Chapter 7 Spread-Spectrum Modulationshannon.cm.nctu.edu.tw/comtheory/chap7.pdf · 1 Chapter 7 Spread-Spectrum Modulation Spread Spectrum Technique simply consumes spectrum in excess

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Chapter 7 Spread-Spectrum Modulation

Spread Spectrum Technique simply consumes spectrum in excess of the minimum spectrumnecessary to send the data.

© Po-Ning [email protected] Chapter 7-2

7.1 Introduction

o Definition of spread-spectrum modulationn Weakly sense

o Occupy a bandwidth that is much larger than the minimum bandwidth (1/2T) necessary to transmit a data sequence.

n Strict senseo Spectrum is spreading by means of a pseudo-white

or pseudo-noise code.

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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 propertyn run propertyn 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.



Exemplified block diagram of PN sequence generators

n Feedback shift register becomes “linear” if the feedback logic consists entirely of modulo-2 adders.

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o Example of linear feedback shift register



o A PN sequence generated by a (possibly non-linear) feedback shift register must eventually become periodic with period at most 2m, 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 2m - 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.

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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 = 2m-1)

o ½ of the runs is of length 1o ¼ of the runs is of length 2o …


n Correlation propertyo Autocorrelation of an

ideal discrete white process = a·d[t], where d[t] is the Kronecker delta function.


c(t)

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o Power spectrum view

Suppose c(t) is perfectly white with c2(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)?


7.3 A notion of spread spectrum

b(t)

c(t)

m(t) b(t)

c(t)

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o Please self-study Example 7.2 for m-sequences.n Its understanding will be part of the exam.



b(t)

c(t)

m(t) b(t)

c(t)

Make the transmitted signal to hide behind the background noise.

Spreading code

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o Example n Recall Slides 6-35 ~ 6-38


n Noting that

we obtain

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9



channel receivertransmitter

Lowpass filter that (only) allows signal b(t) to pass!


7.4 Direct-sequence spread spectrum with coherent binary phase-shift keying

o DSSS system

transmitter

receiver

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(As the conceptual system below.)

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7.5 Signal-space dimensionality and processing gain

o SNR before spreading

o SNR after spreading

o Orthonormal basis used at the receiver end

Assume “coherent detection”.In other words, perfect synchronization and no phase mismatch.



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Mismatch with the text in SNRIo SNR before spreading


o Orthonormal basis used at the receiver end


Assume “coherent detection”.In other words, perfect synchronization but with phase mismatch.

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15




Same as Slides 7-20 and 7-21, we derive

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7.6 Probability of error

The analysis for SNRI and SNRO 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 !


Slide 6-32 said that

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6.3 Coherent phase-shift keying – Error probability

o Error probability of Binary PSKn Based on the decision rule



7.6 Probability of error

o Comparing system performances with/without spreading, we obtain:

o With P = Eb/Tb, where P is the average signal power,

n J/P is termed the jamming margin (required for a specific error rate).

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o Example 7.3n Without spreading, (Eb/N0) required for Pe = 10-5 is

around 10 dB.n PG = 4095n Then, Jamming margin for Pe = 10-5 is

n Information bits can be detected subject to the required error rate, even if the interference level is 409.5 times larger than the received signal power (in the price of the transmission speed is 4095 times slower).



7.7 Frequency-hop spread spectrum

o Basic characterization of frequency hoppingn Slow-frequency hopping

n Fast-frequency hopping

n Chip rate (The smallest unit = Chip)

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o A common modulation scheme for FH systems is the M-ary frequency-shift keying


0

7

6

5

4

3

2

1

Slow-frequency hopping

The smallest unit= Chip

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7

6

5

4

3

2

1

Fast-frequency hopping

The smallest unit= Chip



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 hoppingn 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)

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7.8 Computer experiments: Maximum-length and gold codes

o Code-division multiplexing (CDM)n Each user is assigned a different spreading code.



o So, if

then signal one (i.e., s1) 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

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o Gold sequences



o Gold sequencesn g1(x) and g2(x) are two maximum-length shift-register

sequences of period 2m - 1, whose “cross-correlation” lies in:

n Then, the structure in previous slide can give us 2m - 1 sequences (by setting different initial value in the shift registers).

n Together with the two original m-sequences, we have 2m

+ 1 sequences.

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o Gold’s theoremn The cross-correlation between any pair in the 2m +1

sequences also lies in

o Experiment 1: Correlation properties of PN sequences


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n Autocorrelation


27 - 1 = 127

n Cross-correlation of two maximum-length shift-register (PN) sequences, which are not Gold sequences.


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o Experiment 2: Correlation properties of Gold sequences


n Cross-correlation


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Chapter 7 Spread-Spectrum Modulationshannon.cm.nctu.edu.tw/comtheory/chap7.pdf · 1 Chapter 7 Spread-Spectrum Modulation Spread Spectrum Technique simply consumes spectrum in excess

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