5.4.1. Binary NRZ code (non-return to zero): two-level code, two types of code

5.4.1.1. NRZ-L code "level code"

The NRZ-L code is defined by:

[5.14]

and:

[5.15]

With a_n = -1, the emitted signal is symmetrical; if a_n = 0, the emitted signal is non-symmetrical.

The diagram of the encoder and decoder of the symmetrical NRZ-L code is given in Figure 5.2.

In Figure 5.3, we present an example of a chronogram of the symmetrical NRZ-L code.

Since the symbols a_n are iid (independent and identical distributed), its mean is zero and its variance σ²_a is equal to 1.

The power spectral density of code NRZ-L is given by the following relationships:

[5.16]

Figure 5.4 displays the power spectral density of the symmetrical NRZ-L on-line code. It consists of a main lobe which occupies the frequency band [0, 1/T_b] (for positive frequencies), and by an infinity of secondary lobes whose amplitude tends towards zero as the frequency increases. In addition, the PSD of this code is canceled at frequencies f = (k/T) with k non-zero integer.

5.4.1.2. NRZ-M code “mark” or differential

The NRZ-M code is defined by the relation [5.17] and Table 5.1:

[5.17]

bn	an-1	an
0	0	0
0	1	1
1	0	1
1	1	0

The waveform x(t) and the power spectral density r_NRZ-_M(f) are identical to that of the symmetrical NRZ-L code.

In Figure 5.5, we give the block diagram of the on-line NRZ-M coder and decoder and an example of a chronogram of the symmetrical NRZ-M code is shown in Figure 5.6.

5.4.2. NRZ M-ary code

The NRZ M-ary code is a generalization of the binary NRZ code. Each set of k bits from the binary sequence is associated with a symmetrical M-ary symbol:

[5.18]

The waveform x(t) is a “gate function” of amplitude V but of time duration T = kT_b.

In the case where the symbols a_n are iid, then their mean is zero and their variance is equal to:

[5.19]

The power spectral density of the NRZ M-ary on-line code is:

[5.20]

Figure 5.7 shows an example of a chronogram of the NRZ 4-ary code with T = 2T_b, a_n = {± 1, ±3} and Figure 5.8 displays its power spectral density

5.4.3. Binary RZ code (return to zero)

This refers to a two-level on-line code such as:

[5.21]

and:

[5.22]

with 0 < θ <1, in practice: θ — 1/2.

The binary RZ encoder corresponds to a multiplication operator between the binary sequence and the clock, as shown in Figure 5.9.

An example of a chronogram (for θ — 1/2) of this code is given in Figure 5.10.

The symbols a_n being independent and equiprobable, their mean is equal to 1/2 and their variance — 1/4. The power spectral density Γ_RZ(f) is given by the following relation:

[5.23]

It consists of a continuous spectrum and of a discrete spectrum (discrete frequency components).

In general, one has θ = 1/2, hence:

[5.24]

As the relation [5.24] and Figure 5.11 show, the spectral occupancy of this on-line code is double that of the NRZ code and half of the power is located in the discrete components at frequencies (2k + 1)/T_b. We note in particular a discrete component at the frequency 1 /T_b, which does not have the NRZ code, which allows the recovery of the transmission rhythm (clock) at reception.

For θ = 1, the code is similar to the NRZ code but is asymmetrical, which results in the existence of a discrete component at f = 0 (so a non-null continuous component).

5.4.4. Polar RZ on-line code

This refers to a 3-level on-line code:

[5.25]

and:

[5.26]

The block diagram of the encoder is given in Figure 5.12 for θ = 1/2.

An example of a chronogram for this code is given in Figure 5.13 for θ = 1/2.

The power spectral density of this code is:

[5.27]

The spectrum has no discrete components and its continuous part is identical, except to a factor, to that of the RZ codes.

For θ = 1/2, the PSD is given by the relation [5.28]:

[5.28]

It is displayed in Figure 5.14.

5.4.5. Binary biphase on-line code (Manchester code)

This is a two-level on-line code obtained by transmitting two opposite polarities during the time interval T_b corresponding to a binary symbol, each of these opposite polarities occupying a duration equal to T_b/2.

It is defined by:

[5.29]

and:

[5.30]

This code always has transitions (+V,-V or vice versa) in the middle of each interval of time duration T_b, regardless of the value transmitted (0 or 1). In addition, for the bit “0” there is a transition at the start of the interval. This is not the case for the bit “1”. The transitions are intended to limit the power in the vicinity of frequency zero. They make it possible to recover the clock at reception.

Figure 5.15 displays the block diagram of the binary biphase coder and decoder and Figure 5.16 shows an example of a chronogram of the biphase on-line code.

Since the symbols a_n are independent and equiprobable, of zero mean and of unit variance, the power spectral density of the code is:

[5.31]

This is displayed in Figure 5.17. It vanishes at zero frequency and remains relatively low for frequencies less than 1/4T_b. Therefore, this on-line code can be used for relatively long distance cable transmission.

However, the spectral occupancy of the code is double that of the NRZ code.

5.4.6. Binary biphase mark or differential code (Manchester mark code)

The crossing of the transmission wires for a biphase code is equivalent for the receiver to an inversion of polarity and therefore to the inversion of the value of the binary symbols.

To avoid having to locate the line wires, it is preferred to use its differential variant: the differential biphase binary code. In addition, the latter offers better immunity against noise.

Note that all Ethernet systems use this type of on-line code, mainly in the MAC channel access control layer.

The differential biphase code uses the same symbols as the biphase code, but is coded differentially:

[5.32]

and:

[5.33]

The block diagram of the coder and the decoder is given in Figure 5.18 and an example of a chronogram of this code is given in Figure 5.19.

The power spectral density of the code is identical to that of the biphase code, so its spectral occupancy is therefore twice that of the NRZ code.

5.5.1. Miller’s code

The spectral occupancy of this code derived from the biphase code is relatively reduced compared to the previous codes. Miller’s code is obtained by “dividing by two” the biphase coded signal. The state change of the Miller code is done on rising edges of the biphase code (from the example of the chronogram of Figure 5.21), but it can also be done on falling edges of the biphase signal.

Decoding consists of detecting the presence or absence of transitions in the middle of the symbol interval. The presence of transition corresponds to a state of the symbol. The absence of transition indicates the opposite symbol state.

The encoder block diagram is given in Figure 5.20 and an example of a chronogram for this code is given in Figure 5.21.

The power spectral density of this on-line code is:

[5.34]

It is given in Figure 5.22. It is characterized by a high concentration of power in the vicinity of the frequency 2/5T_b. The power spectral density (which is here a continuous function) has a low value, but not zero at zero frequency, and the tolerance of the signal to the noise is reduced. However, Miller’s code allows the transmission of bitrates up to 5,000 bit/s over telephone cable.

5.5.2. Bipolar RZ code or AMI code (alternate marked inversion)

This is a three-level on-line code such as:

[5.35]

The correlation of the symbols a_n is carried out by alternately assigning the values +1 and -1 to the symbols a_n when the binary symbol b_n = 1.

The mean value of the symbols a_n is zero and their variance is equal to 1/2. The signal x(t) is defined by:

[5.36]

Or by:

[5.37]

to obtain the NRZ bipolar on-line code.

In Figure 5.23, we show the block diagram of the encoder and decoder of the bipolar RZ code and in Figure 5.24 we present an example of a chronogram of this code.

The power spectral density of the bipolar RZ code is:

[5.38]

Equation [5.38] is shown in Figure 5.25. We notice that the PSD is zero at zero frequency and does not have a discrete component at zero frequency (advantage for remote power supply). The spectral occupancy of the code is practically 1/T_b.

Furthermore, by performing a full-wave rectification of the bipolar RZ code, a binary RZ code is obtained which has a discrete component at frequency 1/T_b in its power spectral density and, consequently, an easy recovery of the clock at reception.

This code can be used on cables, but since the coding of the binary symbol “0” corresponds to a null signal emitted, this makes it difficult to recover the clock when sending long sequences of “0”, because in this case the amplitude of the frequency component at 1/T_b decreases quite strongly and the power of the signal collected after filtering may become insufficient to ensure the recovery of the clock. This has led to the development of on-line codes which have the advantages of the bipolar code, and in addition the advantage of allowing a correct recovery of the clock at reception, whatever the sequence of bits sent by the information source. HDB-n on-line codes are good examples of this type of on-line code (n-order high density bipolar code).

5.5.3. CMI code (code mark inversion)

This refers to a two-level code that has the same coding rule as the bipolar code but with different pulse forms:

[5.39]

and:

[5.40]

An example of a CMI code chronogram is given in Figure 5.26.

The power spectral density of this on-line code is:

[5.41]

This is shown in Figure 5.27.

It is low at very low frequencies and has a discrete component at the frequency 1/T_b, so the clock recovery in reception is easy.

The only drawback is its higher spectral occupancy than that of the RZ bipolar code.

The CMI code has been proposed for the interface of 140 Mbit/s digital systems. It is also used in fiber optic transmissions.

5.5.4. HDB-n code (high density bipolar on-line code of order n)

It is derived from the bipolar code in which any sequence of n + 1 consecutive zeros is replaced by a sequence of at most n consecutive zeros. For this, the (n + 1)^th binary element of the n + 1 consecutive zeros sequence is coded by a symbol “7” (violation symbol) equal to ±1, the sign being chosen in such a way that it violates the rule of alternation. Also, in order to avoid the generation of a non-zero average, the violation symbols must themselves respect the rule of alternation between them.

However, it may be that the receiver cannot recognize a violation symbol as such (if it has a sign contrary to the previous 1); in this case, we code the first 0 of the sequence of n + 1 consecutive zeros with a symbol: a_n = ±1, of the same sign as the violation symbol which follows it. This symbol is called “stuffing symbol B".

IN SUMMARY.- A sequence “n + 1" zeros, 0 0..........0 0, is replaced by:

0 0..........0 V or by B 0.........0 V

with:

• – V: a symbol that violates the alternation rule of polarity, but satisfies the alternation rule of violation symbols between them;
• – B: a stuffing symbol that follows the alternation rule of polarity.

We give in Figure 5.28 an example of a chronogram of the code HDB-3.

By doing so, the symbols always have a zero average. The analytical calculation of the power spectral density of the code HDB-n Γ_HDB__n(f), is complicated. It is comparable to the power spectral density of the bipolar RZ code. Figure 5.29 shows the PSD of the HDB-3 code which is used in the DT1 and DT2 digital transmission systems.

In order to easily compare the spectral properties of the different on-line codes studied above, we have represented their power spectral density in Figure 5.30.

5.6.1. Generation and interest of precoding

Any partial response linear code can be generated from the sequence b_n by means of four successive operations given in Figure 5.31.

where:

[5.42]

NOTE.– For convenience and simplicity, we have limited the alphabet of a_n to two symbols. We could quite easily consider an alphabet with M = 2^k symbols.

• – Precoding: the pre-coding operation makes the random sequence b_n correspond to a random sequence b^'_n of the same bitrate and the same number of levels having certain advantages for decoding. Insofar as the symbols contained in b_n are equiprobable and independent, the same goes for the sequence b^'_n.
• – Transcoding: the transcoding operation generates from the random sequence b^'_n a random sequence a_n of the same bitrate having a zero mean (symmetrical amplitude of the symbols with respect to zero).
• – Coding: the partial response linear encoding introduces a controlled inter-symbol dependency, which increases the number of signal levels and changes its spectrum. The encoding is similar to digital filtering. The encoder filter is characterized by its transfer function H(z).
• – Waveform shaping: waveform shaping is a filtering operation defined by an impulse response x(t) that determines the signal carrying the symbol of the transmitted signal s_e(t):

[5.43]

• – Interest of precoding: the advantage of precoding appears in the decoding procedure. Indeed, if the precoder has a transfer function P(z) which is the reverse of the transfer function H(z) modulo 2 of the coder, we will then have the block diagram of Figure 5.32.

Figure 5.32 shows that one goes from sequence b_n to sequence b^'_n by the Z-transform filter: P(z). Then conversely, one goes from sequence b^'_n to sequence b_n by the Z-transform filter: H(z).

so one has:

[5.44]

and:

[5.45]

The coefficients h_k are positive, negative or null integers, and h₀ = 1. The FIR (finite impulse response filter) filtering of the encoder corresponds to algebraic operations. The preceding relationships show that on reception and in the absence of transmission noise, we can decode:

[5.46]

without needing to use the previous b_n–k.

Otherwise, in the absence of precoding, (b'_n — b_n), we would have:

[5.47]

and decoding would be given by the relation:

[5.48]

Therefore, the estimation of b_n would require the estimation of the previous b_n–k. In absolute terms, this would not be a problem, since they have already been estimated before b_n. However, if they contain errors, then these will propagate.

5.6.2. Structure of the coder and precoder

5.6.2.1. Coder

The coder is defined by the following relation:

[5.49]

The Z-transform of c_n is:

[5.50]

Hence, the transfer function of the filter is:

[5.51]

The coder filter is a FIR filter.

In the literature, the coder is often designated by a polynomial in D (delay operator, equivalent to the variable z^–1 of the digital filtering).

The frequency response of the encoder filter is:

[5.52]

H(f) is a periodic function of period 1/T_b (T_b is the clock period).

The general structure of the encoder is given by relation [5.49] and presented in Figure 5.33.

5.6.2.2. Precoder

The precoder is a recursive filter whose transfer function is:

[5.53]

One can then write:

[5.54]

and:

[5.55]

or:

[5.56]

The structure of the precoder is generally given by relation [5.55], and presented in Figure 5.34. But as the alphabet of b_n and b'_n is {0,1}, then, in this case, the general structure of the precoder is also given by relation [5.56].

5.6.2.3. Combined precoder and encoder

By comparing the block diagrams of Figures 5.33 and 5.34, it can be seen that the delay lines of the precoder and encoder can be shared according to the scheme of Figure 5.35.

5.6.3. Power spectral density of partial response linear on-line code

To calculate the power spectral density of partial response linear on-line codes, we use the same approach described in section 5.3, by performing this block-by-block calculation according to the scheme of Figure 5.31, but by considering random signals instead of sequences, as shown in the block diagram of Figure 5.36.

with:

[5.57]

and δ(t) is the Dirac distribution (sometimes referred to as the “Dirac pulse”).

The random signals b(t), b'(t), a(t), c(t), s_e(t) are considered cyclo-stationary and have the power spectral densities: Γ'_b(f), r_b(f), r_a(f), r_c(f), r_Se(f) respectively.

In this case, the autocorrelation function R_b(r) of the random sequence of distributions b'(t) (b_n are assumed independent and equiprobable: iid) is given by:

[5.58]

And its power spectral density Γ_b(f) is given by:

The power spectral density Γ_b(f) of the random sequence of distributions b (t) is identical to that of b (t):

[5.60]

So, the precoding has no influence on the spectral properties of the code.

The autocorrelation function R_a (t) of the random sequence of distributions a(t) is given by:

[5.61]

In fact, the general relationship of R_a(t) with a_ne{—A,A} is:

[5.62]

The power spectral density Γ_a(f) of the random sequence of distributions a(t) is:

[5.63]

The power spectral density Γ_c(f) of the random sequence of distributions c(t) is:

[5.64]

And finally, the power spectral density of the signal emitted s_e(t) is given by:

[5.65]

This relation shows, on the one hand, that it is possible to shape the spectrum of the signal s_e(t) by modifying H(f) and, on the other hand, that the calculation of the spectrum is easy because it depends on the calculation of H(f) and X(f) separately.

5.6.4. Most common partial response linear on-line codes

The most common partial response linear on-line codes, such as duobinary, bipolar NRZ, interlaced bipolar of order 2 and biphase (WAL1 and WAL2), are described hereinafter following the approach used by Michel Stein (Stein 1991). The waveform shaping used is of the NRZ-L type.

5.6.4.1. Duobinary code

The duobinary code is characterized by:

K = 2 ; h_i = 1

In which case, one has:

[5.66]

hence:

[5.67]

The encoder block diagram is given in Figure 5.37.

The precoder is then defined by the following relation and the block diagram of Figure 5.38:

[5.68]

The combined structure of the precoder, transcoder and duobinary encoder followed by the shaping function is shown in Figure 5.39.

The graph of an example of a coded sequence is displayed in Figure 5.40.

Note that mod 2 and the sample s_e(nT) of rank n makes it possible to determine ĉ_n and then b_n (see Chapter 6).

The spectral properties of the duobinary code are deduced from the transfer function of the encoder:

[5.69]

hence:

[5.70]

[5.71]

If ±V are the peak values of s_e (t), and the pulse x(t) has an amplitude V/2 on [0, T_b [, then:

[5.72]

and from [5.65], the power spectral density is:

[5.73]

or:

[5.74]

It is displayed in Figure 5.41.

The power spectral density has a reduced bandwidth equal to 1/2T_b Hz (if we consider the positive frequencies only).

However, with the same peak amplitude, the change from two to three levels increases the sensitivity to noise.

5.6.4.2. NRZ bipolar code

The bipolar code, the principle of which has been explained above, is seen here as a partial response linear code whose coding law is defined by:

K = 2 ; h₁ = -1

In which case, one has:

[5.75]

hence:

[5.76]

The coder scheme is given in Figure 5.42.

The precoder is identical to that of the duobinary code, and the set of precoder, transcoder, encoder and waveform shaping is shown in Figure 5.43.

An example of a coded sequence is shown in Figure 5.44.

The transfer function of the encoder is:

[5.77]

And its power spectral density is calculated as follows:

[5.78]

[5.79]

hence:

Finally, one has:

[5.80]

This is displayed in Figure 5.45.

5.6.4.3. Interleaved order 2 bipolar code

The interleaved bipolar coder of order 2 can be obtained by serializing duobinary and bipolar coders. The structure of the 2nd order interleaved bipolar coding chain is then represented in Figure 5.46, where H₁(z) and H₂(z) respectively represent the transfer functions of the duobinary and bipolar coders.

The transfer function and the impulse response of the encoder are written:

[5.81]

[5.82]

And the output of the encoder is expressed by:

[5.83]

The encoder block diagram is given in Figure 5.47. It is similar to that of the bipolar coder of Figure 5.42 by replacing the delay T_b by 2T_b.

The precoder is defined by the following relation:

[5.84]

Its implementation scheme is shown in Figure 5.48.

The whole chain of two-order interleaved bipolar coding is shown in Figure 5.49.

An example of a coded sequence is displayed in Figure 5.50.

One can easily verify that the coding method illustrated in Figure 5.50 is equivalent to separately carrying out a bipolar coding on the sequence of the even bits {b_2n} and on the sequence of odd bits {b_2n+1} of the message to be transmitted {b_n}. Then, the two coded sequences are interleaved so that the symbol corresponding to the bit of rank n occupies the same rank n in the coded signal.

The frequency response of the encoder is:

[5.85]

[5.86]

[5.87]

This is displayed in Figure 5.51.

5.6.4.4. Biphase codes

The spectral occupancy of the interleaved bipolar code of order 2 is half that of the simple bipolar code.

These codes are characterized by the inversion or not of a periodic carrier signal as a function of the binary element to be transmitted. The most used biphase code is that described previously and designated here by the term “WAL1”. Another variant, called “WAL2” can be used in which the carrier signal has a phase shifted by n/2 with respect to the series of bits. Figure 5.52 shows the two basic possible forms of signals in each variant.

The precoding operation for these codes is replaced by the identity transformation, therefore:

[5.88]

5.6.4.5. Biphase code WAL1

Its transfer function is given by:

[5.89]

and H(f) is given by:

[5.90]

then:

[5.91]

The shaping filter has a rectangular impulse response of width T_b/2. Its transfer function is therefore:

[5.92]

Hence, its power spectral density Γ_s (f) is given by:

[5.93]

The differential biphase code described above is obtained by putting a precoder defined by the following equation upstream of the coder:

[5.94]

5.6.4.6. Biphase code WAL2

The coding and shaping functions are:

[5.95]

[5.96]

[5.97]

and:

[5.98]

So one has:

[5.99]

The same precoder as for the WAL1 code allows differential coding.

The power spectral densities of the two-phase codes WAL1 and WAL2 are displayed in Figure 5.53.

We note that the PSD of code WAL2 has very little energy at low frequencies and this part of the spectrum remains available for useful purposes (adding a voice channel, for example).

Finally, we have also displayed, in Figure 5.54, the power spectral density of the different partial response linear codes analyzed above, in order to allow an easy comparison of the power spectral density properties of these codes.