Filter Design Toolbox

Examples of Adaptive Kalman Filters

Without going into details because the specifics are beyond the scope of this User's Guide, the adaptive filter functions in the toolbox represent variations of Kalman filtering. Thus, Kalman filters are the basis of all the other functions, and perhaps the most effective and efficient since each succeeding filter update in the Kalman algorithm depends only on the most recent input data.

adaptkalman shares many input arguments with the LMS and RLS adaptive functions. To completely specify the Kalman algorithm requires a few additional inputs -- k0, qm, and qp as listed in the following table.

Structure Element	Element Description	initkalman Argument
s.coeffs	Kalman adaptive filter coefficients. Should be initialized with the initial values for the FIR filter coefficients. Updated coefficients are returned when you use s as an output argument.	w0
s.errcov	The state error covariance matrix. Initialize this element with the initial error state covariance matrix. An updated matrix is returned when you use s as an output argument.	k0
s.measvar	Contains the measurement noise variance matrix.	qm
s.procov	Contains the process noise covariance matrix.	qp
s.states	Returns the states of the FIR filter. This is an optional element. If omitted, it defaults to a zero vector of length equal to the filter order.	zi
s.gain	Kalman gain vector. Computed and returned after every iteration. This is a read-only value.	Not applicable
s.iter	Total number of iterations in the adaptive filter run. Although you can set this number in s, you should not.	Not applicable

Befitting the nature of the Kalman approach to adaptive filtering, arguments k0, qm, and qp are matrices that define the known parameters for the algorithm -- the initial conditions. Often you do not know the initial state of the update process equation that defines each filter update. To overcome this fact, we use mean and correlation matrices of the initial state to define the equation.

adaptkalman Example -- System Identification

• x  = 0.1*randn(1,500);
  b  = fir1(31,0.5);
  d  = filter(b,1,x);
  w0 = zeros(1,32);
  k0 = 0.5*eye(32);
  qm = 2; 
  qp = 0.1*eye(32);
  s = initkalman(w0,k0,qm,qp);
  [y,e,s] = adaptkalman(x,d,s);
  stem([b.',s.coeffs.']);
  legend('Actual','Estimated');
  title('System Identification of an FIR Filter via Kalman 
  Filter');
  grid on;
  
  
  

Selected Bibliography

[1] Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, 1996, 493-552.

[2] Haykin, Simon, Adaptive Filter Theory, Prentice-Hall, Inc., 1996

Example of Adaptive Filter That Uses RLS Algorithm

Digital Frequency Transformations