DME is the ranging part of TACAN adopted by the International Civil Organization. As a new international standard ranging system, DME has important applications in navigation, such as positioning service, speed and flight time, airway spacing and regional navigation.

DME consists of two parts, an airborne interrogator and a ground transponder, and calculates the distance by measuring the time of electromagnetic wave transmission in space. The block diagram of its basic working principle is shown in Figure 1.

After the pulse output by the oscillator passes the frequency division, a chatter rectangular pulse with random intervals is obtained; after the coding by the encoder, a Gauss pulse pair with fixed time intervals is generated; after the RF modulation, the RF query signal is transmitted through a non-directional antenna. As shown in Figure 2, after the signal is DME ground receiving platform,

the ground of the transponder decoder will according to the rf frequency and pulse interval between rf interrogation signal of acceptable, if the effective inquiry, decoder can produce a decoding pulse, the radio frequency modulated by omnidirectional antenna radiation also get the corresponding plane to ask signals of question and answer. The oblique distance R from the aircraft to the ground DME platform is

R = 1 2 ( T − T 0 ) C . (1)

where, C is the speed of light, T₀ refers to the time between the ground transponder receiving the question signal and sending the question signal, and T refers to the time difference recorded by the interrogator from sending the question signal to receiving the answer signal.

The coding format of the interrogation signal and the response signal of DME is Gaussian pulse pair. The mathematical model of a baseband DME pulse pair signal is as follows [3]

s ( t ) = e ∂ 2 t 2 + e ∂ 2 ( t − Δ t ) 2 (2)

Among them, the half-amplitude pulse width of both Gauss pulses is 3.5 μs, and the modulated DME signal model can be expressed as

s m ( t ) = ( e α 2 t 2 + e − α 2 ( t − Δ t ) 2 ) e j ( 2 π f DME t + θ DME ) (3)

Figure 3 shows the DME/TACAN standard pulse pair generated according to the mathematical model. The transmitted DME/TACAN signal is composed of such pulse pair continuously [4], and the arrival time of the pulse pair is random.

The main performance indexes of DME beacon transmitter are as follows.

1) Output power: the output power of the terminal DME beacon is more than 100 W, and the output power of the airway DME beacon is more than 1 KW.

2) Signal form: it has a Gaussian envelope. Since the shape of the envelope will affect the signal spectrum, the interference size of the adjacent channel will be affected.

3) Transmission rate: The frontal response pulse repetition frequency range of DME beacon transmitting base is 2700 - 700 pairs/s, the main performance index of DME beacon receiver is sensitivity, which refers to the receiver’s ability to accept weak signals, and the minimum detectable signal power is its measurement data. ICAO’s requirement for DME transponder receiver sensitivity is the need for query pulse pairs with correct intervals and nominal frequencies. When its peak power density at the transponder antenna is −103 dB W/m², the transponder responds with at least 70% response efficiency. Under the condition of the highest sensitivity and good station environment, if the peak pulse power emitted by the DME beacon station is 1 kW, its operating distance can reach 200 nm (370 km).

The transmitting frequency of the response pulse pair in the DME platform working in 64× - 126× channel overlaps with the frequency band range of the signal received by the airborne GNSS receiver [5]. These interference pulses will enter the GNSS receiver and reduce the signal-to-noise ratio of the signal, thus reducing the sensitivity of the user when capturing GNSS signals [6]. This will further increase the GNSS signal tracking failure rate. When it is serious, the availability of observation data will be reduced, and even cycle slip will be caused, resulting in greater errors in positioning results.

The second effect of interference on the GNSS receiver is that it will increase the observed code noise and phase noise, thereby reducing the accuracy of the GNSS user terminal. In addition, the interference will also cause the acquisition time of the GNSS receiver to prolong [7], resulting in GNSS the user’s first positioning time is extended. The mathematical formula of the received signal of a GNSS signal receiver affected by a DME station can be shown as [8]

S R ( t ) = S GNSS ( t ) + ∑ m = 1 M I ∑ u = 1 N u , m P m s ( t − t m , u ) e j ( 2 π f m , u t + θ m , u ) + n ( t ) (4)

Among them, S GNSS ( t ) is the received signal, s ( t ) is the DME baseband signal [9], N u , m is the total logarithm of the impulse transmitted from the m-th DME station received during the observation time, P m is the received power of the m-th DME station signal [10], and t m , u is the arrival time of the U pulse pair generated by the m-th DME platform.

Wavelet analysis and wavelet packet analysis are both suitable for the nonstationary signal analysis. Compared with smaller wave analysis [11], it can not only decompose the low-frequency part of the signal, but also the high-frequency part [12]. The wavelet packet analysis can be used to analyze the signal containing a large amount of medium and high frequency information. Perform better time frequency localization analysis, and adaptively select the best wavelet basis function according to the characteristics of the signal, thereby improving the time frequency resolution.

In the multi-resolution analysis based on wavelet transform, only the low frequency sub-band of the signal is decomposed at each level [13], and the high frequency sub-band is not decomposed. Therefore, the high frequency band resolution is poor. If the space corresponding to the low frequency band in the wavelet analysis is regarded as V j , the space corresponding to the high frequency part is regarded as W j , the high and low frequency bands here are relative to the frequency band of the upper level space; the wavelet transform only divides each space into two. The wavelet packet transform can divide the low-frequency space and the high-frequency space at the same time [6]. Relative to wavelet decomposition, the signal can be decomposed more accurately and the time-frequency resolution can be improved.

Set { V J } as a multi-resolution analysis of L 2 ( R ) , the corresponding scaling function is ϕ , and the wavelet is φ , ϕ satisfying the scaling equation [14]

ϕ ( x ) = ∑ k ∈ Z     h k ϕ − 1 , k ( x ) = 2 ∑ K ∈ Z     h k ϕ ( 2 x − k ) (4)

and

φ ( x ) = 2 ∑ k ∈ Z     g k ϕ ( 2 x − k ) (5)

among them

g k = ( − 1 ) k h ¯ − k + 1 (6)

V j − 1 = V j ⊕ W j , V j − 1 = V j ⊕ W j , V j − 1 = V j + 1 ⊕ W j + 1 ⊕ W j , which called wavelet decomposition of space, f j − 1 = f j + e j , f j − 1 = f j + 1 + e j + 1 + e j , f k ∈ V , f k ∈ V , k ≥ j − 1 etc, which are called wavelet decomposition of function.

Wavelet decomposition refers to decomposing f k , e k ( k ≥ j ) on the basis of decomposing V j − 1 = V j + 1 ⊕ W j + 1 ⊕ W j = V j + 2 ⊕ W j + 2 ⊕ W j + 1 ⊕ W j = ⋯ in a certain way.

Figure 4 shows the three-level wavelet packet decomposition tree diagram of the original signal, where S represents the low-frequency coefficient obtained by the signal through the low-pass filter [14], and D represents the high frequency coefficient obtained by the signal through the high-pass filter. Each layer of wavelet packet can divide the original frequency band into two, and the three-layer wavelet packet can divide the original frequency band into 2^K sub-bands, thereby realizing frequency band subdivision and improving frequency domain resolution. For the entire wavelet packet, it is a binary-organized band-pass filter bank from wide to narrow, and various applications can find the best combination that meets the needs.

The essence of wavelet transform is the similarity between the original signal and

the wavelet basis function. The wavelet coefficient is the coefficient of the wavelet basis function similar to the original signal. In the problem of interference signal suppression, since the GNSS signal is submerged in noise, the similarity between noise and wavelet is less than that between wavelet and DME signal, so the received signal interfered by DME impulse can be passed through wavelet. The packet change is transformed into the wavelet packet coefficient domain. The coefficient with larger absolute value is expressed as DME interference signal [15], and the coefficient with smaller absolute value is expressed as noise and satellite signal. At the same time, setting an appropriate threshold in the wavelet packet domain will make the absolute value larger the coefficient is set to zero, so as to achieve the purpose of suppressing the pulse interference signal. Finally, the processed wavelet packet coefficient is inversely transformed to obtain the signal in the time domain. The basic steps are as follows [16]:

1) Wavelet packet decomposition

Choose the appropriate wavelet and the appropriate decomposition level (denoted as K), and then perform K-level decomposition on the received signal interfered by DME.

2) Threshold quantization of high-frequency coefficients after decomposition

Set an appropriate threshold in the wavelet packet coefficient domain, and zero the coefficients whose absolute value is greater than this threshold to suppress impulse interference signals [17].

3) Inverse wavelet packet transform.

The K-th wavelet packet coefficient is inverse transformed to get the time domain waveform.

Because the satellite signal contains a lot of noise, the satellite signal and the noise can be viewed together as Gaussian noise with zero mean value ε.

Suppose the variance of noise ε is σ 2 , and its power spectral density is S ε ( ω ) = σ 2 . The variance of the wavelet coefficients after the first-level wavelet packet transformation of the filter is G 0 . The frequency response of σ 1 2 is [18]

G 0 ( ω ) = ∑ n = − ∞ ∞ g 0 ( n ) e − j n ω ,     ω ∈ [ − π , π ] (7)

Among them, g 0 ( n ) is the unit impulse response sequence, and the power spectral density of W 1 ε is:

S W 1 ( ω ) = | G 0 ( ω ) | 2 S ε ( ω ) (8)

The autocorrelation function of W 1 ε is:

R W 1 ( τ ) = 1 2 π ∫ − ∞ ∞ S W 1 ( ω ) e − j ω t d ω (9)

Therefore:

σ 2 = R W 1 ( 0 ) = 1 2 π ∫ − ∞ ∞ | G 0 ( ω ) | 2 S ε ( ω ) d ω (10)

= σ 2 2 π ∫ − π π [ ∑ n = − ∞ ∞ g 0 ( n ) e − j n ω ] [ ∑ − ∞ ∞ g 0 ( n ) e j n ω ] d ω (11)

= σ 2 ‖ g 0 ‖ 2 (12)

= σ 2 (13)

Suppose the probability density function of white noise with zero mean variance σ 2 is P s ( x ) , then there is:

P s ( x ) = 1 σ 2 π e − x 2 2 σ 2 (14)

Suppose the probability density function of white noise with zero mean variance Th is P f a , then there is:

P f a = 2 ∫ T h ∞ P s ( x ) d x = 2 ∫ T h ∞ 1 σ 2 π e − x 2 2 σ 2 d x (15)

= 2 π ∫ T h 2 σ ∞ e − t 2 d t (16)

= erfc ( T h 2 σ ) k (17)

According to the above formula, the threshold value is:

T h = σ 2 erfc − 1 ( P f a ) (18)

Each layer of wavelet packet divides the original frequency band into two, and the k-layer wavelet packet can divide the original frequency band into 2^K sub-bands, thereby realizing frequency band subdivision and improving frequency domain resolution. That is, the larger the decomposition level, the finer the division of the signal, and the better the interference suppression effect. When the decomposition gets to a certain level, continuing the decomposition will not make the performance significantly improved. In addition, Due to the wavelet packet filter is not ideal, the increase in the number of decomposition stages will cause aliasing of the subband spectrum. Therefore, the choice of series of the wavelet packet should take into account the bandwidth of the interference signal and the amount of calculation. That is to say, wavelet packet resolution after decomposition is equivalent to the frequency band of the suppressed interference signal.

In numerical analysis, we knowed that the essence of wavelet transform is the similarity between the original signal and the wavelet basis function. The wavelet coefficient is the coefficient of the wavelet basis function similar to the original signal. Therefore, the more similar the generator function is to the analyzed signal, the more local the coefficients after the wavelet transform can be, and the better the interference suppression effect. In the actual process, there is no display expression in a few wavelet functions, and it is difficult to give a certain degree of similarity between the wavelet function and the analyzed function. This experiment adopts the general selection criteria of wavelet function, in order to be able to perform multi-resolution analysis and apply the fast algorithm of wavelet transform.

In the paper, we consider the Daubechies wavelet function family, the Symlet wavelet function family, the Coifle wavelet function family, the bioorthogonal wavelet function family [6], the rbio inverse biorthogonal wavelet function family and the discrete Meyer wavelet function “dmey”. According to the capture performance of the signal after interference suppression, the effects of various wavelet functions in suppressing interference are compared.