TE 386 - Digital Communication Systems
Department Electrical & Electronic Engineering
College of Engineering
Baseband Demodulation & Detection
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Outline
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Baseband Demodulation
Signals and Noise
White Noise Distribution and Determination of Error
Probability
Detection of Binary Signals in Gaussian Noise
Baseband Signal Detection
The Optimal Correlation Receiver
The Matched Filter
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Baseband Signal Detection
The Optimal Correlation Receiver
• In the detection of binary signals, the waveform is already known,
and we wish to determine whether the pulse is present or absent.
• The detector at the receiver must therefore be a decision-making device.
• The decision-making criterion for this type of detector chooses the threshold
level for the binary decision, based on minimizing the probability of error.
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● Consider the transmission of binary symbols s1 (t) or s2 (t) in additive
Gaussian noise. We assume a fixed binary interval T. We desire to
find:
• The optimum receiver structure and
• The optimum signal shapes s1 (t) and s2 (t)
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● From the previous result of Equation(23) or its equivalent
Equation(24), we have:
Choose s1 (t) if:
(30)
● Also, we assume that the a priori probabilities p1 and p2 are equal.
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Finding the number of samples n
• Consider the interval T; we desire to find the number of samples n required.
We assume the noise is band-limited white noise, with two-sided spectral
density:
The total noise power is then, N = N0 B
● We also assume that the two signals s1 (t) and s2 (t) are band-limited over
the same band B.
●
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● For the noise model considered, we recall that the autocorrelation
function, which is the Fourier transform of the spectral density, is given by;
Rn(τ ) = N sin 2πτ B
2πτ B
● Both Gn(f ) and Rn(τ ) are sketched
(31)
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Figure 10: Band-limited white noise (a)Spectral Density
(b)Autocorrelation function
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● It is noted that there is zero correlation between samples spaced multiples
of 1/2B apart.
● This indicates that the samples are independent. Thus the number of
independent samples that we can use, within the interval T, is;
And
n = 2BT samples
(32)
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The Optimum Processor
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• Equation (34) says that we can take the incoming signal v(t), multiply it by
two stored replicas of s1(t) and s2(t), integrate and sample the resultant
output every T sec to see if it exceeds a specific threshold.
• Note: The s1(t) and s2(t) stored in the receiver must be precisely in phase
with the s1(t) and s2(t) portion of the received v(t).
• The resultant operation is called coherent or synchronous detection.
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• In the general form of Equation(34), it is also called correlation
detection.
• A further alternative form of Equation(34) may be obtained by
separating the s1 and s2 terms:
(35)
• The E1 and E2 thus serve as fixed-bias terms to equalize the
detector outputs.
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Implementation of the Optimum Processor
● From Equations(34) and (35) above, we have,
(36)
Or
(37)
● The above Equation(36) is represented by the processor in Figure(11)
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Figure 11: Optimum binary processor - correlation detection
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The Matched Filter
• Another type of detector that can accomplish optimum reception of binary
data is the Matched Filter.
• A match filter is a linear filter designed to provide the maximum signal-tonoise power ratio at its output for a given transmitted symbol waveform.
• The filter for this detector will peak out the signal component at
some instant and suppress noise amplitude at the same time.
• The effect of increasing the signal component and decreasing the
noise component at the same instant is equivalent to maximizing
signal amplitude at the same instant at the output.
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• Consider the proposed matched filter with transfer function H (ω) as shown in
Figure(12).
• Let the input signal be 𝑠(𝑡) + 𝑛(𝑡), where s(t) is the useful signal pulse, n(t) is
the channel noise and 𝑠0 (𝑡) + 𝑛0 (𝑡) is the output of the filter.
Figure 12: Block Diagram of Matched Filter
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• The variance of the output noise (average noise power) is denoted by 𝜎!" , so
that the ratio of the instantaneous signal power to average noise power,
𝑆⁄𝑁 # , at time t = T, out of the sampler in step 1, is
𝑆
𝑎$"
= "
(49)
𝑁 # 𝜎!
• We wish to find the filter transfer function H0(f) that maximizes Equation (49)
• We can express the signal ai(t) at the filter output in terms of the filter transfer
function H(f) (before optimization) and the Fourier transform of the input
signal, as
&
𝑎$ (𝑡) = / 𝐻 𝑓 𝑆(𝑓)𝑒 '"()* 𝑑𝑓
(50)
%&
• where S(f) is the Fourier transform of the input signal, s(t).
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• If the two-sided power spectral density of the input noise is N0/2 watts/Hz, we
can express the output noise power as
&
𝑁
!
"
(51)
𝜎! =
/ 𝐻 𝑓 " 𝑑𝑓
2 %&
• We then combine Equations (49) to (51) to express 𝑆⁄𝑁 # as:
𝑆
𝑁
𝑎$"
= "=
𝜎!
#
"
&
'"()#
𝑑𝑓
∫%& 𝐻 𝑓 𝑆(𝑓)𝑒
𝑁! &
" 𝑑𝑓
𝐻
𝑓
∫
2 %&
(52)
• In order to solve Equation (52), we use the Schwartz inequality which states
that
(53)
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● This inequality holds only if, 𝑓+ 𝑥 = 𝑘𝑓"∗ 𝑥 , where k is an arbitrary constant
and ∗ is complex conjugate
● Let 𝑓+ 𝑥 = 𝐻 𝑓 and 𝑓" 𝑥 = 𝑆(𝑓)𝑒 '"()#
(53)
● Substituting into Equation (52) yields
𝑆
2 &
≤
/ 𝑆(𝑓) " 𝑑𝑓
𝑁 # 𝑁! %&
Or
𝑆
2𝐸
𝑚𝑎𝑥
=
𝑁 # 𝑁!
(54)
(55)
where the energy E of the input signal s(t) is
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• Thus, the maximum output (S/N)T depends on the input signal energy and the
power spectral density of the noise, not on the particular shape of the waveform
that is used.
• The equality in Equation (55) holds only if the optimum filter transfer function
𝐻! (𝑓) is employed, such that
𝐻(𝑓) = 𝐻! 𝑓 = 𝑘𝑆 ∗ 𝑓 𝑒 %'"()#
(56)
ℎ 𝑡 = ℱ %+ 𝑘𝑆 ∗ 𝑓 𝑒 %'"()#
• S(t) is a real-valued signal, hence we can write
𝑘𝑆(𝑇 − 𝑡),
0≤𝑡≤0
(55)
ℎ 𝑡 =>
0,
𝑒𝑙𝑠𝑤𝑒𝑤ℎ𝑒𝑟𝑒
• Thus, the impulse response of a filter that produces the maximum output signalto noise ratio is the mirror image of the message signal s(t), delayed by the
symbol time duration T.
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Decision Threshold
● The matched filter is defined to maximize the signal-to-noise ratio at the
instant 𝑡𝑚, (𝑡𝑚 = 𝑇).
● Whether the signal S(t) is present is therefore decided by the observation of
the output at t = T.
● If r(t) represents the output of the matched filter at t = T, then
𝑟(𝑇) = 𝑆𝑂(𝑇 ) + 𝑛𝑂(𝑇 ).
● And by substituting SQ(tm) from Equation (55) we obtain,
𝑟(𝑇 ) = 𝐸 + 𝑛𝑂(𝑇 )
(67)
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● Case 1: If signal is present, 𝑟(𝑇 ) = 𝐸 + 𝑛𝑂(𝑇 )
● Case 2: If signal is absent, 𝑟(𝑇 ) = 𝑛𝑂(𝑇 )
● We now consider a decision rule such that
1. Signal is present if, 𝑟(𝑇) > 𝑎, and
2. Signal is absent if, 𝑟(𝑇) < 𝑎
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● We now find the optimum decision threshold a, which will minimize
the error probability (likelihood) of the decision.
● The noise is a random signal and its amplitudes have the Gaussian
distribution p(x) which is the probability density function of the
amplitude x and is given by,
(68)
● Where σ2 x is the mean square of the signal.
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● Now from Equation(62),
. Thus we obtain,
(69)
● If we denote the output by r, 𝑟 = 𝑛𝑂(𝑇 ) if the signal is absent,
and
(70)
• If the signal is present, then 𝑟 = 𝐸 + 𝑛𝑂 𝑇 . Thus when the signal is
present,
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Optimum Decision Threshold
● Let a be the decision threshold. The decision is: signal present if r > a
and is signal absent if r < a.
● The second type of error is when a signal is present.
● If S(t) is equally likely to be present and absent, then on the average,
half the time S(τ ) will be absent.
● The sum of the area is minimum if we choose
(71)
● Hence the optimum threshold is given by Equation (71).
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Probability of Error of Matched Filter(Baseband Systems)
• To optimize (minimize) PB in the context of an AWGN channel and
the receiver shown we need to select the optimum receiving filter
in step 1 and the optimum decision threshold in step 2 (Figure A)
• For the binary case, the optimum decision threshold was shown as:
(72)
• Next, for minimizing PB, it is necessary to choose the filter
(matched filter) that maximizes the argument of Q(x)
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Figure 13: Error probability
Conditional probability density functions: p(z|s1) and p(z|s2).
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• Consider that the filter is matched to the input difference signal [s1(t) – s2(t)];
thus, we can write an output SNR at time t = T as:
(73)
• where 𝑁0/2 is the two-sided power spectral density of the noise at the filter
input and
(74)
is the energy of the difference signal at the filter input.
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•
Combining Equations (72) and (73) yields:
•
a more general relationship in terms of received bit energy can be developed by
defining a time cross-correlation coefficient ρ as a measure of similarity
between two signals, 𝑠$ (t) and 𝑠% (t)
and
•
(75)
where – 1 ≤ 𝜌 ≤ 1.
Expanding Equation (74):
(76)
•
Recall each of the first two terms in Equation (76) represents the energy
associated with a bit, Eb: that is,
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(77)
• Substitution:
(78)
• Finally:
(79)
• For the case of 𝜌 = 1, corresponding signals s1(t) and s2(t) are perfectly correlated
over a symbol time. Would such waveforms be used for digital signaling?
• For the case of 𝜌 = −1, corresponding signals s1(t) and s2(t) are “anticorrelated”
over a symbol time and the angle between the signal vectors are 180 degrees.
Signals called antipodal.
• For the case of 𝜌 = 0 corresponding signals s1(t) and s2(t) are zero correlated over
a symbol time and the angle between the signal vectors are 90 degrees. Signals
called orthogonal.
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Error Probability Performance of Binary Signaling:
Unipolar Signaling
• Figure illustrates an example of baseband
orthogonal signaling–namely, unipolar signaling,
where:
𝑠1 𝑡 = 𝐴, 0 ≤ 𝑡 ≤ 𝑇 for binary 1
𝑠2(𝑡) = 0, 0 ≤ 𝑡 ≤ 𝑇 for binary 0
Orthogonal binary signal vector
Detection of unipolar baseband signaling.
(a) Unipolar signaling example. (b)
Correlator detector.
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Error Probability Performance of Binary Signaling:
Bipolar Signaling
• Figure illustrates an example of baseband
antipodal signaling–namely, bipolar signaling,
where:
𝑠1(𝑡) = +𝐴 0 ≤ 𝑡 ≤ 𝑇 for binary 1
𝑠2(𝑡) = −𝐴 0 ≤ 𝑡 ≤ 𝑇 for binary 0
Antipodal binary signal vector
Detection of bipolar baseband signaling.
(a) Bipolar signaling example. (b) Correlator
detector.
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• The lower curve is the betterperforming one
• At 10 dB, we see that the
unipolar signaling yields 𝑃𝐵 in
the order of 10&' but that the
bipolar signaling yields 𝑃𝐵 in
the order of 10&(
Figure 14. Bit-error performance of unipolar and bipolar signaling.
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Example Matched Filter Detection of Antipodal Signals
Consider a binary communication system that receives equally likely signals s1(t)
and s2(t) plus AWGN (see Figure 15 below).
• Assume that the receiving filter is
a matched filter (MF) and that the
noise-power spectral density N0 is
equal to 10–12 watts/Hz.
•
Use the values of received signal
voltage and time shown in Figure
15 to compute the bit-error
probability.
Figure 15 Baseband antipodal waveforms
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Solution:
• We can graphically determine the received energy per bit from the plot of
either s1(t) or s2(t) shown in Figure 15 by integrating to find the energy (area
under the voltage-squared pulse). Doing this in piecewise fashion, we get:
• Since the waveforms depicted in Figure 15 are antipodal and are detected with
a matched filter, we use Equation (79) with ρ = -1 to find the bit-error
probability, as
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Solution:
• From the Q-function table (see appendix), we find that PB =2.7009 × 10–4.
• Alternatively, since the argument of Q(x) is greater than 3, we can also use the
approximate relationship in Equation (10), which yields PB ≈ 2.9 × 10-4.
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THE END
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