About inter-code crosstalk (reprint)

Based on the discussion of crosstalk between codes in the previous section, we can summarize the requirements for the impulse response H (t) of the baseband transmission system as follows: (1) The baseband signal after transmission at the sampling point of the cross-talk, that is, the instantaneous sampling value should meet

(4-18) (2) The tail attenuation should be fast.

(4-18) The cross-talk condition given by the uncensored is based on the sampling judgment at the time of the first code element. is a time delay constant, in order to analyze the simplicity, suppose, so that the condition of the uncensored crosstalk becomes

Order, and taking into account also an integer, can be expressed in the condition that the crosstalk between the uncensored is (4-19)

Formula (4-19) shows that the baseband system impulse response of the crosstalk between the uncensored is not zero, but the sampling value at other sampling time is zero. The customary formula (4-19) is the time domain condition of the crosstalk baseband transmission system for uncensored. Can meet this requirement is can be found, and many, take our more familiar with the sampling function, it is possible to meet this condition. The curve shown in 4-13 is a typical example.

Figure 4-13 of the curve

The above gives the demand for the impulse response of the baseband transmission system without code-based crosstalk, and the following focuses on the requirements for the transmission function of the baseband transmission system and the possible methods for the crosstalk between the uncensored. For the sake of convenience, we start with the simplest single ideal baseband transmission system. (Click here to view Flash) 4.3.1 ideal Baseband transmission system 　　The transmission characteristic of the ideal baseband transmission system has the ideal low-pass characteristic, and its transfer function is

(4-20)

4-14 (a), with its bandwidth   (HZ). The Fourier inverse transformation of the                                                                                                                (4-21)

Fig. 4-14 the and of the ideal baseband transmission system

From the above analysis can be seen, if the signal after the transmission of the entire waveform changes, but as long as its specific point of the sampling value remains unchanged, then the method of re-sampling, still be able to accurately restore the original code. This is the essence of the so-called Nyquist first rule. In the ideal baseband transmission system for the cutoff frequency represented in figure 4-14, the minimum code element interval for the system to transmit the crosstalk between the uncensored is called the Nyquist interval. Accordingly, called the Nyquist rate, it is the maximum code-element transfer rate for the system. Conversely, the minimum transmission bandwidth required for an input sequence to pass an uncensored crosstalk transmission at the code element rate is a (Hz). The 1/2 is usually called the Nyquist bandwidth. Let's look at the issue of frequency band utilization. The so-called frequency band utilization refers to the ratio of the code element rate and the bandwidth, that is, the unit frequency band can transmit the code element rate, its expression is (baud/hz) (4-22) Obviously, the bandwidth utilization of the ideal low-pass transfer function is 2 baud/hz. This is the maximum frequency band utilization, because there will be inter-code crosstalk if the system transmits the signal at a higher code rate. If you reduce the rate of transmission, that is, increase the width of the code element, so that the integer times, by the figure 4-14 (b) is visible, at the sampling point there will be no inter-code crosstalk. However, the frequency band utilization of the system will be reduced correspondingly. From the results discussed above, the ideal low-pass transfer function has the maximum transmission rate and frequency band utilization, which is very good. However, ideal baseband transmission systems are virtually impossible to apply. This is because the ideal low-pass characteristic is not physically achievable at first, and secondly, even if it is possible to approach the ideal low-pass characteristic, the trailing (i.e. attenuation oscillation fluctuation) of the impulse response of this ideal low-pass characteristic is very large, if some deviations occur at the sampling timing, or the external conditions have a slight effect on the transmission characteristics, The frequency drift of the signal will cause the inter-code crosstalk to increase obviously. The following is a further discussion of the equivalent transmission characteristics satisfying the crosstalk conditions of the (4-19) type of uncensored, in order to facilitate the establishment of an actual uncensored crosstalk baseband transmission system. The equivalent characteristics of crosstalk between 4.3.2 Uncensored Because

                                                         ( 4-23) Here we'll remember the variable   as  . The physical meaning of, ,  in the formula is: to translate the segmentation of   into the interval corresponding to the sum of  , we call it "slice superposition". Obviously, it exists only in   and has low-pass characteristics.

Make (4-24) Is the "segment overlay", which we call equivalent transfer functions. Put it into (4-23) type, get (4-25) In the ideal low-pass transmission system, by the formula (4-21), there is Was (4-26) This is a crosstalk between the uncensored. Bashi (4-25) is compared with the ideal low-pass expression (4-26), if the formula (4-25) to meet the cross-talk between the uncensored, it is required (4-27) Formula (4-27) is the equivalent of crosstalk between uncensored. It shows that the transmission characteristic of a baseband transmission system is divided into a width, if each section in the interval can be superimposed into a rectangular frequency characteristics, then it at the rate of transmission baseband signal, it will be able to do the cross-talk between the uncensored. The customary formula (4-27) is the frequency domain condition of the crosstalk baseband transmission system in uncensored. by formula (4-27), if the band limit of the system is not considered, the form of baseband transmission is not unique only from the elimination of crosstalk between the codes.

4.3.3 Practical uncensored inter-crosstalk baseband transmission characteristics Considering that the tail attenuation of the ideal impulse response is slow because the frequency characteristic of the system is too steep, this can enlighten us to design the characteristic according to the structure thought shown in Figure 4-16, that is, to "smooth" the characteristics of the ideal low-pass characteristic as the cutoff frequency, i.e. According to the formula (4-27) the frequency domain condition of the crosstalk baseband transmission system is not difficult to see, as long as the H1 (ω) for the W1 has an odd symmetrical amplitude characteristic, then h (ω) is the uncensored crosstalk. Here, =, equivalent to the angular frequency. The "smoothing" described above is often referred to as "roll-down". The upper and lower cutoff frequencies of the roll-down characteristics are respectively. Defines a roll-down coefficient of (4-28) Obviously.

Fig. 4-16 composition of roll down characteristics (only the positive frequency portion is drawn)

Can be selected according to the actual needs to form a different actual system. The common linear rolling drop, triangle roll drop, rise cosine roll drop and so on. The following is an example of the most used cosine roll-off characteristics for further discussion. Figure 4-17 shows the cosine roll drop characteristics at different time, in the figure =.

(a) Transmission characteristics (only the positive frequency portion is drawn) (b) Impulse response Figure 4-17 Cosine roll-down transmission characteristics

, the cosine roll-down transmission is the ideal low-pass characteristic of the cutoff frequency when there is no roll drop. , is the actual use of the rising cosine roll-down transmission characteristics, can be expressed in the following

(4-29) Accordingly, for

(4-30)

It should be noted that at this time the waveform formed, in addition to the moment the amplitude is zero, at these times its amplitude is also zero. When the normal value is taken, the cosine roll-down transmission characteristic can be expressed as

(4-31) The corresponding impulse response is

(4-32) It is obvious that it is not crosstalk between the code element transmission rate.

From the above analysis of the cosine roll-down transmission characteristics, combined with the different cosine roll-down characteristics given in figure 4-17, the spectrum and waveform, it is not difficult to draw: (1) At that time, for the "roll-off" of the ideal baseband transmission system, the "tail" by the law attenuation. When, that is, the use of cosine rolling, the corresponding still maintain from the beginning, left and right every 0 points of the feature, to meet the sampling of the instant cross-talk between the conditions, but the formula (4-32) in the second factor on the wave attenuation speed is a great influence. On the one hand, there will be a new 0 points, accelerated "tail" attenuation, on the other hand, the waveform "tail" by the law attenuation, much smaller than the ideal low-pass. Attenuation is also related to the speed, the larger, the faster the attenuation, the smaller the inter-code crosstalk, the less likely the wrong verdict. (2) The bandwidth occupied by the output signal spectrum. At that time, the frequency band utilization was 2baud/hz; when, the frequency band utilization was 1baud/hz; in general, = 0 ~ 1 o'clock, the band utilization is 2 ~ 1baud/hz. As you can see, the higher the "tail" attenuation, the wider the bandwidth and the lower the band utilization. Therefore, the use of roll-off characteristics to improve the ideal low-pass, in essence, at the expense of bandwidth utilization at the expense of exchange. The realization of cosine roll-off characteristic is much easier than ideal low-pass, so it is widely used in frequency band utilization, but allows the timing system and transmission characteristics to have a large deviation.

About inter-code crosstalk (reprint)