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Shivkumar KalyanaramanRensselaer Polytechnic Institute
1 : “shiv rpi”
Point-to-Point Wireless Communication (I):Digital Basics, Modulation, Detection, Pulse
Shaping
Shiv KalyanaramanGoogle: “Shiv RPI”
Based upon slides of Sorour Falahati, A. Goldsmith,& textbooks by Bernard Sklar & A. Goldsmith
Shivkumar KalyanaramanRensselaer Polytechnic Institute
2 : “shiv rpi”
The Basics
Shivkumar KalyanaramanRensselaer Polytechnic Institute
3 : “shiv rpi”
Big Picture: Detection under AWGN
Shivkumar KalyanaramanRensselaer Polytechnic Institute
4 : “shiv rpi”
Additive White Gaussian Noise (AWGN)
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
Shivkumar KalyanaramanRensselaer Polytechnic Institute
5 : “shiv rpi”
Effect of Noise in Signal SpaceThe cloud falls off exponentially (gaussian). Vector viewpoint can be used in signal space, with a random noise vector w
Shivkumar KalyanaramanRensselaer Polytechnic Institute
6 : “shiv rpi”
Maximum Likelihood (ML) Detection: Scalar Case
Assuming both symbols equally likely: uA is chosen if
“likelihoods”
Log-Likelihood => A simple distance criterion!
Shivkumar KalyanaramanRensselaer Polytechnic Institute
7 : “shiv rpi”
AWGN Detection for Binary PAM
)(1 tψbEbE−
1s2s
)|( 2mp zz
0
)|( 1mp zz
2/
2/)()(
0
2121 ⎟
⎟⎠
⎞⎜⎜⎝
⎛ −==
NQmPmP ee
ss
2)2(0
⎟⎟⎠
⎞⎜⎜⎝
⎛==
NEQPP b
EB
Shivkumar KalyanaramanRensselaer Polytechnic Institute
8 : “shiv rpi”
Bigger PictureGeneral structure of a communication systems
Formatter Source encoder
Channel encoder Modulator
Formatter Source decoder
Channel decoder Demodulator
Transmitter
Receiver
SOURCEInfo.
Transmitter
Transmittedsignal
Receivedsignal
Receiver
Receivedinfo.
Noise
ChannelSource User
Shivkumar KalyanaramanRensselaer Polytechnic Institute
9 : “shiv rpi”
Digital vs Analog Comm: Basics
Shivkumar KalyanaramanRensselaer Polytechnic Institute
10 : “shiv rpi”
Digital versus analog
Advantages of digital communications:Regenerator receiver
Different kinds of digital signal are treated identically.
DataVoice
Media
Propagation distance
Originalpulse
Regeneratedpulse
A bit is a bit!
Shivkumar KalyanaramanRensselaer Polytechnic Institute
11 : “shiv rpi”
Signal transmission through linear systems
Deterministic signals:Random signals:
Ideal distortion less transmission:All the frequency components of the signal not only arrive with an identical time delay, but also are amplified or attenuated equally.
Input OutputLinear system
Shivkumar KalyanaramanRensselaer Polytechnic Institute
12 : “shiv rpi”
Signal transmission (cont’d)Ideal filters:
Realizable filters:RC filters Butterworth filter
High-pass
Low-pass
Band-pass
Non-causal!
Duality => similar problems occur w/ rectangular pulses in time domain.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
13 : “shiv rpi”
Bandwidth of signal
Baseband versus bandpass:
Bandwidth dilemma:Bandlimited signals are not realizable!Realizable signals have infinite bandwidth!
Baseband signal
Bandpass signal
Local oscillator
Shivkumar KalyanaramanRensselaer Polytechnic Institute
14 : “shiv rpi”
Bandwidth of signal: ApproximationsDifferent definition of bandwidth:
a) Half-power bandwidthb) Noise equivalent bandwidthc) Null-to-null bandwidth
d) Fractional power containment bandwidthe) Bounded power spectral densityf) Absolute bandwidth
(a)(b)
(c)(d)
(e)50dB
Shivkumar KalyanaramanRensselaer Polytechnic Institute
15 : “shiv rpi”
EncodeTransmitPulse
modulateSample Quantize
Demodulate/Detect
Channel
ReceiveLow-pass
filter Decode
PulsewaveformsBit stream
Format
Format
Digital info.
Textual info.
Analog info.
Textual info.
Analog info.
Digital info.
source
sink
Formatting and transmission of baseband signal
Shivkumar KalyanaramanRensselaer Polytechnic Institute
16 : “shiv rpi”
Sampling of Analog SignalsTime domain
)()()( txtxtxs ×= δ
)(tx
Frequency domain
)()()( fXfXfX s ∗= δ
|)(| fX
|)(| fXδ
|)(| fX s)(txs
)(txδ
Shivkumar KalyanaramanRensselaer Polytechnic Institute
17 : “shiv rpi”
Aliasing effect & Nyquist Rate
LP filter
Nyquist rate
aliasing
Shivkumar KalyanaramanRensselaer Polytechnic Institute
18 : “shiv rpi”
Undersampling &Aliasing in Time Domain
Shivkumar KalyanaramanRensselaer Polytechnic Institute
19 : “shiv rpi”
Note: correct reconstruction does not draw straight lines between samples. Key: use sinc() pulses for reconstruction/interpolation
Nyquist Sampling & Reconstruction: Time Domain
Shivkumar KalyanaramanRensselaer Polytechnic Institute
20 : “shiv rpi”
The impulse response of the reconstruction filter has a classic 'sin(x)/xshape.
The stimulus fed to this filter is the series of discrete impulses which are the samples.
Nyquist Reconstruction: Frequency Domain
Shivkumar KalyanaramanRensselaer Polytechnic Institute
21 : “shiv rpi”
Sampling theorem
Sampling theorem: A bandlimited signal with no spectral components beyond , can be uniquely determined by values sampled at uniform intervals of
The sampling rate,
is called Nyquist rate.
In practice need to sample faster than this because the receiving filter will not be sharp.
Sampling process
Analog signal
Pulse amplitudemodulated (PAM) signal
Shivkumar KalyanaramanRensselaer Polytechnic Institute
22 : “shiv rpi”
Quantization
Amplitude quantizing: Mapping samples of a continuous amplitude waveform to a finite set of amplitudes.
In
Out
Qua
ntiz
edva
lues
Average quantization noise power
Signal peak power
Signal power to average quantization noise power
Shivkumar KalyanaramanRensselaer Polytechnic Institute
23 : “shiv rpi”
Encoding (PCM)
A uniform linear quantizer is called Pulse Code Modulation(PCM).Pulse code modulation (PCM): Encoding the quantized signals into a digital word (PCM word or codeword).
Each quantized sample is digitally encoded into an l bits codeword where L in the number of quantization levels and
Shivkumar KalyanaramanRensselaer Polytechnic Institute
24 : “shiv rpi”
Quantization errorQuantizing error: The difference between the input and output of a quantizer )()(ˆ)( txtxte −=
+
)(tx )(ˆ tx
)()(ˆ)(
txtxte
−=
AGC
x
)(xqy =Qauntizer
Process of quantizing noise
)(tx )(ˆ tx
)(te
Model of quantizing noise
Shivkumar KalyanaramanRensselaer Polytechnic Institute
25 : “shiv rpi”
Non-uniform quantizationIt is done by uniformly quantizing the “compressed” signal. At the receiver, an inverse compression characteristic, called “expansion” is employed to avoid signal distortion.
compression+expansion companding
)(ty)(tx )(ˆ ty )(ˆ tx
x
)(xCy = x
yCompress Qauntize
Transmitter ChannelExpand
Receiver
Shivkumar KalyanaramanRensselaer Polytechnic Institute
26 : “shiv rpi”
Baseband transmission
To transmit information thru physical channels, PCM sequences (codewords) are transformed to pulses (waveforms).
Each waveform carries a symbol from a set of size M.Each transmit symbol represents bits of the PCM words.PCM waveforms (line codes) are used for binary symbols (M=2).
M-ary pulse modulation are used for non-binary symbols (M>2). Eg: M-ary PAM.
For a given data rate, M-ary PAM (M>2) requires less bandwidth than binary PCM.For a given average pulse power, binary PCM is easier to detect than M-ary PAM (M>2).
Mk 2log=
Shivkumar KalyanaramanRensselaer Polytechnic Institute
27 : “shiv rpi”
PAM example: Binary vs 8-ary
Shivkumar KalyanaramanRensselaer Polytechnic Institute
28 : “shiv rpi”
Example of M-ary PAM
-B
B
T‘01’
3B
TT
-3B
T‘00’
‘10’
‘1’A.
T
‘0’
T
-A.
Assuming real time Tx and equal energy per Tx data bit for binary-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 =
Shivkumar KalyanaramanRensselaer Polytechnic Institute
29 : “shiv rpi”
Other PCM waveforms: Examples
PCM waveforms category:
Phase encodedMultilevel binary
Nonreturn-to-zero (NRZ)Return-to-zero (RZ)
1 0 1 1 0
0 T 2T 3T 4T 5T
+V-V
+V0
+V0
-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
Shivkumar KalyanaramanRensselaer Polytechnic Institute
30 : “shiv rpi”
PCM waveforms: Selection Criteria
Criteria for comparing and selecting PCM waveforms:Spectral characteristics (power spectral density and bandwidth efficiency)Bit synchronization capabilityError detection capabilityInterference and noise immunityImplementation cost and complexity
Shivkumar KalyanaramanRensselaer Polytechnic Institute
31 : “shiv rpi”
Summary: Baseband Formatting and transmission
Information (data- or bit-) rate:Symbol rate :
Sampling at rate
(sampling time=Ts)
Quantizing each sampledvalue to one of the
L levels in quantizer.
Encoding each q. value to bits
(Data bit duration Tb=Ts/l)
EncodePulse
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 =
Shivkumar KalyanaramanRensselaer Polytechnic Institute
32 : “shiv rpi”
Receiver Structure & Matched Filter
Shivkumar KalyanaramanRensselaer Polytechnic Institute
33 : “shiv rpi”
Demodulation and detection
Major sources of errors:Thermal noise (AWGN)
disturbs the signal in an additive fashion (Additive)has flat spectral density for all frequencies of interest (White)is modeled by Gaussian random process (Gaussian Noise)
Inter-Symbol Interference (ISI)Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared”.
Format Pulse modulate
Bandpassmodulate
Format Detect Demod.& sample
)(tsi)(tgiim
im )(tr)(Tz
channel)(thc
)(tn
transmitted symbol
estimated symbol
Mi ,,1K=M-ary modulation
Shivkumar KalyanaramanRensselaer Polytechnic Institute
34 : “shiv rpi”
Impact of AWGN
Shivkumar KalyanaramanRensselaer Polytechnic Institute
35 : “shiv rpi”
Impact of AWGN & Channel Distortion
)75.0(5.0)()( Tttthc −−= δδ
Shivkumar KalyanaramanRensselaer Polytechnic Institute
36 : “shiv rpi”
Receiver jobDemodulation and sampling:
Waveform recovery and preparing the received signal for detection:
Improving the signal power to the noise power (SNR) using matched filter (project to signal space)Reducing ISI using equalizer (remove channel distortion)Sampling the recovered waveform
Detection:Estimate the transmitted symbol based on the received sample
Shivkumar KalyanaramanRensselaer Polytechnic Institute
37 : “shiv rpi”
Receiver structure
Frequencydown-conversion
Receiving filter
Equalizingfilter
Threshold comparison
For bandpass signals Compensation for channel induced ISI
Baseband pulse(possibly distorted) Sample
(test statistic)Baseband pulseReceived waveform
Step 1 – waveform to sample transformation Step 2 – decision making
)(tr)(Tz
im
Demodulate & Sample Detect
Shivkumar KalyanaramanRensselaer Polytechnic Institute
38 : “shiv rpi”
Baseband vs BandpassBandpass model of detection process is equivalent to baseband model because:
The received bandpass waveform is first transformed to a baseband waveform.
Equivalence theorem:Performing bandpass linear signal processing followed by heterodying the signal to the baseband, …… yields the same results as …… heterodying the bandpass signal to the baseband , followed by a baseband linear signal processing.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
39 : “shiv rpi”
Steps in designing the receiver
Find optimum solution for receiver design with the following goals: 1. Maximize SNR: matched filter2. Minimize ISI: equalizer
Steps in design:Model the received signalFind separate solutions for each of the goals.
First, we focus on designing a receiver which maximizes the SNR: matched filter
Shivkumar KalyanaramanRensselaer Polytechnic Institute
40 : “shiv rpi”
Receiver filter to maximize the SNR
Model the received signal
Simplify the model (ideal channel assumption):Received signal in AWGN
)(thc)(tsi
)(tn
)(tr
)(tn
)(tr)(tsiIdeal channels
)()( tthc δ=
AWGN
AWGN
)()()()( tnthtstr ci +∗=
)()()( tntstr i +=
Shivkumar KalyanaramanRensselaer Polytechnic Institute
41 : “shiv rpi”
Matched Filter Receiver Problem:Design the receiver filter such that the SNR is maximized at the sampling time when is transmitted.
Solution:The optimum filter, is the Matched filter, given by
which is the time-reversed and delayed version of the conjugate of the transmitted signal
)(thMitsi ,...,1 ),( =
T0 t T0 t
)()()( * tTsthth iopt −==)2exp()()()( * fTjfSfHfH iopt π−==
)(tsi )()( thth opt=
Shivkumar KalyanaramanRensselaer Polytechnic Institute
42 : “shiv rpi”
Correlator ReceiverThe matched filter output at the sampling time, can be realized as the correlator output.
Matched filtering, i.e. convolution with si*(T-τ) simplifies to
integration w/ si*(τ), i.e. correlation or inner product!
>=<=
∗=
∫ )(),()()(
)()()(
*
0
tstrdsr
TrThTz
i
T
opt
τττ
Recall: correlation operation is the projection of the received signal onto the signal space!
Key idea: Reject the noise (N) outside this space as irrelevant: => maximize S/N
Shivkumar KalyanaramanRensselaer Polytechnic Institute
43 : “shiv rpi”
Irrelevance Theorem: Noise Outside Signal Space
Noise PSD is flat (“white”) => total noise power infinite across the spectrum.We care only about the noise projected in the finite signal dimensions (eg: the bandwidth of interest).
Shivkumar KalyanaramanRensselaer Polytechnic Institute
44 : “shiv rpi”
Correlation is a maximum when two signals are similar in shape, and are in phase (or 'unshifted' with respect to each other).Correlation is a measure of the similarity between two signals as a function of time shift (“lag”, τ ) between themWhen the two signals are similar in shape and unshifted with respect to each other, their product is all positive. This is like constructive interference, The breadth of the correlation function - where it has significant value - shows for how longthe signals remain similar.
Aside: Correlation Effect
Shivkumar KalyanaramanRensselaer Polytechnic Institute
45 : “shiv rpi”
Aside: Autocorrelation
Shivkumar KalyanaramanRensselaer Polytechnic Institute
46 : “shiv rpi”
Aside: Cross-Correlation &Radar
Figure: shows how the signal can be located within the noise.A copy of the known reference signal is correlated with the unknown signal. The correlation will be high when the reference is similar to the unknown signal. A large value for correlation shows the degree of confidence that the reference signal is detected. The large value of the correlation indicates when the reference signal occurs.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
47 : “shiv rpi”
• 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
Shivkumar KalyanaramanRensselaer Polytechnic Institute
48 : “shiv rpi”
Matched Filter: back to cartoon…
Consider the received signal as a vector r, and the transmitted signal vector as sMatched 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
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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 TT/2
22A−
TA
TA
TA
TA−
TA−
TA
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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 the autocorrelation 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 energy to the PSD of the white noise at the filter input.
4. Two matching conditions in the matched-filtering operation:spectral phase matching that gives the desired output peak at time T.spectral amplitude matching that gives optimum SNR to the peak value.
)2exp(|)(|)( 2 fTjfSfZ π−=
sss ERTzTtRtz ==⇒−= )0()()()(
2/max
0NE
NS s
T
=⎟⎠⎞
⎜⎝⎛
Shivkumar KalyanaramanRensselaer Polytechnic Institute
51 : “shiv rpi”
Implementation of matched filter receiver
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Mz
zM1
z=)(tr
)(1 Tz)(*
1 tTs −
)(* tTsM − )(TzM
zMatched filter output:
Observationvector
Bank of M matched filters
)()( tTstrz ii −∗= ∗ Mi ,...,1=
),...,,())(),...,(),(( 2121 MM zzzTzTzTz ==z
Note: we are projecting along the basis directions of the signal space
Shivkumar KalyanaramanRensselaer Polytechnic Institute
52 : “shiv rpi”
Implementation of correlator receiver
dttstrz i
T
i )()(0∫=
∫T
0
)(1 ts∗
∫T
0
)(ts M∗
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Mz
zM1
z=)(tr
)(1 Tz
)(TzM
zCorrelators output:
Observationvector
Bank of M correlators
),...,,())(),...,(),(( 2121 MM zzzTzTzTz ==z
Mi ,...,1=
Note: In previous slide we “filter” i.e. convolute in the boxes shown.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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
TA
TA−
TA−
TA
0
0
Shivkumar KalyanaramanRensselaer Polytechnic Institute
54 : “shiv rpi”
Matched Filter: Frequency domain View
Simple Bandpass Filter:excludes noise, but misses some signal power
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
56 : “shiv rpi”
Maximal Ratio Combining (MRC) viewpoint
Generalization of this f-domain picture, for combining multi-tap signal
Weight each branch
SNR:
Shivkumar KalyanaramanRensselaer Polytechnic Institute
57 : “shiv rpi”
Examples of matched filter output for bandpassmodulation schemes
Shivkumar KalyanaramanRensselaer Polytechnic Institute
58 : “shiv rpi”
Signal Space Concepts
Shivkumar KalyanaramanRensselaer Polytechnic Institute
59 : “shiv rpi”
Signal space: OverviewWhat is a signal space?
Vector representations of signals in an N-dimensional orthogonal space
Why do we need a signal space?It is a means to convert signals to vectors and vice versa.It is a means to calculate signals energy and Euclidean distances between signals.
Why are we interested in Euclidean distances between signals?For detection purposes: The received signal is transformed to a received vectors. The signal which has the minimum distance to the received signal is estimated as the transmitted signal.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
60 : “shiv rpi”
Schematic example of a signal space
),()()()(),()()()(),()()()(
),()()()(
212211
323132321313
222122221212
121112121111
zztztztzaatatatsaatatats
aatatats
=⇔+==⇔+==⇔+==⇔+=
zsss
ψψψψψψψψ
)(1 tψ
)(2 tψ),( 12111 aa=s
),( 22212 aa=s
),( 32313 aa=s
),( 21 zz=z
Transmitted signal alternatives
Received signal at matched filter output
Shivkumar KalyanaramanRensselaer Polytechnic Institute
61 : “shiv rpi”
Signal space
To form a signal space, first we need to know the inner product between two signals (functions):
Inner (scalar) product:
Properties of inner product:
∫∞
∞−
>=< dttytxtytx )()()(),( *
= cross-correlation between x(t) and y(t)
><>=< )(),()(),( tytxatytax
><>=< )(),()(),( * tytxataytx
><+>>=<+< )(),()(),()(),()( tztytztxtztytx
Shivkumar KalyanaramanRensselaer Polytechnic Institute
62 : “shiv rpi”
Signal space …The distance in signal space is measure by calculating the norm.What is norm?
Norm of a signal:
Norm between two signals:
We refer to the norm between two signals as the Euclidean distance between two signals.
xEdttxtxtxtx ==><= ∫∞
∞−
2)()(),()(
)()( txatax =
)()(, tytxd yx −=
= “length” of x(t)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
63 : “shiv rpi”
Example of distances in signal space
)(1 tψ
)(2 tψ),( 12111 aa=s
),( 22212 aa=s
),( 32313 aa=s
),( 21 zz=z
zsd ,1
zsd ,2zsd ,3
The Euclidean distance between signals z(t) and s(t):
3,2,1
)()()()( 222
211,
=
−+−=−=
i
zazatztsd iiizsi
1E
3E
2E
Shivkumar KalyanaramanRensselaer Polytechnic Institute
64 : “shiv rpi”
Orthogonal signal spaceN-dimensional orthogonal signal space is characterized by N linearly independent functions called basis functions. The basis functions must satisfy the orthogonality condition
where
If all , the signal space is orthonormal.
Constructing Orthonormal basis from non-orthonormal set of vectors:Gram-Schmidt procedure
{ }Njj t
1)(
=ψ
jiij
T
iji Kdttttt δψψψψ =>=< ∫ )()()(),( *
0
Tt ≤≤0Nij ,...,1, =
⎩⎨⎧
≠→=→
=jiji
ij 01
δ
1=iK
Shivkumar KalyanaramanRensselaer Polytechnic Institute
65 : “shiv rpi”
Example of an orthonormal basesExample: 2-dimensional orthonormal signal space
Example: 1-dimensional orthonornal signal space
1)()(
0)()()(),(
0)/2sin(2)(
0)/2cos(2)(
21
20
121
2
1
==
=>=<
⎪⎪⎩
⎪⎪⎨
⎧
<≤=
<≤=
∫tt
dttttt
TtTtT
t
TtTtT
t
T
ψψ
ψψψψ
πψ
πψ
T t
)(1 tψ
T1
0
)(1 tψ
)(2 tψ
0
1)(1 =tψ)(1 tψ
0
Shivkumar KalyanaramanRensselaer Polytechnic Institute
66 : “shiv rpi”
Sine/Cosine Bases: Note!Approximately orthonormal!
These are the in-phase & quadrature-phase dimensions of complex baseband equivalent representations.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
67 : “shiv rpi”
Example: BPSK
Note: two symbols, but only one dimension in BPSK.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
68 : “shiv rpi”
Signal space …Any arbitrary finite set of waveforms where each member of the set is of duration T, can be expressed as a linear combination of N orthogonal waveforms where .
where
{ }Mii ts 1)( =
{ }Njj t
1)(
=ψ MN ≤
∑=
=N
jjiji tats
1)()( ψ Mi ,...,1=
MN ≤
dtttsK
ttsK
aT
jij
jij
ij )()(1)(),(1
0
*∫>=<= ψψ Tt ≤≤0Mi ,...,1=Nj ,...,1=
),...,,( 21 iNiii aaa=s2
1ij
N
jji aKE ∑
=
=Vector representation of waveform Waveform energy
Shivkumar KalyanaramanRensselaer Polytechnic Institute
69 : “shiv rpi”
Signal space …
∑=
=N
jjiji tats
1
)()( ψ ),...,,( 21 iNiii aaa=s
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
iN
i
a
aM1
)(1 tψ
)(tNψ
1ia
iNa
)(tsi
∫T
0
)(1 tψ
∫T
0
)(tNψ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
iN
i
a
aM1
ms=)(tsi
1ia
iNa
ms
Waveform to vector conversion Vector to waveform conversion
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Example of projecting signals to an orthonormal signal space
),()()()(),()()()(
),()()()(
323132321313
222122221212
121112121111
aatatatsaatatats
aatatats
=⇔+==⇔+==⇔+=
sss
ψψψψψψ
)(1 tψ
)(2 tψ),( 12111 aa=s
),( 22212 aa=s
),( 32313 aa=s
Transmitted signal alternatives
dtttsaT
jiij )()(0∫= ψ Tt ≤≤0Mi ,...,1=Nj ,...,1=
Shivkumar KalyanaramanRensselaer Polytechnic Institute
71 : “shiv rpi”
Matched filter receiver (revisited)
)(tr
1z)(1 tT −∗ψ
)( tTN −∗ψNz
Observationvector
Bank of N matched filters
)()( tTtrz jj −∗= ψ Nj ,...,1=
),...,,( 21 Nzzz=z∑
=
=N
jjiji tats
1)()( ψ
MN ≤Mi ,...,1=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Nz
z1
z= z
(note: we match to the basis directions)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
72 : “shiv rpi”
Correlator receiver (revisited)
),...,,( 21 Nzzz=z
Nj ,...,1=dtttrz j
T
j )()(0
ψ∫=
∫T
0
)(1 tψ
∫T
0
)(tNψ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
Nr
rM1
z=)(tr
1z
Nz
z
Bank of N correlators
Observationvector
∑=
=N
jjiji tats
1)()( ψ Mi ,...,1=
MN ≤
Shivkumar KalyanaramanRensselaer Polytechnic Institute
73 : “shiv rpi”
Example of matched filter receivers using basic functions
Number of matched filters (or correlators) is reduced by 1 compared to using matched filters (correlators) to the transmitted signal!
T t
)(1 ts
T t
)(2 ts
T t
)(1 tψ
T1
0
[ ]1z z=)(tr z
1 matched filter
T t
)(1 tψ
T1
0
1z
TA
TA−0
0
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74 : “shiv rpi”
White noise in Orthonormal Signal Space
AWGN n(t) can be expressed as
)(~)(ˆ)( tntntn +=
Noise projected on the signal space (colored):impacts the detection process.
Noise outside on the signal space (irrelevant)
>=< )(),( ttnn jj ψ
0)(),(~ >=< ttn jψ
)()(ˆ1
tntnN
jjj∑
=
= ψ
Nj ,...,1=
Nj ,...,1=
Vector representation of
),...,,( 21 Nnnn=n)(ˆ tn
independent zero-mean Gaussain random variables with variance
{ }Njjn
1=
2/)var( 0Nn j =
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Detection: Maximum Likelihood & Performance Bounds
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Detection of signal in AWGNDetection problem:
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 mim zisn
z z
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Statistics of the observation Vector AWGN channel model:
Signal vector is deterministic.Elements of noise vector are i.i.d Gaussianrandom variables with zero-mean and variance . The noise vector pdf is
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
p N
nnn π
( ) ⎟⎠
⎜⎝ 00 NNπ
⎟⎞
⎜⎛ −−=
2
2/ exp1)|(p iNi
szszz
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Detection
Optimum decision rule (maximum a posteriori probability):
Applying Bayes’ rule gives:
.,...,1 where allfor ,)|sent Pr()|sent Pr(
if ˆSet
Mkikmm
mm
ki
i
=≠≥
=zz
ikp
mpp
mm
kk
i
=
=
allfor maximum is ,)(
)|(if ˆSet
zz
z
z
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79 : “shiv rpi”
Detection …
Partition the signal space into M decision regions, such that MZZ ,...,1
i
kk
i
mm
ikp
mpp
Z
=
=
ˆ meansThat
. allfor maximum is ],)(
)|(ln[
if region inside lies Vector
zz
z
z
z
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80 : “shiv rpi”
Detection (ML rule)For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to:
or equivalently:
which is known as maximum likelihood.
ikmpmm
k
i
==
allfor maximum is ),|(if ˆSet
zz
ikmpmm
k
i
==
allfor maximum is )],|(ln[if ˆSet
zz
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Detection (ML)…Partition the signal space into M decision regions, Restate the maximum likelihood decision rule as follows:
MZZ ,...,1
i
k
i
mm
ikmpZ
=
=
ˆ meansThat
allfor maximum is )],|(ln[if region inside lies Vector
zz
z
ikZ
k
i
=− allfor minimum is ,if region inside lies Vector
szz
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Schematic example of ML decision regions
)(1 tψ
)(2 tψ
1s3s
2s
4s
1Z
2Z
3Z
4Z
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83 : “shiv rpi”
Probability of symbol errorErroneous decision: For the transmitted symbol or equivalently signal vector , an error in decision occurs if the observation vector does not fall inside region .
Probability of erroneous decision for a transmitted symbol
Probability of correct decision for a transmitted symbol
sent) inside lienot does sent)Pr( Pr()ˆPr( iiii mZmmm z=≠
sent) inside liessent)Pr( Pr()ˆPr( iiii mZmmm z==
∫==iZ
iiiic dmpmZmP zz z z )|(sent) inside liesPr()(
)(1)( icie mPmP −=
imis
ziZ
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Example for binary PAM
)(1 tψbEbE−
1s2s
)|( 1mp zz)|( 2mp zz
0
2)2(0
⎟⎟⎠
⎞⎜⎜⎝
⎛==
NEQPP b
EB
2/
2/)()(
0
2121 ⎟
⎟⎠
⎞⎜⎜⎝
⎛ −==
NQmPmP ee
ss
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85 : “shiv rpi”
Average prob. of symbol error …Average probability of symbol error :
For equally probable symbols:
)|(11
)(11)(1)(
1
11
∑ ∫
∑∑
=
==
−=
−==
M
i Zi
M
iic
M
iieE
i
dmpM
mPM
mPM
MP
zzz
)ˆ(Pr)(1
i
M
iE mmMP ≠= ∑
=
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Union bound
Union boundThe 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
Shivkumar KalyanaramanRensselaer Polytechnic Institute
87 : “shiv rpi”
Union bound:
Example of union bound
1ψ
1s
4s
2s
3s
1Z
4Z3Z
2Z r 2ψ
∫∪∪
=432
)|()( 11ZZZ
e dmpmP rrr
1ψ
1s
4s
2s
3s
2A r
1ψ
1s
4s
2s
3s3A
r
1ψ
1s
4s
2s
3s4A
r2ψ 2ψ 2ψ
∫=2
)|(),( 1122A
dmpP rrss r ∫=3
)|(),( 1132A
dmpP rrss r ∫=4
)|(),( 1142A
dmpP rrss r
∑=
≤4
2121 ),()(
kke PmP ss
Shivkumar KalyanaramanRensselaer Polytechnic Institute
88 : “shiv rpi”
Upper bound based on minimum distance
⎟⎟⎠
⎞⎜⎜⎝
⎛=−=
=
∫∞
2/
2/)exp(1
sent) is when , than closer to is Pr(),(
00
2
0
2
NdQdu
Nu
N
P
ik
d
iikik
ikπ
ssszss
kiikd ss −=
⎟⎟⎠
⎞⎜⎜⎝
⎛−≤≤ ∑∑
=≠= 2/
2/)1(),(1)(0
min
1 12 N
dQMPM
MPM
i
M
ikk
ikE ss
ikki
kidd
≠
=,min min
Minimum distance in the signal space:
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89 : “shiv rpi”
Example of upper bound on av. Symbol error prob. based on union bound
)(1 tψ
)(2 tψ
1s3s
2s
4s
2,1d
4,3d
3,2d
4,1d sEsE−
sE−
sE
4,...,1 , === iEE siis
s
ski
Ed
kiEd
2
2
min
,
=
≠
=
Shivkumar KalyanaramanRensselaer Polytechnic Institute
90 : “shiv rpi”
Shivkumar KalyanaramanRensselaer Polytechnic Institute
91 : “shiv rpi”
Eb/No figure of merit in digital communications
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:
Signals are transmitted within a symbol duration and hence, are energy signal (zero power).
A metric at the bit-level facilitates comparison of different DCS transmitting different number of bits per symbol.
b
bb
RW
NS
WNST
NE
==/0
bRW
: Bit rate
: Bandwidth
Note: S/N = Eb/No x spectral efficiency
Shivkumar KalyanaramanRensselaer Polytechnic Institute
92 : “shiv rpi”
Example of Symbol error prob. For PAM signals
)(1 tψ0
1s2s
bEbE−
Binary PAM
)(1 tψ02s3s
52 bE
56 bE
56 bE
−5
2 bE−
4s 1s4-ary PAM
T t
)(1 tψ
T1
0
Shivkumar KalyanaramanRensselaer Polytechnic Institute
93 : “shiv rpi”
Maximum Likelihood (ML) Detection: Vector Case
ps: Vector norm is a natural extension of “magnitude” or length
Nearest Neighbor Rule:
Project the received vector y along the difference vector direction uA- uB is a “sufficient statistic”.
Noise outside these finite dimensions is irrelevant for 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:
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Extension to M-PAM (Multi-Level Modulation)
Note: h refers to the constellation shape/direction
MPAM:
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95 : “shiv rpi”
Complex Vector Space Detection
Error probability:
Note: Instead of vT, use v* for complex vectors (“transpose and conjugate”) for inner products…
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96 : “shiv rpi”
Complex Detection: Summary
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Detection Error => BER
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)
BER is modeled as a uniform probability fAs BER (f) increases, the effects become increasingly intolerablef tends to increase rapidly with lower SNR: “waterfall” curve (Q-function)
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98 : “shiv rpi”
SNR vs BER: AWGN vs Rayleigh
Observe the “waterfall” like characteristic (essentially plotting the Q(x) function)!Telephone lines: SNR = 37dB, but low b/w (3.7kHz)Wireless: Low SNR = 5-10dB, higher bandwidth (upto 10 Mhz, MAN, and 20Mhz LAN)Optical fiber comm: High SNR, high bandwidth ! But cant process w/ complicated codes, signal processing etc
Need diversitytechniquesto deal with Rayleigh (even 1-tap, flat-fading)!
Shivkumar KalyanaramanRensselaer Polytechnic Institute
99 : “shiv rpi”
Better performance through diversity
Diversity the receiver is provided with multiple copies of the transmitted signal. The multiple signal copies should experience uncorrelated fading in the channel.
In this case the probability that all signal copies fade simultaneously is reduced dramatically with respect to the probability that a single copy experiences a fade.
As a rough rule:
0
1e LP
γis proportional to
BERBER Average SNRAverage SNR
Diversity of L:th order
Diversity of L:th order
Shivkumar KalyanaramanRensselaer Polytechnic Institute
100 : “shiv rpi”
SNR
BER
Flat fading channel,Rayleigh fading,
L = 1AWGN channel
(no fading)
( )eP=
0( )γ=
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
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What is Modulation?
Encoding information in a manner suitable for transmission.
Translate baseband source signal to bandpass signalBandpass signal: “modulated signal”
How? Vary amplitude, phase or frequency of a carrier
Demodulation: extract baseband message from carrier
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103 : “shiv rpi”
Digital vs Analog Modulation
Cheaper, faster, more power efficientHigher data rates, power error correction, impairment resistance:
Using coding, modulation, diversity Equalization, multicarrier techniques for ISI mitigation
More efficient multiple access strategies, better security: CDMA, encryption etc
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104 : “shiv rpi”
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)
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Modulation: representation• Any modulated signal can be represented as
s(t) = A(t) cos [ωct + φ(t)]
amplitude phase or frequency
s(t) = A(t) cos φ(t) cos ωct - A(t) sin φ(t) sin ωct
in-phase quadrature
• Linear versus nonlinear modulation ⇒ impact on spectral efficiency
• Constant envelope versus non-constant envelope⇒ hardware implications with impact on power efficiency
Linear: Amplitude or phaseNon-linear: frequency: spectral broadening
(=> reliability: i.e. target BER at lower SNRs)
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106 : “shiv rpi”
Complex Vector Spaces: Constellations
Each signal is encoded (modulated) as a vector in a signal space
MPSK
Circular Square
Shivkumar KalyanaramanRensselaer Polytechnic Institute
107 : “shiv rpi”
Linear Modulation Techniquess(t) = [ Σ an g (t-nT)]cos ωct - [ Σ bn g (t-nT)] sin ωct
I(t), in-phase Q(t), quadrature
LINEAR MODULATIONS
CONVENTIONAL 4-PSK
(QPSK)
OFFSET 4-PSK
(OQPSK)
DIFFERENTIAL 4-PSK
(DQPSK, π/4-DQPSK)
M-ARY QUADRATURE AMPLITUDE MOD.
(M-QAM)
M-ARY PHASE SHIFT KEYING
(M-PSK)
M ≠ 4 M ≠ 4M=4 (4-QAM = 4-PSK)
Square Constellations
CircularConstellations
n n
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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 spectrally
efficient but less power efficient (i.e. BER higher). – M-QAM is more spectrally efficient than M-PSK but
also more sensitive to system nonlinearities.
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Bandwidth vs. Power Efficiency
Source: Rappaport book, chap 6
MPSK:
MQAM:
MFSK:
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MPAM & Symbol Mapping
Note: the average energy per-bit is constantGray coding used for mapping bits to symbols
Why? Most likely error is to confuse with neighboring symbol. Make sure that the neighboring symbol has only 1-bit difference (hamming distance = 1)
)(1 tψ0
1s2s
bEbE−
Binary PAM
)(1 tψ02s3s
52 bE
56 bE
56 bE
−5
2 bE−
4s 1s4-ary PAM
)(1 tψ02s3s
52 bE
56 bE
56 bE
−5
2 bE−
4s 1s4-ary PAM
Gray coding00 01 11 10
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MPAM: Details
Decision Regions
Unequal energies/symbol:
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112 : “shiv rpi”
MPSK:
Constellation points:
Equal energy in all signals:
Gray coding 00
01
11
10
0001
11 10
M-PSK (Circular Constellations)
16-PSK
an
bn 4-PSK
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MPSK: Decision Regions & Demod’ln
4PSK 8PSK
Coherent Demodulator for BPSK.
si(t) + n(t)
cos(2πfct)
g(Tb - t)XZ1: r > 0
Z2: r ≤ 0
m = 1
m = 0
m = 0 or 1
Z1
Z2
Z3
Z4
Z1
Z2
Z3Z4
Z5
Z6 Z7
Z8
4PSK: 1 bit/complex dimension or 2 bits/symbol
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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MQAM:
Unequal symbol energies:
MQAM with square constellations of size L2 is equivalent to MPAM modulation with constellations of size L on each of the in-phase and quadrature signal components
For square constellations it takes approximately 6 dB more power to send an additional 1 bit/dimension or 2 bits/symbol while maintaining the same minimum distance between constellation points
Hard to find a Gray code mapping where all adjacent symbols differ by a single bit
M-QAM (Square Constellations)
16-QAM
4-PSK
an
bn
16QAM: Decision Regions
Z1 Z2 Z3 Z4
Z5 Z6 Z7 Z8
Z9 Z10 Z11 Z12
Z13 Z14 Z15 Z16
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Non-Coherent Modulation: DPSKInformation in MPSK, MQAM carried in signal phase. Requires coherent demodulation: i.e. phase of the transmitted signal carrier φ0 must be matched to the phase of the receiver carrier φMore cost, susceptible to carrier phase drift.Harder to obtain in fading channels
Differential modulation: do not require phase reference.More general: modulation w/ memory: depends upon prior symbols transmitted. Use prev symbol as the a phase reference for current symbolInfo bits encoded as the differential phase between current & previous symbolLess sensitive to carrier phase drift (f-domain) ; more sensitive to dopplereffects: decorrelation of signal phase in time-domain
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Differential Modulation (Contd)DPSK: Differential BPSK A 0 bit is encoded by no change in phase, whereas a 1 bit is encoded as a phase change of π.
If symbol over time [(k−1)Ts, kTs) has phase θ(k − 1) = ejθi , θi = 0, π, then to encode a 0 bit over [kTs, (k + 1)Ts), the symbol would have
phase: θ(k) = ejθi and…… to encode a 1 bit the symbol would have
phase θ(k) = ej(θi+π).
DQPSK: gray coding:
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Quadrature OffsetPhase transitions of 180o can cause large amplitude transitions (through zero point).
Abrupt phase transitions and large amplitude variations can be distorted by nonlinear amplifiers and filters
Avoided by offsetting the quadrature branch pulse g(t) by half a symbol period
Usually abbreviated as O-MPSK, where the O indicates the offsetQPSK modulation with quadrature offset is referred to as O-QPSKO-QPSK has the same spectral properties as QPSK for linear amplification,..
… but has higher spectral efficiency under nonlinear amplification, since the maximum phase transition of the signal is 90 degrees
Another technique to mitigate the amplitude fluctuations of a 180 degree phase shift used in the IS-54 standard for digital cellular is π/4-QPSK
Maximum phase transition of 135 degrees, versus 90 degrees for offset QPSK and 180 degrees for QPSK
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Offset QPSK waveforms
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Frequency Shift Keying (FSK)• 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 [ωct + 2 πkf ∫ d(τ) dτ]−∞
t
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Minimum Shift Keying (MSK) spectra
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Spectral CharacteristicsQPSK/DQPSKGMSK
(MSK)
1.0
1.0 1.5 2.0 2.50.50-120
-100
-80
-60
-40
-20
010
0.25B3-dBTb = 0.16
Normalized Frequency (f-fc)Tb
Pow
er S
pect
ral D
ensi
ty (d
B)
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Bit Error Probability (BER): AWGN
Pb
BPSK, QPSKDBPSK
DQPSK
010-6
2
5
2
5
2
5
2
5
2
510-1
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
γb, SNR/bit, dB
• QPSK is more spectrally efficient than BPSK with the same performance.
• M-PSK, for M>4, is more spectrally efficient but requires more SNR per bit.
• There is ~3 dB power penalty for differential detection.
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Bit Error Probability (BER): Fading Channel
DBPSK
AWGN
BPSK
010-5
2
5
2
5
2
5
2
5
2
51
10-2
10-3
10-4
10-1
5 10 15 20 25 30 35
Pb
γb, SNR/bit, dB
• Pb is inversely proportion to the average SNR per bit.
• Transmission in a fading environment requires about18 dB more power for Pb = 10-3.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
124 : “shiv rpi”
Bit Error Probability (BER): Doppler Effects
0
0 60504030201010-6
10-5
10-4
10-3
10-2
10-1
10
DQPSK
Rayleigh Fading
No FadingfDT=0.003
0.0020.001
0
QPSK
Pb
γb, SNR/bit, dB
• Doppler causes an irreducible error floor when differential detection 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]
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Bit Error Probability (BER): Delay Spread
+
x
+
+
+
++
x
x
x
xx
10-210-4
10-3
10-2
10-1
10-1 100
BPSK QPSK OQPSK MSK
Modulation
Coherent Detection
Irred
ucib
le P
b
τT=rms delay spread
symbol period
• ISI causes an irreducible error floor.
• The rms delay spread imposes a limit on the maximum bit rate in a multipath environment.
For example, for QPSK,τ Maximum Bit Rate
Mobile (rural) 25 μsec 8 kbps Mobile (city) 2.5 μsec 80 kbps Microcells 500 nsec 400 kbps Large 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]
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Summary of Modulation Issues• 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
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128 : “shiv rpi”
Recall: Impact of AWGN only
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Impact of AWGN & Channel Distortion
)75.0(5.0)()( Tttthc −−= δδ
Multi-tap, ISI channel
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ISI Effects: Band-limited Filtering of Channel
ISI due to filtering effect of the communications channel (e.g. wireless channels)
Channels behave like band-limited filters
)()()( fjcc
cefHfH θ=
Non-constant amplitude
Amplitude distortion
Non-linear phase
Phase distortion
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Inter-Symbol Interference (ISI)ISI in the detection process due to the filtering effects of the systemOverall equivalent system transfer function
creates echoes and hence time dispersioncauses ISI at sampling time
)()()()( fHfHfHfH rct=
iki
ikkk snsz ∑≠
++= α
ISI effect
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Inter-symbol interference (ISI): Model
Baseband system model
Equivalent model
Tx filter Channel
)(tn
)(tr Rx. filterDetector
kz
kTt =
{ }kx{ }kx1x
2x
3xT T)(
)(fHth
t
t
)()(fHth
r
r
)()(fHth
c
c
Equivalent system
)(ˆ tn
)(tzDetector
kz
kTt =
{ }kx{ }kx1x
2x
3xT T)(
)(fHth
filtered noise)()()()( fHfHfHfH rct=
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Nyquist bandwidth constraintNyquist bandwidth constraint (on equivalent system):
The theoretical minimum required system bandwidth to detect Rs [symbols/s] without ISI is Rs/2 [Hz]. Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz] can support a maximum transmission rate of 2W=1/T=Rs[symbols/s] without ISI.
Bandwidth efficiency, R/W [bits/s/Hz] : An important measure in DCs representing data throughput per hertz of bandwidth.Showing how efficiently the bandwidth resources are used by signaling techniques.
Hz][symbol/s/ 222
1≥⇒≤=
WRWR
Tss
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Equiv System: Ideal Nyquist pulse (filter)
T21
T21−
T
)( fH
f t
)/sinc()( Ttth =1
0 T T2T−T2−0
TW
21
=
Ideal Nyquist filter Ideal Nyquist pulse
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Nyquist pulses (filters)Nyquist pulses (filters):
Pulses (filters) which result in no ISI at the sampling time.Nyquist filter:
Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even-symmetric frequency function
Nyquist pulse: Its shape can be represented by a sinc(t/T) function multiply by another time function.
Example of Nyquist filters: Raised-Cosine filter
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Pulse shaping to reduce ISI
Goals and trade-off in pulse-shapingReduce ISIEfficient bandwidth utilizationRobustness to timing error (small side lobes)
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Raised Cosine Filter: Nyquist Pulse Approximation
2)1( Baseband sSB
sRrW +=
|)(||)(| fHfH RC=
0=r5.0=r
1=r1=r
5.0=r
0=r
)()( thth RC=
T21
T43
T1
T43−
T21−
T1−
1
0.5
0
1
0.5
0 T T2 T3T−T2−T3−
sRrW )1( Passband DSB +=
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Raised Cosine FilterRaised-Cosine Filter
A Nyquist pulse (No ISI at the sampling time)
⎪⎪⎩
⎪⎪⎨
⎧
>
<<−⎥⎦
⎤⎢⎣
⎡−
−+−<
=
Wf
WfWWWW
WWfWWf
fH
||for 0
||2for 2||4
cos
2||for 1
)( 00
02
0
π
Excess bandwidth:0WW − Roll-off factor
0
0
WWWr −
=10 ≤≤ r
20
000 ])(4[1
])(2cos[))2(sinc(2)(tWW
tWWtWWth−−
−=
π
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139 : “shiv rpi”
Pulse Shaping and Equalization Principles
Square-Root Raised Cosine (SRRC) filter and Equalizer
)()()()()(RC fHfHfHfHfH erct=No ISI at the sampling time
)()()()(
)()()(
SRRCRC
RC
fHfHfHfH
fHfHfH
tr
rt
===
=Taking care of ISI caused by tr. filter
)(1)(
fHfH
ce = Taking care of ISI
caused by channel
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Pulse Shaping & Orthogonal Bases
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Virtue of pulse shaping
PSD of a BPSK signalSource: Rappaport book, chap 6
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Example of pulse shapingSquare-root Raised-Cosine (SRRC) pulse shaping
t/T
Amp. [V]
Baseband tr. Waveform
Data symbol
First pulseSecond pulse
Third pulse
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Example of pulse shaping …Raised Cosine pulse at the output of matched filter
t/T
Amp. [V]
Baseband received waveform at the matched filter output(zero ISI)
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Eye pattern
Eye pattern:Display on an oscilloscope which sweeps the system response to a baseband signal at the rate 1/T (T symbol duration)
time scale
ampl
itude
scal
e Noise margin
Sensitivity to timing error
Distortiondue to ISI
Timing jitter
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Example of eye pattern:Binary-PAM, SRRC pulse
Perfect channel (no noise and no ISI)
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Example of eye pattern:Binary-PAM, SRRC pulse …
AWGN (Eb/N0=20 dB) and no ISI
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Example of eye pattern:Binary-PAM, SRRC pulse …
AWGN (Eb/N0=10 dB) and no ISI
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Summary
Digital BasicsModulation & Detection, Performance, BoundsModulation Schemes, ConstellationsPulse Shaping
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Extra Slides
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150 : “shiv rpi”
Bandpass Modulation: I, Q Representation
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Analog: Frequency Modulation (FM) vsAmplitude Modulation (AM)
FM: all information in the phase or frequency of carrierNon-linear or rapid improvement in reception quality beyond a minimum received signal threshold: “capture” effect. Better noise immunity & resistance to fadingTradeoff bandwidth (modulation index) for improved SNR: 6dB gain for 2x bandwidthConstant envelope signal: efficient (70%) class C power amps ok.
AM: linear dependence on quality & power of rcvd signalSpectrally efficient but susceptible to noise & fadingFading improvement using in-band pilot tones & adapt receiver gain to compensateNon-constant envelope: Power inefficient (30-40%) Class A or AB power amps needed: ½ the talk time as FM!
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152 : “shiv rpi”
Example Analog: Amplitude Modulation