Point-to-Point Wireless Communication (I):
Digital Basics, Modulation, Detection, Pulse
Shaping
Shiv Kalyanaraman
Google: “Shiv RPI”
shivkuma@ecse.rpi.edu
Based upon slides of Sorour Falahati, A. Goldsmith,
& textbooks by Bernard Sklar & A. Goldsmith
Shivkumar Kalyanaraman
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The Basics
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Big Picture: Detection under AWGN
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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”
Its PSD is flat, hence, it is called white noise.
 Autocorrelation is a spike at 0: uncorrelated at any non-zero lag
[w/Hz]
Power spectral
Density
(flat => “white”)
Probability density function
(gaussian)
Autocorrelation
Function
(uncorrelated)
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Effect of Noise in Signal Space


The cloud falls off exponentially (gaussian).
Vector viewpoint can be used in signal space, with a random noise vector w
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Maximum Likelihood (ML) Detection: Scalar Case
“likelihoods”
Assuming both symbols equally likely: uA is chosen if
A simple distance criterion!
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Log-Likelihood =>
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AWGN Detection for Binary PAM
pz (z | m2 )
pz (z | m1 )
s2
 Eb
s1
0
 1 (t )
Eb
 s1  s 2 / 2 

Pe (m1 )  Pe (m2 )  Q
 N /2 
0


 2 Eb
PB  PE (2)  Q
 N0
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



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Bigger Picture
General structure of a communication systems
Noise
Info.
SOURCE
Source
Received
Transmitted
Received
info.
signal
signal
Transmitter
Receiver
Channel
User
Transmitter
Formatter
Source
encoder
Channel
encoder
Modulator
Receiver
Formatter
Source
decoder
Channel
decoder
Demodulator
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Digital vs Analog Comm: Basics
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Digital versus analog

Advantages of digital communications:
 Regenerator receiver
Original
pulse
Regenerated
pulse
Propagation distance
 Different
kinds of digital signal are treated
identically.
Voice
Data
A bit is a bit!
Media
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Signal transmission through linear systems
Input
Output
Linear system
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.
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Signal transmission (cont’d)

Ideal filters:
Non-causal!
Low-pass
Band-pass

High-pass
Duality => similar problems
occur w/
rectangular pulses in time domain.
Realizable filters:
RC filters
Butterworth filter
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Bandwidth of signal

Baseband versus bandpass:
Baseband
signal
Bandpass
signal
Local oscillator

Bandwidth dilemma:
 Bandlimited signals are not realizable!
 Realizable signals have infinite bandwidth!
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Bandwidth of signal: Approximations

Different definition of bandwidth:
a) Half-power bandwidth
b) 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)
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(e)50dB
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Formatting and transmission of baseband signal
Digital info.
Format
Textual
source info.
Analog
info.
Sample
Quantize
Pulse
modulate
Encode
Bit stream
Format
Analog
info.
sink
Low-pass
filter
Decode
Pulse
waveforms
Demodulate/
Detect
Textual
info.
Transmit
Channel
Receive
Digital info.
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Sampling of Analog Signals
Time domain
Frequency domain
xs (t )  x (t )  x(t )
X s ( f )  X ( f )  X ( f )
x(t )
| X(f )|
| X ( f ) |
x (t )
xs (t )
| Xs( f ) |
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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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Nyquist Sampling & Reconstruction:
Time Domain
Note: correct reconstruction does not draw straight lines between samples.
Key: use sinc() pulses for reconstruction/interpolation
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Nyquist Reconstruction: Frequency Domain
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.
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Sampling theorem
Analog
signal
Sampling
process
Pulse amplitude
modulated (PAM) signal

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.
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Quantization

Amplitude quantizing: Mapping samples of a continuous amplitude
waveform to a finite set of amplitudes.
Out
Quantized
values
In
Average quantization noise power
Signal peak power
Signal power to average
quantization noise power
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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
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Quantization error

Quantizing error: The difference between the input and output of a
e(t )  xˆ (t )  x(t )
quantizer
Process of quantizing noise
Qauntizer
Model of quantizing noise
y  q (x )
AGC
xˆ (t )
x(t )
xˆ (t )
x(t )
x
e(t )
+
e(t ) 
xˆ (t )  x(t )
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Non-uniform quantization


It 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
y  C (x)
x(t )
companding
x̂
yˆ (t )
y (t )
xˆ (t )
x
Compress
ŷ
Qauntize
Expand
Channel
Transmitter
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Receiver
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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 k  log 2 M 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 Mary PAM (M>2).
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PAM example: Binary vs 8-ary
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Example of M-ary PAM
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
2
2
A

10B
• Energy per symbol in binary-PAM:
Binary PAM
(rectangular pulse)
4-ary PAM
(rectangular pulse)
3B
A.
‘1’
B
T
T
T
‘10’
‘0’
-A.
T
-B
‘00’
‘11’
‘01’
T
T
-3B
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Other PCM waveforms: Examples

PCM waveforms category:
 Phase encoded
 Multilevel binary
 Nonreturn-to-zero (NRZ)
 Return-to-zero (RZ)
NRZ-L
+V
-V
1
0 1 1 0
+V
Manchester -V
Unipolar-RZ +V
0
Miller +V
-V
+V
Bipolar-RZ 0
-V
0
+V
Dicode NRZ 0
-V
2T 3T 4T 5T
0
T
1
0 1 1 0
T
2T 3T 4T 5T
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PCM waveforms: Selection Criteria

Criteria for comparing and selecting PCM waveforms:
 Spectral characteristics (power spectral density and
bandwidth efficiency)
 Bit synchronization capability
 Error detection capability
 Interference and noise immunity
 Implementation cost and complexity
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Summary: Baseband Formatting and transmission
Digital info.
Format
Textual
source info.
Analog
info.
Bit stream
(Data bits)
Sample
Sampling at rate
f s  1 / Ts
(sampling time=Ts)
Quantize
Encode
Pulse waveforms
(baseband signals)
Pulse
modulate
Encoding each q. value to
l  log 2 L bits
(Data bit duration Tb=Ts/l)
Mapping every m  log 2 M data bits to a
symbol out of M symbols and transmitting
a baseband waveform with duration T
Quantizing each sampled
value to one of the
L levels in quantizer.
Information (data- or bit-) rate: Rb  1 / Tb [bits/sec]
 Symbol rate : R  1 / T [symbols/s ec]

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Rb  mR
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Receiver Structure & Matched Filter
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Demodulation and detection
g i (t ) Bandpass si (t )
Pulse
modulate
modulate
channel
transmitted symbol
hc (t )
Format
mi
estimated symbol
Format

m̂i
Detect
M-ary modulation
i  1,, M
n(t )
Demod.
z (T ) & sample r (t )
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”.
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Impact of AWGN
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Impact of AWGN & Channel Distortion
hc (t )   (t )  0.5 (t  0.75T )
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Receiver job

Demodulation 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
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Receiver structure
Step 1 – waveform to sample transformation
Step 2 – decision making
Demodulate & Sample
Detect
z (T )
r (t )
Frequency
down-conversion
For bandpass signals
Received waveform
Receiving
filter
Equalizing
filter
Threshold
comparison
m̂i
Compensation for
channel induced ISI
Baseband pulse
(possibly distorted)
Baseband pulse
Sample
(test statistic)
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Baseband vs Bandpass

Bandpass 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.
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Steps in designing the receiver

Find optimum solution for receiver design with the
following goals:
1. Maximize SNR: matched filter
2. Minimize ISI: equalizer

Steps in design:
 Model the received signal
 Find separate solutions for each of the goals.

First, we focus on designing a receiver which maximizes the
SNR: matched filter
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Receiver filter to maximize the SNR

Model the received signal
si (t )
r (t )  si (t ) h c (t )  n(t )
r (t )
hc (t )
n(t )
AWGN

Simplify the model (ideal channel assumption):
 Received signal in AWGN
Ideal channels
hc (t )   (t )
r (t )
si (t )
r (t )  si (t )  n(t )
n(t )
AWGN
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

Matched Filter Receiver
Problem:
 Design the receiver filter h(t ) such that the SNR is
maximized at the sampling time when si (t ), i  1,..., M
is transmitted.
Solution:
 The optimum filter, is the Matched filter, given by
h(t )  hopt (t )  si (T  t )
*
H ( f )  H opt ( f )  Si ( f ) exp(  j 2fT )
*
which is the time-reversed and delayed version of the conjugate of the
transmitted signal
h(t )  hopt (t )
si (t )
0
T
t
0
T
t
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Correlator Receiver

The matched filter output at the sampling time, can be realized
as the correlator output.
 Matched filtering, i.e. convolution with si*(T-) simplifies to
integration w/ si*(), i.e. correlation or inner product!
z (T )  hopt (T )  r (T )
T
  r ( )si ( )d  r (t ), s(t ) 
*
0
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
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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).

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Aside: Correlation Effect




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”,  ) between them
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,
The breadth of the correlation function - where it has significant value - shows for how long
the signals remain similar.
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Aside: Autocorrelation
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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.
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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
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rpi”
Source: Bores Signal
Processing
Matched Filter: back to cartoon…


Consider the received signal as a vector r, and the transmitted signal vector as s
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)
y (t )  si (t ) h opt (t )
h opt (t )
si (t )
A
T
A2
A
T
T
t
T
si (t )
h opt (t )
A
T
A
T
T/2 T
t
A
T
0
t
2T
t
0 T/2 T 3T/2 2T
t
T
y (t )  si (t ) h opt (t )
A2
T/2 T
t

A
T
A2
2
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Properties of the Matched Filter
1.
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.
Z ( f ) | S ( f ) |2 exp(  j 2fT )
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.
z (t )  Rs (t  T )  z (T )  Rs (0)  Es
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.
Es
S
max   
 N T N 0 / 2
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.
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Implementation of matched filter receiver
Bank of M matched filters
s (T  t )
*
1
z1 (T )
r (t )
sM (T  t )
*
zM
 z1 
   z
 
 zM 
(T )
Matched filter output:
z Observation
vector
zi  r (t )  s i (T  t ) i  1,..., M
z  ( z1 (T ), z2 (T ),..., zM (T ))  ( z1 , z2 ,..., zM )
Note: we are projecting along the basis directions of the signal space
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Implementation of correlator receiver
Bank of M correlators
s 1 (t )

T
z1 (T )
 z1 

 
 zM 
0
r (t )
s

M
(t )

T
0
z
Correlators output:
z Observation
vector
zM (T )
z  ( z1 (T ), z2 (T ),..., zM (T ))  ( z1 , z2 ,..., zM )
T
zi   r (t )si (t )dt
i  1,..., M
0
Note: In previous slide we “filter” i.e. convolute inShivkumar
the boxesKalyanaraman
shown.
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Implementation example of matched filter receivers
s1 (t )
Bank of 2 matched filters
A
T
0
T
z1 (T )
A
T
t
r (t )
0
T
s2 (t )
T
0
A
T
0
T
t
 z1 
 
 
 z2 
z
z
z2 (T )
A
T
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Matched Filter: Frequency domain View
Simple Bandpass Filter:
excludes noise, but misses some signal power
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Matched Filter: Frequency Domain View (Contd)
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!
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Maximal Ratio Combining (MRC) viewpoint

Generalization of this f-domain picture, for combining
multi-tap signal
Weight each branch
SNR:
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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

What is a signal space?


Why do we need a signal space?



Vector representations of signals in an N-dimensional orthogonal 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 Kalyanaraman
Rensselaer Polytechnic Institute
59
: “shiv rpi”
Schematic example of a signal space
 2 (t )
s1  (a11, a12 )
 1 (t )
z  ( z1 , z2 )
s3  (a31, a32 )
s2  (a21, a22 )
Transmitted signal
alternatives
Received signal at
matched
output
Rensselaerfilter
Polytechnic
Institute
s1 (t )  a11 1 (t )  a12 2 (t )  s1  (a11 , a12 )
s2 (t )  a21 1 (t )  a22 2 (t )  s 2  (a21 , a22 )
s3 (t )  a31 1 (t )  a32 2 (t )  s 3  (a31 , a32 )
z (t )  z1 1 (t )  z 2 2 (t )  z  ( z1 , z 2 )
Shivkumar Kalyanaraman
60
: “shiv rpi”
Signal space

To form a signal space, first we need to know the
inner product between two signals (functions):
 Inner (scalar) product:

 x(t ), y (t ) 
*
x
(
t
)
y
(t )dt


= cross-correlation between x(t) and y(t)
 Properties
of inner product:
 ax(t ), y (t )  a  x(t ), y (t ) 
 x(t ), ay(t )  a*  x(t ), y(t ) 
 x(t )  y (t ), z (t )  x(t ), z (t )    y (t ), z (t ) 
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
61
: “shiv rpi”
Signal space …


The distance in signal space is measure by calculating the norm.
What is norm?
 Norm of a signal:
x(t )   x(t ), x(t )  
= “length” of x(t)



x(t ) dt  Ex
2
ax(t )  a x(t )

Norm between two signals:
d x, y  x(t )  y(t )

We refer to the norm between two signals as the Euclidean distance between
two signals.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
62
: “shiv rpi”
Example of distances in signal space
 2 (t )
s1  (a11, a12 )
d s1 , z
E1
 1 (t )
z  ( z1 , z2 )
E3
s3  (a31, a32 )
d s3 , z
E2
d s2 , z
s2  (a21, a22 )
The Euclidean distance between signals z(t) and s(t):
d si , z  si (t )  z (t )  (ai1  z1 ) 2  (ai 2  z 2 ) 2
i  1,2,3
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
63
: “shiv rpi”
Orthogonal signal space

N-dimensional orthogonal signal space is characterized by N
linearly independent functions  j (t )Nj1 called basis functions.
The basis functions must satisfy the orthogonality condition
T
  i (t ), j (t )   i (t ) (t )dt  K i ji
*
j
0
where

0t T
j , i  1,..., N
1  i  j
0  i  j
 ij  
If all K i  1 , the signal space is orthonormal.

Constructing Orthonormal basis from non-orthonormal set of vectors:
 Gram-Schmidt procedure
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
64
: “shiv rpi”
Example of an orthonormal bases

Example: 2-dimensional orthonormal signal space

2

(
t
)

cos( 2t / T )
 1
T


 (t )  2 sin( 2t / T )
 2
T
0t T
 2 (t )
0t T
T
0
  1 (t ), 2 (t )   1 (t ) 2 (t )dt  0
 1 (t )
0
 1 (t )   2 (t )  1

Example: 1-dimensional orthonornal signal space
 1 (t )
 1 (t )  1
1
T
0
0
T
t
 1 (t )
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
65
: “shiv rpi”
Sine/Cosine Bases: Note!

Approximately orthonormal!

These are the in-phase & quadrature-phase dimensions of
complex baseband equivalent representations.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
66
: “shiv rpi”
Example: BPSK

Note: two symbols, but only one dimension in BPSK.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
67
: “shiv rpi”
Signal space …

Any arbitrary finite set of waveforms si (t )iM1
where each member of the set is of duration T, can be expressed
as a linear combination of N orthogonal waveforms
where  .(t )N
NM
j
j 1
N
si (t )   aij j (t )
i  1,..., M
NM
j 1
where
T
1
1
*
aij 
 si (t ), j (t ) 
s
(
t
)

i
j (t )dt
Kj
K j 0
j  1,..., N
0t T
i  1,..., M
N
Ei   K j aij
si  (ai1 , ai 2 ,..., aiN )
2
j 1
Vector representation of waveform
Waveform energy
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
68
: “shiv rpi”
Signal space …
N
si (t )   aij j (t )
si  (ai1 , ai 2 ,..., aiN )
j 1
Waveform to vector conversion
Vector to waveform conversion
 1 (t )
si (t )

T
ai1
0
 N (t )

T
0
aiN
 1 (t )
 ai1 
    sm
 
aiN 
sm
 ai1 

 
aiN 
ai1
 N (t )
si (t )
aiN
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
69
: “shiv rpi”
Example of projecting signals to an
orthonormal signal space
 2 (t )
s1  (a11, a12 )
 1 (t )
s3  (a31, a32 )
s2  (a21, a22 )
Transmitted signal
alternatives
s1 (t )  a11 1 (t )  a12 2 (t )  s1  (a11, a12 )
s2 (t )  a21 1 (t )  a22 2 (t )  s 2  (a21, a22 )
s3 (t )  a31 1 (t )  a32 2 (t )  s 3  (a31, a32 )
T
Rensselaer Polytechnic Institute
aij   si (t ) j (t )dt
0
j  1,..., N
70
i  1,..., M Shivkumar
0  t  TKalyanaraman
: “shiv rpi”
Matched filter receiver (revisited)
(note: we match to the basis directions)
Bank of N matched filters
 (T  t )

1
z1
r (t )
  N (T  t )
zN
 z1 
  z
 
 z N 
Observation
vector
z
N
si (t )   aij j (t )
j 1
i  1,..., M
NM
z  ( z1 , z2 ,..., z N )
z j  r (t )  j (T  t )
j  1,..., N
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
71
: “shiv rpi”
Correlator receiver (revisited)
Bank of N correlators
 1 (t )

z1
T
0
r (t )
 N (t )

T
zN
0
N
si (t )   aij j (t )
 r1 

 
rN 
z
z
Observation
vector
i  1,..., M
j 1
z  ( z1 , z2 ,..., z N )
T
z j   r (t ) j (t )dt
NM
j  1,..., N
Shivkumar Kalyanaraman
0 Institute
Rensselaer Polytechnic
72
: “shiv rpi”
Example of matched filter receivers using
basic functions
s1 (t )
 1 (t )
s2 (t )
A
T
1
T
T
0
0
T
t
A
T
t
0
T
t
1 matched filter
 1 (t )
r (t )
1
T
0

z1
z1 
z
z
T t
Number of matched filters (or correlators) is reduced by 1 compared to
using matched filters (correlators) to the transmitted signal!
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
73
: “shiv rpi”
White noise in Orthonormal Signal Space

AWGN n(t) can be expressed as
n(t )  nˆ (t )  n~ (t )
Noise outside on the signal space
(irrelevant)
Noise projected on the signal space
(colored):impacts the detection process.
N
nˆ (t )   n j j (t )
Vector representation of nˆ (t )
j 1
n j  n(t ), j (t ) 
 n~ (t ), (t )  0
j
j  1,..., N
j  1,..., N
n  (n1 , n2 ,..., nN )
n 
N
independent zero-mean
Gaussain random variables with
variance var( n j )  N 0 / 2
j j 1
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
74
: “shiv rpi”
Detection: Maximum Likelihood &
Performance Bounds
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
75
: “shiv rpi”
Detection of signal in AWGN

Detection problem:
 Given the observation vector z , perform a mapping from z
to an estimate m̂ of the transmitted symbol, mi , such that
the average probability of error in the decision is
minimized.
n
mi
Modulator
z
si
Decision rule
m̂
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
76
: “shiv rpi”
Statistics of the observation Vector

z  si  n
AWGN channel model:
Signal vector si  (ai1 , ai 2 ,..., aiN )
is deterministic.
 Elements of noise vector n  (n1 , n2 ,..., nN ) are i.i.d Gaussian
random variables with zero-mean and variance N 0 / 2 . The
noise vector pdf is
2


n
1


pn (n) 
exp
N 0 N / 2  N 0 
 The elements of observed vector z  ( z1 , z 2 ,..., z N )
are
independent Gaussian random variables. Its pdf is
2


z

s
1
i

pz (z | si ) 
exp  
N /2


N
N 0 
0



Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
77
: “shiv rpi”
Detection

Optimum decision rule (maximum a posteriori probability):
Set mˆ  mi if
Pr( mi sent | z )  Pr( mk sent | z), for all k  i
where k  1,..., M .

Applying Bayes’ rule gives:
Set mˆ  mi if
pz (z | mk )
pk
, is maximum for all k  i
pz ( z )
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
78
: “shiv rpi”
Detection …

Partition the signal space into M decision regions,
such that Z1 ,..., Z M
Vector z lies inside region Z i if
pz (z | mk )
ln[ pk
], is maximum for all k  i.
pz ( z )
That means
mˆ  mi
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
79
: “shiv rpi”
Detection (ML rule)

For equal probable symbols, the optimum decision rule
(maximum posteriori probability) is simplified to:
Set mˆ  mi if
pz (z | mk ), is maximum for all k  i
or equivalently:
Set mˆ  mi if
ln[ pz (z | mk )], is maximum for all k  i
which is known as maximum likelihood.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
80
: “shiv rpi”
Detection (ML)…


Partition the signal space into M decision regions, Z1 ,..., Z M
Restate the maximum likelihood decision rule as follows:
Vector z lies inside region Z i if
ln[ pz (z | mk )], is maximum for all k  i
That means
mˆ  mi
Vector z lies inside region Z i if
z  s k , is minimum for all k  i
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
81
: “shiv rpi”
Schematic example of ML decision regions
 2 (t )
Z2
s2
s3
s1
Z3
Z1
 1 (t )
s4
Z4
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
82
: “shiv rpi”
Probability of symbol error

Erroneous decision: For the transmitted symbol s i or equivalently signal
vector mi , an error in decision occurs if the observation vector z does not
fall inside region Z i.
 Probability of erroneous decision for a transmitted symbol
ˆ  mi )  Pr(mi sent)Pr( z does not lie inside Zi mi sent)
Pr(m

Probability of correct decision for a transmitted symbol
ˆ  mi )  Pr(mi sent)Pr( z lies inside Zi mi sent)
Pr(m
Pc (mi )  Pr( z lies inside Z i mi sent) 
Pe (mi )  1  Pc (mi )
 p (z | m )dz
z
i
Zi
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
83
: “shiv rpi”
Example for binary PAM
pz (z | m2 )
pz (z | m1 )
s2
 Eb
s1
0
 1 (t )
Eb
 s1  s 2 / 2 

Pe (m1 )  Pe (m2 )  Q
 N /2 
0


 2 Eb
PB  PE (2)  Q
 N0
Rensselaer Polytechnic Institute
84




Shivkumar Kalyanaraman
: “shiv rpi”
Average prob. of symbol error …

Average probability of symbol error :
M
ˆ  mi )
PE ( M )   Pr ( m
i 1

For equally probable symbols:
1
PE ( M ) 
M
M
1
Pe (mi )  1 

M
i 1
1
 1
M
M
 P (m )
i 1
c
i
M
  p (z | m )dz
i 1 Z i
z
i
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
85
: “shiv rpi”
Union bound
Union bound
The probability of a finite union of events is upper bounded
by the sum of the probabilities of the individual events.
M
1
PE ( M ) 
M
Pe (mi )   P2 (s k , s i )
k 1
k i
M
M
 P (s
i 1 k 1
k i
2
k
, si )
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
86
: “shiv rpi”
Example of union bound
Pe (m1 ) 
 p (r | m )dr
r
Z 2 Z3 Z 4
r
Z2
2
Z1
s2
1
s1
1
Union bound:
s3
4
s4
Z4
Z3
Pe (m1 )   P2 (s k , s1 )
k 2
2
A2 r
2
r
s2
r
s2
s1
s3
P2 (s 2 , s1 ) 
s4
 p (r | m )dr
r
A
1
2
Rensselaer Polytechnic
Institute
s2
s1
1
2
s1
1
s3
P2 (s 3 , s1 ) 
A3
 p (r | m )dr
r
A3
87
1
s4
1
s3
A4
s4
P2 (s 4 , s1 )   pr (r | m1 )dr
A4
Shivkumar Kalyanaraman
: “shiv rpi”
Upper bound based on minimum distance
P2 (s k , s i )  Pr( z is closer to s k than s i , when s i is sent)
 d ik / 2
1
u2
exp(  )du Q
 N /2
N0
N 0
0




d ik




dik  si  s k
1
PE ( M ) 
M
 d min / 2 

P2 (s k , si )  ( M  1)Q

 N /2 
i 1 k 1
0


k i
M
M
Minimum distance in the signal space:
d min  min d ik
i ,k
ik
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
88
: “shiv rpi”
Example of upper bound on av. Symbol error prob.
based on union bound
 2 (t )
si  Ei  Es , i  1,...,4
d i ,k  2 Es
ik
Es
d min  2 Es
s2
d1, 2
d 2,3
s3
 Es
s1
d 3, 4
d1, 4
Es
 1 (t )
s4
 Es
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
89
: “shiv rpi”
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
90
: “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.
Eb
STb
S W


N 0 N / W N Rb
Rb
: Bit rate
W
: Bandwidth
Note: S/N = Eb/No x spectral efficiency
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
91
: “shiv rpi”
Example of Symbol error prob. For PAM signals
Binary PAM
s2
s1
 Eb
s4
E
6 b
5
0
Eb
4-ary PAM
s3
s2
E
2 b
5
0
 1 (t )
E
2 b
5
s1
E
6 b
5
 1 (t )
 1 (t )
1
T
0
T t
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
92
: “shiv rpi”
Maximum Likelihood (ML) Detection: Vector Case
Nearest Neighbor Rule:
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:
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)
ps: Vector norm is a natural extension of “magnitude” or length
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
93
: “shiv rpi”
Extension to M-PAM (Multi-Level Modulation)

Note: h refers to the constellation shape/direction
MPAM:
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
94
: “shiv rpi”
Complex Vector Space Detection
Error probability:

Note: Instead of vT, use v* for complex vectors (“transpose and conjugate”)
for inner products…
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
95
: “shiv rpi”
Complex
Detection:
Summary
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
96
: “shiv rpi”
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 f
 As BER (f) increases, the effects become increasingly intolerable
 f tends to increase rapidly with lower SNR: “waterfall” curve (Q-function)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
97
: “shiv rpi”
SNR vs BER: AWGN vs Rayleigh
Need
diversity
techniques
to deal with
Rayleigh
(even 1-tap,
flat-fading)!




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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
98
: “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:
Pe is proportional to
BER
1
 0L
Diversity of
L:th order
Average SNR
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
99
: “shiv rpi”
BER vs. SNR (diversity effect)
BER
(  Pe )
AWGN
channel
(no fading)
Flat fading channel,
Rayleigh fading,
L=1
SNR
L=4
L=3
( 0)
L=2
We will explore this story later… slide set part II
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
100
: “shiv rpi”
Modulation Techniques
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
101
: “shiv rpi”
What is Modulation?

Encoding information in a manner suitable for
transmission.
 Translate baseband source signal to bandpass signal
 Bandpass signal: “modulated signal”

How?
 Vary amplitude, phase or frequency of a carrier

Demodulation: extract baseband message from carrier
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
102
: “shiv rpi”
Digital vs Analog Modulation
Cheaper, faster, more power efficient
 Higher 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

Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
103
: “shiv rpi”
Goals of Modulation Techniques
• High Bit Rate
• High Spectral Efficiency (max Bps/Hz)
• High Power Efficiency (min power to achieve a target BER)
• Low-Cost/Low-Power Implementation
• Robustness to Impairments
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
104
: “shiv rpi”
Modulation: representation
• Any modulated signal can be represented as
s(t) = A(t) cos [wct + f(t)]
amplitude
phase or frequency
s(t) = A(t) cos f(t) cos wct
- A(t) sin f(t) sin wct
quadrature
in-phase
• Linear versus nonlinear modulation  impact on spectral efficiency
Linear: Amplitude or phase
Non-linear: frequency: spectral broadening
• Constant envelope versus non-constant envelope
hardware implications with impact on power efficiency
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(=> reliability: i.e. target BER at lower SNRs)
Shivkumar Kalyanaraman
105
: “shiv rpi”
Complex Vector Spaces: Constellations
MPSK
Circular

Square
Each signal is encoded (modulated) as a vector in a signal space
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
106
: “shiv rpi”
Linear Modulation Techniques
s(t) = [ S an g (t-nT)]cos wct - [ S bn g (t-nT)] sin wct
n
n
I(t), in-phase
Q(t), quadrature
LINEAR MODULATIONS
M-ARY QUADRATURE
AMPLITUDE MOD.
(M-QAM)
Square
Constellations
M 4
M-ARY PHASE
SHIFT KEYING
(M-PSK)
M=4
(4-QAM = 4-PSK)
CONVENTIONAL
4-PSK
(QPSK)
Circular
Constellations
M 4
OFFSET
DIFFERENTIAL
4-PSK
4-PSK
(OQPSK) (DQPSK, /4-DQPSK)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
107
: “shiv rpi”
M-PSK and M-QAM
M-PSK (Circular Constellations)
bn
M-QAM (Square Constellations)
bn
4-PSK
16-QAM
16-PSK
4-PSK
an
an
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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
108
: “shiv rpi”
Bandwidth vs. Power Efficiency
MPSK:
MQAM:
MFSK:
Rensselaer Polytechnic Institute
Source: Rappaport book, chap 6
Shivkumar Kalyanaraman
109
: “shiv rpi”
MPAM & Symbol Mapping
Binary PAM
s2
s1
Note: the average energy
per-bit is constant
 Gray coding used for
mapping bits to symbols

 Eb
Why? Most likely error is to
confuse with neighboring
symbol.
 Make sure that the
neighboring symbol has only
1-bit difference (hamming
distance = 1)
6
Eb
5
Eb
4-ary PAM
s3
s2
s4

0
 1 (t )
Eb
5
2
0
2
Eb
5
s1
6
Eb
5
 1 (t )
Gray coding
00
01
11
10
s4
4-ary PAM
s3
s2
s1
E
6 b
5
E
2 b
5
0
E
2 b
5
E
6 b
5
 1 (t )
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
110
: “shiv rpi”
MPAM: Details
Unequal energies/symbol:
Rensselaer Polytechnic Institute
Shivkumar Kalyanaraman
Decision111
Regions
: “shiv rpi”
M-PSK (Circular Constellations)
MPSK:
bn
4-PSK
16-PSK
an

Constellation points:

Equal energy in all signals:
01

01
Gray coding
00
11
Rensselaer Polytechnic Institute
00
10
11 Shivkumar Kalyanaraman
10
112
: “shiv rpi”
MPSK: Decision Regions & Demod’ln
Z2
Z3
Z4
Z3
Z2
Z5
Z1
Z1
Z6
Z4
4PSK
Z8
Z7
8PSK
4PSK: 1 bit/complex dimension or 2 bits/symbol
m=1
si(t) + n(t)
X
Z1: r > 0
g(Tb - t)
m = 0 or 1
Z2: r ≤ 0
m=0
cos(2πfct)
Coherent Demodulator for BPSK.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
113
: “shiv rpi”
M-QAM (Square Constellations)
MQAM:
bn
16-QAM
4-PSK
an

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
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114
Z1
Z2
Z3
Z4
Z5
Z6
Z7
Z8
Z9
Z10
Z11
Z12
Z13
Z14
Z15
Z16
16QAM: Decision Regions
Shivkumar Kalyanaraman
: “shiv rpi”
Non-Coherent Modulation: DPSK





Information 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 φ
More 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 symbol
 Info bits encoded as the differential phase between current & previous symbol
 Less sensitive to carrier phase drift (f-domain) ; more sensitive to doppler
effects: decorrelation of signal phase in time-domain
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
115
: “shiv rpi”
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 Kalyanaraman
Rensselaer Polytechnic Institute
116
: “shiv rpi”
Quadrature Offset

Phase 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 offset
QPSK modulation with quadrature offset is referred to as O-QPSK
O-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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
117
: “shiv rpi”
Offset QPSK waveforms
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
118
: “shiv rpi”
Frequency Shift Keying (FSK)
• Continuous Phase FSK (CPFSK)
– digital data encoded in the frequency shift
– typically implemented with frequency
modulator to maintain continuous phase
s(t) = A cos [wct + 2 kf  d() d]
t

– 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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
119
: “shiv rpi”
Minimum Shift Keying (MSK) spectra
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
120
: “shiv rpi”
Power Spectral Density (dB)
Spectral Characteristics
10
0
QPSK/DQPSK
GMSK
-20
-40
-60
B3-dBTb = 0.16
0.25
-80
1.0
-100
(MSK)
-120
0
0.5
1.0
1.5
2.0
2.5
Normalized Frequency (f-fc)Tb
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
121
: “shiv rpi”
Bit Error Probability (BER): AWGN
10-1
5
2
10-2
5
For Pb = 10-3
2
BPSK 6.5 dB
QPSK 6.5 dB
DBPSK ~8 dB
DQPSK ~9 dB
DBPSK
10-3
BPSK, QPSK
5
Pb
2
DQPSK
10-4
• QPSK is more spectrally efficient than BPSK
5
with the same performance.
2
• M-PSK, for M>4, is more spectrally efficient
10-5
but requires more SNR per bit.
5
• There is ~3 dB power penalty for differential
2
10-6
detection.
0
2
4
6
8
10
12
14
b, SNR/bit, dB
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
122
: “shiv rpi”
Bit Error Probability (BER): Fading Channel
1
5
2
10-1
5
2
10-2
5
Pb
DBPSK
• Pb is inversely proportion to the average SNR per bit.
2
BPSK
10-3
5
• Transmission in a fading environment requires about
18 dB more power for Pb = 10-3.
AWGN
2
10-4
5
2
10-5
0
5
10
15
20
25
30
35
b, SNR/bit, dB
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
123
: “shiv rpi”
Bit Error Probability (BER): Doppler Effects
• Doppler causes an irreducible error floor when differential
detection is used decorrelation of reference signal.
100
QPSK
DQPSK
10 -1
Rayleigh Fading
• The irreducible Pb depends on the data rate and the Doppler.
For fD = 80 Hz,
10 -2
data rate
-3
Pb 10
fDT=0.003
No Fading
10 -4
0.002
0.001
10 -5
0
10 -6
0
10
20
30
40
50
T
Pbfloor
10 kbps
10-4s
3x10-4
100 kbps
10-5s
3x10-6
1 Mbps
10-6s
3x10-8
The implication is that Doppler is not an issue for high-speed
wireless data.
[M. D. Yacoub, Foundations of Mobile Radio Engineering ,
CRC Press, 1993]
60
b, SNR/bit, dB
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
124
: “shiv rpi”
Bit Error Probability (BER): Delay Spread
• ISI causes an irreducible error floor.
10-1
Irreducible Pb
Coherent Detection
+ BPSK
QPSK
OQPSK Modulation
x MSK
x
10-2
• The rms delay spread imposes a limit on the
maximum bit rate in a multipath environment.
For example, for QPSK,

Maximum Bit Rate
Mobile (rural)
25 msec
8 kbps
Mobile (city)
2.5 msec
80 kbps
Microcells
500 nsec
400 kbps
Large Building
100 nsec
2 Mbps
x
x
+
+
x
+
10-3
+
x
[J. C.-I. Chuang, "The Effects of Time Delay Spread on Portable Radio
Communications Channels with Digital Modulation,"
IEEE JSAC, June 1987]
+
10-4
10-2
10-1
rms delay spread 
=
symbol period
T
100
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
125
: “shiv rpi”
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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
126
: “shiv rpi”
Pulse Shaping
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
127
: “shiv rpi”
Recall: Impact of AWGN only
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
128
: “shiv rpi”
Impact of AWGN & Channel Distortion
hc (t )   (t )  0.5 (t  0.75T )
Multi-tap, ISI channel
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
129
: “shiv rpi”
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
Hc ( f )  Hc ( f ) e
j c ( f )
Non-constant amplitude
Non-linear phase
Amplitude distortion
Phase distortion
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
130
: “shiv rpi”
Inter-Symbol Interference (ISI)
ISI in the detection process due to the filtering effects
of the system
 Overall equivalent system transfer function

H ( f )  Ht ( f )Hc ( f )H r ( f )
 creates
echoes and hence time dispersion
 causes ISI at sampling time
ISI effect
zk  sk  nk    i si
ik
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
131
: “shiv rpi”
Inter-symbol interference (ISI): Model
Baseband system model

xk 
x1 x2
T

xk 
ht (t )
Channel
hc (t )
Ht ( f )
Hc ( f )
Tx filter
x3
hr (t )
Hr ( f )
zk
t  kT
Detector
x̂k 
n(t )
Equivalent model
x1 x2
T
T
r (t ) Rx. filter
Equivalent system
x3
zk
z (t )
h(t )
H( f )
t  kT
T
Detector
x̂k 
nˆ (t )
H ( f )  Ht ( f )Hc ( f )H r ( f )
filtered noise
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
132
: “shiv rpi”
Nyquist bandwidth constraint

Nyquist 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.
Rs
Rs
1

W 
 2 [symbol/s/ Hz]
2T
2
W

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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
133
: “shiv rpi”
Equiv System: Ideal Nyquist pulse (filter)
Ideal Nyquist filter
Ideal Nyquist pulse
H( f )
h(t )  sinc( t / T )
T
1
2T
1
0
W
Rensselaer Polytechnic Institute
1
2T
 2T  T
f
1
2T
0
T 2T
t
Shivkumar Kalyanaraman
134
: “shiv rpi”
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 evensymmetric 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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
135
: “shiv rpi”
Pulse shaping to reduce ISI

Goals and trade-off in pulse-shaping
 Reduce ISI
 Efficient bandwidth utilization
 Robustness to timing error (small side lobes)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
136
: “shiv rpi”
Raised Cosine Filter: Nyquist Pulse Approximation
h(t )  hRC (t )
| H ( f ) || H RC ( f ) |
r 0
1
1
r  0.5
0.5
1  3 1
T 4T 2T
0
r 1
1 3
2T 4T
1
T
Rs
Baseband W sSB (1  r )
2
r 1
0.5
 3T  2T  T
0
r  0.5
r 0
T
2T
3T
Passband W DSB (1  r ) Rs
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
137
: “shiv rpi”
Raised Cosine Filter

Raised-Cosine Filter
 A Nyquist pulse (No ISI at the sampling time)
for | f | 2W0  W
1

2   | f | W  2W0 
H ( f )  cos 
 for 2W0  W | f | W
W  W0
4


0
for | f | W
cos[ 2 (W  W0 )t ]
h(t )  2W0 (sinc( 2W0t ))
1  [4(W  W0 )t ]2
Roll-off factor r 
0  r 1
Excess bandwidth: W  W0
W  W0
W0
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
138
: “shiv rpi”
Pulse Shaping and Equalization Principles
No ISI at the sampling time
H RC ( f )  H t ( f ) H c ( f ) H r ( f ) H e ( f )

Square-Root Raised Cosine (SRRC) filter and Equalizer
H RC ( f )  H t ( f ) H r ( f )
H r ( f )  H t ( f )  H RC ( f )  H SRRC ( f )
1
He ( f ) 
Hc ( f )
Taking care of ISI
caused by tr. filter
Taking care of ISI
caused by channel
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
139
: “shiv rpi”
Pulse Shaping & Orthogonal Bases
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
140
: “shiv rpi”
Virtue of pulse shaping
Rensselaer Polytechnic Institute
Source: Rappaport book, chap 6
PSD of a BPSK signal Shivkumar Kalyanaraman
141
: “shiv rpi”
Example of pulse shaping

Square-root Raised-Cosine (SRRC) pulse shaping
Amp. [V]
Baseband tr. Waveform
Third pulse
t/T
First pulse
Second pulse
Data symbol
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
142
: “shiv rpi”
Example of pulse shaping …

Raised Cosine pulse at the output of matched filter
Amp. [V]
Baseband received waveform at
the matched filter output
(zero ISI)
t/T
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
143
: “shiv rpi”
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)
Distortion
due to ISI
amplitude scale
Noise margin
Sensitivity to
timing error
Timing jitter
timeShivkumar
scale
Kalyanaraman
Rensselaer Polytechnic Institute
144
: “shiv rpi”
Example of eye pattern:
Binary-PAM, SRRC pulse

Perfect channel (no noise and no ISI)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
145
: “shiv rpi”
Example of eye pattern:
Binary-PAM, SRRC pulse …

AWGN (Eb/N0=20 dB) and no ISI
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
146
: “shiv rpi”
Example of eye pattern:
Binary-PAM, SRRC pulse …

AWGN (Eb/N0=10 dB) and no ISI
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
147
: “shiv rpi”
Summary
Digital Basics
 Modulation & Detection, Performance, Bounds
 Modulation Schemes, Constellations
 Pulse Shaping

Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
148
: “shiv rpi”
Extra Slides
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
149
: “shiv rpi”
Bandpass Modulation: I, Q Representation
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
150
: “shiv rpi”
Analog: Frequency Modulation (FM) vs
Amplitude Modulation (AM)

FM: all information in the phase or frequency of carrier





Non-linear or rapid improvement in reception quality beyond a minimum
received signal threshold: “capture” effect.
Better noise immunity & resistance to fading
Tradeoff bandwidth (modulation index) for improved SNR: 6dB gain for 2x
bandwidth
Constant envelope signal: efficient (70%) class C power amps ok.
AM: linear dependence on quality & power of rcvd signal



Spectrally efficient but susceptible to noise & fading
Fading improvement using in-band pilot tones & adapt receiver gain to
compensate
Non-constant envelope: Power inefficient (30-40%) Class A or AB power amps
needed: ½ the talk time as FM!
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
151
: “shiv rpi”
Example Analog: Amplitude Modulation
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
152
: “shiv rpi”