Introduction to Analog And Digital Communications

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Introduction to Analog And Digital Communications

• Second Edition
• Simon Haykin, Michael Moher

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Chapter 10 Noise in Digital Communications

• 10.1 Bit Error Rate
• 10.2 Detection of a Single Pulse in Noise
• 10.3 Optimum Detection of Binary PAM in Noise
• 10.4 Optimum Detection of BPSK
• 10.5 detection of QPSK and QAM in Noise
• 10.6 Optimum Detection of Binary FSK
• 10.7 Differential Detection in Noise
• 10.8 Summary of Digital Performance
• 10.9 Error Detection and Correction
• 10.10 Summary and Discussion

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• Two strong external reasons for the increased
  dominance of digital communication
• The rapid growth of machine-to-machine
  communications.
• Digital communications gave a greater noise
  tolerance than analog
• Broadly speaking, the purpose of detection is to
  establish the presence of an information-bearing
  signal in noise.

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• Lesson 1 the bit error rate is the primary
  measure of performance quality of digital
  communication systems, and it is typically a
  nonlinear function of the signal-to-noise ratio.
• Lesson 2 Analysis of single-pulse detection
  permits a simple derivation of the principle of
  matched filtering. Matched filtering may be
  applied to the optimum detection of many linear
  digital modulation schemes.
• Lesson 3 The bit error rate performance of
  binary pulse-amplitude modulation (PAM) improves
  exponentially with the signal-to-noise ratio in
  additive white Gaussian noise.
• Lesson 4 Receivers for binary and quaternary
  band-pass linear modulation schemes are
  straightforward to develop from matched-filter
  principles and their performance is similar to
  binary PAM.
• Lesson 5 Non-coherent detection of digital
  signals results in a simpler receiver structure
  but at the expense of a degradation in bit error
  rate performance.
• Lesson 6 The provision of redundancy in the
  transmitted signal through the addition of
  parity-check bits may be used for forward-error
  correction. Forward-error correction provides a
  powerful method to improve the performance of
  digital modulation schemes.

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10.1 Bit Error Rate

• Average bit error rate (BER)
• Let n denote the number of bit errors observed
  in a sequence of bits of length N then the
  relative frequency definition of BER is
• Packet error rate (PER)
• For vocoded speech, a BER of to is often
  considered sufficient.
• For data transmission over wireless channels, a
  bit error rate of to is often the
  objective.
• For video transmission, a BER of to is often the
  objective, depending upon the quality desired and
  the encoding method.
• For financial data, a BER of or better is
  often the requirement.

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• SNR
• The ratio of the modulated energy per information
  bit to the one-sided noise spectral density
  namely,
• The analog definition was a ratio of powers. The
  digital definition is a ratio of energies, since
  the units of noise spectral density are watts/Hz,
  which is equivalent to energy. Consequently, the
  digital definition is dimensionless, as is the
  analog definition.
• The definition uses the one-sided noise spectral
  density that is, it assumes all of the noise
  occurs on positive frequencies. This assumption
  is simply a matter of convenience.
• The reference SNR is independent of transmission
  rate. Since it is a ratio of energies, it has
  essentially been normalized by the bit rate.

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10.2 Detection of a Single Pulse in Noise

• The received signal consists solely
  of white Gaussian noise
• The received signal consists of
  plus a signal of known form.
• For the single-pulse transmission scheme
• The received signal
• The first integral on the right-hand side of Eq.
  (10.5) is the signal term, which will be zero if
  the pulse is absent, and the second integral is
  the noise term which is always there.

Fig. 10.1
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Fig. 10.1
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• The variance of the output noise

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• The noise sample at the output of the linear
  receiver has
• A mean of zero.
• A variance of
• A Gaussian distribution, since a filtered a
  Gussian process is also Gaussian (see Section
  8.9).
• The signal component of Eq.(10.5)
• Schwarz inequality for integrals is

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• With defined by Eq.(10.1), the signal
  component of this correlation is maximized at
  This emphasizes the importance of
  synchronization when performing optimum detection.

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10.3 Optimum Detection of Binary PAM in Noise

• Consider binary PAM transmission with on-off
  signaling
• The output of the matched filter receiver at the
  end of the kth symbol interval is
• Since the matched filter wherever it is
  nonzero, we have

Fig. 10.2
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Fig. 10.2
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• BER performance
• Consider the probability of making an error with
  this decision rule based on conditional
  probabilities. If a 1 is transmitted, the
  probability of an error is

Fig. 10.3
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Fig. 10.3
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• Type II error.
• An error can also occur if a 0 is transmitted and
  a 1 is detected
• The probability regions associated with Type I
  and Type II errors are illustrated in Fig. 10.5.
  The combined probability of error is given by
  Bayes rule (see Section 8.1)

Fig. 10.4
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Fig. 10.4
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• A priori probability
• The average probability of error is given
• Average probability of error

Fig. 10.5
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Fig. 10.5
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• Our next step is to express this probability of
  bit error in terms of the digital reference
  model.
• To express the variance in terms of the noise
  spectral density, we have, from Eq.(10.9), that
• To express the signal amplitude A in terms of the
  energy per bit , we assume that 0 and 1 are
  equally likely to be transmitted. Then the
  average energy per bit at the receiver input is

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• Nonrectangular pulse shapes
• The received signal
• P(t) is a normalized root-raised cosine pulse
  shape

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• Applying the matched filter for the kth symbol
  of to Eq.(10.31), we get
• Under these conditions, and the BER
  performance es the same as with rectangular pulse
  shaping

Fig. 10.6
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Fig. 10.6
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10.4 Optimum Detection of BPSK

• One of the simplest forms of digital band-pass
  connunications is binary phase-shift keying. With
  BPSK, the transmitted signal is

Fig. 10.7
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Fig. 10.7
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• Detection of BPSK in noise
• The signal plus band-pass noise at the input to
  the coherent BPSK detector
• The output of the product modulator
• The matched filter

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• The output of the integrate-and-dump detector in
  this
• The noise term in Eq.(10.40) is given by

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• In digital communications, the objective is to
  recover the information, 0s and1s, as possible.
  Unlike analog communications, there is no
  requirement that the transmitted waveform should
  be recovered be recovered with minimum
  distortion.
• Performance analysis
• If we assume a 1 was transmitted and then the
  probability of error is
• The bit error rate

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• The analysis of BPSK can be extended to
  nonrectangular pulse shaping in a manner similar
  to what occurred at baseband. For nonrectangular
  pulse shaping, we represent the transmitted
  signal as

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10.5 detection of QPSK and QAM in Noise

• Detection of QPSK in niose
• That QPSKmodulated signal

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• Using in-phase and quadrature representation for
  the band-pass noise, we find that the QPSK input
  to the coherent detector of Fig.10.8 is described
  by
• The intermediate output of the upper branch of
  Fig.10.8 is
• The output of the lower branch of the quadrature
  detector is

Fig. 10.8
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Fig. 10.8
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• For the in-phase component, then the mean output
  is
• The first bit of the dibit is a 1 then
• The probability of error on the in-phase branch
  of the QPSK signal is

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• The average energy by bit may be determined from
• The bit error rate with after matched filtering
  is given by

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• In terms of energy per bit, the QPSK performance
  is exactly the same as BPSK
• Double-sideband and single-sideband transmission,
  we found that we could obtain the same quality of
  performance but with half of the transmission
  bandwidth. With QPSK modulation, we use the same
  transmission bandwidth as BPSK but transmit twice
  as many bits with the same reliability.
• Offset-QPSK or OQPSK is a variant of QPSK
  modulation
• The OQPSK and QPSK, the bit error rate
  performance of both schemes is identical if the
  transmission path does not distort the single.
• One advantage of OQPSK is its reduced phase
  variations and potentially less distortion if the
  transmission path includes nonlinear components
  such as an amplifier operating near or at
  saturation. Under such nonlinear conditions,
  OQPSK may perform better than QPSK.

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• Detection of QAM in noise
• The baseband modulated signal be represented by

Fig. 10.9
Fig. 10.10
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Fig. 10.9
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Fig. 10.10
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• There are two important differences that should
  be noted
• Equation (10.57) represents a symbol error rate.
• With binary transmission with levels of A and
  A, the average transmitted power is
• The probability of symbol error in terms of the
  digital reference SBR is
• We may use independent PAM schemes on the
  in-phase and quadrature components. That is to
  say, one PAM signal modulates the
  in-phase carrier and the second
  PAM signal modulates the quadrature carrier
• Due to the orthogonality of the in-phase and
  quadrature components, the error rate is the same
  on both and the same as the baseband PAM system.

Fig. 10.11
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Fig. 10.11
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10.6 Optimum Detection of Binary FSK

• The transmitted signal for is
• The two matched filters are
• The corresponding waveforms are orthogonal

Fig. 10.12
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Fig. 10.12
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• Let the received signal be
• Then the output of the matched filter
  corresponding to a 0 is
• The noise component at the output of the matched
  filter for a 0 is

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• The output of the filter matched to a 1 is
• The noise component of Eq.(10.65)
• By analogy with bipolar PAM in Section 10.3, the
  probability of error for binary FSK is

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• The BER in terms of the digital reference model
  is
• In general, we see that antipodal signaling
  provides a dB advantage over orthogonal
  signaling in bit error rate performance.

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10.7 Differential Detection in Noise

• band-pass signal at the output of this filter
• The output of the delay-and-multiply circuit is

Fig. 10.13
Fig. 10.14
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Fig. 10.13
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Fig. 10.14
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• Since in a DPSK receiver decisions are made on
  the basis of the signal received in two
  successive bit intervals, there is a tendency for
  bit errors to occur in pairs.

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10.8 Summary of Digital Performance

• Gray encoding
• Gray encoding of symbols where there is only a
  one-bit difference between adjacent symbols.
• Performance comparison

Fig. 10.15
Table. 10.1
Fig. 10.16
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Fig. 10.15
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Table. 10.1
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Fig. 10.16
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• Noise in signal-space models
• QPSK may be represented
• The basis functions are

Fig. 10.17
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Fig. 10.17
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• The remaining two orthonormal functions are given
  by
• Under the band-pass assumption, the functions
  and are clearly orthogonal with each other
  and, since they are zero whenever and
  are nonzero, they are also orthogonal to these
  functions.

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10.9 Error Detection and Correction

• In radio (wireless) channels, the received signal
  strength may vary with time due to fading.
• In satellite applications, the satellite has
  limited transmitter power.
• In some cable transmission systems, cables may be
  bundled together so closely that there may be
  crosstalk between the wires.
• We can achieve the goal of improved bit error
  rate performance by adding some redundancy into
  the transmitted sequence. The purpose of this
  redundancy is to allow the receiver to detect
  and/or correct errors that are introduced during
  transmission.
• Forward-error correction (FEC)
• The incoming digital message (information bits)
  are encoded to produce the channel bits. The
  channel bits include the information bits,
  possibly in a modified form, plus additional bits
  that are used for error correction.
• The channel bits are modulated and transmitted
  over the channel.
• The received signal plus noise is demodulated to
  produce the estimated channel bits.
• The estimated channel bits are then decoded to
  provide an estimate of the original digital
  message.

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• Let bits be represented by 0 and 1 values.
• Let represent mudulo-2 addition. When this
  operation is applied to pairs of bits we obtain
  the results and
• The operator may also be applied to blocks
  of bits, where it means the modulo-2 sum of the
  respective elements of the blocks. For example,
• Suppose we add the parity-check bit p such that
• The error detection capability of a code as the
  maximum number of errors that the receiver can
  always detect in the transmitted code word.

Fig. 10.18
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Fig. 10.18
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• Error detection with block codes
• Hamming weight of a binary block as the number of
  1s in the block.
• Hamming distance between any two binary blocks is
  the number of places in which they differ.
• The single parity check code can always detect a
  single bit error in the received binary block.

Table. 10.2
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Table. 10.2
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• Error correction
• The objective when designing a forward
  error-correction code is to add parity bits to
  increase the minimum Hamming distance, as this
  improves both the error detection and
  error-correction capabilities of the code.
• Hamming codes
• Hamming coeds are a family of codes with block
  lengths
• There are parity bits and information bits.

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• The first step
• The encoding that occurs at the transmitter this
  requires the calculation of the parity bits based
  on the information bits. The second step is the
  decoding that occurs at the receiver this
  requires the evaluation of the parity checksums
  to determine if, and where, the parity equations
  have been violated.
• K-by-n generator matrix

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• The elements of C are given by
• Hamming code we may define the (n - k)-by-n
  parity check matrix H as

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• We compute the product of the received vector and
  the parity-check matrix
• It satisfies the parity-check equations and
• More powerful codes
• Reed-Solomon and Bose-Chaudhuri-Hocquenghem (BCH)
  block codes.
• These are (n, k) codes where there are k
  information bits and a total of n bits including
  n - k parity bits.
• Convolutional codes.
• the result of the convolution of one or more
  parity-check equations with the information bits.
• Turbo codes
• Block codes but use a continuous encoding
  strategy similar to convolutional codes.

Table. 10.4
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Table. 10.4
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Fig. 10.19
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Fig. 10.19
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• Signal-space interpretation of forward-error-corre
  ction codes
• We could construct a four-dimensional vector with
  a signal-space representation of
• We can explain some of the concepts of coding
  theory geometrically as shown in Fig.10.20.
• For a linear code we defined the minimum Hamming
  distance as the number of locations where the
  binary code words differ.
• Euclidean distance

Fig. 10.20
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Fig. 10.20
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10.10 Summary and Discussion

• Analysis of the detection of a single pulse in
  noise shows the optimality of matched filtering.
  We showed that the bit error rate performance
  using matched filtering was closely related to
  the Q-function that was defined in Chapter 8.
• We showed how the principle of matched filtering
  may be extended to the detection of
  pulse-amplitude modulation, and that bit error
  rate performance may be determined in a manner
  similar to single-pulse detection.
• We discussed how the receiver structure for the
  coherent detection of band-pass modulation
  strategies such as BPSK, QPSK, and QAM was
  similar to coherent detection of corresponding
  analog signals.

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• We showed how quadrature modulation schemes such
  as QPSK and QAM provide the same performance as
  their one-dimensional counterparts of BPSK and
  PAM, due to the orthogonality of the in-phase and
  quadrature components of the modulated signals.
• Antipodal strategies such as BPSK are more power
  efficient than orthogonal transmission strategies
  such as on-off signaling and FSK.
• We introduced the concept of non-coherent
  detection when we illustrated that BPSK could be
  detected using a new approach where the
  transmitted bits are differentially encoded. The
  simplicity of this detection technique results in
  a small BER performance penalty.