Ntoshiba Lecture Ofr Ciit

8/6/2019 Ntoshiba Lecture Ofr Ciit

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Shivkumar KalyanaramanRensselaer Polytechnic Institute

1 : shiv rpi

Point-to-Point Wireless Communication (I):

Digital Basics, Modulation, Detection, Pulse

Shaping

Shiv Kalyanaraman

Google: Shiv [email protected]

Based upon slides of Sorour Falahati, A. Goldsmith,

& textbooks by Bernard Sklar & A. Goldsmith
mailto:[email protected]:[email protected]


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Shivkumar KalyanaramanRensselaer Polytechnic Institute2 : shiv rpi

The Basics


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Big Picture: Detection under AWGN


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Additive White Gaussian Noise (AWGN)

u

Thermal noise is described by a zero-mean Gaussian random process,n(t) that ADDS on to the signal => additive

Probability density function

(gaussian)

[w/Hz]

Power spectral

Density

(flat => white)

Autocorrelation

Function

(uncorrelated)

Its PSD is flat, hence, it is called white noise. Autocorrelation is a spike at 0: uncorrelated at any non-zero lag


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Effect of Noise in Signal Space

u

The cloud falls off exponentially (gaussian).u Vector viewpoint can be used in signal space, with a random noise vectorw


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Maximum Likelihood (ML) Detection:ScalarCase

Assuming both symbols equally likely: uA is chosen if

likelihoods

Log-Likelihood => A simple distance criterion!


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AWGN Detection for Binary PAM

)(1 t

bEbE1

s2

s

)|( 2mp zz

0

)|( 1mp zz

2/

2/)()(

0

21

21

==

NQmPmP

ee

ss

2

)2(0

==

N

EQPP bEB


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Bigger Picture

General structure of a communication systems

FormatterSource

encoder

Channel

encoderModulator

FormatterSource

decoder

Channel

decoderDemodulator

Transmitter

Receiver

SOURCE

Info.Transmitter

Transmitted

signal

Received

signalReceiver

Received

info.

Noise

ChannelSource User


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Digital vs Analog Comm: Basics


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Digital versus analog

u Advantages of digital communications:u Regenerator receiver

u Different kinds of digital signal are treated

identically.

Data

Voice

Media

Propagation distance

Original

pulse

Regenerated

pulse

A bit is a bit!


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Signal transmission through linear systems

u Deterministic signals:u

Random signals:

u Ideal distortion less transmission:

All the frequency components of the signal not only arrive with

an identical time delay, but also are amplified or attenuatedequally.

Input Output

Linear system


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Signal transmission (contd)u Ideal filters:

u

Realizable filters:RC filters Butterworth filter

High-pass

Low-pass

Band-pass

Non-causal!

Duality => similar problems occur w/

rectangular pulses in time domain.


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Bandwidth of signal

u

Baseband versus bandpass:

u Bandwidth dilemma:

uBandlimited signals are not realizable!uRealizable signals have infinite bandwidth!

Baseband

signal

Bandpass

signal

Local oscillator


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Bandwidth of signal: Approximations

u

Different definition of bandwidth:a) Half-power bandwidthb) Noise equivalent bandwidth

c) Null-to-null bandwidth

d) Fractional power containment bandwidth

e) Bounded power spectral density

f) Absolute bandwidth

(a)

(b)

(c)

(d)

(e)50dB


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EncodeTransmit

Pulse

modulateSample Quantize

Demodulate/

Detect

Channel

Receive

Low-pass

filterDecode

Pulse

waveformsBit stream

Format

Format

Digital info.

Textual

info.

Analog

info.

Textualinfo.

Analog

info.

Digital info.

source

sink

Formatting and transmission of baseband signal


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Sampling of Analog Signals

Time domain)()()( txtxtxs =

)(tx

Frequency domain)()()( fXfXfXs =

|)(| fX

|)(| fX

|)(| fXs)(txs

)(tx


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Aliasing effect & Nyquist Rate

LP filter

Nyquist rate

aliasing


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Undersampling &Aliasing in

Time Domain


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Note: correct reconstruction does not draw straight lines between samples.

Key: use sinc() pulses for reconstruction/interpolation

Nyquist Sampling & Reconstruction:

Time Domain


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The impulse response of the

reconstruction filter has a classic 'sin(x)/x

shape.

The stimulus fed to this filter is the series

of discrete impulses which are the

samples.

Nyquist Reconstruction: Frequency Domain


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Sampling theorem

u Sampling theorem: A bandlimited signal with no spectral

components beyond , can be uniquely determined by values

sampled at uniform intervals of

u The sampling rate,

is calledNyquist rate.

u In practice need to sample faster than this because the receiving

filter will not be sharp.

Samplingprocess

Analogsignal

Pulse amplitudemodulated (PAM) signal


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Quantization

u

Amplitude quantizing:Mapping samples of a continuous amplitudewaveform to a finite set of amplitudes.

In

Out

Quan

tiz

ed

valu

es

Average quantization noise power

Signal peak power

Signal power to average

quantization noise power


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Encoding (PCM)

u A uniform linear quantizer is called Pulse Code Modulation

(PCM).

u Pulse code modulation (PCM): Encoding the quantized signals

into a digital word (PCM word or codeword).

u Each quantized sample is digitally encoded into an lbits

codeword whereL in the number of quantization levels and


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Quantization error

u

Quantizing error:The difference between the input and output of aquantizer )()()( txtxte =

+

)(tx )( tx

)()(

)(

txtx

te

=

AGC

x

)(xqy =Qauntizer

Process of quantizing noise

)(tx )( tx

)(te

Model of quantizing noise


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Non-uniform quantization

u It is done by uniformly quantizing the compressed signal.

u At the receiver, an inverse compression characteristic, called expansion is

employed to avoid signal distortion.

compression+expansion companding

)(ty)(tx )( ty )( tx

x

)(xCy = x

yCompress Qauntize

Channel

Expand

Transmitter Receiver


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Baseband transmission

u To transmit information thru physical channels, PCM sequences(codewords) are transformed to pulses (waveforms).u Each waveform carries a symbol from a set of size M.

u Each transmit symbol represents bits of the PCM words.

u

PCM waveforms (line codes) are used for binary symbols (M=2).

u M-ary pulse modulation are used for non-binary symbols (M>2).

Eg: M-ary PAM.u For a given data rate, M-ary PAM (M>2) requires less bandwidth than

binary PCM.

u For a given average pulse power, binary PCM is easier to detect than M-

ary PAM (M>2).

Mk 2lo g=


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PAM example: Binary vs 8-ary


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Example of M-ary PAM

-B

B

T

01

3B

T

T

-3B

T

0010

1

A.

T

0

T

-A.

Assuming real time Tx and equal energyper Tx data bitforbinary-PAM and 4-ary PAM:

4-ary: T=2Tb and Binary: T=Tb

Energy per symbol in binary-PAM:

4-ary PAM

(rectangular pulse)

Binary PAM

(rectangular pulse)

11

22 10BA =


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Other PCM waveforms: Examples

u PCM waveforms category:

Phase encoded Multilevel binary

Nonreturn-to-zero (NRZ) Return-to-zero (RZ)

1 0 1 1 0

0 T 2T 3T 4T 5T

+V

-V

+V

0

+V

0

-V

1 0 1 1 0

0 T 2T 3T 4T 5T

+V

-V

+V

-V

+V

0

-V

NRZ-L

Unipolar-RZ

Bipolar-RZ

Manchester

Miller

Dicode NRZ


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PCM waveforms: Selection Criteria

u

Criteria for comparing and selecting PCM waveforms:u Spectral characteristics (power spectral density and

bandwidth efficiency)

u Bit synchronization capability

u Error detection capability

u Interference and noise immunity

u Implementation cost and complexity


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Summary: Baseband Formatting and transmission

u Information (data- or bit-) rate:

u Symbol rate :

Sampling at rate

(sampling time=Ts)

Quantizing each sampled

value to one of theL levels in quantizer.

Encoding each q. value to

bits

(Data bit duration Tb=Ts/l)

Encode

Pulse

modulateSample Quantize

Pulse waveforms

(baseband signals)

Bit stream

(Data bits)Format

Digital info.

Textual

info.

Analog

info.

source

Mapping every data bits to a

symbol out of M symbols and transmitting

a baseband waveform with duration T

ss Tf /1= Ll 2log=

Mm 2log=

[bits/sec]/1 bb TR =ec][symbols/s/1 TR =

mRRb =


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Receiver Structure & Matched Filter


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Demodulation and detection

u Major sources of errors:

u Thermal noise (AWGN)

u disturbs the signal in an additive fashion (Additive)

u has flat spectral density for all frequencies of interest (White)

u is modeled by Gaussian random process (Gaussian Noise)

u Inter-Symbol Interference (ISI)u Due to the filtering effect of transmitter, channel and receiver, symbols

are smeared.

FormatPulse

modulate

Bandpass

modulate

Format DetectDemod.

& sample

)(tsi)(tgiim

im )(tr)(Tz

channel)(thc

)(tn

transmitted symbol

estimated symbol

Mi ,,1

=

M-ary modulation


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Impact of AWGN


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Impact of AWGN & Channel Distortion

)75.0(5.0)()( Tttthc =


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Receiver job

u

Demodulation and sampling:u Waveform recovery and preparing the received signal for

detection:

u Improving the signal power to the noise power (SNR)

using matched filter (project to signal space)uReducing ISI using equalizer (remove channel distortion)

u Sampling the recovered waveform

u Detection:u Estimate the transmitted symbol based on the received sample


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Receiver structure

Frequencydown-conversion

Receivingfilter

Equalizingfilter

Threshold

comparison

For bandpass signals Compensation for

channel induced ISI

Baseband pulse

(possibly distorted)Sample

(test statistic)Baseband pulse

Received waveform

Step 1 waveform to sample transformation Step 2 decision making

)(tr

)(Tz

i

m

Demodulate & Sample Detect


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Baseband vs Bandpass

u Bandpass model of detection process is equivalent to baseband model

because:

u The received bandpass waveform is first transformed to a baseband

waveform.

u Equivalence theorem:

u Performing bandpass linear signal processing followed by

heterodying the signal to the baseband,

u yields the same results as

u heterodying the bandpass signal to the baseband , followed by abaseband linear signal processing.


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39 : shiv rpi

Steps in designing the receiver

u

Find optimum solution for receiver design with the followinggoals:

1. Maximize SNR: matched filter

2. Minimize ISI: equalizer

u Steps in design:

u Model the received signal

u Find separate solutions for each of the goals.

u First, we focus on designing a receiver which maximizes theSNR: matched filter


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40 : shiv rpi

Receiver filter to maximize the SNR

u

Model the received signal

u Simplify the model (ideal channel assumption):

u Received signal in AWGN

)(thc)(tsi

)(tn

)(tr

)(tn

)(tr)(tsiIdeal channels)()( tthc =

AWGN

AWGN

)()()()( tnthtstr ci +=

)()()( tntstr i +=


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41 : shiv rpi

Matched Filter Receiveru Problem:

u Design the receiver filter such that the SNR is

maximized at the sampling time when

is transmitted.

u Solution:

u

The optimum filter, is the Matched filter, given by

which is the time-reversedand delayed version of the conjugate of the

transmitted signal

)(th

Mitsi ,...,1),( =

T0 t T0 t

)()()(*

tTsthth iopt ==

)2exp()()()(*

fTjfSfHfH iopt ==

)(tsi )()( thth opt=


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42 : shiv rpi

Correlator Receiveru The matched filter output at the sampling time, can be realized

as the correlator output.u Matched filtering, i.e. convolution withsi

*(T- ) simplifies

to integration w/si*( ), i.e. correlation or inner product!

>= maximize S/N


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Irrelevance Theorem: Noise Outside Signal Space

u Noise PSD is flat (white) => total noise power

infinite across the spectrum.u We care only about the noise projected in the finite

signal dimensions (eg: the bandwidth of interest).


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u Correlation is a maximum when two signals are similar in shape, and are in phase (or 'unshifted' with respect to each other).

u Correlation is a measure of thesimilarity between two signals as a function oftime shift (lag, ) between them

u When the two signals are similar in shape and unshifted with respect to each other, their product is all positive. This is like constructive interference,

u The breadth of the correlation function - where it has significant value - shows forhow longthe signals remain similar.

Aside: Correlation Effect


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Aside: Autocorrelation

A id C C l i &R d


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Aside: Cross-Correlation &Radar

u Figure: shows how the signal can be located within the noise.u A copy of the known reference signal is correlated with the unknown signal.

u The correlation will be high when the reference is similar to the unknown signal.

u A large value for correlation shows the degree of confidence that the reference signal is detected.

u The large value of the correlation indicates when the reference signal occurs.


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A copy of a known reference signal is correlated with the unknown signal. The correlation will be high if the reference is similar to the unknown signal. The unknown signal is correlated with a number of known reference functions. A large value for correlation shows the degree of similarity to the reference. The largest value for correlation is the most likely match.

Same principle in communications: reference signals corresponding to

symbols The ideal communications channel may have attenuated, phase shifted the

reference signal, and added noise

Source: Bores Signal Processing


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48 : shiv rpi

Matched Filter: back to cartoon

u Consider the received signal as a vectorr, and the transmitted signal vector as su Matched filter projects the r onto signal space spanned by s (matches it)

Filtered signal can now be safely sampled by the receiver at the correct sampling instants,

resulting in a correct interpretation of the binary message

Matched filter is the filter that maximizes the signal-to-noise ratio it can be

shown that it also minimizes the BER: it is a simple projection operation


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49 : shiv rpi

Example of matched filter (real signals)

T t T t T t0 2T

)()()( thtsty opti =2A)(tsi )(thopt

T t T t T t0 2T

)()()( thtsty opti =2A)(tsi )(thopt

T/2 3T/2T/2 T T/2

2

2A

T

A

T

A

T

A

T

AT

A

T

A


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50 : shiv rpi

Properties of the Matched Filter1. The Fourier transform of a matched filter output with the matched signal as input

is, except for a time delay factor, proportional to the ESD of the input signal.

2. The output signal of a matched filter is proportional to a shifted version of theautocorrelation function of the input signal to which the filter is matched.

3. The output SNR of a matched filter depends only on the ratio of the signal energyto the PSD of the white noise at the filter input.

4. Two matching conditions in the matched-filtering operation:

u spectral phase matching that gives the desired output peak at time T.

u spectral amplitude matching that gives optimum SNR to the peak value.

)2exp(|)(|)( 2 fTjfSfZ =

sss ERTzTtRtz === )0()()()(

2/max

0N

E

N

S s

T

=


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51 : shiv rpi

Implementation of matched filter receiver

Mz

z

1

z=)(tr

)(1 Tz)(

*

1 tTs

)(*

tTsM )(TzM

z

Bank of M matched filters

Matched filter output:

Observation

vector

)()( tTstrz ii = Mi ,...,1=),...,,())(),...,(),(( 2121 MM zzzTzTzTz ==z

Note: we are projecting along the basis directions of the signal space


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52 : shiv rpi

Implementation of correlator receiver

dttstrz i

T

i )()(0

=

T

0

)(1 ts

T

0

)(ts M

Mz

z

1

z

=

)(tr

)(1 Tz

)(TzM

z

Bank of M correlators

Correlators output:

Observation

vector

),...,,())(),...,(),(( 2121 MM zzzTzTzTz ==z

Mi ,...,1=

Note: In previous slide we filter i.e. convolute in the boxes shown.


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53 : shiv rpi

Implementation example of matched filter receivers

2

1

z

zz=

)(tr

)(1 Tz

)(2

Tz

z

Bank of 2 matched filters

T t

)(1 ts

T t

)(2 tsT

T0

0

T

A

T

AT

A

T

A

0

0


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54 : shiv rpi

Matched Filter: Frequency domain View

Simple Bandpass Filter:

excludes noise, but misses some signal power

(C )


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55 : shiv rpi

Multi-Bandpass Filter: includes more signal power, but adds more noise also!

Matched Filter: includes more signal power, weighted according to size

=> maximal noise rejection!

Matched Filter: Frequency Domain View (Contd)


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Maximal Ratio Combining (MRC) viewpoint

u

Generalization of this f-domain picture, for combiningmulti-tap signal

Weight each branch

SNR:

Examples of matched filter output for bandpass


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Examples of matched filter output for bandpass

modulation schemes


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58 : shiv rpi

Signal Space Concepts


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Signal space: Overview

u What is a signal space?u Vector representations of signals in an N-dimensional orthogonal space

u Why do we need a signal space?u It is a means to convert signals to vectors and vice versa.

u It is a means to calculate signals energy and Euclidean distances between

signals.

u Why are we interested in Euclidean distances between signals?

u For detection purposes: The received signal is transformed to a receivedvectors.

u The signal which has the minimum distance to the received signal is

estimated as the transmitted signal.


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60 : shiv rpi

Schematic example of a signal space

),()()()(

),()()()(

),()()()(

),()()()(

212211

323132321313

222122221212

121112121111

zztztztz

aatatats

aatatats

aatatats

=+==+==+==+=

z

s

s

s

)(1 t

)(2 t

),( 12111 aa=s

),( 22212 aa=s

),( 32313 aa=s

),( 21 zz=z

Transmitted signal

alternatives

Received signal at

matched filter output


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61 : shiv rpi

Signal space

u

To form a signal space, first we need to know theinner product between two signals (functions):

u Inner (scalar) product:

u Properties of inner product:

>=< dttytxtytx )()()(),(*

= cross-correlation between x(t) and y(t)

>=< )(),()(),( tytxatytax

>=< )(),()(),( * tytxataytx

>>=


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62 : shiv rpi

Signal space

u The distance in signal space is measure by calculating the norm.

u What is norm?

u Norm of a signal:

u Norm between two signals:

u We refer to the norm between two signals as the Euclidean distance between two

signals.

xEdttxtxtxtx ==>


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63 : shiv rpi

Example of distances in signal space

)(1 t

)(2 t

),( 12111 aa=s

),( 22212 aa=s

),( 32313 aa=s

),( 21 zz=z

zsd ,1

zsd ,2zsd ,3

The Euclidean distance between signalsz(t) ands(t):

3,2,1

)()()()( 2222

11,

=

+==

i

zazatztsd iiizsi

1E

3E

2E


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64 : shiv rpi

Orthogonal signal space

u N-dimensional orthogonal signal space is characterized by N linearly

independent functions called basis functions. The basis functionsmust satisfy the orthogonality condition

where

u If all , the signal space is orthonormal.

u Constructing Orthonormal basis from non-orthonormal set of vectors:

u Gram-Schmidt procedure

{ }Njj t 1)( =

jiij

T

iji Kdttttt =>=< )()()(),(*

0

Tt0Nij ,...,1, =

=

=ji

jiij

0

1

1=iK

E l f th l b


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65 : shiv rpi

Example of an orthonormal basesu Example: 2-dimensionalorthonormal signal space

u Example: 1-dimensionalorthonornal signal space

1)()(

0)()()(),(

0)/2sin(2

)(

0)/2cos(2

)(

21

2

0

121

2

1

==

=>==


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69 : shiv rpi

Signal space

==

N

j jiji

tats1

)()( ),...,,( 21 iNiii aaa=

s

iN

i

a

a

1

)(1 t

)(tN

1ia

iNa

)(tsiT

0

)(1 t

T

0

)(tN

iN

i

a

a

1

ms=)(tsi

1ia

iNa

ms

Waveform to vector conversion Vector to waveform conversion

Example of projecting signals to an


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70 : shiv rpi

Example of projecting signals to an

orthonormal signal space

),()()()(

),()()()(

),()()()(

323132321313

222122221212

121112121111

aatatats

aatatats

aatatats

=+==+==+=

s

s

s

)(1 t

)(2 t

),( 12111 aa=s

),( 22212 aa=s

),( 32313 aa=s

Transmitted signal

alternatives

dtttsaT

jiij )()(0

= Tt

0Mi ,...,1=Nj ,...,1=

Matched filter receiver (revisited)


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71 : shiv rpi

Matched filter receiver (revisited)

)(tr

1z)(1 tT

)( tTN Nz

Bank of N matched filters

Observation

vector

)()( tTtrz jj = Nj ,...,1=

),...,,( 21 Nzzz=z==N

j

jiji tats1

)()(

MNMi ,...,1=

Nz

z1

z= z

(note: we match to the basis directions)

Correlator recei er (re isited)


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72 : shiv rpi

Correlator receiver (revisited)

),...,,( 21 Nzzz=z

Nj ,...,1=dtttrz jT

j )()(0

=

T

0

)(1 t

T

0

)(tN

Nr

r

1

z=)(tr

1z

Nz

z

Bank of N correlators

Observation

vector

==

N

j jiji

tats1

)()( Mi ,...,1=

MN

Example of matched filter receivers using


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73 : shiv rpi

p e o c ed e ece ve s us g

basic functions

u Number of matched filters (or correlators) is reduced by 1 compared to using matched filters (correlators) to thetransmitted signal!

T t

)(1 ts

T t

)(2 ts

T t

)(1 t

T1

0

[ ]1

z z=)(tr z

1 matched filter

T t

)(1 t

T

1

0

1z

TA

T

A0

0

White noise in Orthonormal Signal Space


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74 : shiv rpi

White noise in Orthonormal Signal Space

u AWGN n(t) can be expressed as

)(~)()( tntntn +=

Noise projected on the signal space

(colored):impacts the detection process.

Noise outside on the signal space

(irrelevant)

>= < )(),( ttnn jj

0)(),(~ >=< ttn j

)()(1

tntnN

j

jj=

=

Nj ,...,1=

Nj ,...,1=

Vector representation of

),...,,( 21 Nnnn=n

)( tn

independent zero-meanGaussain random variables with

variance

{ }N

jjn 1=

2/)var( 0Nnj =


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Detection: Maximum Likelihood &

Performance Bounds


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76 : shiv rpi

Detection of signal in AWGN

u Detection problem:

u Given the observation vector , perform a mapping from

to an estimate of the transmitted symbol, , such that

the average probability of error in the decision is

minimized.

m im

Modulator Decision rule mimz

is

n

z z

S i i f h b i V


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Statistics of the observation Vector

u AWGN channel model:u Signal vector is deterministic.

u Elements of noise vector are i.i.d Gaussian

random variables with zero-mean and variance . The

noise vector pdf is

u The elements of observed vector are

independent Gaussian random variables. Its pdf is

),...,,( 21 iNiii aaa=s

),...,,( 21 Nzzz=z

),...,,( 21 Nnnn=n

nsz += i

2/0N

( )

=

0

2

2/

0

exp1

)(NN

pN

n

nn

( )

=

0

2

2/

0

exp1

)|(NN

pi

Ni

sz

szz

D t ti


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78 : shiv rpi

Detection

u

Optimum decision rule (maximum a posteriori probability):

u Applying Bayes rule gives:

.,...,1where

allfor,)|sentPr()|sentPr(

ifSet

Mk

ikmm

mm

ki

i

=

=

zz

ikp

mpp

mm

kk

i

=

=

allformaximumis,)(

)|(

ifSet

z

z

z

z

D t ti


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79 : shiv rpi

Detection

u

Partition the signal space into Mdecision regions,such thatMZZ ,...,1

i

kk

i

mm

ikp

mpp

Z

=

=

meansThat

.allformaximumis],)(

)|(ln[

ifregioninsideliesVector

z

z

z

z

z

i ( )


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80 : shiv rpi

Detection (ML rule)

u For equal probable symbols, the optimum decision rule

(maximum posteriori probability) is simplified to:

or equivalently:

which is known as maximum likelihood.

ikmp

mm

k

i

==

allformaximumis),|(

ifSet

zz

ikmp

mm

k

i

=

=

allformaximumis)],|(ln[

ifSet

zz

D t ti (ML)


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Detection (ML)

u Partition the signal space into Mdecision regions,

u Restate the maximum likelihood decision rule as follows:

MZZ ,...,1

i

k

i

mm

ikmp

Z

=

=

meansThat

allformaximumis)],|(ln[


z

z

z

ik

Z

k

i

= allforminimumis,


sz

z

S h ti l f ML d i i i


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Schematic example of ML decision regions

)(1 t

)(2 t

1s3s

2s

4s

1Z

2Z

3Z

4Z

P b bilit f b l


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Probability of symbol erroru Erroneous decision:For the transmitted symbol or equivalently signal

vector , an error in decision occurs if the observation vector does notfall inside region .

u Probability of erroneous decision for a transmitted symbol

u Probability of correct decision for a transmitted symbol

sent)insidelienotdoessent)Pr(Pr()Pr( iiii mZmmm z=

sent)insideliessent)Pr(Pr()Pr( iiii mZmmm z==

==iZ

iiiic dmpmZmP zzz z )|(sent)insideliesPr()(

)(1)( icie mPmP =

im

is

ziZ

E l f bi PAM


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Example for binary PAM

)(1 tbEbE 01s2s

)|( 1mp zz)|( 2mp zz

2

)2(0

==

N

EQPP bEB

2/

2/)()(

0

21

21

==

N

QmPmP eess

Average prob. of symbol error


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Average prob. of symbol error

u Average probability of symbol error :

u For equally probable symbols:

)|(11

)(1

1)(1

)(

1

11

=

==

=

==

M

i Z

i

M

i

ic

M

i

ieE

i

dmpM

mPM

mPM

MP

zzz

)(Pr)(1

i

M

i

E mmMP ==

Union bo nd


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Union bound

Union boundhe probability of a finite union of events is upper bounded

by the sum of the probabilities of the individual events.

=

M

ikk

ikie PmP1

2 ),()( ss ==

M

i

M

ikk

ikE PM

MP1 1

2 ),(1)( ss

Example of union bound


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Union bound:

Example of union bound

1

1s

4s

2s

3s

1Z

4Z3Z

2Z r2

= 432 )|()(11

ZZZ

e dmpmP rrr

1

1s

4s

2s

3s

2Ar

1

1s

4s

2s

3s

3A

r

1

1s

4s

2s

3s

4A

r2 2

2

=2

)|(),( 1122A

dmpP rrssr =

3

)|(),( 1132A

dmpP rrssr =

4

)|(),( 1142A

dmpP rrssr

=4

2121),()(

kke

PmP ss


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Upper bound based on minimum distance

==

=

2/

2/)exp(

1

sent)iswhen,thancloser toisPr(),(

00

2

0

2

N

dQdu

N

u

N

P

ik

d

iikik

ik

ssszss

kiikd ss =

= = 2/

2/)1(),(

1)(

0

min

1 1

2

N

dQMP

M

MPM

i

M

ikk

ikE ss

ik

kiki

dd

=,

min minMinimum distance in the signal space:

Example of upper bound on av. Symbol error prob.


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based on union bound

)(1 t

)(2 t

1s3s

2s

4s

2,1d

4,3d

3,2d

4,1dsEsE

sE

sE

4,...,1, === iEE siis

s

ski

Ed

ki

Ed

2

2

min

,

=

=


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Eb/No figure of merit in digital


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g g

communicationsu SNR or S/N is the average signal power to the average noise

power. SNR should be modified in terms of bit-energy in

digital communication, because:

u Signals are transmitted within a symbol duration and hence,

are energy signal (zero power).

u A metric at the bit-levelfacilitates comparison of differentDCS transmitting different number of bits per symbol.

b

bb

R

W

N

S

WN

ST

N

E==

/0

bR

W

: Bit rate

: Bandwidth

Note: S/N = Eb/No x spectral efficiency

Example of Symbol error prob. For PAM signals


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Example of Symbol error prob. For PAM signals

)(1 t

0

1s2s

bEbE

Binary PAM

)(1 t0

2s3s

52 b

E

56 b

E

56 b

E

52 b

E

4s 1s

4-ary PAM

T t

)(1 t

T

1

0

M i Lik lih d (ML) D t ti V t C


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Maximum Likelihood (ML) Detection: VectorCase

ps: Vector norm is a natural extension of magnitude or length

Nearest Neighbor Rule:

Project the received vectory along the

difference vector direction uA- uB is a

sufficient statistic.

Noise outside these finite dimensions is

irrelevantfor detection. (rotational

invariance of detection problem)

By the isotropic property of the Gaussian noise, we expect the error

probability to be the same for both the transmit symbols uA, uB.

Error probability:

Extension to M PAM (Multi Level Modulation)


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: shiv rpi

Extension to M-PAM (Multi-Level Modulation)

u Note: h refers to the constellation shape/direction

MPAM:

Complex Vector Space Detection


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Complex Vector Space Detection

Error probability:

u Note: Instead ofvT, use v* for complex vectors (transpose and conjugate)for inner products


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Complex

Detection:

Summary

Detection Error => BER


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Detection Error => BER

u If the bit error is i.i.d (discrete memoryless channel) over the sequence of bits,then you can model it as a binary symmetric channel (BSC)u BER is modeled as a uniform probabilityfu As BER (f) increases, the effects become increasingly intolerable

u ftends to increase rapidly with lower SNR: waterfall curve (Q-function)

SNR vs BER: AWGN vs Rayleigh


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SNR vs BER: AWGN vs Rayleigh

u Observe the waterfall like characteristic (essentially plotting the Q(x) function)!u Telephone lines: SNR = 37dB, but low b/w (3.7kHz)u Wireless: Low SNR = 5-10dB, higher bandwidth (upto 10 Mhz, MAN, and 20Mhz LAN)u Optical fiber comm: High SNR, high bandwidth ! But cant process w/ complicated codes, signal processing etc

Need

diversitytechniques

to deal with

Rayleigh

(even 1-tap,

flat-fading)!


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Better performance through diversity

Diversity the receiver is provided with multiple copiesof the transmitted signal. The multiple signal copiesshould experience uncorrelated fading in the channel.

In this case the probability that allsignal copies fadesimultaneously is reduced dramatically with respect to

the probability that a single copy experiences a fade.

As a rough rule:

0

1e L

P

is proportional to

BERBER Average SNRAverage SNR

Diversity ofL:th order

Diversity ofL:th order


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SNR

BER

Flat fading channel,Rayleigh fading,

L = 1AWGNchannel

(no fading)

( )eP=

0( )=

BER vs. SNR (diversity effect)

L = 2L = 4 L = 3

We will explore this story later slide set part II


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101 : shiv rpi

Modulation Techniques

What is Modulation?


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What is Modulation?

u Encoding information in a manner suitable for

transmission.

u Translate baseband source signal to bandpass signal

u Bandpass signal: modulated signal

u How?

u Vary amplitude, phase or frequency of a carrier

u Demodulation: extract baseband message from carrier

Digital vs Analog Modulation


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Digital vs Analog Modulation

u Cheaper, faster, more power efficient

u Higher data rates, power error correction, impairment

resistance:

u Using coding, modulation, diversity

u Equalization, multicarrier techniques for ISI

mitigation

u More efficient multiple access strategies, better

security: CDMA, encryption etc

Goals of Modulation Techniques


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Goals of Modulation Techniques

High Bit Rate

High Spectral Efficiency

High Power Efficiency

Low-Cost/Low-Power Implementation

Robustness to Impairments

(min power to achieve a target BER)

(max Bps/Hz)

Modulation: representation


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Modulation: representation Any modulated signal can be represented as

s(t) = A(t) cos [ ct + (t)]

s(t) = A(t) cos (t) cos ct- A(t) sin (t) sin ctamplitude

in-phase quadrature

phase or frequency

Linear versus nonlinear modulation impact on spectral efficiency

Constant envelope versus non-constantenvelope hardware implications with impact on power efficiency

Linear: Amplitude or phase

Non-linear: frequency: spectral broadening

(=> reliability: i.e. target BER at lower SNRs)

Complex Vector Spaces: Constellations


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Complex Vector Spaces: Constellations

u Each signal is encoded (modulated) as a vector in a signal space

MPSK

Circular Square

Linear Modulation Techniques


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107 : shiv rpi

s(t) = [ an g (t-nT)]cos ct - [ bn g (t-nT)] sin ctI(t), in-phase

Q(t), quadrature

LINEAR MODULATIONS

CONVENTIONAL

4-PSK

(QPSK)

OFFSET

4-PSK

(OQPSK)

DIFFERENTIAL

4-PSK

(DQPSK, /4-DQPSK)

M-ARY QUADRATURE

AMPLITUDEMOD.

(M-QAM)

M-ARY PHASE

SHIFT KEYING

(M-PSK)

M 4 M 4M=4(4-QAM = 4-PSK)

Square

Constellations

Circular

Constellations

n n

M-PSK and M-QAM


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108 : shiv rpi

M PSK and M QAM

M-PSK (Circular Constellations)

16-PSK

an

bn 4-PSK

M-QAM (Square Constellations)

16-QAM

4-PSK

an

bn

Tradeoffs

Higher-order modulations (M large) are more spectrallyefficientbut less power efficient (i.e. BER higher).

M-QAM is more spectrally efficient than M-PSK but

also more sensitive to system nonlinearities.

Bandwidth vs. Power Efficiency


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109 : shiv rpi

Bandwidth vs. Power Efficiency

Source: Rappaport book, chap 6

MPSK:

MQAM:

MFSK:

MPAM & Symbol Mapping


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MPAM & Symbol Mapping

u Note: the average energy

per-bit is constant

u Gray coding used for

mapping bits to symbols

u Why? Most likely error is toconfuse with neighboring

symbol.

u Make sure that the

neighboring symbol has only1-bit difference (hamming

distance = 1)

)(1 t

01s2s

bEbE

Binary PAM

)(1 t0

2s3s

52 bE

56 bE

56 bE

52 bE

4s 1s

4-ary PAM

)(1 t0

2s3s

52 b

E

56 b

E

56 b

E

52 b

E

4s 1s4-ary PAM

Gray coding00 01 11 10

MPAM: Details


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111 : shiv rpiDecision Regions

Unequal energies/symbol:

MPSK:

M-PSK (Circular Constellations)

bn 4-PSK


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u Constellation points:

u Equal energy in all signals:

u Gray coding00

01

11

10

0001

11 10

16-PSK

an

S

MPSK: Decision Regions & Demodln


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113 : shiv rpi

g

4PSK 8PSK

Coherent Demodulator for BPSK.

si(t) + n(t)

cos(2fct)

g(Tb - t)XZ1: r > 0

Z2: r 0

m = 1

m = 0

m = 0 or 1

Z1

Z2

Z3

Z4

Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

4PSK: 1 bit/complex dimension or 2 bits/symbol

MQAM:M-QAM (Square Constellations)

bn


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114 : shiv rpi

Q

u Unequal symbol energies:

u MQAM with square constellations of sizeL2 is equivalent to

MPAM modulation with constellations of sizeL on each ofthe in-phase and quadrature signal components

u For square constellations it takes approximately 6 dB morepower to send an additional 1 bit/dimension or 2bits/symbol while maintaining the same minimum distancebetween constellation points

u

Hard to find a Gray code mapping where all adjacentsymbols differ by a single bit

16-QAM

4-PSK

an

16QAM: Decision Regions

Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

Z10

Z11

Z12

Z13

Z14

Z15

Z16

Non-Coherent Modulation: DPSK


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u Information in MPSK, MQAM carried in signal phase.u Requires coherent demodulation: i.e. phase of the transmitted signal carrier

0

mustbe matched to the phase of the receiver carrier

u More cost, susceptible to carrier phase drift.u Harder to obtain in fading channels

u Differential modulation: do not require phase reference.u

More general: modulation w/ memory: depends upon prior symbols transmitted.u Use prev symbol as the a phase reference for current symbolu Info bits encoded as the differential phase between current & previous symbolu Less sensitive to carrier phase drift (f-domain) ; more sensitive to doppler

effects: decorrelation of signal phase in time-domain

Differential Modulation (Contd)


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116 : shiv rpi

( )

u DPSK: Differential BPSK

u A 0 bit is encoded by no change in phase, whereas a 1 bit is encoded as aphase change of.

u If symbol over time [(k1)Ts, kTs) has phase (k 1) = eji, i= 0, ,

u then to encode a 0 bit over [kTs, (k+ 1)Ts), the symbol would have

u

phase: (k) = eji

andu to encode a 1 bit the symbol would have

u phase (k) = ej(i+).

u DQPSK: gray coding:

Quadrature Offset


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Qu Phase transitions of 180o can cause large amplitude transitions (through zero

point).

u Abrupt phase transitions and large amplitude variations can be distorted bynonlinear amplifiers and filters

u Avoided by offsetting the quadrature branch pulseg(t) by half a symbol periodu Usually abbreviated as O-MPSK, where the O indicates the offset

u QPSK modulation with quadrature offset is referred to as O-QPSK

u O-QPSK has the same spectral properties as QPSK for linear amplification,..u but has higher spectral efficiency under nonlinear amplification,u since the maximum phase transition of the signal is 90 degrees

u Another technique to mitigate the amplitude fluctuations of a 180 degree phaseshift used in the IS-54 standard for digital cellular is /4-QPSK

u Maximum phase transition of 135 degrees, versus 90 degrees for offset QPSKand 180 degrees for QPSK

Offset QPSK waveforms


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Frequency Shift Keying (FSK)


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q y y g ( )

Continuous Phase FSK (CPFSK)

digital data encoded in the frequency shift typically implemented with frequency

modulator to maintain continuous phase

nonlinear modulation but constant-envelope

Minimum Shift Keying (MSK)

minimum bandwidth, sidelobes large

can be implemented using I-Q receiver

Gaussian Minimum Shift Keying (GMSK) reduces sidelobes of MSK using a premodulation filter

used by RAM Mobile Data, CDPD, and HIPERLAN

s(t) = A cos [ ct + 2 kf d( ) d ]t

Minimum Shift Keying (MSK) spectra


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y g ( ) p

Spectral Characteristics


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121 : shiv rpi

p

QPSK/DQPSK

GMSK

(MSK)

1.0

1.0 1.5 2.0 2.50.50-120

-100

-80

-60

-40

-20

0

10

0.25

B3-dB Tb = 0.16

Normalized Frequency (f-fc)Tb

P

owerSpect ra

lDensity( d

B)

Bit Error Probability (BER): AWGN10-1


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Pb

BPSK, QPSK

DBPSK

DQPSK

010-6

2

5

2

5

2

5

2

5

2

5

10-3

10-4

10-5

10-2

2 4 6 8 10 12 14

For Pb = 10-3

BPSK 6.5 dB

QPSK 6.5 dB

DBPSK ~8 dB

DQPSK ~9 dB

b, SNR/bit, dB

QPSK is more spectrally efficient than BPSKwith the same performance.

M-PSK, for M>4, is more spectrally efficientbut requires more SNR per bit.

There is ~3 dB power penalty for differential detecti

Bit Error Probability (BER): Fading Channel


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DBPSK

AWGN

BPSK

010-5

2

5

2

5

2

5

2

5

2

5

1

10-2

10-3

10-4

10-1

5 10 15 20 25 30 35

Pb

b, SNR/bit, dB

Pb is inversely proportion to the average SNR per bit.

Transmission in a fading environment requires about

18 dB more power for Pb

= 10-3 .

Bit Error Probability (BER): Doppler Effects


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124 : shiv rpi

0

0 60504030201010-6

10-5

10-4

10-3

10-2

10-1

10

DQPSK

Rayleigh Fading

No FadingfDT=0.003

0.002

0.001

0

QPSK

Pb

b, SNR/bit, dB

Doppler causes an irreducible error floor when differentialdetection is used decorrelation of reference signal.

The implication is that Doppler is not an issue for high-speed

wireless data.

data rate T Pbfloor

10 kbps 10-4s 3x10-4

100 kbps 10-5s 3x10-6

1 Mbps 10-6s 3x10-8

The irreducible Pb depends on the data rate and the Doppler.

For fD = 80 Hz,

[M. D. Yacoub, Foundations of Mobile Radio Engineering,

CRC Press, 1993]

Bit Error Probability (BER): Delay Spread


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125 : shiv rpi

+

x

+

+

+

+

+

x

x

x

x

x

10-210-4

10-3

10-2

10-1

10-1 100

BPSKQPSKOQPSKMSK

Modulation

Coherent Detection

IrreducibleP

b

T

=rms delay spread

symbol period

ISI causes an irreducible errorfloor.

The rms delay spread imposes a limit on themaximum bit rate in a multipath environment.

For example, for QPSK,

Maximum Bit RateMobile (rural) 25 sec 8 kbpsMobile (city) 2.5 sec 80 kbpsMicrocells 500 nsec 400 kbpsLarge Building 100 nsec 2 Mbps

[J. C.-I. Chuang, "The Effects of Time Delay Spread on Portable Radio

Communications Channels with Digital Modulation,"

IEEE JSAC, June 1987]

Summary of Modulation Issues


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Tradeoffs

linear versus nonlinear modulation

constant envelope versus non-constant

envelope

coherent versus differential detection

power efficiency versus spectral efficiency

Limitations

flat fading

doppler

delay spread


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Pulse Shaping

Recall: Impact of AWGN only


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Impact of AWGN & Channel Distortion


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)75.0(5.0)()( Tttthc =

Multi-tap, ISI channel

ISI Effects: Band-limited Filtering of Channel


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u ISI due to filtering effect of the communications channel

(e.g. wireless channels)

u Channels behave like band-limited filters

)()()( fjcc cefHfH =

Non-constant amplitude

Amplitude distortion

Non-linear phase

Phase distortion

Inter-Symbol Interference (ISI)


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u ISI in the detection process due to the filtering effects

of the systemu Overall equivalent system transfer function

u creates echoes and hence time dispersion

u causes ISI at sampling time

)()()()( fHfHfHfH rct=

i

ki

ikkk snsz

++= ISI effect

Inter-symbol interference (ISI): Model


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132 : shiv rpi

u Baseband system model

u Equivalent model

Tx filter Channel

)(tn

)(tr Rx. filterDetector

kz

kTt=

{ }kx{ }kx1x 2x

3xT

T)(

)(

fH

th

t

t

)(

)(

fH

th

r

r

)(

)(

fH

th

c

c

Equivalent system

)( tn

)(tz

Detector

kz

kTt=

{ }kx{ }kx1x 2x

3xT

T)(

)(

fH

th

filtered noise

)()()()( fHfHfHfH rct=

Nyquist bandwidth constraint


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u Nyquist bandwidth constraint (on equivalent system):

u

The theoretical minimum required system bandwidth todetectRs [symbols/s] without ISI isRs/2 [Hz].

u Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz]

can support a maximum transmission rate of2W=1/T=Rs[symbols/s] without ISI.

u Bandwidth efficiency,R/W[bits/s/Hz] :

u An important measure in DCs representing data throughputper hertz of bandwidth.

u Showing how efficiently the bandwidth resources are usedby signaling techniques.

Hz][symbol/s/222

1=

W

RW

R

T

ss

Equiv System: Ideal Nyquist pulse (filter)


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134 : shiv rpi

T21

T21

T

)( fH

f t

)/sinc()( Ttth =

1

0 T T2TT20

TW

2

1=

Ideal Nyquist filter Ideal Nyquist pulse

Nyquist pulses (filters)


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u Nyquist pulses (filters):

u Pulses (filters) which result in no ISI at the sampling time.

u Nyquist filter:

u Its transfer function in frequency domain is obtained by

convolving a rectangular function with any real even-

symmetric frequency function

u Nyquist pulse:

u Its shape can be represented by a sinc(t/T) function multiply

by another time function.

u Example of Nyquist filters: Raised-Cosine filter

Pulse shaping to reduce ISI


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u Goals and trade-off in pulse-shaping

u Reduce ISI

u Efficient bandwidth utilization

u Robustness to timing error (small side lobes)

Raised Cosine Filter: Nyquist Pulse Approximation


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137 : shiv rpi

2)1(Baseband sSB

sRrW +=

|)(||)(| fHfH RC=

0=r

5.0=r

1=r

1=r

5.0=r

0=r

)()( thth RC=

T2

1

T4

3

T

1T4

3T2

1T

1

1

0.5

0

1

0.5

0 T T2 T3TT2T3

sRrW )1(Passband DSB +=

Raised Cosine Filter


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u Raised-Cosine Filter

u A Nyquist pulse (No ISI at the sampling time)

>


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u Square-Root Raised Cosine (SRRC) filter and Equalizer

)()()()()(RC fHfHfHfHfH erct=

No ISI at the sampling time

)()()()()()()(

SRRCRC

RC

fHfHfHfHfHfHfH

tr

rt

==== Taking care of ISI

caused by tr. filter

)(

1)(

fH

fH

c

e = Taking care of ISI

caused by channel

Pulse Shaping & Orthogonal Bases


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Virtue of pulse shaping


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PSD of a BPSK signal

Source: Rappaport book, chap 6

Example of pulse shaping


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u Square-root Raised-Cosine (SRRC) pulse shaping

t/T

Amp. [V]

Baseband tr. Waveform

Data symbol

First pulse

Second pulse

Third pulse

Example of pulse shaping


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u Raised Cosine pulse at the output of matched filter

t/T

Amp. [V]

Baseband received waveform at

the matched filter output

(zero ISI)

Eye pattern


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u Eye pattern:Display on an oscilloscope which sweeps the systemresponse to a baseband signal at the rate 1/T(Tsymbol duration)

time scale

amplitu

descale

Noise margin

Sensitivity totiming error

Distortion

due to ISI

Timing jitter

Example of eye pattern:Binary-PAM, SRRC pulse


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u Perfect channel (no noise and no ISI)



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u AWGN (Eb/N0=20 dB) and no ISI



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u AWGN (Eb/N0=10 dB) and no ISI

Summary


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u Digital Basics

u Modulation & Detection, Performance, Bounds

u Modulation Schemes, Constellations

u Pulse Shaping


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Extra Slides

Bandpass Modulation: I, Q Representation


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Analog: Frequency Modulation (FM) vs

Amplitude Modulation (AM)


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u FM: all information in the phase or frequency of carrieru

Non-linear or rapid improvement in reception quality beyond a minimum received signalthreshold: capture effect.

u Better noise immunity & resistance to fading

u Tradeoff bandwidth (modulation index) for improved SNR: 6dB gain for 2x bandwidth

u Constant envelope signal: efficient (70%) class C power amps ok.

u AM: linear dependence on quality & power of rcvd signalu Spectrally efficient but susceptible to noise & fading

u Fading improvement using in-band pilot tones & adapt receiver gain to compensate

u Non-constant envelope: Power inefficient (30-40%) Class A or AB power amps needed:

the talk time as FM!

Example Analog: Amplitude Modulation


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