Polynomial Manipulation
The most fundamental function in classical signal processing is the rational transfer function:
which is a ratio of two polynomials in z–1. The ability to manipulate polynomials is therefore one of the most fundamental operations in signal processing. The PV‑WAVE Signal Processing Toolkit provides a rich set of polynomial manipulation functions that greatly enhance your ability to extend the functionality of the Signal Processing Toolkit.
The polynomial manipulation functions belong to one of two classes. Those that manipulate standard polynomials of the form
c(x) = c0 + c1x + c2x2 + ... + cNxN
are summarized in Table 1-10: Standard Polynomial Manipulation Routines. For detailed information, see Chapter 2: Reference (A to F), Chapter 3: Reference (G to P), and Chapter 4: Reference (Q to Z).
Routine | Description |
|---|---|
CONVOL1D | One dimensional convolution |
DBLPOLY | Polynomial function evaluation |
PAIRCONJ | Pair conjugate complex numbers |
PAIRINV | Pair reciprocal complex numbers |
P_DEG | Numerical determination of polynomial degree |
P_DIV | Polynomial division |
P_MULT | Polynomial multiplication |
P_SQRT | Polynomial spectral factorization |
P_SUM | Polynomial sum |
ROOT2POLY | Compute coefficients of polynomial with specified roots |
ZEROPOLY | Polynomial root finding |
Those functions that manipulate polynomials in z–1 of the form:
D(z) = d0 + d1z–1 + d2z–2 + ... + dNz–N
are summarized in Table 1-11: Functions for Polynomials in z–1. For more information, see Chapter 3: Reference (G to P) and Chapter 4: Reference (Q to Z).
Function | Description |
|---|---|
P_STAB | Polynomial stabilization |
SCHURCOHN | Schur-Cohn stability test |
A common signal processing problem is having to simultaneously manipulate polynomials in both z and z–1. For example, the magnitude frequency response of a real-coefficient FIR filter:
B(z) = b0 + b1z–1 + b2z–2 + ... + bMz–M
may be found by evaluating B(z)B(z–1) for values of z on the unit circle.
The REVERSE function in PV‑WAVE Advantage is useful for manipulating polynomials in z and z–1. The mathematical equivalent of reversing the coefficients of a polynomial is given by:
One way to compute D(z) = B(z)B(z–1) is to compute:
which has the same coefficient ordering as D(z).
Using PV‑WAVE Signal Processing Toolkit commands, the operation:
D(z) = B(z)B(z–1)
is simply:
d = P_MULT(b, REVERSE(b))
where the arrays b and d contain the coefficients of the polynomials B(z) and D(z), respectively.
