Aufgaben:Exercise 4.1: About the Gram-Schmidt Process

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Specification for the Gram-Schmidt process

For the four signals  $s_1(t), \, \text{...} \, , s_4(t)$  defined by the figure, the three resulting basis functions  $\varphi_1(t)$,  $\varphi_2(t)$  and  $\varphi_3(t)$  are to be determined by applying the Gram-Schmidt process, so that for the signals with   $i = 1, \, \text{...} \, , 4$  can be written:

$$s_i(t) = s_{i1} \cdot \varphi_1(t) + s_{i2} \cdot \varphi_2(t) + s_{i3} \cdot \varphi_3(t)\hspace{0.05cm}.$$

• In subtask (1), let  $A^2 = 1 \ \rm mW$  and  $T = 1 \ \rm µ s$.
• In the later subtasks, the amplitude and the time are normalized quantities, respectively:   $A = 1$,  $T = 1$.
• Thus, both the coefficients  $s_{\it ij}$  and the basis functions  $\varphi_{\it j}(t)$  – with  $j = 1, 2, 3$  – are dimensionless quantities.

Notes:

• The exercise belongs to the chapter  "Signals, Basis Functions and Vector Spaces".
• Reference is made in particular to the sections  "Orthonormal basis functions"  and  "Gram-Schmidt process".

Questions

1

What are the units of the following quantities with  $A^2 = 1 \, \rm mW$  and  $T = 1 \, {\rm µ s}$?

	The basis functions  $\varphi_j(t)$  are dimensionless.
	The basis functions  $\varphi_j(t)$  have the unit  $\rm \sqrt{\rm s}$.
	The coefficients  $s_{\it ij}$  are dimensionless.
	The coefficients  $s_{\it ij}$  have the unit  $\rm \sqrt{\rm Ws}$.

2

Perform the first step of the Gram-Schmidt process. As for the other tasks, let  $A = 1$  and  $T = 1$ hold.

$s_{\rm 11} \ = \ $

$s_{\rm 12} \ = \ $

$s_{\rm 13} \ = \ $

3

What are the coefficients of the signal  $s_2(t)$  with  $A = 1$  and  $T = 1$?

$s_{\rm 21} \ = \ $

$s_{\rm 22} \ = \ $

$s_{\rm 23} \ = \ $

4

What are the coefficients of the signal  $s_3(t)$  with  $A = 1$  and  $T = 1$?

$s_{\rm 31} \ = \ $

$s_{\rm 32} \ = \ $

$s_{\rm 33} \ = \ $

5

What are the coefficients of the signal  $s_4(t)$  with  $A = 1$  and  $T = 1$?

$s_{\rm 41} \ = \ $

$s_{\rm 42} \ = \ $

$s_{\rm 43} \ = \ $

Solution

Solution

(1)  Solutions 2 and 4 are correct:

• Every orthonormal basis function should have energy 1, that is, it must hold:

$$||\varphi_j(t)||^2 = \int_{-\infty}^{+\infty}\varphi_j(t)^2\,{\rm d} t = 1 \hspace{0.05cm}.$$

• For this condition to be satisfied, the basis function must have unit $\rm \sqrt{\rm s}$. Another equation to be considered is

$$s_i(t) = \sum\limits_{j = 1}^{N}s_{ij} \cdot \varphi_j(t).$$

• Like $A$, the signals themselves have the unit $\rm \sqrt{\rm W}$. Because of the unit $\rm \sqrt{\rm 1/s}$ of $\varphi_{ j}(t)$, this equation can be satisfied with the correct dimension only if the coefficients $s_{\it ij}$ are given with the unit $\rm \sqrt{\rm Ws}$.

(2)  The energy of the signal $s_1(t)$ is equal to $E_1 = 2$. It follows for the norm, the basis function $\varphi_1(t)$ and the coefficient $s_{\rm 11}$:

$$||s_1(t)|| = \sqrt{2},\hspace{0.9cm}\varphi_1(t) = \frac{s_1(t)}{||s_1(t)||},\hspace{0.9cm} s_{11} = \sqrt{E_1} = \sqrt{2} \hspace{0.1cm}\hspace{0.15cm}\underline { {\approx 1.414} } \hspace{0.05cm}.$$

The other coefficients are $\underline {s_{\rm 12} = s_{\rm 13} = 0}$, since the associated basis functions have not been found at all yet, while $\varphi_1(t)$ is equal in form to $s_1(t)$.

(3)  Since at most two basis functions are found after considering $s_2(t)$, $s_{\rm 23} \hspace{0.15cm} \underline{= 0}$ holds with certainty. On the other hand one obtains for the coefficient

$$||s_1(t)|| = \sqrt{2},\hspace{0.9cm}\varphi_1(t) = \frac{s_1(t)}{||s_1(t)||},\hspace{0.9cm} s_{11} = \sqrt{E_1} = \sqrt{2} \hspace{0.1cm}\hspace{0.15cm}\underline { {\approx 1.414} } \hspace{0.05cm};$$

for the auxiliary function $\theta_2(t)$:

$$\theta_2(t) = s_2(t) - s_{21} \cdot \varphi_1(t) = \left\{ \begin{array}{c} 1 - 0.707 \cdot 0.707 = 0.5\\ 0 - 0.707 \cdot (-0.707) = 0.5 \end{array} \right.\quad \begin{array}{*{1}c} 0 \le t < 1 \\ 1 \le t < 2 \\ \end{array} \hspace{0.05cm}; $$

for the second basis function:

$$\varphi_2(t) = \frac{\theta_2(t)}{||\theta_2(t)||},\hspace{0.2cm} ||\theta_2(t)|| = \sqrt{0.5^2 + 0.5^2} = \sqrt{0.5} \approx 0.707$$

$$\Rightarrow \hspace{0.3cm} \varphi_2(t) = \left\{ \begin{array}{c} 0.5/0.707 = 0.707\\ 0 \end{array} \right.\quad \begin{array}{*{1}c} 0 \le t < 2 \\ 2 \le t < 3 \\ \end{array} \hspace{0.05cm}; $$

and finally for the second coefficient

$$s_{22} = \hspace{0.1cm} < \hspace{-0.1cm} s_2(t), \hspace{0.1cm}\varphi_2(t) \hspace{-0.1cm} > \hspace{0.1cm} = 1 \cdot 0.707 + 0 \cdot 0.707 \hspace{0.1cm}\hspace{0.15cm}\underline { = 0.707} \hspace{0.05cm}.$$

The calculations are illustrated in the graph below.

Gram-Schmidt calculations

(4)  It can be seen immediately that $s_3(t)$ can be expressed as a linear combination of $s_1(t)$ and $s_2(t)$.

$$s_{3}(t) = -s_{1}(t) + s_{2}(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}s_{31} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} - s_{11} + s_{21} = -1.414 + 0.707 = \hspace{0.1cm}\hspace{0.15cm}\underline {-0.707}\hspace{0.05cm},$$

$$s_{32} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} - s_{12} + s_{22} = 0 + 0.707 \hspace{0.1cm}\underline {= 0.707}\hspace{0.05cm},$$

$$s_{33} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} - s_{13} + s_{23} = 0 + 0 \hspace{0.1cm}\underline {= 0}\hspace{0.05cm}. $$

(5)  The range $2 ≤ t ≤ 3$ is not covered by either $\varphi_1(t)$ or $\varphi_2(t)$. Therefore, $s_4(t)$ provides the new basis function $\varphi_3(t)$. Furthermore, since $s_4(t)$ has components only in the range $2 ≤ t ≤ 3$ and $||s_4(t)|| = 1$, we obtain $\varphi_3(t) = s_4(t)$ as well as

$$s_{41} \hspace{0.1cm}\hspace{0.15cm}\underline {= 0}, \hspace{0.2cm}s_{42} \hspace{0.1cm}\hspace{0.15cm}\underline {= 0}, \hspace{0.2cm}s_{43} \hspace{0.1cm}\hspace{0.15cm}\underline { = 1} \hspace{0.05cm}. $$