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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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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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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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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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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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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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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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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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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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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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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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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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Matched Filter: Frequency domain View
Simple Bandpass Filter:
excludes noise, but misses some signal power
(C )
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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[
ifregioninsideliesVector
z
z
z
ik
Z
k
i
= allforminimumis,
ifregioninsideliesVector
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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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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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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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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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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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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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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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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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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( )
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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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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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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+
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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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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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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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)
Example of eye pattern:Binary-PAM, SRRC pulse
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u AWGN (Eb/N0=20 dB) and no ISI
Example of eye pattern:Binary-PAM, SRRC pulse
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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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