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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

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.

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.

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.

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.

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

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

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.

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

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

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

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

Nonrectangular pulse shapes

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

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

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

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

The output of the integrate-and-dump detector in this

- The noise term in Eq.(10.40) is given by

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

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

10.5 detection of QPSK and QAM in Noise

- Detection of QPSK in niose
- That QPSK–modulated signal

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

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

The average energy by bit may be determined from

- The bit error rate with after matched filtering is given by

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.

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

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

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

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

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.

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

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.

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

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.

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.

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

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

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.

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

The elements of C are given by

- Hamming code we may define the (n - k)-by-n parity check matrix H as

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

Signal-space interpretation of forward-error-correction 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

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.

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.

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