Modulation, Demodulation and Coding Course

1
Modulation, Demodulation and Coding Course

• Period 3 - 2005
• Sorour Falahati
• Lecture 3

2
Last time we talked about

• Transforming the information source to a form
  compatible with a digital system
• Sampling
• Aliasing
• Quantization
• Uniform and non-uniform
• Baseband modulation
• Binary pulse modulation
• M-ary pulse modulation
• M-PAM (M-ay Pulse amplitude modulation)

3
Formatting and transmission of baseband signal
Digital info.
Bit stream (Data bits)
Pulse waveforms (baseband signals)

• Information (data) rate
• Symbol rate
• For real time transmission

Format
Textual info.
source
Pulse modulate
Encode
Sample
Quantize
Analog info.
Sampling at rate (sampling timeTs)
Encoding each q. value to
bits (Data bit duration TbTs/l)
Quantizing each sampled value to one of the L
levels in quantizer.
Mapping every data bits to a
symbol out of M symbols and transmitting a
baseband waveform with duration T
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Qunatization example
amplitude x(t)
111 3.1867
110 2.2762
101 1.3657
100 0.4552
011 -0.4552
010 -1.3657
001 -2.2762
000 -3.1867
Ts sampling time
t
PCM codeword
110 110 111 110 100 010 011 100
100 011
PCM sequence
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Example of M-ary PAM
3B
0 Ts
2Ts
4-ary PAM (rectangular pulse)
2.2762 V 1.3657 V
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B
01
T
T
0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb
T
T
00
10
-B
1 1 0 1 0 1
-3B
T
A.
1
0
0 T 2T 3T 4T 5T 6T
Binary PAM (rectangular pulse)
-A.
T

• Assuming real time tr. and equal energy per tr.
  data bit for binary-PAM and 4-ary PAM
• 4-ary T2Tb and Binay TTb

0 T 2T 3T
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Today we are going to talk about

• Receiver structure
• Demodulation (and sampling)
• Detection
• First step for designing the receiver
• Matched filter receiver
• Correlator receiver
• Vector representation of signals (signal space),
  an important tool to facilitate
• Signals presentations, receiver structures
• Detection operations

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Demodulation and detection
Format
Pulse modulate
Bandpass modulate
M-ary modulation
channel
transmitted symbol

• 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.

estimated symbol
Format
Detect
Demod. sample
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Example Impact of the channel
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Example Channel impact
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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
• Reducing ISI using equalizer
• 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
Threshold comparison
Frequency down-conversion
Receiving filter
Equalizing filter
Compensation for channel induced ISI
For bandpass signals
Baseband pulse (possibly distored)
Received waveform
Sample (test statistic)
Baseband pulse
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Baseband and 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
• Maximize SNR
• Minimize ISI
• 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.

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Design the receiver filter to maximize the SNR

• Model the received signal
• Simplify the model
• Received signal in AWGN

AWGN
Ideal channels
AWGN
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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

T
0
t
T
0
t
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Example of matched filter
0
2T
T
t
T
t
T
t
0
2T
T/2
3T/2
T
t
T
t
T
t
T/2
T
T/2
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Properties of the matched filter

• 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.
• 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.
• 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.
• 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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Correlator receiver

• The matched filter output at the sampling time,
  can be realized as the correlator output.

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Implementation of matched filter receiver
Bank of M matched filters
Matched filter output Observation vector
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Implementation of correlator receiver
Bank of M correlators
Correlators output Observation vector
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Example of implementation of matched filter
receivers
Bank of 2 matched filters
0
T
t
T
0
T
0
T
t
0
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Signal space

• What 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.

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Schematic example of a signal space
Transmitted signal alternatives
Received signal at matched filter output
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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

cross-correlation between x(t) and y(t)
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Signal space contd

• 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.

length of x(t)
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Example of distances in signal space
The Euclidean distance between signals z(t) and
s(t)
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Signal space - contd

• N-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.
• Orthonormal basis
• Gram-Schmidt procedure

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Example of an orthonormal basis functions

• Example 2-dimensional orthonormal signal space
• Example 1-dimensional orthonornal signal space

0
0
0
T
t
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Signal space contd

• 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
  orthonogal waveforms where .
• where

Vector representation of waveform
Waveform energy
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Signal space - contd
Waveform to vector conversion
Vector to waveform conversion
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Example of projecting signals to an orthonormal
signal space
Transmitted signal alternatives
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Signal space contd

• To find an orthonormal basis functions for a
  given set of signals, Gram-Schmidt procedure can
  be used.
• Gram-Schmidt procedure
• Given a signal set , compute an
  orthonormal basis
• Define
• For compute
• If let
• If , do not assign any basis function.
• Renumber the basis functions such that basis is
• This is only necessary if for any i
  in step 2.
• Note that

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Example of Gram-Schmidt procedure

• Find the basis functions and plot the signal
  space for the following transmitted signals
• Using Gram-Schmidt procedure

T
t
0
0
T
t
1
2
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Implementation of matched filter receiver
Bank of N matched filters
Observation vector
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Implementation of correlator receiver
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Example of matched filter receivers using basic
functions
T
t
0
T
t
0
1 matched filter
0
T
t

• Number of matched filters (or correlators) is
  reduced by 1 compared to using matched filters
  (correlators) to the transmitted signal.
• Reduced number of filters (or correlators)

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White noise in orthonormal signal space

• AWGN n(t) can be expressed as

Noise projected on the signal space which
impacts the detection process.
Noise outside on the signal space
Vector representation of
independent zero-mean Gaussain
random variables with variance