Mathematical Analysis

We analyze the performance of the joint detection algorithms via the mathematical derivations and the simulation comparisons. We mainly discuss the false detection probability where a false detection happens if the transmitted symbol is different the detected one. Let P_e be the false error probability when jth symbol is sent and ith symbol is detected, and assume P_e be nearly the same for different pairs of i → j. If there are total J candidates, the entire FDP can be expressed

P_f = 1 − (1 − P_e)^J−1 ≈ (J − 1)P_e (6.35) where the last approximation holds if the P_e is small enough.

Property of χ² distribution

Before the analysis of the FDE, we introduce some used properties and distribution in the probability. First, we consider the detection error probability in χ² distribution. If a random variable x has a χ²distribution with 2L degree of freedom (DOF), its probability density function (PDF) is respectively given by

f_χ²(x; 2L) = (1/2)^L

Γ (L) x^L−1e^−x/2, (6.36)

and cumulative distribution function (CDF) is

F_χ2(x; 2L) = γ (L, x/2)

Γ(L) , (6.37)

where γ(L, x) is the lower incomplete Gamma function defined by γ(L, x) =

Z _x

0

t^L−1e^−tdt, (6.38)

and Γ(L) = γ(L, ∞) is the Gamma function.

Now, we consider the general case of the error detection probability in χ² distribution. Let two random variable X₁ and X₂ have the χ²_2L distribution, M be known number and V₁ and V₂ are two positive value, the probability of “M + V₁X₁ ≤ V₂X₂ ” can be calculated via the

However, this integration may not exist the closed-form; thus, to evaluate the probability, we may perform the integrations via the numerical programs; for examples, the C language, the MATLAB or the Mathematical.

Distribution of |y_k,j|²

Next, we consider the statistic property and distribution of the cross correlation of r and x_j. Recall the definition of the correlation, we refine the correlation as

y_n,j = 1 N

N −1X

t=0

r(t)x^∗_j(t + n). (6.40)

For simplified derivation, we assume the candidate symbol is white where its autocorrelation is a delta function and the cross correlations between two symbols are uncorrelated zero-mean complex Gaussian random variable; thus, we have

R_j,i(n) = 1 N

N −1X

t=0

x_j(t)x^∗_i(t − n) =

½ δ(n) if j = i

η otherwise , (6.41)

and η has zero-mean (complex) Gaussian distribution. Additionally, without lost of generality, we let that the averaged sample power of the candidates is normalized to 1, which yields the symbol energy being N . Based on this assumption, the variance of η is given by

σ_η²=

P_{N −1}

k=0 |X_j(k)X_i^∗(k)|²

N² = 1

N. (6.42)

The statistic property of y_n,j depends on the correctness of the testing symbols and the existence of the delay path. If the testing symbol is the correct one, we have

y_n,c= y_n,j⁰_=j = h(n) + w_a(n), (6.43) and w_a(n) = _N¹ P_{N −1}

t=0 w(t)x^∗_j(t + n) is a zero-mean complex Gaussian variable with ^σ_N^w2 variance.

If h(n) is an non-zero path gain, |y_n,c|² has the non-central χ² distribution. Moreover, if |h_k|² is much larger than the w_a(n), |y_n,c|² can be expressed as

|y_n,c|² ≈ |h(n)|²+ σ_w²

2NX_a(n) (6.44)

where X_a(n) has a χ² distribution with 2 DOF. On contrast, if h(n) is zero or not significant, we have

|y_n,c|² = σ²_w

2NX_a(n). (6.45)

If the testing symbol is incorrect, y_n,j contains only the noise term as given by

y_n,j⁰_6=j = Y_n,e= XL l=1

α_lη(d_l− n) + w_a(n) = w_e(n) (6.46)

where w_e(n) is the zero-mean complex Gaussian variable with ^σ2h+σ_N ^2w variance and the σ²_h = P_L

l=1|α_l|² herein is the channel energy. Similarly,

|y_n,e|² = σ_h²+ σ_w²

2N X_e(n) (6.47)

where X_e(n) also has the same distribution as X_a(n).

Detector Using Uniform PDP Assumption

To evaluate the false detection probability, we need to analyze the error probability P_e. In the following, we first evaluate the P_e in given channel energy σ_h² and then extend to the fading channel by averaging all possible σ_h². Let that the observation window T is larger than the multipath delay spread and the delays are located at the sample spacing. The decision metric of the correct and incorrect events are given by

M_c=

where X_C and X_E both the random variables that have the χ² distribution with 2T degrees since they are both the summations of T χ² random variables with 2 DOF.

Now, if h is deterministic and given, by substituting M = σ_h², V₁ = ^σ_N^w2 and V₂= ^σh2+σ_N ^2w into (6.39), we have the error probability that

P_e(σ²_h) = 1 − 1 When h(n) is a fading channel, we may average the previous given probability over all possible h(n) to get the averaged error probability. Alternatively, if the PDF of σ_h² is given by f_h(x), the averaged false detection probability is given by

P_f = 1 − Z _∞

0

(1 − P_e(x))^J−1f_h(x)dx. (6.51) Actually, both the integrations in P_e(σ_h²) and P_f may not exist the closed form solutions; there-fore, to evaluate the theoretical performance, we should consider the numerical solution via the programs.

Detector Using Frequency Domain Filter

Now, we consider the performance of the detector using the frequency domain filter concept.

Likewise, we first derive the error probability in the given channel condition; then, extend to the case of the fading channel.

If Z_k,j = F (f ) ~ Y_j(f − k) is the filter output, we first evaluate the probability distribution of

|Z_k,j|². Consider the cases of correctness and incorrectness of the testing symbols. If the symbol is correct, we have

Z_k,j = F (f ) ~ H(f − k) + W_a(k) (6.52) where W_a(k) =P_Ntap−1

f =0 F (f )w_a(f − k), which is a zero-mean complex Gaussian variable with variance = (P

the χ² distribution with 2N_d degrees. Besides, from Parseval’s theorem, we furthermore have where A_n is the time domain weight of F (k) as given in (6.24). Moreover, if we consider the partial usages of the filter outputs, we approximately have the same result whereas N_dmeanwhile is the number of used samples.

On contrast, when the testing symbol is incorrect, we similarly have that M_e= C(σ²_h+ σ_w²)

2 X_E (6.55)

where X_E is the random variable that has the same distribution as X_C. Now, if h(n) is given, the error probability is given by

P_e(σ_h², σ²_h|F) = 1 − 1

In additional, when the N_tapmoving average filter is adopted, we furthermore have that_C¹ = N_tap and When h(n) is a fading channel, to derive the average false detection probability, we further-more require the PDF of σ²_h and the mapping function from σ_h² to σ_h|F² . If f_h(x) is the PDF of Also, the integrations should not exist the closed form solution and we still run the numerical integration to derive the theoretical result.

Performance in Single Path Channel

We now consider the case over the simplest single path channel. We firstly evaluate the per-formance of the cases over AWGN channel; then, average the error probability with respect to the channel energy in χ² distribution with 2 DOF to derive the performance of the cases over Rayleigh channel.

Assume ideal synchronization in symbol time and fractional CFO. For the cases of the detec-tors adopting the uniform PDP assumption and adopting the frequency domain filter, the error probabilities are derived via the integrations in (6.50) and (6.56) correspondingly, and the false error probability is obtained via substituting the derived error probability into (6.35). Further-more, the performances over the single-path Rayleigh channel are calculated via the integrations in (6.51) and (6.58) correspondingly with respect

f_h(x) = 2

is the average channel energy.

Additionally, we consider the performance in the case when the PDP is aware. Straightfor-wardly, letting T = 1 in (6.50), we can obtain the error probability in this case. Performing the integration, it exists the closed-form solution as given by

P_e(σ_h²) = σ²_w+ σ_h²

if σ_w² is larger than σ_h²; i.e., in the extreme low SNR condition. When the channel response is a single-path Rayleigh channel, by averaging the false detection probability with respect to f_h(σ_h²) = _Eσ²2 averaged SNR reciprocal. If σ²_w is much larger than Eσ_h², we can obtain the approximated P_f given by