[)()( 0 tttAttAts


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Continuous Carrier

Carrier: A sin[t +]

A = const

= const = const

Amplitude modulation(AM)

A = A(t) carries information = const

= const

Frequency modulation(FM)

A = const

= (t) carries information

= const

Phase modulation (PM)

A = const = const

= (t) carries information


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There are three major classes of digital modulation techniquesused for transmission of digitally represented data:

1. Amplitude Shift Keying (ASK)1. Frequency Shift Keying (FSK)

2. Phase-shift keying (PSK)

All convey data by changing some aspect of a base signal, thecarrier wave (usually a sinusoid), in response to a data signal.

In the case of PSK, the phase is changed to represent the datasignal.

There are two fundamental ways of utilizing the phase of asignal in this way:

1. By viewing the phase itself as conveying theinformation, in which case the demodulator must have areference signal to compare the received signal's phaseagainst; or

2. By viewing the changein the phase as conveyinginformation differential schemes, some of which do

not need a reference carrier (to a certain extent).

Digital Bandpass Modulation Techniques


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Angle Modulation

Angle modulation is a class of analog modulation.

In this type of modulation, the frequency or phase ofcarrier is varied in proportion to the amplitude of themodulating signal.

Frequency and phase modulation are also known asAngle Modulation.

These techniques are based on altering the angle (orphase) of a sinusoidal carrier wave to transmit data,as opposed to varying the amplitude, such as in AMtransmission.

Figure:

An angle modulated signal


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0( ) ( ) cos( ( )) ( ) cos( )s t A t t A t t

0

)()(

dt

tdti

dtt

i )()(

Angle Modulation

Consider a signal with constant frequency:

Frequency is the derivative of phase or the rate of change ofphase. The instantaneous frequency can be written as:

or


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Then considering a message signal

m(t) = Amcos (2fmt),

we can write the phase modulation as

0( ) ( )pt t K m t

0( ) cos[ ( )]PM ps t A t K m t

0( ) ( )i pd

t K m t dt

Phase Shift Keying (PSK) or PM

If we consider an angle modulated signal s(t) = A cos (t)

or a phase modulated signal as

where i= 2fc and


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In case of frequency modulation for a message signal

m(t) = Amcos (2fmt)

0( ) ( )i ft K m t

0( ) [ ( )]t

ft K m t d

0 ( )

t

ft K m d

0

0

( ) cos[ ( ) ]

cos[ ( )]

t

FM f

f

s t A t K m d

A t K a t

where:

( ) ( )

t

a t m d

Frequency Shift Keying (FSK) or FM


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0 0.05 0.1 0.15

-1

-0.5

0

0.5

1

The message signal

0 0.05 0.1 0.15

-1

-0.5

0

0.5

1

Time

The modulated signal

Example


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% Matlab script for generating FM via Phase modulation

clear all;close all;

t0=.15; % signal duration

ts=0.0005; % sampling interval

fc=200; % carrier frequency

kf=100; % Modulation index

fs=1/ts; % sampling frequency

t=[0:ts:t0]; % time vector

df=0.25; % required frequency resolution%message signal

m=[ones(1,t0/(3*ts)),-1*ones(1,t0/(3*ts)),ones(1,t0/(3*ts)+1)];

int_m(1)=0;

for i=1:length(t)-1 % Integral of m

int_m(i+1)=int_m(i)+m(i)*ts;

end

u=cos(2*pi*fc*t+2*pi*kf*int_m); % phase modulating with the% integral of the signal


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Phasor Representation of AmplitudeModulation

Consider the AM signal in phasor form:

0( ) Re 12 2

m mj t j tj t e e

s t e


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Phasor Representation of FM

Consider the FM signal in phasor form:

0( ) Re 12 2

m mj t j t j ts t e e e

Di it l d l ti ( ) PSK (b) FSK ( ) ASK (d) ASK/PSK (APK)


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Digital modulations, (a) PSK (b) FSK (c) ASK (d) ASK/PSK (APK)


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Amplitude Shift Keying Modulation Process

In Amplitude Shift Keying (ASK), the amplitude of the carrier is switchedbetween two (or more) levels according to the digital data

For BASK (also called ON-OFF Keying (OOK)), one and zero are representedby two amplitude levels A1and A0


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Analytical Expression:

where Ai=peak amplitude

Hence,

where

00,0

10),cos()(

binaryTt

binaryTttAts

ci

)cos(2)cos(2)cos()( 02

00 tAtAtAts rmsrms

R

V

PtT

E

tP

2

00 )cos(

2

)cos(2

1,......2,0,00,0

10),cos()(2

)( Mi

binaryTt

binaryTttT

tEts i

i

i

1,......2,0,)(0

2 MidttsET

i


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Where for binary ASK (also known as ON OFF Keying (OOK))

Mathematical ASK Signal Representation

The complex envelopeof an ASK signal is:

The magnitudeand phaseof an ASK signal are:

The in-phase and quadrature components are:

the quadrature component is wasted.

1( ) ( ) cos( ), 0 1c cs t A m t t t T binary

0 ( ) 0, 0 0s t t T binary

)()( tmAtg c

0)(),()( ttmAtA c

)()( tmAtx c

,0)( ty


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It can be seen that thebandwidth of ASKmodulated is twice thatoccupied by the source

baseband stream

Bandwidth of ASK

Bandwidth of ASK can be found from its power spectral density

The bandwidth of an ASK signal is twice that of the unipolar NRZline code used to create it., i.e.,

This is the null-to-null bandwidth of ASK

b

bT

RB 22


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If raised cosine rolloff pulse shaping is used, then the bandwidthis:

Spectral efficiency of ASK is half that of a baseband unipolar NRZline code

This is because the quadrature component is wasted

(1 )b

B r R


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Detectors for ASK (Coherent Receiver)

Coherent detection requires the phase information

A coherent detector mixes the incoming signal with a locallygenerated carrier reference

Multiplying the received signal r(t)by the receiver local oscillator

(say Accos (wct))yields a signal with a baseband component plusa component at 2fc

Passing this signal through a low pass filter eliminates the highfrequency component


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The output of the LPF is sampled once per bit period This sample z(T)is applied to a decision rule

z(T)is called the decision statistic

Matched filter receiver of OOK signal

A MF pair such as the rootraised cosine filter can thus beused to shape the source and

received baseband symbols In fact this is a very common

approach in signal detection inmost bandpass data modems

N h R i


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Noncoherent Receiver

Does not require a phase reference at the receiver

If we do not know the phase and frequency of the

carrier, we can use a noncoherent receiverto recoverASK signal

Envelope Detector:

The simplest implementation of an envelope detector

comprises a diode rectifier and smoothing filter


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Phase Shift Keying (PSK)


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Phase Shift Keying (PSK)

These signals are called antipodal.

The reason that they are chosen is that they have acorrelation coefficient of -1, which leads to theminimum error probability for the same Eb/No.

These two signals have the same frequency andenergy.


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Phase Shift Keying (PSK)BINARY PSK

All PSK signals can be graphically represented by asignal constellation in a two-dimensional coordinatesystem with 1(t) and 2(t) as its horizontal andvertical axis, respectively under:

Figure: BPSK signal constellation

and


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Phase Shift Keying (PSK) In PSK, the phase of the carrier signal is switched

between 2 (for BPSK) or more (for MPSK) in response to

the baseband digital data

With PSK the information is contained in theinstantaneous phase of the modulated carrier

Usually this phase is imposed and measured with respect

to a fixed carrier of known phase Coherent PSK For binary PSK, phase states of 0o and 180o are used

Waveform:

PSK Generation


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PSK Generation


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Phase shift keying (PSK) was developed during the earlydays of the deep-space program. PSK is now widely usedin both military and commercial communications

systems.

The general analytical expression for PSK can be writtenas

where

g(t)is signal pulse shape

A = amplitude of the signal. It is a peak value of the

waveform. = carrier phase

c= 2fc

( ) ( ) cos[ ( )], 0 , 1, 2,....,i c i bs t A g t t t t T i M


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Waveform Amplitude Coefficient


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Waveform Amplitude CoefficientThe waveform amplitude coefficient appearing in the equation of PSKhas same general form modulation formats. The derivation ofthis expression begins with

where A is the peak value of the waveform. Since the peak value ofthe sinusoidal waveform equals times the root-mean-square(rms) value, we can write

assuming the signal to be a voltage or a current waveform, A2rmsrepresents average power P (normalized to 1). Therefore, we can

write

Replacing P watts by E joules/T seconds, we get

So, we shall use either the amplitude A in above equation or

tcosAts )(

TE/2

2

tcosAts rms 2)(

tcos rms2 2

tcosPts 2)(

t)( cos T

2Ets

TE/2


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We can also write a PSK signal as:

Furthermore, s1(t) may be represented as a linear combination oftwo orthogonal functions 1(t) and 2(t) as follows

Where

M

it

T

Ets ci

)1(2cos

2)(

tM

it

M

i

T

Ecc

cossin

)1(2sincos

)1(2cos

2

)()1(2

sin)()1(2

cos)( 21 tM

iEt

M

iEtsi

]sin[2

)(]cos[2

)( 21 tT

tandtT

t cc

yxyxyx sinsincoscos)cos(


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Using the concept of the orthogonal basis function, wecan represent PSK signals as a two dimensional vector

For M-ary phase modulation M = 2k, where kis the numberof information bits per transmitted symbol

In an M-ary system, one of M 2possible symbols, s1(t),, sm(t), is transmitted during each Ts- second signalinginterval

The mapping or assignment of kinformation bits into

M = 2kpossible phases may be performed in many ways,e.g. for M = 4

21

)1(2sin,

)1(2cos)(

M

iE

M

iEts bbi


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A preferred assignment is to use Gray code in which

adjacent phases differ by only one binary digit such thatonly a single bit error occurs in a k-bit sequence.

It is also possible to transmit data encoded as the phasechange (phase difference) between consecutive symbols

This technique is known as Differential PSK (DPSK)

There is no non-coherent detection equivalent for PSKexcept for DPSK


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M-ary PSK In MPSK, the phase of the carrier takes on one of Mpossible

values

Thus, MPSK waveform is expressed as

Each si(t)may be expanded in terms of two basis function 1(t)and 2(t)defined as

MiM

iti ,.....,2,1,)1(2)(

M

itTEtsi

)1(2cos2)( 0

M

ittgtsi

)1(2cos)()( 0

...........

1616

884

2

2

PSK

PSKQPSK

BPSK

MPSKM k

,cos2

)(1 tT

t cs

,sin2

)(2 tT

t cs


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Quadrature PSK (QPSK) Two BPSK in phase quadrature

QPSK (or 4PSK) is a modulation technique that transmits 2-bitof information using 4 states of phases

For example

General expression:

2-bit Information

00 0

01 /2

10

11 3/2

Each symbol correspondsto two bits

sc

s

sQPSK Tti

M

itf

T

Ets

04,3,2,1,

)1(22cos

2)(


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We can also have:


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sc

s

sQPSK Tti

M

it

T

Ets

04,3,2,1,

4

)1(2cos

2)(

We can also have:


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One of 4 possible waveforms is transmitted during each signalinginterval Ts

i.e., 2 bits are transmitted per modulation symbol Ts=2Tb)

In QPSK, both the in-phase and quadrature components are used

The I and Q channels are aligned and phase transition occur onceevery Ts= 2Tbseconds with a maximum transition of 180 degrees

From

As shown earlier we can use trigonometric identities to show that

Mitf

TEts c

s

sQPSK )1(22cos2)(

)sin()1(2sin2)cos()1(2cos2)( tM

iTEt

Mi

TEts c

s

sc

s

sQPSK


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In terms of basis functions

we can write sQPSK(t) as

With this expression, the constellation diagram can easily be drawn For example:

tfT

tandtfT

t cs

c

s

2sin2

)(2cos2

)( 21

)(

)1(2sin)(

)1(2cos)( 21 t

M

iEt

M

iEts ssQPSK

Coherent Detection


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Coherent Detection

1. Coherent Detection of PSK

Coherent detection requires the phase information

A coherent detector operates by mixing the incoming data signalwith a locally generated carrier reference and selecting thedifference component from the mixer output

Multiplying r(t) by the receiver LO (say A cos(ct)) yields a signalwith a baseband component plus a component at 2fc

The LPF eliminates the high frequency component The output of the LPF is sampled once per bit period

The sampled value z(T) is applied to a decision rule

z(T) is called the decision statistic


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Matched filter receiver

A MF pair such as the root raised cosine filter can thus be used toshape the source and received baseband symbols

In fact this is a very common approach in signal detection in mostbandpass data modems

2 Coherent Detection of MPSK


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2. Coherent Detection of MPSK

QPSK receiver is composed of 2 BPSK receivers

one that locks on to the sine carrier and

the other that locks onto the cosine carrier

tAt 01 cos)(

tAt 02 sin)(

2

0 0 1 0 0 00 0

( ) ( ) ( ) ( cos ) ( cos )2

s sT T sA Tz t s t t dt A t A t dt L

1 0 2 0 00 0

( ) ( ) ( ) ( cos ) ( sin ) 0s sT T

z t s t t dt A t A t dt

)cos( 00 tAs


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If

Decision:

1. Calculate zi(t) as

2. Find the quadrant of (Z0, Z1)

Output S0(t) S1(t) S2(t) S3(t)

Z0 Lo 0 -Lo 0

Z1 0 -Lo 0 Lo

)45cos()()45cos()( 0201oo

tAtandtAt

Output S0(t) S1(t) S2(t) S3(t)

Z0 Lo -Lo -Lo Lo

Z1 Lo Lo -Lo -Lo

dtttrtz iT

i )()()( 0

4cos

2

2

0

sTAL


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A coherent QPSK receiver requires accurate carrier recovery usinga 4th power process, to restore the 90ophase states to modulo 2


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Correlation Receiver


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T

0

T

0

comparator selectssi(t)

with max zi(t)

Decision Stage

reference signal

.

.

.

.

..

.

.

.

.

.

.

)(1 t

)(tM

dtttrTzT

)()()( 10

1

dtttrTz MT

M )()()(0

T

0

T

0

comparator selectssi(t)

with max zi(t)

)( tsi

Decision Stage)(1 ts

reference signal

)(tsM

.....

.

......

dttstrTzT

)()()( 10

1

dttstrTz MT

M )()()(0

)()()( tntstr i

)( ts

i)()()( tntstr i


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)()()( 11 ttats ii

Transmitted signals si(t) in terms of 1(t) and coefficients ai1(t) are

Assume that s1 was transmitted, then values of product integratorswith reference to 1 are

)()()()( 11111 tEttats

)()()()( 11212 tEttats

T

dtttntEEszE0

1

2

11)()()(|

1

T

dtttntEEszE0

12

12)()()(|

1

22


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where E{n(t)}=0

Decision stage determines the the location of the transmitted signalwithin the signal space

For antipodal case choice of 1(t) = 2/T cosw0t normalizes E{zi(T)}to E

Prototype signals si(t) are the same as reference signals j(t) exceptfor normalizing scale factor

Decision stage chooses signal with largest value of zi(T)

EdttT

tntET

EszET

0 002

11 c os2

)(c os2

|

Edtt

TtntE

TEszE

T

0 00

2

12cos

2)(cos

2|

Sampled Matched Filter


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Sampled Matched Filter

The impulse response h(t) of a filter matched to s(t) is:

Let the received signal r(t) comprise a prototype signal si(t) plusnoise n(t)

Bandwidth of the signal is W =1/2T where T is symbol time then Fs= 2W = 1/T

Sample at t =kTs . This allows us to use discrete notation:

Let ci(n) be the coefficients of the MF where n is the time index andN represents the samples per symbol

elsewhere

TttTsth 0

0)()(

,...1,02,1)()()( kiknkskr i

])1[()( nNsnc ii

(eq 4.26)

(eq 4.27)


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Discrete form of convolution integral suggests

Since noise is assumed to have zero mean, so the expected valueof a received sample is:

Therefore, if si(t) is transmitted, the expected MF output is:

Combining eq (4.27) and (eq 4.29) to express the correlator outputs

at time k = N1 = 3:

Nmodulo,.....,1,0)()()(1

0

Kncnkrkz i

N

n

i

2,1)()( ikskrEi

Nmodulo,.....,1,0)()()(1

0

KncnkskzE i

N

n

ii

(eq 4.28)

(eq 4.29)

3

1 1 1

0

( 3) (3 ) ( ) 2n

z k s n c n

3

2 1 2

0

( 3) (3 ) ( ) 2n

z k s n c n

(eq 4.30a) (eq 4.30b)

Sampled


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p

Matched

Filter

Fig 4.10

Coherent Detection of MPSK


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The signal space for a multiple phase-shift keying (MPSK) signal setis illustrated for a four-level (4-ary) PSK or quadriphase shiftkeying(QPSK)

Fig 4.11


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At the transmitter, binary digits are collected two at a time for eachsymbol interval

Two sequential digits instruct the modulator as to which of the four

waveforms to produce si(t) can be expressed as:

where:E: received energy of waveform over each symbol duration T

w0: carrier frequency

Assuming an ortho-normal signal space, the basis functions are:

tT

t 01 cos2

)(

Mi

Tt

M

it

T

Etsi

,...1

0)

2cos(

2)( 0

tT

t 02 sin2

)(

si(t) can be written in terms of these orthonormal coordinates:


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si(t) can be written in terms of these orthonormal coordinates:

The decision rule for the detector is:

Decide that s1(t) was transmitted if received signal vector fall inregion 1

Decide that s2(t) was transmitted if received signal vector fall inregion 2 etc

i.e choose ith waveform if zi(T) is the largest of the correlatoroutputs

The received signal r(t) can be expressed as:

Mi

Tttatats iii

,...1

0)()()( 2211

)(2sin)(2cos 21 tM

iEtM

iE

Mi

Tttntt

T

Etr ii

,...1

0)(sinsincoscos

2)( 00

The upper corelator computes dtttrXT

)()(


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The upper corelator computes

The lower corelator computes

dtttrX )()( 10

dtttrYT

)()( 20

The computation of the received


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pphase angle can beaccomplished by computing thearctan of Y/X

Where:X: is the inphase component of thereceived signal

Y: is the quadrature component ofthe received signal

: is the noisy estimate of thetransmitted i

The demodulator selects the ithat is closest to the angle

Or it computes | i - | for each iprototypes and chooses i yieldingsmallest output

Fig 4.13

Coherent Detection of FSK


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FSK modulation is characterized by the information in the frequencyof the carrier

Typical set of FSK signal waveform:

Where : is an arbitrary constant

E: is the energy content of si(t) over each symbol duration T

(wi+1- wi): is typically assumed to be an integral multiple of /T

Assuming the basis functions form an orthonormal set:

Amplitude 2/T normalizes the expected output of the MF

Mi

Ttt

T

Ets ii

,...1

0)cos(

2)(

NjtT

t jj ,....,1cos2

)(

dttT

tT

Ea ji

T

ij )cos(2

)cos(2

0


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Therefore

This implies, the ith prototype signal vector is located on the ith

coordinate axis at a displacement E from the origin of the symbolspace

For general M-ary case and given E, the distance between any twoprototype signal vectors si and sj is constant:

otherwise

jiforEaij

0

jiforEssssd jiji 2||||),(

Signal space partitioning for 3-ary FSK


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NONCOHERENT DETECTION


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Detection of Differential PSK DPSK refers to a detection scheme classified as noncoherent

because it does not require a reference in phase with the received

carrier Therefore if the transmitted waveform is:

The received signal can be characterized by:

is an arbitrary constant ; assumed to be uniform random variabledistributed between 0and 2n(t):AWGN process

Mi

Tttt

T

Ets ioi

,....1

0)](cos[

2)(

Mi

Tttntt

T

Etr io

,....1

0)(])(cos[

2)(


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If we assume that varies slowly relative to two period times(2T),the phase difference between two successive waveforms j(T1)andk(T2) is independent of :

Basis for differentially coherent detection of differentially encoded

PSK is: Carrier phase of previous signaling interval is used as phase

reference for demodulation

Its use requires differentially encoded message signal at thetransmitter since information is carried by difference in phase

between successive waveforms

)()()()()( 21212 TTTTT ijkjk


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Detector in general calculates thecoordinates of the incoming signal

by correlating it with locallygenerated waveforms such as:2/T cosw0t and 2/T sinw0t

The detector then measures the

angle between the currentlyreceived signal vector and thepreviously received signal vector.

Fig 4.16 Signal space for DPSK

Detection of Binary PSK


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Sample Index,k 0 1 2 3 4 5 6 7 8 9 10Information Message,ak 1 1 0 1 0 1 1 0 0 1

Differentially Encodedmessage (first bitarbitrary), dk

1 1 1 0 0 1 1 1 0 1 1

Corresponding phaseshift, k

0 0 0 0 0 0 0 0

Differential Encoding

kkk add 1

kkk

add 1

Encoding Schemes

Decoding Scheme


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Sample Index,k 0 1 2 3 4 5 6 7 8 9 10

r(k) 0 0 0 1 1 0 0 0 1 0 0

1 1 0 1 0 1 1 0 0 1

)( ka

1 kkk rra

Advantages1) Phase ambiguity can be resolved

2) Non-coherent detection techniques can be used

)( ka

Sample Index,k 0 1 2 3 4 5 6 7 8 9 10r(k) 1 1 1 0 0 1 1 1 0 1 1

1 1 0 1 0 1 1 0 0 1

Drawback of Differential Encoding/Decoding:


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)( ka

Sample Index,k 0 1 2 3 4 5 6 7 8 9 10

r(k) 1 1 1 1 0 1 1 1 0 1 1

1 1 1 0 0 1 1 0 0 1

2-bits in error

When single bit errors occur in the received data sequence dueto noise, they tend to propagate as double bit errors

Since the decoder is comparing the logic state of current bit withprevious bit, and if the previous bit is in error, the next decoded bitwill also be in error

Differentially Encoded PSK (DEPSK) Modulation

Th d d {d } i d t h hift i ith


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The encoded sequence {dk} is used to phase-shift a carrier withphase angle 0 and representing symbols 1 and 0respectively

Method for the detection of DEPSK

Coherent detection of PSK followed by differential decoder (dk isequivalent to rk in the previous slides)


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In DPSK, the carrier phase of the previous data bit can be usedas a reference

Detection of FSK


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The demodulator must be configured as an energy

detector, without exploiting phase measurements

For this reason noncoherent detector requires twice asmany channel branches as the coherent detector.

One implementation can be obtained by using the in-phase (I) and quadrature (Q) channels to detect a binaryFSK (BFSK) as shown in the next slide.

{Correlation Squaring

I and Q energy

Summation Test statistic and

decision


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T

0

T

0

Decision Stage

I channel

Q channel

2

(.)

2(.)

tT 1cos/2

tT1sin/2

)(1 Tz

)(2 Tz

)(21

Tz

)(2

2Tz

T

0

T

0

I channel

Q channel

2(.)

2(.)

tT 2cos/2

tT 2sin/2

)(3 Tz

)(4 Tz

)(23

Tz

)(24 Tz

TzTz ()( 242

3

)()(2

2

2

1 TzTz

)(tr )(Tz

2H

1H

0)(Tz

)( tsiDecision Stage

{ { { {

Fig 4.18

One of the simplest ways of detecting binary FSK is to pass the


94/114

One of the simplest ways of detecting binary FSK is to pass thesignal through 2 Band Pass Filters(BPF) tuned to the 2 signalingfreqs and detect which has the larger output averaged over a

symbol period

An envelope detector consists of a rectifier and a low pass filter


95/114

An envelope detector consists of a rectifier and a low-pass filter

Detectors are matched to signal envelopes and not to signalsthemselves

For BFSK, decision is made on the basis of maximum amplitudeamongst the two envelope detectors

For MFSK, decision is made on the basis of which of the Menvelope detectors has the maximum output

Fig 4.19

Required Tone Spacing for Noncoherent


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Orthogonal FSK Signaling

FSK is usually implemented as orthogonal signaling, but not all FSKsignaling is orthogonal

How can we tell

If the two tones f1 and f2 are orthogonal to each other

Or they are uncorrelated over symbol time T

Property: any pair of tones in the set must have a frequencyseparation that is a multiple of 1/T Hz

A tone with frequency f1 that is switched on for a symbol duration Tseconds and then switched off can be analytically described as:

))/()2(cos()( Ttrecttfts ii

2/||0

2/2/1)/(

Ttfor

TtTforTtrect

Fourier transform of si(t):


97/114

The spectra of two such adjacent tones: tone1 with frequency f1 andtone 2 with frequency f2 are shown:

TffcTts ii )(sin)}({

Fig 4.20

F FSK th b d idth i t l t d t th t l


98/114

For FSK the bandwidth requirements are related to the spectralspacing between the two tones

The frequency difference between the center of the spectral main

lobe and the first zero crossing represents the minimum requiredspacing

With noncoherent detection, this corresponds to a minimum toneseparation of 1/T Hz.

For coherent detection, this corresponds to a minimum toneseparation of 1/2T Hz.

See Example 4.3, Page 202.

Example 4.3: Non-coherent FSK signaling


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Two waveforms for: where f1>f2, thesymbol rate 1/T and = a constant arbitrary angle

(a) Prove that the minimum tone spacing for non coherentlydetected orthogonal FSK signaling is 1/T

1 2cos(2 ) cos 2f t and f t

1 2

0

cos(2 )cos 2 0

T

f t f t dt

1 2 1 2

0 0

cos cos 2 cos 2 sin sin 2 cos2 0

T T

f t f t dt f t f t dt

1 2 1 2

0

cos [cos 2 ( ) cos 2 ( ) ]T

f f t f f t dt 1 2 1 2

0

sin [sin 2 ( ) sin 2 ( ) ] 0T

f f t f f t dt

(eq 4.45)

T


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1 2 1 2

1 2 1 2 0

sin 2 ( ) sin 2 ( )cos

2 ( ) 2 ( )

T

f f t f f t

f f f f

1 2 1 2

1 2 1 2 0

cos 2 ( ) cos 2 ( )sin 02 ( ) 2 ( )

T

f f t f f tf f f f

1 2 1 2

1 2 1 2

sin 2 ( ) sin 2 ( )cos

2 ( ) 2 ( )

f f T f f T

f f f f

1 2 1 2

1 2 1 2

cos 2 ( ) 1 cos 2 ( ) 1sin 0

2 ( ) 2 ( )

f f T f f T

f f f f

Assume that f1+f2 >> 1

1 2 1 2

1 2 1 2

sin 2 ( ) cos 2 ( )0

2 ( ) 2 ( )

f f T f f T

f f f f

(eq 4.49)

(eq 4.50)

C bi i 4 49 & 4 50


101/114

1 2 1 2cos sin 2 ( ) sin cos2 ( ) 1 0f f T f f T

1 2

1 2

Note that for arbitrary , the termseq4.51cansum to0 only when sin 2 ( ) 0

andsimultaneouslycos2 ( ) 1

f f T

f f T

sin 0x for x n and

cos 1 2 integersx for x k where n and k are

:for arbitrary

1 22 ( ) 2f f T k or 1 2k

f fT

1 2

1(minimum 1)f f tone spacing at k

T

Combining eq-4.49 & eq-4.50

(eq 4.51)

(eq 4.52)


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(b) What is the minimum tone spacing for coherently detectedorthogonal FSK signaling?

For coherent detection we need to satisfy eq 4.45 for = 0instead of any arbitrary phase. Rewriting eq 4.51 for = 0

Or

1 2sin 2 ( ) 0f f T

1 22

nf f

T

1 2

1(minimum 1)

2f f tone spacing at n

T

Complex Envelope


103/114

A real bandpass waveform s(t) can be represented using complexnotation

Where g(t) is known as the complex envelope, expressed as:

The magnitude of the complex envelope is:

And its phase:

g(t) is called the baseband message or data in complex form and

, the carrier wave in complex form

})(Re{)(0tjw

etgts

)()()(|)(|)()()(

tjtjetRetgtjytxtg

)()(|)(|)( 22 tytxtgtR

)(

)(

tan)(1

tx

ty

t

tjwe 0

Then the modulation can be expressed as the product of the two


104/114

p p

Quadrature Implementation of a Modulator

Consider a baseband waveform g(t) appearing at discrete times k=1,2,, Then g(t), x(t) and y(t) can be written as gk, xk,yk. Let xk=yk=0.707A; Then complex envelope can be expressed in discrete form

as:

The modulation process suggests:

]}sin[cos)]()(Re{[)( 00 tjttjytxts

ttyttx00 sin)(cos)(

AjAjyxg kkk 707.0707.0

}Re{)( 0tjw

k egts ]}sin[cos]Re{[ 00 tjtjyx kk

tytx kk 00 sincos

tAtA 00 sin707.0cos707.0

4cos 0

tA

Quadrature Type Modulator


105/114

Lead/Lagrelationships

of sinusoids

Error Performance For Binary Systems

4 7 1 Probability of Bit error for coherently detected BPSK


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4.7.1 Probability of Bit error for coherently detected BPSK

Whereas the received signal is :r(t) = si(t)+n(t)

The antipodal signals s1(t) and s2(t) can be characterized in a one-dimensional signal space as:

Decision is made on the basis:

Probability of bit error PB,

TttEts

tEts

0

)()(

)()(

22

11

otherwisets

Tzifts

)(

0)()(

2

01

0

21

)2/(

2

22exp

2

1

021

aaQduP

aaB

For equal-energy antipodal signaling the receiver output components are


107/114

For equal energy antipodal signaling, the receiver output components are

Then

2211 sforEaandsforEa bb

duPNE

Bb

0/2

2

2exp

2

1

0

2

N

EQP bB

Probability of Bit error for coherently detected Differentially

Encoded BPSK


108/114

Encoded BPSK

00

2

1

2

2 N

E

QN

E

QPbb

B

Fig 4.25

Probability of Bit error for coherently detected Binary

Orthogonal FSK


109/114

Orthogonal FSK

A general treatment for binary coherent signals (not limited to

antipodal signals) yields

For orthogonal, BFSK =/2; thus =0 and:

duPNE

Bb

0/)1(

2

2exp

2

1

duPNE

Bb

0/

2

2exp

2

1

0NEQP bB

Probability of Bit error for Noncoherently detected Binary


110/114

Orthogonal FSK

02exp2

1

N

E

Pb

B

Eb/N0 penalty of the simplernoncoherent detection is onlyabout 1dB at practical bit error

rates As a result, the simpler,

noncoherent FSK forms thebasis of many low end (e.g.1200 bps) telephone and radiomodems in the market-place

Note that, non-coherentperformance of FSK is notnearly as bad as that for ASK

Probability of Bit error for Binary DPSK


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Theoretical performance forCOPSK and DPSK is shownhere for an AWGN channel

BER for COPSK is exactly thesame as that derived forbipolar (antipodal) basebandtransmission

0

exp2

1

N

EP bB

Advantages and Disadvantages of FSK


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Advantages

FSK is a constant envelope modulation

hence insensitive to amplitude (gain) variations in the channel hence compatible with non-linear transmitter and receiver

systems

Detection of FSK can be based on relative frequency changesbetween symbol states and thus does not require absolute

frequency accuracy in the channel (FSK is thus relatively tolerant toLO drift and Doppler Shift)

Disadvantages

FSK is less bandwidth efficient than ASK or PSK The bit/symbol error rate performance of FSK is worse than PSK

In case of FSK, increasing the number of frequencies can increasethe occupied bandwidth

Advantages and Disadvantages of PSK


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Advantage:

Bandwidth Efficiency

In order to improve on the bandwidth efficiency of bandpass datatransmission, we can increase the number of symbol states

A reduction in bandwidth by a factor of k

Mkk

BB

Binary

aryM 2_ log

b

aryM

b

BinarykT

BthenT

Bif1

,1

_

Disadvantages:


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Reduced immunity to noise

As a general rule, we know that as the number ofsymbol states is increased, the tolerance to noise isreduced

Two exceptions to this rule, QPSK and orthogonalMFSK

Decreased immunity to noise compared to binary

Increased transmission power compared to binary

Increased complexity compared to binary Lower transmission quality compared to binary

Lec 9 DC_ Spring 2012_Bandpass Modulation

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Transcript of Lec 9 DC_ Spring 2012_Bandpass Modulation