deconv

Deconvolution or polynomial division

Usage

[q, r] = deconv(b, a)

• It performs the polynomial division:

• b is a vector containing the coefficients of $b(x)$, which can be real or complex.

• a is a vector containing the coefficients of $a(x)$, which can be real or complex. The first element of a, i.e., a(1), must be non-zero.

• The outputs q and r contain the coefficients of the polynomials $q(x)$ and $r(x)$, respectively, such that

  where $r(x)$ is the remainder of the division with order $<a(x)$.

• When n < m, the output q is 0 and r is the same as b.

💡 When representing a polynomial by a vector p, p(1) and p(end) are the highest and the zeroth order coefficients, respectively.

💡 conv(a,q) + r gives the same vector as b.

Examples

• Polynomial division. The numerator has lower order, so that r is the same as the numerator u:

Input

u = 1:5;
v = 1:6;
[q,r] = deconv(u,v)

Output

q =
 0.000

r =
 1.000   2.000   3.000   4.000   5.000

• Polynomial division. The numerator has higher order than the denominator:

Input

u = 1:7;
v = 1:6;
[q,r] = deconv(u,v)

Output

q =
 1.000   0.000

r =
Columns 1 through 5:
 0.000   0.000   0.000   0.000   0.000
Columns 6 through 7:
 0.000   7.000

• Convolution conv gives back u of the previous example.

Input

u = 1:7;
v = 1:6;
[q,r] = deconv(u,v)
conv(v,q) + r

Output

q =
 1.000   0.000

r =
Columns 1 through 5:
 0.000   0.000   0.000   0.000   0.000
Columns 6 through 7:
 0.000   7.000

ans =
Columns 1 through 5:
 1.000   2.000   3.000   4.000   5.000
Columns 6 through 7:
 6.000   7.000