Application of wavelet packet technology in suppressing narrowband interference

Wavelet analysis is a new time-frequency analysis method developed in the past decade. It overcomes the short-time Fourier transform's single-resolution defect, has the characteristics of multi-resolution analysis, and has the ability to characterize local information of signals in both time and frequency domains. Wavelet packet analysis is an extension of wavelet analysis. Its basic idea is to concentrate information energy, find order in details, filter out the rules, and provide a more sophisticated analysis method for signals. It divides the frequency band into multiple levels, further decomposes the high-frequency part that is not subdivided by multi-resolution analysis, and can adaptively select the corresponding frequency band according to the characteristics of the analyzed signal to match it with the signal spectrum, thereby improving the time-frequency resolution. We can use wavelet packet decomposition technology to filter out interference signals based on the decomposition characteristics of wavelet packets.

1 Basic principles of wavelet packet analysis

1.1 Wavelet transform

The continuous wavelet transform of the signal x(t) is defined as:

This formula is equivalent to the signal x(t) passing through a

finite impulse bandpass filter (FIR) with a transfer function of (ω). Selecting different m values ​​is equivalent to the signal passing through different bandpass filters, so that signals of different frequency bands can be separated.

The process of using filter banks to implement wavelet transform WT is analyzed as follows: After the analyzed signal passes through the mirror filter, the signal band is divided into two frequency bands, low frequency and high frequency. The low frequency signal is down-sampled, decomposed by the next mirror filter, and divided again. This process is repeated continuously. The signal band can be divided into (ω/2j, ω/2j+1) through the filter bank. The decomposition process can be represented by Figure 1.

In Figure 1, Ai and Di represent the approximation and details of the signal respectively, and LPi and HPi represent low-pass and high-pass filters of different scales respectively.

1.2 Wavelet Packet Transform

Wavelet packet transform is based on wavelet transform and is defined as:

h0 and h1 in the formula are equivalent to low-pass and high-pass filters of length 2N.

The process of using filter banks to implement wavelet packet transform WPT is similar to WT transform. The difference between the two is that the WT filter bank continuously divides the low-frequency band into two, while WPT divides the high-frequency and low-frequency bands into two at the same time, and finally the entire band is divided into uniform bands.

The decomposition process can also be represented by a wavelet packet decomposition tree, as shown in Figure 2.

In Figure 2, LPi, HPi, L\'Pi, H\'Pi represent low-pass and high-pass filters of different scales and branches respectively. Here, the signal S can be expressed in many decomposition ways, such as:

The actual processing process generally determines the strategy of further decomposition based on the problem to be solved and the energy distribution of the signal. There are too many ways to decompose wavelet packets. For one-dimensional signals, each decomposition divides the original coefficients into two groups of coefficients. For a signal of length N, if it is decomposed to L layers, there are a total of α=2L decomposition methods. Each decomposition method corresponds to a wavelet tree. If a certain discrimination method is introduced to select the optimal wavelet tree that meets certain standards from multiple wavelet trees, the amount of calculation can also be greatly reduced. The characteristic function that guides the decomposition of wavelet packets is mainly the entropy of information. Entropy is a quantity that measures the regularity of information. The main entropies are Shannon entropy, P-order standard entropy, logarithmic energy entropy, threshold entropy, and SURE entropy. Their definitions are not detailed here.

2 Simulation analysis experiment

In Matlab environment, the voice signal bluetooth.wav is pulse coded modulated ADPCM using Simulink toolbox, and then BPSK modulated and directly multiplied by the m sequence with a period of 64 for spread spectrum. The bandwidth of the spread spectrum signal S1(t) is 896 kHz; the carrier frequency of the single tone noise S2(t) is 45 kHz; the signal-to-noise ratio is -15 dB, and the two signals are superimposed to obtain W(t), W(t)=S1(t)+S2(t), as shown in Figure 3.

The power spectral density PSD of the voice spread spectrum signal before and after mixing with the single tone noise is shown in Figure 4, and the time domain waveform before and after interference is shown in Figure 5.

[page] The bit error rate of the speech spread spectrum sequence mixed with single-tone noise after despreading and demodulation is 3.92e-1. Subjectively, it is impossible to distinguish the speech content. The following uses wavelet packet decomposition technology to remove single-tone noise, selects Shannon entropy, and uses the principle formula (3), (4) and the wavelet tree decomposition block diagram shown in Figure 2. In the Matlab simulation environment, the wpdencmp function is used, and the "db43" wavelet packet is used for 4-layer decomposition. The global domain value is 5.035, and the soft decision criterion is applied to extract the spread spectrum sequence de.mat. Since the wavelet tree can divide the high and low frequency bands, it can more effectively lock the narrowband interference component.

After despreading and demodulating the de.mat data, the bit error rate is 1.429e-4, which is 3 orders of magnitude higher and the performance is greatly improved. The denoised digital audio signal is decoded by ADPCM, and the resulting time domain diagram is shown in Figure 7. Listening again, the speech content can be clearly distinguished, with only a very small amount of background noise. If the speech information needs to be further enhanced, signal feature extraction and other processing methods can be used to remove other noises, which will not be described in detail in this article.

In order to further highlight the denoising effect of wavelet packets, we use the short-time Fourier transform method, which also has the characteristics of time domain localization, into the experimental program. The time domain waveform of the denoised speech is shown in Figure 8.

Comparing Figures 7 and 8 with the time domain waveform of the original speech signal, Figure 8 loses more speech details. From the subjective listening effect, it is not as good as the restoration effect after wavelet denoising. Although the short-time Fourier transform can describe the frequency information in a certain local time period, since only the same window function is added in the whole process, it is not suitable for the different requirements of high and low frequency changes of the signal.

3 Conclusion

This paper discusses the principle and application of removing single audio interference of speech spread spectrum signal based on wavelet packet analysis technology. The arbitrary multi-scale decomposition characteristics and good time and frequency domain localization characteristics of wavelet packet transform can be used to quickly track and determine the time and frequency domain positions of signal components, especially for the case where the spread spectrum signal is mixed with narrowband interference. The simulation experimental results show that the use of wavelet packet analysis can obtain satisfactory denoising effect.