自适应傅立叶分解法：高效ECG信号去噪及比较研究

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本文主要探讨了一种创新的心电图信号去噪方法，即基于自适应傅立叶分解（Adaptive Fourier Decomposition，AFD）的ECG信号处理技术。AFD的核心原理是依据信号的能量分布特性来进行信号分解，这使得它在处理频率范围重叠但能量分布差异较大的纯ECG信号和噪声时具备显著优势。传统的心电信号处理方法如小波变换、Stockwell变换、经验模式分解（Empirical Mode Decomposition，EMD）和集成经验模式分解（Integrated EMD，IEMD）往往难以有效分离这两种信号类型。文章首先介绍了AFD的基本概念，它不同于传统的频域滤波方法，能够更精确地适应信号的局部特性。在AFD过程中，通过对信号进行迭代分解，算法可以根据噪声信号的信噪比（Signal-to-Noise Ratio，SNR）动态调整分解策略，以达到最优的去噪效果。这一停止准则的引入，使得AFD能够根据不同噪声水平自动调整分解的复杂度，提高了去噪的灵活性和准确性。为了验证AFD方法的有效性和性能，研究者利用了一个ECG模型生成了合成ECG信号，并将其与MIT-BIH心律失常数据库中的真实ECG信号进行了对比测试。结果显示，与上述提到的传统去噪方法相比，基于AFD的ECG去噪方法在抑制噪声的同时，对QRS波群的保留更加完好，从而改善了QRS检测的准确性。此外，实验数据还显示，当面对不同强度的高斯噪声时，AFD表现出更强的鲁棒性和适应性，尤其是在低信噪比条件下，其去噪效果更为明显。总结来说，这篇文章提出了一种高效且灵活的ECG信号去噪策略，通过自适应的傅立叶分解，能够在不破坏重要生理信息的前提下，有效地去除心电信号中的噪声成分。这对于医疗设备的信号分析、生物电信号处理以及心率监测等领域具有重要意义，为进一步提升心电图信号的质量和分析精度提供了新的思路和技术支持。

signal is expressed by

()= ()+ () ( )st ht wt 10

where h(t) denotes the pure signal to be recovered, and w(t) de-

notes the independent noise which energy is smaller than that of h

(t), then the AFD will extract MCs of the pure signal ﬁrst during the

decomposition. Therefore, such MCs in the ﬁrst N decomposition

levels can be selected to approximate the pure signal. Based on the

decomposition components of G(t), the reconstructed ECG signal

(

can be expressed as

⎪⎪

⎧

⎨

⎩

⎫

⎬

⎭

∑

()= ( )

()

{}

ht G e B eRe , .

na n

The key problem is to determine a suitable maximum decom-

position level N adaptively. To solve this problem, a stop criterion

of the AFD is proposed based on the estimated SNR of the noisy

signal. Since the denoising aims to make the MSE deﬁned as

−

()

−

LEh hMSE

as small as possible where L denotes the total data length [31,32],

the reconstructed ECG signal at the suitable decomposition level N

should be able to minimize

⎛

⎝

⎜

⎞

⎠

⎟

()

−+

()

SNR /10

where

deﬁned as

()

() ( )

SNR 10log

denotes the estimated SNR of the noisy signal. According to

(6) and (11), the energy of the reconstructed ECG signal can be

represented as

∑

() =

()

{}

ht G e

Then, combining (13) and (15), the suitable maximum decom-

position level N can be determined by solving



⎧

⎨

⎪

⎩

⎪

⎛

⎝

⎜

⎞

⎠

⎟

⎫

⎬

⎪

⎭

⎪

⎫

⎬

⎪

⎭

⎪

{

()

∑

−+

()

>∈

min

SNR /10

Since the standard remainder R

(t)in(1) satisﬁes

() ≤

()

N 17

where M is a constant that is related to the energy of the de-

composed signal [33], the energy of the standard remainder R

(t)

decreases as the decomposition level N increases. In other words,

the energy ratio of the noisy signal to the reconstructed signal

decreases monotonically when the decomposition level N in-

creases, which means that (16) only has one optimization solution.

Therefore, (16) can be used as the stop criterion of the iterative

process of the AFD in the ECG denoising process. The objective

function value in (16) is evaluated at each decomposition level. At

ﬁrst decomposition levels, the objective function values will de-

crease. When the objective function achieves its minimum point,

the decomposition iteration should be stopped. The ﬁltered result

can be reconstructed by using all extracted MCs based on (11).

Algorithm 1. Computational process of the AFD-based ECG de-

noising method.

Input: s(t): noisy ECG signal; SNR

: estimated SNR of s(t).

{

}

()← ()− ()

tst stmean

;

{

}

()← ()+ ()

tst st

Initialize

, N¼ 1,

()= (

tGt

{}

e 1

, and

=B 1

;

4: repeat

⎛

⎝

⎜

⎞

⎠

⎟

←()−

+ {}{}

−

Gt G e e,

NNNaa

;

←

{}

−

;



⎧

⎨

⎩

⎫

⎬

⎭

←∈

++{}+

a arg max G , e : a

NNaN11

N 1

;

←

−

;

←+NN

;

10:

⎛

⎝

⎜

⎞

⎠

⎟

←−+

()

∑

{}

FV 1

SNR /10

;

11: until OFV achieves its minimum value

12:

{

}

←

∑

{}

Ge BRe ,

na n

;

Output:

: reconstructed ﬁltered result; N: ﬁnal decomposition

level.

In each decomposition level, one value in the a

array is de-

termined from the dictionary based on the MPP. Although the

dictionary of a

can be adaptively generated according to the

processed signals [21], the optimization problem shown in (7)

cannot be solved efﬁciently using the exhaustive searching

method. Therefore, to improve the computational efﬁciency of the

MPP implementation, the Nelder–Mead algorithm is applied in the

process of searching a

[34]. The main computational process of

the proposed method is shown in Algorithm 1.

4. Simulation results

Simulations are carried out in the MATLAB using two types of

ECG signals: the synthetic ECG signal based on an ECG model

proposed in [23] and real ECG signals from the MIT-BIH Ar-

rhythmia Database. The signal generated by the ECG model is

applied to verify the signal denoising principle shown in Section 3,

while real ECG signals are used to verify the ECG denoising per-

formance of the proposed method. The performance of the pro-

posed method is evaluated based on the SNR and the MSE. In the

following evaluations, the SNR of the reconstructed ECG signal is

calculated by

∑

()

∑

()

−

SNR 10log

where h(t) is the synthetic ECG signal or the real ECG signal, and

(

)

is the noise of the reconstructed signal which is estimated by

()=

()− () ( )wt ht ht 19

where

(

is the reconstructed signal. The MSE is calculated by

Z. Wang et al. / Computers in Biology and Medicine 77 (2016) 195–205 197

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