KR100720861B1 - Difference threshold test method for m-fsk signals - Google Patents

Abstract

The present invention relates to a M- ary frequency shift key modulation signal detection method using a difference threshold test (DTT).

The proposed algorithm shows better performance than ratio threshold test (RTT) when detecting M- ary frequency shift signals if the modulation magnitude and diversity order are finite. In addition, we obtain asymptotic performance when the modulation magnitude and diversity order are infinitely large, and show that the smallest signal-to-noise ratio that can be communicated using the differential threshold test, the ratio threshold test, or without error is the same.

Differential threshold test, ratio threshold test, frequency shift modulation, Reed-Solomon code, diversity

Description

Difference threshold test method for M-FSK signals

FIG. 1 is a diagram showing the probability that a hard decision, a differential threshold test, and a ratio threshold test incorrectly decode when writing a (7,3) Reed-Solomon code in a Rayleigh attenuation channel.

FIG. 2 is a diagram showing the probability that the hard decision, the ratio threshold test, and the differential threshold test decode incorrectly when L = 2 in the rice attenuation channel and (7,3) Reed-Solomon code is used.

The present invention relates to a method for detecting M- ary frequency shift key modulation signals using a difference threshold test (DTT).

When detecting the M- ary frequency shift modulation signal in mobile communication, since the block code can correct up to twice as many as those deleted due to an error, the reliability of the received symbol is determined and the unreliable symbol is deleted before the decoding process. It is good.

Thus, side information is used to identify and delete symbols corrupted by channel effects such as fading, jamming, background noise, and the like.

Therefore, various methods for generating such side information have been proposed and their performance has been analyzed.

For example, in the ratio threshold test, if the ratio of the largest output and the second largest output among the energy detector outputs is smaller than the threshold, the symbol is considered lost. In addition, the loss of using the Bayesian test is the smallest of the Chernoff limits of the probability of false decoding.

However, although the base test performs considerably better than the ratio threshold test, the algorithm is complicated and disadvantageous in that channel state information (CSI) can be used.

In view of the above problems of the base test method, the present invention provides a differential threshold test method that does not write channel state information and is easier to implement than the ratio threshold test.

In the differential threshold test, the difference between the largest output and the second largest output among the energy detector outputs is considered to be lost. The difference threshold test is an approximation of the base test, but shows almost the same performance as the base test. Moreover, the differential threshold test does not use channel state information and is easier to implement than the ratio threshold test. Therefore, the differential threshold test can be easily used in any system that uses the ratio threshold test.

In the present invention, we will examine the differential threshold test suitable for asynchronous M- ary quadrature frequency shift modulation using L- order diversity reception and Reed-Solomon code when there are several channels.

When the modulation magnitude M and the diversity order L are finite, the proposed differential threshold test outperforms the ratio threshold test. Determine the asymptotic performance of the differential threshold test, and when M is infinitely large, the smallest signal-to-noise ratio that can communicate without error in the differential threshold test is the same as the smallest signal-to-noise ratio that can communicate without error in the ratio threshold test. see. Consider an asynchronous M- ary quadrature frequency shift modulated signal using Reed-Solomon code and L- order diversity reception. Suppose that an information source is encoded with a ( n , k ) Reed-Solomon code whose code length is n = M -1 and code rate is r = k / n . After the replacement is ideally replaced, the sign symbol consisting of log ₂ M bits is replaced with s _i ( t ) = Acos (2π f _i t ), which is one of the M orthogonal signals. Where i = 1, 2,... , M −1 and 0 ≦ t ≦ T. Where A is the amplitude of the signal, f _i is the i- th tone frequency, and T is the symbol spacing. Tone frequency { f _i } is equal to signal { s _i ( t )} i = 1, 2,... , When M -1 | Choose f _{i +1} - f _i | = 1 / T to be asynchronous orthogonal.

Consider L independent channels that suffer from nonselective frequency and slowly changing Rayleigh decay. When the signal s ₁ ( t ) is sent, the signal r _l ( t ) received by the l- th diversity branch is

[Equation 1]

Where l = 0, 1,... , L −1 and 0 ≦ t ≦ T. On the other hand, g _l and θ _l represent the amplitude and phase caused by the l- th diversity channel, respectively, and n _l ( t ) is white normal noise with one power density N ₀ . It is assumed that { g _l } and { θ _l } do not change during the symbol interval, g _l is a Rayleigh random variable, and θ _l is evenly spread between [0, 2π).

Consider an asynchronous receiver with M energy detectors per diversity channel. Then, the mth energy detector output of the lth diversity branch is

[Equation 2]

이고

이며, 이 둘은 모두 m=1, 2, …, M일 때 평균이 0, 분산이 N ₀/2인 독립 정규 확률변수이다. 여기서, g _l cosθ _l 과 g _l sinθ _l 은 평균 이 0이고 분산이 E[g ² _l ]/2인 독립 정규 확률변수임을 새겨두자. 평균 채널 전력 이득 {E[g ² _l ]}은 어느 다양성 가지에서도 같으며, l=0, 1, 2, …, L-1일 때 E[g ² _l ]=1/L이라 둔다.to be. here,

ego

Are both m = 1, 2,... , A stand-normal random variable with zero mean, variance of N _0/2 when the M. Note that g _l cosθ _l and g _l sinθ _l are independent normal random variables with mean 0 and variance E [ g ² _l ] / 2. The average channel power gain { E [ g ² _l ]} is the same for any of the various branches, where l = 0, 1, 2,. , L L = 1 E [ g ² _l ] = 1 / L.

Consider an equal gain combining (EGC), which is not known channel gain and is best known when attenuation { g _l } is an independent Rayleigh random variable. The m- th decision statistic of the vowel, like gain, is the sum of the energy detector outputs { Z _{l, m} } of the variance branches, where m = 1, 2, When M

[Equation 3]

to be. Z _m is then the central chi-square random variable with 2 L degrees of freedom. Now, when s ₁ ( t ) is sent, the conditional probability density function f _Zm ( z | 1) of Z _m and the conditional cumulative inclusion function F _Zm ( z | 1) are each z = 0,

[Equation 4]

Wow

[Equation 5]

라 할 때, m=1이면

이고, m≠1이면

이다. 부호화율이 r=k/n인 리드-솔로몬 부호를 썼기 때문에, 받은 정보비트 에너지의 평균은

이다.to be. Here, the average of the symbol symbol energy received from one diversity channel

If m = 1

If m ≠ 1

to be. Since we used a Reed-Solomon code with a code rate of r = k / n , the average of the received information bit energy is

to be.

In the decoding of Reid-Solomon's code as wrong and lost, let's say we use the differential threshold test as a way to see as lost. The differential threshold test considers the detected symbol to be lost if the difference between the largest and second largest of the energy detector outputs is not greater than the threshold. Therefore, when s ₁ ( t ) is sent, the difference threshold test is γ ≥ 0, m ∈ {1, 2,. , M }

[Equation 6]

Is lost, and when m ≠ 1

[Equation 7]

, You get a symbol wrong,

[Equation 8]

와 N ₀ 따로 따로에 의존하는 것이 아니라

에 의존하도록 문턱값 γN ₀를 정했다. 널리 알려진 굳은 판정은 γ=0일 때 제안한 시스템과 같음을 새겨두자.If you get a symbol correctly. Where performance

And N _{0 do} not depend on

The threshold γN ₀ was determined to depend on. Note that the well-known hard decision is the same as the proposed system when γ = 0.

Because of the symmetry of the signals, the probability of getting the symbol correctly, the probability of getting the symbol wrong, and the loss of the symbol if the signals are sent with the same probability are independent of the sent signal. Therefore, we can say that we sent s ₁ ( t ). Then, { Z _m , m = 2, 3,... , M } are independent variables and have the same distribution, so when the threshold is γ , the probability of getting the symbol correctly is

[Equation 9]

The probability of getting the symbol wrong is

[Equation 10]

And the probability of losing a symbol is

[Equation 11]

to be. In other words, the probability P _E that the ( n , k ) Reed-Solomon code system decodes incorrectly

[Equation 12]

to be.

의 함수로 보여준다. 여기서, 차분 문턱값 검정과 비율 문턱값 검정의 문턱값은

마다 최적값으로 하였고, P _E 는 수치적분법으로 계산하였다. 차분 문턱값 검정은 비율 문턱값 검정보다 성능이 좋지만 M과 L이 클수록 성능 이득이 줄어든다.1 shows P _E in a system with the (7,3) Reed-Solomon code.

Show as a function of Here, the thresholds of the difference threshold test and the ratio threshold test are

The optimum value for each was calculated and P _E was calculated by the numerical integration method. The differential threshold test performs better than the ratio threshold test, but the larger M and L , the lower the performance gain.

FIG. 2 shows the probability that the differential threshold test and the ratio threshold test would decode incorrectly when L = 2 in the rice attenuation channel and (7,3) Reed-Solomon code. The rice factor K is the ratio of the reflection and diffusion components of the rice attenuation channel and is the same for all L diversity channels. Note that if K = 0, it is Rayleigh attenuation channel. In FIG. 2, it can be seen that the differential threshold test performs better than the ratio threshold test when K = 0, 10 ^0.3 (3dB), and 10 ^0.6 (6dB). The higher the K factor, the closer the normal noise channel is to the rice attenuation channel, so the higher the K , the smaller the performance gain.

,

와

를 얻어 보자. 결정 통계량의 조건부 확률밀도함수 수학식 4와 조건부 누적분포함수 수학식 5를 수학식 9에 넣고

라 하면Now, the asymptotic probability when M is infinitely large

,

Wow

Let's get it. Conditional Probability Density Function Equation 4 and Conditional Cumulative Distribution Function Equation 5 of the Decision Statistics

If

[Equation 13]

이기 때문에ego,

Because

[Equation 14]

라 두면 수학식 14는to be. now,

Equation 14 is

[Equation 15]

이다. 수학식 13에서 C(u)는 오직 M에 의존하는 식임을 새겨두자. 르베그의 지배수렴 정리와,

을 쓰면, 심벌을 바르게 얻을 점근 확률은Can be written as

to be. Note that in Equation 13, C ( u ) is dependent only on M. Rebek's converging theorem,

, The asymptotic probability of getting the symbol right

[Equation 16]

이면

임을 새기고,

을 쓰면, M→∞일 때 χ _M →∞이므로, C(u)의 자연대수의 극한은 아래와 같다.to be. Now, when b is an extended real system,

Back side

Carved Im,

To write, because the days when M → ∞ χ _M → ∞, the limit of the natural logarithm of C (u) is as follows:

[Equation 17]

Therefore, the limit of C ( u ) is

Equation 18

Where I (·) is an indication function where I (x) = 1 if x is true, or I ( x ) = 0. Then, the asymptotic probability of correctly obtaining a symbol in Equation 16 is as follows.

[Equation 19]

는 불완전 감마함수이며,

는 감마함수이다. 차분 문턱값 검정을 쓸 때 심벌을 바르게 얻을 점근 확률은 문턱값 γ와 독립적이라는 점을 새겨두자. 한편, 비율 문턱값 검정을 쓰면 문턱값이 클수록, 심벌을 바르게 얻을 점근 확률은 작아진다. 또한, L=1로 두면, 수학식 19는 채널이 하나일 때의 결과임을 새겨두자.here,

Is the incomplete gamma function,

Is the gamma function. Note that when using the difference threshold test, the asymptotic probability of getting the symbol correctly is independent of the threshold γ . On the other hand, when the ratio threshold test is used, the larger the threshold value, the smaller the asymptotic probability of correctly obtaining the symbol. In addition, if L = 1, the equation (19) is a result of one channel.

을 넣으면, 심벌을 잘못 얻을 확률은 Now, in Equation 10

, The probability of getting the symbol wrong is

[Equation 20]

을 수학식 20의 둘째 식에 넣으면Becomes In the last step, we used partial integration. On the right side of Equation 20, it can be easily seen that the first term is e ^{- γ} . Next,

Into the second expression of Equation 20

[Equation 21]

이 되는데,

이므로, 수학식 21의 극한은

이다. 끝으로 실수 z가 어떤 값이더라도 Fz _l(z|1)(1+γN ₀/z) ^{L -2} γN ₀/z ²≤B를 만족시키는 실수 B가 존재하 므로 수학식 20의 마지막 항은

보다 작거나 같다. 여기서, 실수 z가 어떤 값이더라도 Fz ₂(z|1)＜1이므로 지배수렴정리를 쓰면 수학식 20의 마지막 항은 M이 무한히 클 때 0이 된다.

This is

Since the limit of (21) is

to be. Finally, no matter what value of real z is, there is a real number B that satisfies Fz _l ( z | 1) (1+ γN ₀ / z ) ^{L -2} γN ₀ / z ² ≤ B , so the last term in Equation 20

Is less than or equal to Here, since any value of real z is Fz ₂ ( z | 1) <1, the governing convergence theorem makes the last term of Equation 20 become 0 when M is infinitely large.

Therefore, the asymptotic probability of getting the symbol incorrectly is

[Equation 22]

In other words, even if the modulation size M is infinitely large, the probability of getting a symbol wrong in the differential threshold test is not zero. In contrast, the ratio threshold test has an asymptotic probability of getting a symbol wrong when the threshold is greater than a certain value. On the other hand, since p _er ( γ ) = 1- p _c ( γ ) -p _e ( γ ), the asymptotic probability of losing a symbol is given by

[Equation 23]

to be.

Next, let's look at the asymptotic performance of getting the symbol correctly when M → ∞ and L → ∞. If t = u / L is written in Equation 19, the asymptotic probability of getting the symbol correctly can be expressed as follows.

[Equation 24]

The latter part of the function in the integral of Equation 24 is the chi-square probability density function with 2 L degrees of freedom and an average of 2 L (1/2 L ) = 1, so that when L is infinitely large, the impact function δ ( t -1) Becomes So the asymptotic probability of getting the symbol right

[Equation 25]

to be. In other words, if r = 1, the performance at infinitely large diversity orders and modulation sizes approaches asymptotically additive white light normal noise channels.

Now, if we have chosen the threshold correctly, show that the asymptotic performance of the differential threshold test and the ratio threshold test when M is infinitely large is the same. First, when satisfying the relationship of 2 t + e ≤ n - k , the ( n , k ) Reed-Solomon code can correct wrong symbols t and lost symbols e . On the other hand, when M → ∞, the block length is infinite, so the condition

[Equation 26]

를 얻을 수 있다. 점근 확률 수학식 22와 23을 써서 수학식 26을 다시 나타내면Satisfies the difference threshold test for error-free communication. Under these conditions, the smallest signal-to-noise ratio that can communicate without error

Can be obtained. Representing Equation 26 using the asymptotic probability Equations 22 and 23

[Equation 27]

를 수치적으로 계산할 수 있다. 한편,

가 수렴하도록 γ를 충분히 크게 골랐다고 두면,

의 수렴값

을 얻을 수 있다. 곧, 수학식 27에서 γ=∞라 하면,

은

에서 얻을 수 있는데, 이것은 비율 문턱값 검정에서와 같은 결과이다. 그러므로, 차분 문턱값 검정의 점근 확률은 비율 문턱값 검정과 다르지만, 차분 문턱값 검정에서 잘못 없이 통신할 수 있는 가장 작은 신호대잡음비와 비율 문턱값 검정에서 잘못 없이 통신할 수 있는 가장 작은 신호대잡음비는 같다.to be. Using Equation 27, the smallest possible communication

Can be calculated numerically. Meanwhile,

If we choose γ large enough to converge,

Convergence of

Can be obtained. In other words, γ = ∞ in Equation 27,

silver

This is the same result as in the ratio threshold test. Therefore, the asymptotic probability of the differential threshold test is different from the ratio threshold test, but the smallest signal-to-noise ratio that can communicate without error in the differential threshold test and the smallest signal-to-noise ratio that can communicate without error in the ratio threshold test are the same. .

The difference threshold test of this invention is an approximation of the base test, but shows almost the same performance as the base test. Moreover, the differential threshold test does not write channel state information and is easier to implement than the ratio threshold test. Therefore, the differential threshold test can be easily used in any system that uses the ratio threshold test. If the modulation magnitude and diversity order are finite, better performance can be obtained than ratio threshold testing when detecting M- ary frequency shift signals. In addition, when the modulation magnitude and diversity order are infinitely large, the asymptotic performance of the differential threshold test is obtained, and if the threshold is properly selected, the smallest signal-to-noise ratio that can communicate without error in the differential threshold test and the ratio threshold test is mutually negligible. It can be seen that the same.

Claims (12)

s ₁ ( t ), γ ≥ 0, m ∈ {1, 2,. , M }

If it's a symbol missing, when m ≠ 1

, You get a symbol wrong,

A differential threshold test method for detecting M- ary frequency shifted signals that follows the decision rule to obtain symbols correctly. A differential threshold test method having a finite modulation magnitude and diversity order and having a probability of correctly obtaining a symbol shown in the following equation when detecting an M- ary frequency shift signal.

A differential threshold test method having a finite modulation magnitude and diversity order and having a probability of incorrectly obtaining a symbol represented by the following equation when detecting an M- ary frequency shift signal.

A differential threshold test method having a finite modulation magnitude and diversity order and having a probability of losing a symbol represented by the following equation when detecting an M- ary frequency shift signal.

A differential threshold test method having a finite modulation magnitude and diversity order and having a probability of incorrectly decoding a symbol shown in the following equation when detecting an M- ary frequency shift signal.

delete A differential threshold test method having an asymptotic probability of correctly obtaining the symbol shown in Equation 9 below when the modulation magnitude is infinitely large and the M- ary frequency shift signal is detected.

A differential threshold test method having an asymptotic probability of erroneously obtaining a symbol represented by the following equation when the modulation magnitude is infinitely large and the M- ary frequency shift signal is detected.

A differential threshold test method having an asymptotic probability to see when a modulation magnitude is infinitely large and an M- ary frequency shift signal is lost by losing a symbol represented by the following equation.

A differential threshold test method having an asymptotic probability of correctly obtaining a symbol shown in the following equation when detecting a magnitude and diversity order of infinitely large M -shifted frequency shift signal.

11. The method of claim 10, wherein the performance approaches asymptotically additive white light normal noise channel when r = 1. 9. The method of claim 8, wherein the asymptotic probability is when the diversity order is infinitely large.

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