Compute the inverse of the forgetting factor required for streaming input data
Compute Forgetting Factor Required for Streaming Input Data
The growth in the QR decomposition can be seen by looking at the magnitude of the first element of the upper-triangular factor , which is equal to the Euclidean norm of the first column of matrix ,
To see this, create matrix as a column of ones of length and compute of the economy-size QR decomposition.
n = 1e4; A = ones(n,1);
R = fixed.qlessQR(A)
R = 100.0000
ans = 100
ans = 100
The diagonal elements of the upper-triangular factor of the QR decomposition may be positive, negative, or zero, but
fixed.qrAB always return the diagonal elements of as non-negative.
In a real-time application, such as when data is streaming continuously from a radar array, you can update the QR decomposition with an exponential forgetting factor where . Use the
fixed.forgettingFactor function to compute a forgetting factor that acts as if the matrix were being integrated over rows to maintain a gain of about . The relationship between and is .
m = 16; alpha = fixed.forgettingFactor(m); R_alpha = fixed.qlessQR(A,alpha)
R_alpha = 3.9377
ans = 4
If you are working with a system and have been given a forgetting factor , and want to know the effective number of rows that you are integrating over, then you can use the
fixed.forgettingFactorInverse function. The relationship between and is .
ans = 16
alpha — Forgetting factor
Forgetting factor, specified as a scalar.
m — Number of rows in matrix A
positive integer-valued scalar
Number of rows in matrix A with the equivalent gain, returned as a positive integer-valued scalar.
In real-time applications, such as when data is streaming continuously from a radar array , the QR decomposition is often computed continuously as each new row of data arrives. In these systems, the previously computed upper-triangular matrix, R, is updated and weighted by forgetting factor ɑ, where 0 < ɑ < 1. This computation treats the matrix A as if it is infinitely tall. The series of transformations is as follows.
Without the forgetting factor ɑ, the values of R would grow without bound.
With the forgetting factor, the gain in R is
The gain of computing R without a forgetting factor from an m-by-n matrix A is . Therefore,
 Rader, C.M. "VLSI Systolic Arrays for Adaptive Nulling." IEEE Signal Processing Magazine (July 1996): 29-49.
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