chap7

Chapter 7 Spread-Spectrum Modulation

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

7.1 Introduction

Definition of spread-spectrum modulationWeakly sense

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

Strict senseSpectrum is spreading by means of a pseudo-white or pseudo-noise code.

7.2 Pseudo-noise sequences

A (digital) code sequence that mimics the (second-order) statistical behavior of a white noise.For example,

balance propertyrun propertycorrelation property

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

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

Example of linear feedback shift register

A PN sequence generated by a feedback shift register must eventually become periodic with period at most 2m, where m is the number of shift registers.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.A PN sequence whose period reaches its maximum value is named the maximum-length sequence or simply m-sequence.

A maximum-length sequence generated from a linear shift register satisfies all three properties:

Balance propertyThe number of 1s is one more than that of 0s.

Run property (total number of runs = 2m−1)½ of the runs is of length 1¼ of the runs is of length 2…

Correlation propertyAutocorrelation of an ideal discrete white process = a·δ[τ], where δ[τ] is the Kroneckerdelta function.

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)?

m(t) channel b(t)

Please self-study Example 7.2 for examples of the m-sequences.

Its understanding will be part of the exam.

7.3 A notion of spread spectrum

m(t) channel b(t)

Make the transmitted signal to blend (and hide behind) the background noise.

Spreading code

7.3 A notion of spread spectrum

channel receivertransmitter

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

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

DSSS system

transmitter

receiver

(As the conceptual system below.)

7.5 Signal-space dimensionality and processing gain

SNR before spreading

SNR after spreading

Orthonormal basis used at the receiver end

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

SNR after spreading

Mismatch with the text in SNRI

SNR after spreading

Orthonormal basis used at the receiver end

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

The factor 2 enters when the “phase” is assumed unknown to the before-spreading receiver; hence, both cosine and sinedomains must be “inner-producted”. If the “phase” is also assumed unknown to the after-spreading receiver, then the fact 2 in SNRI/SNRO formula will (again) disappear.

7.6 Probability of error

Gaussian assumptionj is Gaussian distributed due to central limit theorem

Similar to what the textbook has assumed, let J denote the interference power experiencing at the channel with “phase mismatch”.

32 said that

6.3 Coherent phase-shift keying – Error probability

Error probability of Binary PSKBased on the decision rule

Comparing system performances with/without spreading, we obtain:

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

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

Example 7.3Without spreading, (Eb/N0) required for Pe = 10−5 is around 10 dB.PG = 4095Then, Jamming margin for Pe = 10−5 is

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

Basic characterization of frequency hoppingSlow-frequency hopping

Fast-frequency hopping

Chip rate (The smallest unit = Chip)

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

Slow-frequency hopping

The smallest unit= Chip

Fast-frequency hopping

The smallest unit= Chip

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.Two detection rules are generally used in fast-frequency hopping

Make decision separately for each chip, and do majority vote based on these chip-based decisions (Simple)Make maximum-likelihood decision based on all chip receptions (Optimal)

7.8 Computer experiments: Maximum-length and gold codes

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

So, if

then signal one (i.e., s1) can be exactly reconstructed.In practice, it may not be easy to have a big number of PN sequences satisfying the above equality. Instead, we desire

Gold sequences

Gold sequencesg1(x) and g2(x) are two maximum-length shift-register sequences of period 2m − 1, whose “cross-correlation”lies in:

Then, the structure in previous slide can gives us 2m − 1 sequences (by setting different initial value in the shift registers).Together with the two original m-sequences, we have 2m

+ 1 sequences.

Gold’s theoremThe cross-correlation between any pair in the 2m +1 sequences also lies in

Experiment 1: Correlation properties of PN sequences

Autocorrelation

27 − 1 = 127

Cross-correlation (of general PN sequences, not necessarily Gold sequences)

Experiment 2: Correlation properties of Gold sequences

Cross-correlation

chap7

noise sequences5 poning

code7 poning

pseudonoise code

white noise

background noise

data sequence

linear feedback shift

linear shift register

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