# Performance Analysis of Shrinkage Linear Complex-Valued LMS Algorithm

## Summary (1 min read)

### Introduction

- It has been successfully applied in the system identification, beamforming and frequency estimation [1]–[5].
- This letter provides the theoretical analysis of the SL-CLMS algorithm proposed in [13].
- The symbols E(·) and Tr(·) stand for the mathematical expectation and trace of a matrix, respectively.

### II. REVIEW OF THE SL-CLMS ALGORITHM

- T is the input vector, and η(k) accounts for the background noise with zero-mean and variance σ2η = E[|η(k)| 2].
- In the SL-CLMS algorithm, the weight update is given by w(k +.

### III. PERFORMANCE ANALYSIS OF THE SL-CLMS ALGORITHM

- The authors make the following assumptions, which are widely used for analyzing VSS adaptive algorithms.
- The background noise η(k) is zero-mean circular white Gaussian and statistically independent of the noise-free a priori error signal ea(k) = w̃ H(k)x(k) and input vector x(k), where w̃ = w(k)−wo is the weight error vector, also known as A1.
- Assumption A1 is one of the most common assumptions in the adaptive filtering theory [1], [16].
- This assumption might not be very accurate for fast varying step-size, see simulation results below.

### IV. SIMULATION RESULTS

- The correlated signal is used as the input.
- Lines without marks: simulation results; lines with marks: theoretical results.
- It is seen that the theoretical prediction is accurate in all the cases, apart from the transient period when the step- size varies very quickly.

### V. CONCLUSION

- The authors have presented the theoretical analysis of the transient and steady-state EMSE performance of the SL- CLMS adaptive algorithm for the case of non-circular input signal and circular Gaussian noise.
- Comparison of simulation and theoretical results for identification scenarios with dif- ferent parameters have shown that the theoretical prediction provided by their analysis is very accurate.

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##### Citations

29 citations

### Cites background or methods from "Performance Analysis of Shrinkage L..."

...8) turns into the form given in our previous work [36]....

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...3) Since the variances of the real and imaginary parts of the error signal can be different, the magnitude of the error signal may not follow the Rayleigh distribution used in [36]....

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...Therefore, we cannot use results from [36], and we need to find the PDF of |e(k)| in the following....

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9 citations

7 citations

6 citations

### Cites background from "Performance Analysis of Shrinkage L..."

...Since signals are often expressed in complex forms in many practical scenarios [17,18], adaptive filtering in complex domain is of great significance....

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5 citations

##### References

2,776 citations

### "Performance Analysis of Shrinkage L..." refers background or methods in this paper

...where σ(2)(k) is the variance of the real (imaginary) part of e(k) [30], i....

[...]

...Then, z = |e(k)| obeys the Rayleigh distribution [30] with the probability density function...

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1,987 citations

### "Performance Analysis of Shrinkage L..." refers background or methods in this paper

...It has been successfully applied in the system identification, beamforming and frequency estimation [1]–[5]....

[...]

...Assumption A1 is one of the most common assumptions in the adaptive filtering theory [1], [16]....

[...]

1,629 citations

### Additional excerpts

...Note that a more accurate secondorder approximationE{2 ea (k) σ2 e(k) } ≈ γ E[σ(2) ea (k)] E[σ2 e(k)] requires computing the factor γ = 1− = 1− cov(σ(2) ea (k),σ(2) e(k)) E[σ2 ea (k)]E[σ2 e(k)] + var(σ(2) e(k)) E[σ2 e(k)] 2 , where cov(·) denotes the covariance, and var(·) is the variance [28], [29]....

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966 citations

811 citations