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• the bandwidth of the transmitted signal  need to be much greater than that of the message .
• the relatively wide bandwidth of  must be caused by an independent modulating waveform  called the spreading signal, and this signal must be known by the receiver in order for the message signal  to be detected.

• direct sequence (DS). here, a DSB-SC type of spreading modulation is used [i.e., ], and  is a polar NRZ waveform.

(FYI) 2. spread spectrum techniques: direct sequence systems

• frequency hopping (FH). here  is of the FM type where there are  hop frequencies determined by the  words obtained from the spreading code waveform .

(FYI)

• hybrid techniques that include both DS and FH.

DS system is usually used with BPSK information signal.^[각주:1]

data(bit) rate is

, where  is bit interval.

• spreading signal,  which is generated by pseudonoise(PN) code generator has values of , and use the  as a complex envelope.

chip rate is

, where  is chip interval which corresponding to pulse width

• spread spectrum system assumes  .

BPSK/DS transmitter

• section의 목적상 (spread spectrum system의 이해), information signal,  이 BPSK로 modulation 되는 과정에 대한 설명은 생략하도록 한다. digital signal인 는 modulation 과정을 거쳐서, bandpass signal의 형태인 로 주어진다고 가정한다.
• pseudorandom sequence or code generator 에 대한 내용은 중요성을 고려하여 별도의 글을 작성하였다.

• 참고 posting

pseudorandom sequence generator
pseudorandom noise (PRN), PN code, maximum-length sequence

• PRN generator-maximum-length sequence characteristic

has period of

• PSD for an m-sequence PN waveform

(FYI) BPSK/DS signal의 spectrum 에 대한 유도는 pseudorandom noise (PRN), PN code, maximum-length sequence 을 참고한다.

• signal spectrum analysis: bandwidth change in system

we see that the PSDs of both

• pulse width가 증가하면, spectrum shape은 sharpe, and vice versa.

to simplify the mathematics, approximate these PSDs by rectangular spectra where the heights of the PSD are selected so that the areas under the curves are unity because the powers of

• bandpass BPSK signal은 spreading signal과 multiplication함으로서 DS signal이 완성된다. 굳이 complex envelope의 개념을 이용하여 연산의 순서을 생각하면서 문제를 복잡하게 생각할 필요가 없이, bandpass BPSK signal과 spreading signal이 곱하여 DS signal이 완성된다고 이해한다. 

• 위 block diagram에 나타난 system processing 과정 중 '완성된 bandpass BPSK signal'에 단순히 'spreading signal'을 multiplication함으로 DS spread signal이 완성된다는 점을 remind.

BPSK/DS receiver

• system이 동작되는 환경에 따라 발생할 수 있는 많은 경우의 jamming case^[각주:2] 중에서, narrowband sine wave jamming signal (tones) case 에 따른 processing에 대해서 알아본다.

(FYI) tone jamming 에 대한 비슷한 수학적 분석, 11. spread spectrum communications: applications of spread spectrum - military: AJ system

• input to the receiver consists of the SS signal plus a noise (jammer) signal.

assume that noise (jamming) signal is a narrowband (sine wave) jammer signal which is set to  for the worst-case jamming effect,

• output of the despreader is

since

• with the multiplication of , the input to the LPF is

and the terms

• final detected signal form will be

• computing processing gain which defined as the ratio of 'the noise power out without SS' divided by 'the noise power out with SS'. this is equivalent to the ratio  where  is the SNR power into the receiver and  is the SNR power out of the LPF.

jamming power at the receiver output without SS (conventional BPSK system)

jamming power at the receiver output with SS

• one canonical form of a 'binary LFSR' is a 'simple shift register generator (SSRG)'

the shift register consists of binary storage elements (boxes) which transfer their contents to the right after each clock pulse. the contents of the register are linearly combined with the binary (0, 1) coefficients

 and are fed back to the first stage.

• for spread spectrum, we are looking for a maximal length cycles (maximum length sequence, MLLFSR)^[각주:2]

• modular shift register generator (MSRF), which is commonly used implementation of a LFSR

• PRN is a signal which is similar to noise

although it seems to lack any definite pattern, PRN consists of a deterministic sequence of pulses that will repeat itself after its period.

pseudonoise code (PN code)

• PN code is one that has a spectrum similar to a random sequence of bits but is deterministically generated.
• the most commonly used sequences in direct-sequence spread spectrum systems are maximal length sequences, gold codes, kasami codes, and barker codes.

maximum length sequence^{[각주:1] [각주:2]}

• one of the simplest and most effective devices for generating deterministic sequences of pseudorandom Ones and Zeros is the shift register.

shift register of degree n is a device consisting of n consecutive binary storage positions, which shifts the contents of each position to the next position down the line at the rate set by a timing device, or clock. 

• a shift register can be converted into a pseudorandom sequence generator (more specific, maximum length sequence) by including a feedback loop, which computes a new term for the first stage based on the previous n terms.

• given any linear shift register of degree n, the ouput sequence is always ultimately periodic, with a period  at most. any output sequence achieving a period of  is called a maximum length sequence.

• correlation function and its power spectra of maximum length sequence
• correlation function

(FYI) for general signal,

for binary sequence auto-correlation,

based on above formula, general auto-correlation function for maximum length sequence looks like,

• power spectrum of correlation function

• numeric methods can be formulated as algorithms. an algorithm is a step-by-step procedure that states a numeric method in a form (a pseudocode) understandable to humans.

pseudocode is a compact and informal high-level description of a computer programming algorithm that uses the structural conventions of some programming language, but is intended for human reading rather than machine reading.