Chapter 7 Spread-Spectrum Modulation
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1 Chapter 7 Spread-Spectrum Modulation Spread Spectrum Technique simply consumes spectrum in excess of the minimum spectrum necessary to send the data. 7.1 Introduction o Definition of spread-spectrum modulation n Weakly sense o Occupy a bandwidth that is much larger than the minimum bandwidth (1/2T) necessary to transmit a data sequence. n Strict sense o Spectrum is spreading by means of a pseudo-white or pseudo-noise code. Po-Ning Chen@cm.nctu Chapter 7-2 1
2 7.2 Pseudo-noise sequences o A (digital) code sequence that mimics the (second-order) statistical behavior of a white noise. o For example, n balance property n run property n correlation property o From implementation standpoint, the most convenient way to generate a pseudo-noise sequence is to employ several shift-registers and a feedback through combinational logic. Po-Ning Chen@cm.nctu Chapter Pseudo-noise sequences Exemplified block diagram of PN sequence generators n Feedback shift register becomes linear if the feedback logic consists entirely of modulo-2 adders. Po-Ning Chen@cm.nctu Chapter 7-4 2
3 7.2 Pseudo-noise sequences o Example of linear feedback shift register Po-Ning Chen@cm.nctu Chapter Pseudo-noise sequences o A PN sequence generated by a (possibly non-linear) feedback shift register must eventually become periodic with period at most 2 m, where m is the number of shift registers. o A PN sequence generated by a linear feedback shift register must eventually become periodic with period at most 2 m - 1, where m is the number of shift registers. o A PN sequence whose period reaches its maximum value is named the maximum-length sequence or simply m-sequence. Po-Ning Chen@cm.nctu Chapter 7-6 3
4 7.2 Pseudo-noise sequences o A maximum-length sequence generated from a linear shift register satisfies all three properties: n Balance property o The number of 1s is one more than that of 0s. n Run property (total number of runs = 2 m-1 ) o ½ of the runs is of length 1 o ¼ of the runs is of length 2 o Po-Ning Chen@cm.nctu Chapter Pseudo-noise sequences n Correlation property o Autocorrelation of an ideal discrete white process = a d[t], where d[t] is the Kronecker delta function. c(t) Po-Ning Chen@cm.nctu Chapter 7-8 4
5 7.2 Pseudo-noise sequences o Power spectrum view Suppose c(t) is perfectly white with c 2 (t) = 1. Then, m(t) = b(t)c(t) and b(t) = m(t)c(t). Question: What will be the power spectrum of m(t)? Po-Ning Chen@cm.nctu Chapter A notion of spread spectrum b(t) m(t) b(t) c(t) c(t) Po-Ning Chen@cm.nctu Chapter
6 7.2 Pseudo-noise sequences o Please self-study Example 7.2 for m-sequences. n Its understanding will be part of the exam. Po-Ning Chen@cm.nctu Chapter A notion of spread spectrum Make the transmitted signal to hide behind the background noise. b(t) m(t) b(t) Spreading code c(t) c(t) Po-Ning Chen@cm.nctu Chapter
7 o Example n Recall Slides 6-35 ~ 6-38 Po-Ning Chen@cm.nctu Chapter 7-13 n Noting that we obtain Po-Ning Chen@cm.nctu Chapter
8 Po-Ning Chapter 7-15 Po-Ning Chapter
9 7.3 A notion of spread spectrum transmitter channel Lowpass filter that (only) allows signal b(t) to pass! receiver Po-Ning Chen@cm.nctu Chapter Direct-sequence spread spectrum with coherent binary phase-shift keying o DSSS system transmitter receiver Po-Ning Chen@cm.nctu Chapter
10 Po-Ning Chapter 7-19 (As the conceptual system below.) Po-Ning Chapter
11 7.5 Signal-space dimensionality and processing gain o SNR before spreading o SNR after spreading Assume coherent detection. In other words, perfect synchronization and no phase mismatch. o Orthonormal basis used at the receiver end Po-Ning Chen@cm.nctu Chapter 7-21 o SNR before spreading Po-Ning Chen@cm.nctu Chapter
12 o SNR after spreading Po-Ning Chapter 7-23 Po-Ning Chapter
13 Po-Ning Chapter 7-25 Mismatch with the text in SNR I o SNR before spreading o SNR after spreading Assume coherent detection. In other words, perfect synchronization but with phase mismatch. o Orthonormal basis used at the receiver end Po-Ning Chen@cm.nctu Chapter
14 o SNR before spreading Po-Ning Chapter 7-27 Po-Ning Chapter
15 o SNR after spreading Po-Ning Chapter 7-29 Same as Slides 7-20 and 7-21, we derive Po-Ning Chapter
16 7.6 Probability of error The analysis for SNR I and SNR O assumes perfectly white noise j(t) for simplicity. Note that a white noise j(t) has infinite power such that j(t) and c(t)j(t) will have the same PSD! Po-Ning Chen@cm.nctu Chapter 7-31 Slide 6-32 said that Po-Ning Chen@cm.nctu Chapter
17 6.3 Coherent phase-shift keying Error probability o Error probability of Binary PSK n Based on the decision rule Po-Ning Chen@cm.nctu Chapter Probability of error o Comparing system performances with/without spreading, we obtain: o With P = E b /T b, where P is the average signal power, n J/P is termed the jamming margin (required for a specific error rate). Po-Ning Chen@cm.nctu Chapter
18 o Example 7.3 n Without spreading, (E b /N 0 ) required for P e = 10-5 is around 10 db. n PG = 4095 n Then, Jamming margin for P e = 10-5 is n Information bits can be detected subject to the required error rate, even if the interference level is times larger than the received signal power (in the price of the transmission speed is 4095 times slower). Po-Ning Chen@cm.nctu Chapter Frequency-hop spread spectrum o Basic characterization of frequency hopping n Slow-frequency hopping n Fast-frequency hopping n Chip rate (The smallest unit = Chip) Po-Ning Chen@cm.nctu Chapter
19 7.7 Frequency-hop spread spectrum o A common modulation scheme for FH systems is the M-ary frequency-shift keying Po-Ning Chen@cm.nctu Chapter The smallest unit = Chip Slow-frequency hopping Po-Ning Chen@cm.nctu Chapter
20 The smallest unit = Chip Fast-frequency hopping Po-Ning Chen@cm.nctu Chapter Frequency-hop spread spectrum o Fast-frequency hopping is popular in military use because the transmitted signal hops to a new frequency before the jammer is able to sense and jam it. o Two detection rules are generally used in fast-frequency hopping n Make decision separately for each chip, and do majority vote based on these chip-based decisions (Simple) n Make maximum-likelihood decision based on all chip receptions (Optimal) Po-Ning Chen@cm.nctu Chapter
21 7.8 Computer experiments: Maximum-length and gold codes o Code-division multiplexing (CDM) n Each user is assigned a different spreading code. Po-Ning Chen@cm.nctu Chapter Computer experiments: Maximum-length and gold codes o So, if then signal one (i.e., s 1 ) can be exactly reconstructed. o In practice, it may not be easy to have a big number of PN sequences satisfying the above equality. Instead, we desire Po-Ning Chen@cm.nctu Chapter
22 7.8 Computer experiments: Maximum-length and gold codes o Gold sequences Po-Ning Chen@cm.nctu Chapter Computer experiments: Maximum-length and gold codes o Gold sequences n g 1 (x) and g 2 (x) are two maximum-length shift-register sequences of period 2 m - 1, whose cross-correlation lies in: n Then, the structure in previous slide can give us 2 m - 1 sequences (by setting different initial value in the shift registers). n Together with the two original m-sequences, we have 2 m + 1 sequences. Po-Ning Chen@cm.nctu Chapter
23 7.8 Computer experiments: Maximum-length and gold codes o Gold s theorem n The cross-correlation between any pair in the 2 m +1 sequences also lies in Po-Ning Chen@cm.nctu Chapter 7-45 o Experiment 1: Correlation properties of PN sequences Po-Ning Chen@cm.nctu Chapter
24 n Autocorrelation = 127 Po-Ning Chen@cm.nctu Chapter 7-47 n Cross-correlation of two maximum-length shift-register (PN) sequences, which are not Gold sequences. Po-Ning Chen@cm.nctu Chapter
25 o Experiment 2: Correlation properties of Gold sequences Po-Ning Chapter 7-49 n Cross-correlation Po-Ning Chen@cm.nctu Chapter
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