1. Modulation, Demodulation and Coding Course
Period
Falahati

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)
Lecture 3

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:
Textual
info.
Format
source
Pulse
modulate
Sample
Quantize
Encode
Analog
info.
Sampling at rate
(sampling time=Ts)
Encoding each q. value to
bits
(Data bit duration Tb=Ts/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
Lecture 3

4. Quantization example Quant. levels boundaries x(nTs): sampled values
amplitude
x(t)
x(nTs): sampled values
xq(nTs): quantized values
boundaries
Quant. levels
Ts: sampling time t
PCM
codeword
PCM sequence
Lecture 3

5. Example of M-ary PAM ‘11’ ‘01’ ‘10’ ‘00’ ‘0’ ‘1’ 2.2762 V 1.3657 V
Ts Ts
4-ary PAM
(rectangular pulse) 3B
V V
‘11’ B T T
‘01’
0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb
‘10’
‘00’ T T -B
-3B T A.
‘0’
‘1’
0 T T 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: T=2Tb and Binay: T=Tb
T T T
Lecture 3

6. 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
Lecture 3

7. 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
Lecture 3

8. Example: Impact of the channel
Lecture 3

9. Example: Channel impact …
Lecture 3

10. Receiver job Demodulation and sampling: Detection:
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
Lecture 3

11. 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
Received waveform
Baseband pulse
(possibly distored)
Baseband pulse
Sample
(test statistic)
Lecture 3

12. 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.
Lecture 3

13. 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.
Lecture 3

14. Design the receiver filter to maximize the SNR
Model the received signal
Simplify the model:
Received signal in AWGN
AWGN
Ideal channels
AWGN
Lecture 3

15. 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 t T t
Lecture 3

16. Example of matched filterT 2T t
T/2 T t
T/2 T T t
T/2 T
3T/2 2T t
Lecture 3

17. 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.
Lecture 3

18. Correlator receiver
The matched filter output at the sampling time, can be realized as the correlator output.
Lecture 3

19. Implementation of matched filter receiver
Bank of M matched filters
Matched filter output:
Observation
vector
Lecture 3

20. Implementation of correlator receiver
Bank of M correlators
Correlators output:
Observation
vector
Lecture 3

21. Example of implementation of matched filter receivers
Bank of 2 matched filters T t T T T t
Lecture 3

22. Signal space What is a signal space? Why do we need 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.
Lecture 3

23. Schematic example of a signal space
Transmitted signal
alternatives
Received signal at
matched filter output
Lecture 3

24. 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)
Lecture 3

25. Signal space – cont’d
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)
Lecture 3

26. Example of distances in signal space
The Euclidean distance between signals z(t) and s(t):
Lecture 3

27. Signal space - cont’d
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
Lecture 3

28. Example of an orthonormal basis functions
Example: 2-dimensional orthonormal signal space
Example: 1-dimensional orthonormal signal space T t
Lecture 3

29. Signal space – cont’d 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
Lecture 3

30. Signal space - cont’d Waveform to vector conversion
Vector to waveform conversion
Lecture 3

31. Example of projecting signals to an orthonormal signal space
Transmitted signal
alternatives
Lecture 3

32. Signal space – cont’d
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
Lecture 3

33. 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 T t T t 1 2 -A A
Lecture 3

34. Implementation of matched filter receiver
Bank of N matched filters
Observation
vector
Lecture 3

35. Implementation of correlator receiver
Bank of N correlators
Observation
vector
Lecture 3

36. Example of matched filter receivers using basic functionsT t T t
1 matched filter 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)
Lecture 3

37. 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
Lecture 3