An improved adaptive pulse compression method based on MMSE criterion

Abstract

The present invention discloses an improved adaptive pulse compression method based on the MMSE criterion, which is used in the field of radar signal processing technology. The method of the present invention comprises: firstly, establishing an echo signal model, and regarding the echo signal sequence corresponding to a certain distance unit as the superposition of the signal component of the current distance unit and the signal component and noise of the adjacent distance unit; then, on the basis of the MMSE criterion, introducing the constraint term of the minimum filtering result of the adjacent distance unit, and establishing an improved filter cost function based on the MMSE criterion; then, optimizing and solving the adaptive filter cost function, and obtaining an improved expression of the optimal filter based on the MMSE criterion; finally, obtaining the improved filter estimation value based on the MMSE criterion and the pulse compression result of the echo signal by an iterative operation method. The present invention can suppress the range sidelobes of surrounding targets in a targeted manner during the pulse compression process, thereby improving the radar's detection capability for weak targets.

Description

Technical Field

[0001] The present invention relates to the technical field of radar signal processing, and in particular to an improved adaptive pulse compression method based on a minimum mean-square error (MMSE) criterion. Background Art

[0002] Pulse compression technology is a key component of radar signal processing. It can extend the signal's range and improve range resolution, providing effective performance support for radar systems. Currently, the most widely used pulse compression method is the matched filter, an optimal linear filter based on the maximum output signal-to-noise ratio criterion. However, in practical applications, when there are multiple adjacent point targets in the range direction, the sidelobes of the matched filter may mask weaker targets, leading to weaker target obscuration.

[0003] To suppress the sidelobe effects of matched filters, various pulse compression methods have been proposed, including least squares (LS), mismatched filters, the CLEAN algorithm, adaptive pulse compression methods based on the MMSE criterion, and pulse compression repair (PCR). LS is the optimal filter in the sense of least mean squares, but it is a fixed-weight filter determined by the transmitted signal and does not account for the influence of strong scattering points outside the processing window. Mismatched filters reduce the impact of sidelobes to a certain extent at the expense of signal-to-noise ratio, but their sidelobe suppression effectiveness is limited. The CLEAN algorithm sequentially removes large targets and their sidelobe responses by estimating their locations, but is not ideal for extracting weak targets.

[0004] Therefore, based on the shortcomings of non-adaptive pulse compression techniques, MMSE-based adaptive pulse compression techniques, such as those described in Reference 1 (Blunt SD, Gerlach K. Adaptive pulse compression via MMSE estimation [J]. IEEE Transactions on Aerospace and Electronic Systems, 2006, 42 (2): 572-584.), have attracted widespread attention. This method uses the results of non-adaptive pulse compression as prior knowledge and obtains the optimal filter under the MMSE criterion through iterative adaptive means, which can effectively suppress range sidelobes. For example, PCR, such as that described in Reference 2 (Gerlach K, Blunt SD. Radar pulse compression repair [J]. IEEE Transactions on Aerospace and Electronic Systems, 2007, 43 (3): 1188-1195.), is a post-pulse compression method that mainly performs adaptive pulse compression on the results of matched filtering, and can suppress range sidelobes with a smaller number of iterations. However, PCR, like adaptive pulse compression techniques based on the MMSE criterion, uses the complete echo signal to construct the filter cost function, failing to consider the influence of surrounding pixels on the filtering results. This leaves room for improvement in range sidelobe suppression. Therefore, to more effectively suppress range sidelobes and improve weak target detection, it is necessary to improve existing adaptive pulse compression methods based on the MMSE criterion and further enhance the effectiveness of radar signal pulse compression results. Summary of the Invention

[0005] To address the existing pulse compression problems, the present invention provides an improved adaptive pulse compression method based on the MMSE criterion. First, an echo signal model is constructed. Then, the suppression term of the adjacent range unit filtering results is combined with the MMSE criterion to construct a filter cost function. Finally, specific steps for filter optimization are given. This method can suppress range sidelobes during the pulse compression process and improve the radar's detection capability for weak targets.

[0006] The improved adaptive pulse compression method based on the MMSE criterion of the present invention comprises the following steps:

[0007] Step 1: Establish an echo signal model and regard the echo signal sequence y(l) corresponding to the lth range unit as the echo signal component y of the current range unit t (l) The echo signal component y of the adjacent range unit c (l) and the superposition of noise n(l);

[0008] Step 2: Based on the MMSE criterion, the constraint term of the minimum filtering result of adjacent distance units is introduced to establish an improved filter cost function based on the MMSE criterion;

[0009] First, the filter cost function of the lth distance unit based on the MMSE criterion is constructed as follows:

[0010] P=E[x(l)-w(l) H y(l) 2 ];

[0011] Where x(l) is the impulse response of the sampled l-th distance unit, w(l) is the filter coefficient, the superscript H indicates the conjugate transpose, and E[·] indicates the expectation;

[0012] On the basis of MMSE criterion, the constraint term |w(l) of the minimum filtering result of adjacent distance units is introduced. H y c (l) 2 , add the constraint term to the filter cost function, and establish the improved filter cost function based on the MMSE criterion as follows:

[0013] J=E[x(l)-w(l) H y(l) 2 +λw(l) H y c (l) 2 ];

[0014] Where λ is a regularization parameter, which is used to control the degree of suppression of the filtering results of adjacent distance units;

[0015] Step 3: Optimize and solve the adaptive filter cost function J to obtain the improved expression of the optimal filter based on the MMSE criterion;

[0016] Step 4: Calculate the improved MMSE-based filter estimation value and range pulse compression result through iterative calculation.

[0017] In step 1, the radar transmission signal is s=[s1 s2 … s N ] T , N is the signal length, and the superscript T indicates transposition; let the echo signal of length N starting from the lth range unit be y(l) = [y(l)y(l+1)…y(l+N-1)] T , l=1,2,…L, L is the number of sampling points of the distance scene; the specific expression of the echo signal is: y(l)=Sx(l)+n(l), where x(l) is the distance unit impulse response vector with a length of 2N-1, and x(l) is decomposed into:

[0018]

[0019] Among them, the lth distance unit is the current distance unit, x t (l) represents the impulse response of the lth distance unit, x c (l) represents the impulse response of the adjacent range unit; then the echo signal y(l) can be decomposed into:

[0020] y(l)=S(x t (l)+x c (l))+n(l)=x(l)s+Sx c (l)+n(l)=y t (l)+y c (l)+n(l);

[0021] Among them, the matrix composed of the transmitted signal shift n(l) is the noise signal; y t (l) = x(l)s is the echo signal component on the lth distance unit, y c (l) = Sx c (l) is the echo signal component on the adjacent distance unit.

[0022] The step three includes: first, finding the complex gradient of the filter coefficient w for the filter cost function J, as follows:

[0023]

[0024] make The expression for w is: w(l)={E[y(l)y H (l)+λy c (l)y c H (l)]} -1 E[y(l)x H (l)];

[0025] Then, substitute y(l)=Sx(l)+n(l) into the expression of w, and assume that the range impulse responses of each target are uncorrelated, and the noise is also uncorrelated with the range impulse response. Then, the expression of the adaptive filter of the lth range unit is obtained as follows:

[0026] w(l)=ρ(l)[Sdiag(ρ c )S H +ρ(l)ss H +R] -1 s;

[0027] Among them, the power expectation of the impulse response of the lth distance unit ρ(l) =| x(l) 2 ; Power matrix ρ in the distance direction c =(1+λ)[ρ(l-N+1) … ρ(l-1) 0 ρ(l+1) … ρ(l+N-1)] T ; The correlation matrix of white noise R=E[n(l)n(l) H ].

[0028] In the fourth step, an estimated value of w(l) is obtained by an iterative operation. During the iterative process, the value of λ is set from small to large, and the correlation matrix R of the white noise is ignored. The iterative operation includes the following steps:

[0029] Step 1) Consider the radar transmission signal as the initial value of the filter: w i (l) = s, i = 1, where i is the number of iterations. The result of the matched filtering of the lth distance unit is The matched filtering results of each distance unit are regarded as the estimated values ​​of each distance unit:

[0030] Step 2) The estimated value of the distance unit of the i-th pulse compression is padded with zeros, and a zero-value sequence of length N-1 is extended before and after it: Calculate the corresponding power estimate On this basis, the adaptive filter value w at the i+1th iteration is calculated i+1 (l) and then update the estimated value of the range cell with the new adaptive filter: Get the pulse compression results of each distance unit

[0031] Step 3) Repeat step 2 until a preset number of iterations or a sidelobe suppression effect is achieved, terminate the iteration, and output the final adaptive filter coefficients and the estimated pulse compression results of each range unit.

[0032] Compared with the prior art, the present invention has the following beneficial effects: the improved adaptive pulse compression method based on the MMSE criterion of the present invention first models the echo model, then introduces the minimum filtering result of adjacent distance units as a constraint term into the MMSE cost function, then optimizes and solves the cost function, and finally obtains the estimated value of the filter and the estimated value of the distance unit through an iterative update method. By introducing the minimum filtering value of adjacent units, the method of the present invention can specifically suppress the range sidelobes of surrounding targets, thereby improving the radar's detection capability for weak targets. The core algorithm of the method of the present invention is simple to implement and has high practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1This is a flow chart of an improved adaptive pulse compression method based on the MMSE criterion according to the present invention;

[0034] Figure 2 1 is a graph comparing the results of using different pulse compression methods on three target signals of equal strength according to an embodiment of the present invention;

[0035] Figure 3 This is a comparison diagram of the results of using different pulse compression methods on echo signals with weak targets according to an embodiment of the present invention. DETAILED DESCRIPTION

[0036] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0037] The present invention is an improved adaptive pulse compression method based on the MMSE criterion. Figure 1 As shown, the specific steps include the following steps 1 to 4.

[0038] Step 1: Construct the echo signal model. Assume that the radar transmission signal is s=[s1 s2 … s N ] T , N is the length of the radar transmission signal, and the impulse response of the range scene is L sampling points x=[x(1) x(2) … x(L)] T The superscript T indicates transposition. The ideal radar echo range profile is L+N-1 in length. Assume that starting from the lth range unit, the radar echo vector of length N is y(l) = [y(l) y(l+1) … y(l+N-1)] T ,l=1,2,…L, its specific expression is: y(l)=Sx(l)+n(l). Among them:

[0039] Noise component n(l) = [n(l-N+1) … n(l) … n(l+N-1)] T ;

[0040] The impulse response vector of the range cell x(l) = [x(l-N+1) …x(l) …x(l+N-1)] T ;

[0041] The matrix composed of the transmitted signal shift

[0042] By observing the impulse response vector x(l) of the range unit with a length of 2N-1, it can be seen that in addition to the impulse response x(l) of the lth range unit at the center point, the impulse responses of the surrounding 2N-2 range units also participate in the construction of the echo signal. Therefore, x(l) is decomposed into x t (l) and x c (1) Two parts:

[0043] x(l)=[0 … 0 x(l) 0 … 0] T +[x(l-N+1) … x(l-1) 0 x(l+1) … x(l+N-1)] T

[0044] =x t (l)+x c (l)

[0045] Among them, the lth distance unit can be regarded as the current distance unit, x t (l) represents the impulse response of the current distance unit, x c (l) represents the impulse response of the adjacent distance unit.

[0046] The echo signal can be decomposed into:

[0047] y(l)=S(x t (l)+x c (l))+n(l)=x(l)s+Sx c (l)+n(l)=y t (l)+y c (l)+n(l);

[0048] Therefore, the echo signal y(l) can be regarded as the echo signal component y on the lth range unit t (l) = x(l)s and the interference signal component y on the adjacent distance unit c (l) = Sx c (l) and the superposition of the noise signal n(l).

[0049] Step 2: Establish an improved filter cost function based on the MMSE criterion.

[0050] The echo signal y(l) is pulse compressed, and the corresponding pulse compression in the lth range direction is: Where w(l)=[w(l) w(l+1) … w(l+N-1)] T is the filter coefficient, the superscript H represents the conjugate transpose, is the estimated value of the impulse response x(l) of the lth range unit.

[0051] Based on the MMSE criterion, the filter cost function of the lth distance unit can be constructed as follows:

[0052]

[0053] Where E[·] represents the expectation.

[0054] Because the MMSE-based filter cost function only considers the true value x(l) and the estimated value of the l-th distance unit In order to achieve sidelobe suppression more effectively, the method of the present invention adds the minimum filtering result of adjacent distance units as a constraint term to the filter cost function, and establishes an improved filter cost function based on the MMSE criterion as follows:

[0055] J=E[x(l)-w(l) H y(l) 2 +λw(l) H y c (l) 2 ]

[0056] Where λ is a regularization parameter, which is used to control the degree of suppression of the filtering results of adjacent distance units.

[0057] As can be seen from the above equation, the cost function J proposed by the method of the present invention is composed of a data fidelity term based on MMSE and a filtering result suppression term for adjacent range bins. The purpose of this cost function is to restore the impulse response of the lth range bin while suppressing the influence of surrounding range bins on the filtering result.

[0058] Step 3: Optimize and solve the filter cost function J to obtain the improved expression of the optimal filter based on the MMSE criterion.

[0059] First, find the complex gradient of w with respect to J:

[0060] make The expression for w is: w(l)={E[y(l)y H (l)+λy c (l)y c H (l)]} -1 E[y(l)x H (l)].

[0061] Substitute y(l)=Sx(l)+n(l) into the expression of w, assuming that the noise and the range impulse response are uncorrelated, and the range impulse responses of each target are also uncorrelated, then:

[0062] E[y(l)x H (l)] = ρ(l)S, where the power expectation of the impulse response of the lth range cell is: ρ(l) = |x(l)| 2 ;

[0063] E[y(l)y H (l)+λy c (l)y cH (l)]=Sdiag(ρ c )S H +ρ(l)ss H +R;

[0064] in:

[0065] Correlation matrix of white noise: R = E[n(l)n(l) H ];

[0066] Power matrix in the distance direction: ρ c =(1+λ)[ρ(l-N+1) … ρ(l-1) 0 ρ(l+1) … ρ(l+N-1)] T .

[0067] Therefore, the adaptive filter coefficient of the lth distance unit is obtained as:

[0068] w(l)=ρ(l)[Sdiag(ρ c )S H +ρ(l)ss H +R] -1 s;

[0069] Where diag represents the diagonal function.

[0070] Step 4: Calculate the improved MMSE-based filter estimation value and range pulse compression result through iterative calculation.

[0071] Since R is the correlation matrix of white noise, it cannot be obtained specifically during the calculation process, and ρ cThe power data contains noise components, so the calculation can be ignored in the specific process. Furthermore, in the improved MMSE-based adaptive filter expression, the impulse response of each range cell is unknown, making it difficult to obtain accurate adaptive filter coefficients. Therefore, an iterative calculation is used to obtain an estimated value of w(l). Because the calculation of filter w(l) requires the pulse compression results of the N-1 range cells before and after the current range gate, the existing pulse compression results need to be zero-padded during the iteration process. This means that a sequence of zeros with a length of N-1 is added to the head and tail of the existing pulse compression results. During the iteration process, the value of λ represents the degree of suppression of the adjacent distance unit filtering results. The value range of λ is [0, ∞]. When the value of λ approaches 0, the degree of suppression of the adjacent distance unit filtering results is small. When the value of λ approaches infinity, the degree of suppression of the adjacent distance unit filtering results is large but may cause a decrease in the main lobe intensity. Therefore, during the iterative calculation process, the value of λ should be determined based on experience and increase with the increase in the number of iterations to gradually enhance the degree of suppression of the adjacent distance unit filtering results.

[0072] Step 1) Consider the radar transmission signal as the initial value of the filter: w i (l) = s, i = 1, where i is the number of iterations. The result of the matched filtering of the lth distance unit is The matched filtering results of each distance unit are regarded as the estimated values ​​of each distance unit:

[0073] Step 2) The estimated value of the distance unit of the i-th pulse compression is padded with zeros, and a zero-value sequence of length N-1 is extended before and after it: Calculate the corresponding power estimate On this basis, the adaptive filter value w at the i+1th iteration is calculated according to the formula obtained in step 3. i+1 (l) and then update the estimated value of the range cell with the new adaptive filter: Get the pulse compression results of each distance unit

[0074] Step 3) Repeat step 2 until a preset number of iterations or a sidelobe suppression effect is achieved, terminate the iteration, and output the final adaptive filter coefficients and the estimated pulse compression results of each range unit.

[0075] Example

[0076] To verify the effectiveness of the method of the present invention in sidelobe suppression, this embodiment conducts a pulse compression experiment under multiple targets and compares it with the existing matched filtering method (MF), MMSE method, and PCR method.

[0077] Assume that the transmitted signal is a linear frequency modulation signal with a signal bandwidth of 50MHz, a time width of 1μs, and a sampling rate of 150MHz. There are three equally strong target signals with a spacing of 5 range gates in the radar echo. The pulse compression results using different methods are as follows: Figure 2 The result of using the matched filter method to compress the signal pulse is shown in Figure 2 As shown by the black solid line in the middle, it can be seen that the matched filter can obtain the main lobes of the three target signals, but due to the large amplitude of the side lobes of the distance targets at each point, the main lobes of the three targets are not well distinguished. The PCR method is used to perform pulse compression on the echo signal to obtain the result as shown in the figure below. Figure 2 As shown by the green solid line, it can be seen that the PCR method can effectively distinguish the main lobes of the three target signals, and the main-to-side lobe ratio is reduced to 107dB. The echo signal is compressed by adaptive pulse compression based on MMSE, and the results are shown in the figure below. Figure 2 As shown by the blue solid line, the method is iterated twice. It can be seen that the MMSE-based adaptive pulse compression method can well compress the main lobes of the three target signals, and the main-to-sidelobe ratio is reduced to 151dB. The improved MMSE-based adaptive pulse compression method of the present invention is used for pulse compression. The filter is iterated twice. The values ​​of λ in the two iterations are 100 and 500 respectively. The results are shown in FIG. Figure 2 As shown by the red solid line in the figure, the main-to-sidelobe ratio is 198 dB. It can be seen that compared with the matched filtering, MMSE and PCR methods, the method of the present invention can more effectively suppress the target sidelobe while detecting the target signal strength.

[0078] Assume that there are two targets with a distance of 3 range gates in the radar echo, but the target echo energy differs by 20dB. Use each method to perform pulse compression on the radar echo with weak targets. The results are as follows: Figure 3 The result of matched filtering is shown in Figure 3 As shown by the black solid line in the figure, it can be seen that due to the large side lobe amplitude of the strong point target, the weak target signal is masked by the range side lobe of the strong target signal, making it impossible to effectively detect the information of the weak target signal. The results of the PCR pulse compression method are shown in Figure 3 As shown by the green solid line, it can be seen that the PCR method can effectively detect the main lobe of the weak target, and its main-to-side lobe ratio is reduced to 134dB. The filtering results of the adaptive pulse compression method based on MMSE are shown in Figure 3As shown by the blue solid line, the method is iterated twice. It can be seen that the method can effectively compress the main lobe of the weak target signal, and the main-to-side lobe ratio is reduced to 137dB. The improved adaptive pulse compression method based on the MMSE criterion of the present invention is used for pulse compression. The filter is iterated twice. The values ​​of λ in the two iterative updates are 5 and 1000 respectively. The results are shown in FIG. Figure 3 As shown by the red solid line in the figure, its main-to-sidelobe ratio is 173 dB. It can be seen that compared with the matched filtering, MMSE and APC methods, the method of the present invention can more effectively suppress the range sidelobe of strong point targets while detecting the signal strength of weak targets, thereby improving the radar's detection capability for weak targets.

[0079] Therefore, through the pulse compression experimental results of multi-target strong point signals and the pulse compression experimental results with weak targets, it can be proved that the method of the present invention is a pulse compression method that effectively suppresses range sidelobes and detects target signals.

[0080] Except for the technical features described in the specification, all other technical features are known to those skilled in the art. The present invention omits descriptions of well-known components and well-known technologies to avoid redundancy and unnecessary limitation of the present invention. The implementation methods described in the above embodiments do not represent all implementation methods consistent with the present application. Based on the technical solution of the present invention, various modifications or variations that can be made by those skilled in the art without creative effort are still within the scope of protection of the present invention.

Claims

1. An improved adaptive pulse compression method based on MMSE criterion, characterized in that: The following steps are involved: Step 1: Establish an echo signal model and regard the echo signal sequence y(l) corresponding to the current l-th distance unit as the echo signal component y of the current distance unit t (l) The echo signal component y of the adjacent range unit c (l) and the superposition of the noise signal n(l); Step 2: Based on the MMSE criterion, the constraint term of the minimum filtering result of adjacent distance units is introduced to establish an improved filter cost function based on the MMSE criterion; First, the filter cost function of the lth distance unit based on the MMSE criterion is constructed as follows: P=E[|x(l)-w(l) H y(l)| 2 ]; where w(l) is the filter coefficient, x(l) is the impulse response of the l-th distance unit, the superscript H indicates the conjugate transpose, and E[·] indicates the expectation; Then the constraint term |w(l) of the minimum filtering result of the adjacent distance unit is used H y c (l)| 2 Add it to the filter cost function and establish an improved filter cost function based on the MMSE criterion J = E[|x(l)-w(l) H y(l)| 2 +λ|w(l) H y c (l)| 2 ], where λ is a regularization parameter used to control the degree of suppression of the filtering results of adjacent distance units; Step 3: Optimize and solve the adaptive filter cost function J to obtain an improved expression of the optimal filter based on the MMSE criterion; Step 3 includes: First, the complex gradient of the filter coefficient w is calculated for the filter cost function J as follows: ▽ w J(w)=E{-y(l)x H (l)+y(l)y H (l)w(l)+λy c (l)y c H (l)w(l)}; Order ▽ w J(w)=0, the expression of w is: w(l)={E[y(l)y H (l)+λy c (l)y c H (l)]} -1 E[y(l)x H (l)]; Then, substitute y(l)=Sx(l)+n(l) into the expression of w, and assume that the range impulse responses of each target are uncorrelated, and the noise is also uncorrelated with the range impulse response. Then, the expression of the adaptive filter of the lth range unit is obtained as follows: w(l)=ρ(l)[Sdiag(ρ c )S H +ρ(l)ss H +R] -1 s; Where diag represents the diagonal function; the power expectation of the impulse response of the lth distance unit is ρ(l)=|x(l)| 2 ; Power matrix ρ in the distance direction c =(1+λ)[ρ(l-N+1)…ρ(l-1) 0 ρ(l+1) … ρ(l+N-1)] T ; The correlation matrix of white noise R=E[n(l)n(l) H ]; S is a matrix composed of the shifts of the radar transmission signal s; Step 4: Calculate the improved filter estimate based on the MMSE criterion and the pulse compression result of the echo signal through an iterative operation method; wherein MMSE stands for minimum mean square error; said step 4 uses an iterative operation method to obtain the estimated value of w(l), during the iterative process, setting the value of λ from small to large, and setting the correlation matrix R that ignores the calculation of white noise; said iterative operation includes the following steps: Step 1) Consider the radar transmission signal s as the initial value of the filter coefficient: w i (l) = s, i = 1, where i is the number of iterations. The result of the matched filtering of the lth distance unit is The result of matched filtering of each distance unit is regarded as the estimated value of the distance unit: Step 2) The estimated value of the distance unit of the i-th pulse compression is padded with zeros, and a zero-value sequence of length N-1 is extended before and after it: Calculate the corresponding power estimate Then calculate the adaptive filter coefficient w at the i+1th iteration i+1 (l), and then use the new adaptive filter coefficients w i+1 (l) Update the estimated value of the distance unit: Get the pulse compression results of each distance unit Step 3) Repeat step 2 until a preset number of iterations or a sidelobe suppression effect is achieved, terminate the iteration, and output the final adaptive filter coefficients and the estimated pulse compression results of each range unit.

2. The method according to claim 1, characterized in that In the step 1, the radar transmission signal is represented as s=[s1 s2…s N ] T , N is the signal length, and the superscript T indicates transposition; let the echo signal of length N starting from the lth range unit be y(l) = [y(l) y(l+1) … y(l+N-1)] T ,l=1,2,…L, L is the number of sampling points of the distance scene, and the echo signal is expressed as: y(l)=Sx(l)+n(l); Matrix composed of radar transmission signal shifts The impulse response vector of the range cell x(l) = [x(l-N+1)...x(l)...x(l+N-1)] T ; Noise component n(l) = [n(l-N+1) … n(l) … n(l+N-1)] T ; Decompose the impulse response vector x(l) of the distance unit with the lth distance unit located at the center and a length of 2N-1 into the impulse response x t (l) and the impulse response component x of the adjacent 2N-2 distance units c (l); then the echo signal y(l)=S(x t (l)+x c (l))+n(l)=x(l)s+Sx c (l)+n(l)=y t (l)+y c (l)+n(l).

Images