[PPT] - Lecture 10 Pulses Chapter 8 and 9 Detection of a Binary Signal PowerPoint Presentation

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Lecture 10

Chapter 8 and 9

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Posterior probability distribution

Discrete memoryless information channel

depends or is conditioned by the corresponding channel input symbol and by only that symbol.

For a well-designed encoder output of the encoder, will appear to be random and independent The probability of each codeword symbol, seen in isolation, at the input to the channel from an ideal encoder is described by a prior probability distribution ps(s),

called simply a prior, w/components called prior probabilities

In constrast, the probability of each channel output symbol, seen in isolation, is described by a posterior probability distribution pr(r) on the output, called simply the posterior The conditional probability distribution ps|r(s|r) on the input for a fixed r at the output is called the posterior probability distribution on the input symbol.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Detection Statistic

Detection process at Rx converts a discrete-time sample of the electrical waveform, real or complex, into a senseword symbol that is sent to the decoder This sample value is called a detection statistic A detection statistic can have many forms It is generated by transforming a received waveform into a sequence of samples The decoder then determines the transmitted codeword or, equivalently, determines the user dataword represented by that codeword

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Different Viewpoints of an Information Channel

The aspects of a communication system that constitute an information channel are described using a conditional probability distribution This conditional distribution depends on the channel model The information channel can be viewed from either the transmitter or from the receiver When viewed from the transmitter, the memoryless information channel is modeled as a conditional probability distribution pr|s(r|s), abbreviated p(r|s), that the symbol r will be received when the symbol s is transmitted When viewed from the receiver, the information channel is modeled as a conditional probability distribution ps|r(s|r), abbreviated p(s|r), that when the symbol r is received, the symbol s was transmitted.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Conditional, Joint and Prior Probalities

Using the Bayes rule, the combination of the information channel model and the prior distribution of symbols can be expressed as a joint probability distribution p(s, r) p(s, r) = ps(s)p(r|s) = pr(r)p(s|r), (1) where ps(s) is the prior for the transmitted input symbol s, and pr(r) is the posterior probability distribution for the received output symbol r It follows immediately from (1) that (sum out one to get the other) pr(r) =

s

ps(s)p(r|s) (2) and so p(s|r) = ps(s)p(r|s)

s ps(s)p(r|s).

(3) For the same physical channel, the expression for the joint probability p(s, r) based on continuous wave optics is different than the expression based on discrete photon optics because these are different models and use different methods of detection.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Soft and Hard Decision Detection

The detected sequence of symbols form the senseword at the output of the information channel, which becomes the input to the decoder In soft-decision detection, each component of the senseword is a sample rk (real or complex) or a quantized form of that sample In hard-decision detection, the detection process decides on a symbol from a discrete output alphabet based on the received sample and on prior knowledge about the possible inputs The output of hard-decision detection is a sequence of logical symbols generated by a hypothesis-testing procedure that is used to form the senseword.

Hypothesis testing is quantified by the probability of a detection error pe

The probability of a detection error pe is not meaningful for soft-decision detection because the quantized samples are not generated by hypothesis testing Accordingly, the information channel defined using hard-decision detection is not the same as an information channel defined using soft-decision detection

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Methods of Coded Modulation

At the Tx, the information channel receives a sequence of logical symbols from the encoder and converts this sequence into a sequence of real or complex electrical pulses for the electrical channel At the Rx, the information channel receives a sequence of real or complex samples taken from the electrical waveform and converts this sequence into a sequence of logical symbols for the decoder This back and forth conversion is the task of modulation and demodulation This sequence of real numbers can be described as a waveform w(t) on continuous time using Dirac impulses as given by w(t) =

∞

j=−∞

sj δ t − jT , (4) where T is the symbol interval, and the symbol sj in the jth interval is a point of the L-point signal constellation {s0, s1, ..., sL−1} that is specified by the user data.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Signal Constellations

(a) (b) (c) (d)

dmin φ dmin dmin dmin

Figure: Signal constellations. (a) A pulse amplitude modulation constellation. (b) A square quadrature amplitude modulation (QAM) constellation. (c) A nonsquare slightly irregular QAM constellation. (d) A phase-shift keyed

constellation. Also shown is the minimum euclidean distance dmin.

The minimum distance dmin is the minimum euclidean distance between any two signal points in the constellation

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Signal Constellations -cont.

The elements sj of the encoded sequence must come from the chosen signal constellation, but there may be constraints on the allowable sequence patterns to control or eliminate errors at the receiver The abstract representation of the datastream given in (4) must appear as a continuous waveform at the transmitter, and must then appear at the receiver as a corresponding continuous waveform that is sampled An obvious way to form a continuous waveform is to replace each impulse by the transmit pulse shape x(t)

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Pulse vs. Waveform Description of Channels

w(t) Datastream Pulse shaping x(t) s(t) Transmitted waveform Physical Channel Received waveform r(t) Detection filter Sequence of Samples Sampler Dirac Impulse Pulse shaping x(t) Transmitted pulse Physical Channel Received pulse Detection filter Filtered pulse q(t) p(t) δ(t) x(t)

(a) Pulse Description (b) Waveform Description

h(t) h(t) y(t) y(t) rk

Figure: Channel response: (a) to an impulse (b) to a datastream.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Target Pulses

For a modulated datastream, shown in Figure 2b, this replacement can be expressed mathematically as s(t) = w(t) ⊛ x(t) =

∞

j=−∞

sj x(t − jT). (5) The waveform at the transmitter s(t) in (5) is not the waveform that is eventually of interest

it is the waveform at the receiver from which the datastream must be recovered

For a linear system, let q(t) represent the composite of the transmitted pulse x(t), the natural dispersion h(t) in the physical channel, and the intentional filtering f(t) in the receiver shown in Figure 2 so that q(t) = x(t) ⊛ h(t) ⊛ f(t). (6) The desired impulse response q(t) at the location where the waveform is sampled is called the target pulse.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Rx Waveform Before Sampling

The noiseless electrical waveform r(t) used for sampling in the receiver can be written in terms of the target pulse as r(t) = w(t) ⊛ q(t) =

∞

j=−∞

sj q t − jT . (7) Figure 2 shows the generation of r(t) from q(t). The sampler at time kT will see only the desired sample sk if q(t) is a Nyquist pulse

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Nyquist Pulses

A sample of the received filtered waveform r(t) has no interference from

ther symbol intervals if the target pulse q(t) is a (scaled) Nyquist pulse

Working backwards from the receiver to the transmitter, using a Nyquist pulse as the target pulse q(t) implies that the Fourier transform of X(f)

f the transmitted pulse x(t) must be X(f) = Q(f)/H′(f)

H′(f) is the transfer function of the complete linear system given by h′(t) = h(t) ⊛ f(t) (cf. (6))

Note that the transmitted pulse x(t) and the transmitted waveform s(t) s(t) =

∞

j=−∞

sj x t − jT , (8) are not seen in the receiver where the waveform is sampled Only the corresponding waveform r(t) based on the target pulse q(t) is seen at the sampler Common practice to speak of the transmitted pulse x(t) as if it were a Nyquist pulse However, the requirement that the sample has no interference from other symbols means that the Nyquist property is actually required at the receiver.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Raised Cosine Spectra Nyquist Pulses

A sinc pulse is an often-mentioned Nyquist pulse because it leads to the smallest bandwidth of the interpolated baseband waveform However, interpolation using a sinc pulse is computationally intensive because sinc(t) decays slowly as 1/t

large number of terms must be summed to accurately produce the baseband waveform

To reduce the pulse spread and also reduce the instability, a Nyquist pulse with a more confined time duration is used, but at the cost of a larger bandwidth One class of Nyquist pulses with various timewidths is the set of pulses with raised cosine spectra given by q(t) = sin(πt) cos(βπt) (πt) 1 − (2βt)2, (9) where β is a parameter in the range [0, 1] that controls the temporal duration of the pulse. For large t and a nonzero value of β, the pulse q(t) eventually decays as t−3.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Raised Cosine Spectra Nyquist Pulses - cont.

The spectrum Q(f) is Q(f) =

  

1 for |f| ≤ 1 − β

1 2

1 − sin
π

2β (|f| − 1)

for

1 − β ≤ |f| ≤ 1 + β

therwise

. The pulse has a one-sided total bandwidth (1 + β)/2 in constrast to 1/2 for a sinc pulse.

3 2 1 3

Time

0.5 1

Response Frequency

1

Response

β1 β1/2 β0

1 2 1

2

1

2

1
1

Figure: Time and frequency plots of a raised-cosine Nyquist pulse as a function of β.

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Detection of a Binary Signal

Study the detection of a binary antipodal signaling waveform (eg. BPSK) with no intersymbol interference in additive white gaussian noise n(t) We will see with additive white gaussian noise with variance N0/2 = σ2, the optimal detector, known as a matched-filter detector, has a probability of error given by pe =

1 2erfc

Eb/N0
,

(10) where the average energy per bit Eb is defined as Eb . = p0E0 + p1E1, (11) with prior probabilities p0 and p1 satisfying p0 + p1 = 1 For antipodal signaling E0 = E1, so the average energy per bit Eb for any prior is equal to the average energy per pulse Ep with Eb = E0 = E1 = Ep This statement is not true for other modulation formats.

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Detection of a Binary Signal

2 4 6 8 10

5
4
3
2
1

Log10 pe Eb/N0 (dB)

Figure: Probability of a detection error for antipodal signaling in additive white gaussian noise using a matched filter.

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Figure of Merit

The term Eb/N0 or Ep/N0 in (10) is an appropriate parameter for matched-filter detection A corresponding parameter inside the erfc function for other situations will be expressed in a variety of other ways according to the situation Alternative notations are

2Eb

N0 = A σ =

γ

4 = dmin 2σ = Q, (12) where for binary phase-shift keying dmin = 2A is the euclidean distance between the two antipodal signal points ±A, and where γ = d2

min/σ2 is

the sample signal-to-noise ratio The term Q, here equal to √γ/2, is a summary notation used for the general case for the argument of the erfc function in the form Q/ √ 2 A modified form, γ, of the sample signal-to-noise ratio appropriate for signal-dependent noise is described later The expressions in (12) hold, as such, only for a matched filter For other detection filters, noise models, or signal constellations, the argument of the erfc function in (10) may still be expressed using suitably redefined versions of these terms.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Detection of a Binary Wave-Optics Signal

The noisy electrical waveform is passed through a detection filter f(t), and sampled In the simplest case, the noisy electrical waveform consists of the sum of the modulated signal and additive white gaussian noise n(t) The sample values rk for this noisy waveform are rk =

∞

j=−∞

sjq kT − jT + n′(kT), (13) where n′(t) is the filtered noise Will show that when the detection filter is matched to the received pulse p(t) such that the target pulse q(t) is a Nyquist pulse, (13) reduces to rk = sk + nk, (14) where the nk = n′(kT) are uncorrelated gaussian random variables and are thus independent Then each sk must be recovered from the corresponding noisy replica rk Derive the minimum the probability of symbol detection error pe.

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Hypothesis Testing

The real-number discrete-time memoryless channel has the real value r at the channel output given that the real value sℓ is transmitted at the channel input Described by the conditional probability density function f(r|sℓ), where ℓ indexes the L points of the signal constellation A binary modulation format has two transmitted values: s0 and s1 Hard-decision detection on each channel output uses the single real number r to decide between two hypotheses:

Hypothesis H0 is that s0 was transmitted Hypothesis H1 is that s1 was transmitted

A deterministic detection rule divides the set of real numbers R into two detection regions, R1 and R0 If r ∈ R1, then hypothesis H1 is chosen. If r ∈ R0, then hypothesis H0 is chosen.

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Decision Regions

Suppose that s0 is transmitted If r ∈ R0, we correctly decide in favor of H0, but if r ∈ R1, we incorrectly decide in favor of H1 The conditional probability p1|0 of this incorrect decision is given by the integration of the conditional probability density function f(r|0) over all possible values r ∈ R1 p1|0 =

R1

f(r|0)dr = 1 − p0|0 = 1 −

R0

f(r|0)dr, (15) where p0|0 is the conditional probability of a correct decision, and p0|0 + p1|0 = 1 If s1 is transmitted, then the conditional probability p0|1 of detecting s0 is p0|1 =

R0

f(r|1)dr = 1 − p1|1 = 1 −

R1

f(r|1)dr, (16) where f(r|1) is the conditional probability density function on r given that s1 is transmitted, and p1|1 is the conditional probability of a correct decision

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Maximum Single Symbol Posterior Detection

For binary modulation, the prior is given by p = p(s0, s1)

ne component being the prior probability of transmitting symbol s1

the second component being the prior probability of transmitting symbol s0, with p0 + p1 = 1

The total probability of a detection error pe is determined by weighting the conditional error probability by the prior probability with pe = p0p1|0 + p1p0|1 (17) = 1 − p0p0|0 − p1p1|1 (18) = 1 − pc, where pc . = p0p0|0 + p1p1|1 is the probability that the correct hypothesis is chosen.

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Choosing the Correct Region

Substituting the last expression for p1|0 given in (15) and the first expression for p0|1 given in (16) into (17) gives pe = p0 −

R0
p0f(r|0) − p1f(r|1)

dr. (19) To minimize the probability of a bit error, maximize the integral in (19) by the choice of R0. The maximum occurs if the region R0 includes every r for which p0f(r|0) > p1f(r|1), and the region R1 includes every r for which p0f(r|0) < p1f(r|1) The condition defining R0 can be written as p0f(r|0) p1f(r|1) > 1 (20) Values of r satisfying the opposite inequality are placed in R1

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Posterior Probability Ratio

Using the Bayes rule given in (cf. Chapter 2) , the numerator and denominator of (20) are, respectively

p0f(r|0) = p(r)f(0|r) (21) p1f(r|1) = p(r)f(1|r). (22)

The conditional probability density function f(sℓ|r) on the transmitted value sℓ, given that the value r is received, is called the posterior probability distribution Substituting (22) into (20) and canceling the common term p(r), define u(r) as the ratio of the posterior probability density functions on the left side of (20)

u(r) . = p0f(r|0) p1f(r|1) (23) = f(0, r) f(1, r) , (24)

where f(sℓ, r) = f(sℓ|r)p(r) is the joint probability density function The optimal detection rule in terms of u(r) is then

choose H0 if u(r) ≥ 1 choose H1 if u(r) < 1, (25)

where H0 is the assertion “s0 was transmitted” and H1 is the assertion “s1 was transmitted”.

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Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Pulses Detection

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Decision Regions

R0 R1

f(r|0) f(r|1) f(r|0)=f(r|1) u(r)>0 u(r)<0

Figure: The decision regions R0 and R1 for binary hypothesis testing for an equiprobable prior.

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