Analysis of wireless English multimedia communication based on spatial state model equation

In 1990, Perona and Malik proposed a nonlinear anisotropic diffusion equation, namely the P-M model, in order to maintain image edge information. Definitions are as follows:

RTP / RTCP protocol is a network transmission protocol specially designed for real-time media transmission by the wireless English transmission working group of Internet Engineering Task Force (ETF). A synchronization control algorithm between wireless English media based on RTP / RTCP is proposed. The algorithm needs to process each RTCPSR (sender reports) message. The algorithm has high complexity and needs the support of network time protocol (NTP). In addition, the synchronization control is carried out by dropping frames or pausing decoding before wireless English decoding at the receiver, which not only ignores the delay jitter introduced by wireless English decoding, but also affects the normal decoding and reconstruction of video to a certain extent. According to the characteristics of space multimedia communication environment and communication equipment, this paper proposes a wireless English synchronous QoS control algorithm with low complexity and easy implementation [9].

System model is a mathematical model that reflects the causal relationship and transformation relationship between system variables. The commonly used mathematical models of discrete systems mainly include difference equation, transfer function model and state space model. Difference equation is a common mathematical model in one-dimensional system, but this model is not suitable for multi-dimensional system. The more complex the multidimensional system is, the higher the dimension of the system is. The higher the order of the difference equation (Discrete System) or differential equation (continuous time or space system) describing the system by the relationship between input and output is, the more difficult the system analysis is. Transfer function is the main mathematical model describing the system characteristics in classical control theory. It is suitable for the linear constant system of single input and single output (SISO). It can easily deal with the transient response analysis or frequency method analysis and design of this kind of system. However, this model is not suitable for multi input and multi output (MIMO) systems, time-varying systems and nonlinear systems. In addition, the transfer function is only an incomplete description of the system. It can only reflect the linear dynamic characteristics of the transfer between the input and output of the system, but can not reflect the dynamic change characteristics within the system, which greatly limits the application of the transfer function model in practice. The state space model is based on the analysis of the internal structure of the system, which is composed of the state equation describing the dynamic characteristics of the system and the output equation describing the transformation relationship between the system output variables and the state variables. It can reflect the changes of all independent variables in the multi-dimensional system, and then determine all internal motion states of the system [15]. Therefore, the state space model reflects all the information of the system dynamic behavior and is a complete description of the system behavior. The state space model has a wide range of applications. It is not only suitable for SISO linear constant systems, but also suitable for nonlinear systems, time-varying systems, MIMO systems and stochastic systems. For different systems, the mathematical expression is simple and unified. The more outstanding advantage is that the state space model adopts local calculation and uses the state variables inside the system to describe its dynamic characteristics. This property of local calculation not only greatly simplifies the mathematical expression of multi-dimensional system, but also facilitates the analysis and research of the system.

At present, the most representative state space models are Roesser model and Fornasini Marchesini model. Multidimensional Roesser state space model is based on the multidimensional system corresponding to the one-dimensional state space model of discrete-time system. It is a model established after the evolution of multidimensional system. The current input and the previous state of the system are associated with this control. Roesser state space model of one-dimensional discrete system is: (7) xX(i+1)=Ax(i)+Bu(i) xX\left({i + 1} \right) = Ax\left(i \right) + Bu\left(i \right) (8) y(i)=Cx(i)+DU(I) y\left(i \right) = Cx\left(i \right) + DU\left(I \right)

Where, u and y represent the input and output of the system respectively, and A, B, C and D are coefficient matrices. Formula and formula are the state equation describing the dynamic characteristics of the system and the output equation describing the transformation relationship between the system output variables and state variables. The two-dimensional Roesser state space model was first proposed by Roesser when he studied multi-dimensional linear filter networks in the 1970s. The Roesser state space model of single input and single output of two-dimensional linear discrete system is: (9) [xh(i+1,j)xv(i,j+1)]=A[Xh(I,J)Xv(I,J)]+BU(i,j) \left[{\matrix{{{x^h}\left({i + 1,j} \right)} \hfill \cr {{x^v}\left({i,j + 1} \right)} \hfill \cr}} \right] = A\left[{\matrix{{{X^h}\left({I,J} \right)} \hfill \cr {{X^v}\left({I,J} \right)} \hfill \cr}} \right] + BU\left({i,j} \right) (10) y(i,j)=C[Xh(I,J)Xv(I,J)]+DU(i,j) y\left({i,j} \right) = C\left[{\matrix{{{X^h}\left({I,J} \right)} \hfill \cr {{X^v}\left({I,J} \right)} \hfill \cr}} \right] + DU\left({i,j} \right)

Where, x^h (i, j) ∈ R^r₁ and x^v (i, j) ∈ R^r₂ represent the state vectors in the horizontal and vertical directions respectively; u(i, j) and y(i, j) represent input signal and output signal respectively; A, B, C, D are real matrices, and A ∈ R^r×r, B ∈ R^r×1, C ∈ R^1×r, D ∈ R^1×1, r = r₁ + r₂.

Extending the two-dimensional Roesser model in the horizontal and vertical directions to the multi-dimensional system, the Roesser state space model with multi-dimensional single input and single output is obtained as follows: (11) [X1(i1+1,…,in)…xn(i1,…,in+1)]=A[x1(i1,…,in)…xn(i1,…,in)]+BU(i1,…,in) \left[{\matrix{{{X_1}\left({{i_1} + 1, \ldots,{i_n}} \right)} \hfill \cr \ldots \hfill \cr {{x_n}\left({{i_1}, \ldots,{i_n} + 1} \right)} \hfill \cr}} \right] = A\left[{\matrix{{{x_1}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr \ldots \hfill \cr {{x_n}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr}} \right] + BU\left({{i_1}, \ldots,{i_n}} \right) (12) y(i1,…,in)=c[X1(i1,…,in)…xn(i1,…,in)]+DU(i1,…,in) y\left({{i_1}, \ldots,{i_n}} \right) = c\left[{\matrix{{{X_1}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr \ldots \hfill \cr {{x_n}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr}} \right] + DU\left({{i_1}, \ldots,{i_n}} \right)

Where, x_k(i₁,…,i_n) ∈ R^rk, k = 1,…, n represents the k-th sub state vector; DU(i₁,…,i_n) and y(i₁,…,i_n) represent input signal and output signal respectively; A, B, C and D are real matrices with sizes of r×r, r×1, 1×r, 1×1 respectively, and r=∑k=1nrk r = \sum\limits_{k = 1}^n {{r_k}} . The local state x(i₁,…,i_n) is composed of sub state vectors, which are recorded as: (13) X(i1,…,in)=[x1(i1,…,in)…xn(i1,…,in)] X\left({{i_1}, \ldots,{i_n}} \right) = \left[{\matrix{{{x_1}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr \ldots \hfill \cr {{x_n}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr}} \right]

The number of rows of local state X(i₁,…,i_n) (equal to the order of matrix A) is called the order of n-dimensional Roesser model, and the dimension r_k of sub state vector x(i₁,…,i_n) is called pair variable z_k, local order of k=l,…, n. The system is also simplified as a system (A, B, C, D). The transfer function corresponding to the system is: (14) G(Z1,…,Zn)=CZ(Ir−AZ)−1B+D G\left({{Z_1}, \ldots,{Z_n}} \right) = CZ{\left({{I_r} - AZ} \right)^{- 1}}B + D

Where, Z = diag{z₁ I_r1,…,z_n I_rn}, z₁,…,z_n represents unit delay operation. When the unit forward transfer operation si,si=zi−1 {s_i},{s_i} = z_i^{- 1} , i = 1,…,n is adopted, the corresponding transfer function of the system is: (15) G¯(s1,…,sn)=G(S1−1,…,Sn−1)=C(S−A)−1 B+D \bar G\left({{s_1}, \ldots,{s_n}} \right) = G\left({S_1^{- 1}, \ldots,S_n^{- 1}} \right) = C{\left({S - A} \right)^{- 1}}\,B + D

Where S = diag{s₁ z_n I_r1,…,s_n I_rn}. Block diagonal matrices S and Z are also usually represented by direct sum. For example, the matrix S can be expressed as S = s₁ I_r1 ⊕ ⋯ ⊕ s_n I_rn.

For n-dimensional transfer function: (16) G(Z1,…,Zn)=n(z1,…,zn)d(z1,…,zn) G\left({{Z_1}, \ldots,{Z_n}} \right) = {{n\left({{z_1}, \ldots,{z_n}} \right)} \over {d\left({{z_1}, \ldots,{z_n}} \right)}}

Or (17) G˜(S1,…,Sn)=n˜(z1,…,zn)d˜(z1,…,zn) \tilde G\left({{S_1}, \ldots,{S_n}} \right) = {{\tilde n\left({{z_1}, \ldots,{z_n}} \right)} \over {\tilde d\left({{z_1}, \ldots,{z_n}} \right)}}

If matrices A, B, C and d make the formula or formula valid, then a, B, C and D are the Roesser model implementation matrix of the given transfer function.

In the Roesser model with single input and single output, input u(i₁,…,i_n) and output y(i₁,…,i_n) are single signals. When the input and output become multiple signals, the Roesser model of single input and single output can be extended to the Roesser model of multiple input and multiple output. The Roesser state space model of multiple input and multiple output of linear discrete system with q-dimensional input and p-dimensional output is as follows: (18) [X1(i1+1,…,in)…xn(i1,…,in+1)]=A[X1(i1,…,in)…xn(i1,…,in)]+BU(i1,…,in) \left[{\matrix{{{X_1}\left({{i_1} + 1, \ldots,{i_n}} \right)} \hfill \cr \ldots \hfill \cr {{x_n}\left({{i_1}, \ldots,{i_n} + 1} \right)} \hfill \cr}} \right] = A\left[{\matrix{{{X_1}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr \ldots \hfill \cr {{x_n}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr}} \right] + BU\left({{i_1}, \ldots,{i_n}} \right) (19) Y(i1,…,in)=C[X1(i1,…,in)…xn(i1,…,in)]+Du(i1,…,in) Y\left({{i_1}, \ldots,{i_n}} \right) = C\left[{\matrix{{{X_1}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr \ldots \hfill \cr {{x_n}\left({{i_1}, \ldots,{i_n}} \right)} \hfill \cr}} \right] + Du\left({{i_1}, \ldots,{i_n}} \right)

Wherein, u(i₁,…,i_n) ∈ R^q and y(i₁,…,i_n) ∈ R^p represent input signals and output signals respectively; x_k(i₁,…,i_n) ∈ R^r_k, k = 1,…,n) represents the k-th sub state vector; A. B, C and D are real matrices with sizes of r×r, r×q, p×r, p×q respectively, and r=∑k=1nrk r = \sum\limits_{k = 1}^n {{r_k}} . The transfer function is applicable to the single input single output system, while for the multi input multi output system, it corresponds to the transfer function matrix. The transfer function matrix of the system is: (20) H(z1,…,zn)=cz(Ir−AZ)−1B+D H\left({{z_1}, \ldots,{z_n}} \right) = cz{\left({{I_r} - AZ} \right)^{- 1}}B + D

Where H(z₁,…,z_n) is a matrix of order p×q. In fact, each term in the transfer function matrix is a transfer function. For the rational transfer function matrix C, if each transfer function in the matrix is a (strict) causal transfer function, then this matrix is called a (strict) causal transfer function matrix. A multidimensional transfer function matrix exists in Roesser model if and only if it is a causal transfer function matrix. For a causal transfer function matrix H(z₁,…,z_n) of order p×q, the matrix right fraction description is expressed as: (21) H(z1,…,zn)=NR(Z1,…,Zn)Dr−1(Z1,…,Zn) H\left({{z_1}, \ldots,{z_n}} \right) = {N_R}\left({{Z_1}, \ldots,{Z_n}} \right)D_r^{- 1}\left({{Z_1}, \ldots,{Z_n}} \right)

Where N_R(Z₁,…,Z_n) and D_r(Z₁,…,Z_n) are n-dimensional polynomial matrices of p×q and q×q, respectively. If det D_r(0,…,0) ≡ / 0. H(z₁,…,z_n) is called causal transfer function matrix. If N_R(0,…,0) = 0 and D_r(0,…,0) = I_q, H(z₁,…,z_n) is called a strict causal transfer function matrix.