Convolution and solving for the Toeplitz and Hankel matrices

Question

Some linear functions associated with a convolution system. Suppose that 𝑢 and 𝑦 are scalar-valued iscrete-time signals (i.e., sequences) related via convolution:

𝑦(𝑘) = ∑_j ℎ_j𝑢(𝑘 − 𝑗), 𝑘 ∈ ℤ, where ℎ𝑘 ∈ ℝ. You can assume that the convolution is causal, i.e., ℎ_j = 0 when 𝑗 < 0.

(a) The input/output (Toeplitz) matrix. Assume that 𝑢(𝑘) = 0 for 𝑘 < 0, and define

𝑈 = [𝑢(0); 𝑢(1); ...; 𝑢(𝑁)] , 𝑌 = [𝑦(0); 𝑦(1); ...; 𝑦(𝑁)].

Thus 𝑈 and 𝑌 are vectors that give the first 𝑁 + 1 values of the input and output signals,

respectively. Find the matrix 𝑇 such that 𝑌 = 𝑇𝑈. The matrix 𝑇 describes the linear mapping

from (a chunk of) the input to (a chunk of) the output. 𝑇 is called the input/output or Toeplitz

matrix (of size 𝑁 + 1) associated with the convolution system.

(b) The Hankel matrix. Now assume that 𝑢(𝑘) = 0 for 𝑘 > 0 or 𝑘 < −𝑁 and let 𝑈 = [𝑢(0); 𝑢(−1); ...; 𝑢(−𝑁)],

𝑌 = [𝑦(0); 𝑦(1); ...;𝑦(𝑁)].

Here 𝑈 gives the past input to the system, and 𝑌 gives (a chunk of) the resulting future output.

Find the matrix 𝐻 such that 𝑌 = 𝐻𝑈. 𝐻 is called the Hankel matrix (of size 𝑁 + 1) associated

with the convolution system.

Answer 1

Start by using the given information to figure the summation limits:

y(k) = ∑_{j=max(0,k–N)}^kh_ju(k–j)

i.e.,

y(0) = h₀u(0), y(1) = h₀u(1) + h₁u(0), ..., y(N) = h₀u(N) + h₁u(N–1) + ... + h_N–1u(1) + h_Nu(0).

Therefore T = [

h₀, 0, ..., 0, 0;

h₁, h₀, ..., 0, 0;

...

h_N, h_N–1, ..., h₁, h₀].