Question

In Exercies 7-12, assume the signals listed are solutions of the given difference equation. Determine if the signals form a basis for the solution space of the equation. Justify your answers using appropriate theorems.

\({{\bf{1}}^k}\), \({{\bf{3}}^k}\cos \frac{{k\pi }}{{\bf{2}}}\), \({{\bf{3}}^k}{\bf{sin}}\frac{{k\pi }}{{\bf{2}}}\), \({y_{k + {\bf{3}}}} - {y_{k + {\bf{2}}}} + {\bf{9}}{y_{k + {\bf{1}}}} - {\bf{9}}{y_k} = {\bf{0}}\)

Short answer

The dimension of H is 3, so the three linearly independent signals form a basis of H.

Step-by-step solution

Write the Casorati matrix

The Casorati matrixof the solution is

\({A_k} = \left[ {\begin{array}{*{20}{c}}{{1^k}}&{{3^k}\cos \frac{{k\pi }}{2}}&{{3^k}\sin \frac{{k\pi }}{2}}\\{{1^{k + 1}}}&{{3^{k + 1}}\cos \frac{{\left( {k + 1} \right)\pi }}{2}}&{{3^{k + 1}}\sin \frac{{\left( {k + 1} \right)\pi }}{2}}\\{{1^{k + 2}}}&{{3^{k + 2}}\cos \frac{{\left( {k + 2} \right)\pi }}{2}}&{{3^{k + 2}}\sin \frac{{\left( {k + 2} \right)\pi }}{2}}\end{array}} \right]\).

Check the Casorati matrix for \(k = {\bf{0}}\)

Substitute 0 for k in the Casorati matrix.

\(\begin{aligned} {A_0} &= \left[ {\begin{array}{*{20}{c}}{{1^0}}&{{3^0}\cos \frac{{\left( 0 \right)\pi }}{2}}&{{3^0}\sin \frac{{\left( 0 \right)\pi }}{2}}\\{{1^{0 + 1}}}&{{3^{0 + 1}}\cos \frac{{\left( {0 + 1} \right)\pi }}{2}}&{{3^{0 + 1}}\sin \frac{{\left( {0 + 1} \right)\pi }}{2}}\\{{1^{0 + 2}}}&{{3^{0 + 2}}\cos \frac{{\left( {0 + 2} \right)\pi }}{2}}&{{3^{0 + 2}}\sin \frac{{\left( {0 + 2} \right)\pi }}{2}}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}1&1&0\\1&0&3\\1&{ - 9}&0\end{array}} \right]\\ &\sim \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\end{aligned}\)

The Casorati matrix is row equivalent to theidentity matrix. Therefore, it is invertible.

Hence, the set of signals \(\left\{ {{1^k},\,{3^k}\cos \frac{{k\pi }}{2},\;{3^k}\sin \frac{{k\pi }}{2}} \right\}\) is linearly independent. The dimension of H is 3, so the three linearly independent signals form a basis of H.