Orthogonal Signal Generation: An Analytical Approach

Abstract

Nowadays, many signal processing activities are related to the orthogonal properties of specific signals. However, only a few methods offer an analytical solution for generating orthogonal signals, especially when only one input to the generating system is available. These methods are often related to very specific applications and lack generalization. In this paper, the use of the Gram–Schmidt orthogonalization process combined with simple transformation operators is proposed as a new framework for generating orthogonal signals. The objective is to provide a rigorous, clear and simple procedure capable of deriving multiple orthogonal signals from a single input. Many examples are discussed to better illustrate the novelty of the method and the main results.

Appendix A. Appendix

1.1 Appendix A.1. Gram–Schmidt (GS) procedure

The Gram–Schmidt (GS) procedure [9, 20, 26, 29] is used to make orthogonal an input dictionary ${\textbf{V}}^T=[v_1(t), v_2(t),..., v_K(t)]$ composed of K signals, the output dictionary ${\textbf{W}}^T=[w1(t), w_2(t),..., w_K(t)]$ being composed of K mutually orthogonal signals. The orthogonal signals $w_k(t)$ are obtained iteratively using the following relationship for k ranging from 1 to K:

$$\begin{aligned} w_k(t)= v_{k}(t) - \sum _{j=1}^{k-1} \rho _{kj} w_j(t) \end{aligned}$$

with the coefficient

$$\begin{aligned} \rho _{kj}= \frac{ \langle v_{k}, w_j\rangle _t}{\langle w_j, w_j\rangle _t}, \end{aligned}$$

and $< \bullet , \bullet >_t$ defining the scalar product in the time domain ranging from $t_1$ to $t_2$. Note however that our approach is not limited to signals, the method can also be applied to spectra [12] and in this case, the scalar product will relate to the variable f over an integration space ranging from $f_1$ to $f_2$. As an illustration, a signal approximation example using the GS procedure is reported in Appendix A.2.

Fig. 12

a Rectangular signal x(t) to be approximated and its approximation ${\hat{x}}(t)$ (in dashed line). b Exponential dictionary with $v_1(t), v_2(t), v_3(t)$ and its orthogonal basis $w_1(t)=v_1(t), w_2(t), w_3(t)$ (dashed line)

1.2 Appendix A.2. Signal Approximation

As an illustration, let us consider the signal approximation issue for which the signal to be approximated is $x(t)=\text{ Rect}_T(t-T/2)$. The signal x(t) is a non-oscillating signal of finite energy, of compact support with 2 discontinuities at $t=0$ and $t=T=1$ s. This signal can be decomposed into a dictionary whose generative signal is $v_k(t)=\exp (-k t) u(t)$, u(t) being the Heaviside signal (see Fig. 12 ). In this case, the different scalar products are calculated from $t=0$ to $+\infty$. With $K=3$ and after calculation, the approximated signal is a linear combination ${\hat{x}}(t)\approx -1.31 v_1(t) +9.58 v_2(t) -7.64 v_3(t)$ and the mean square error is $MSE=0.029$. From the input dictionary ${\textbf{V}}^T=[v_1(t), v_2(t), v_3(t)]$, we construct a dictionary of mutually orthogonal signals ${\textbf{W}}^T=[w_1(t), w_2(t), w_3(t)]$. For $K=3$, the approximated signal ${\hat{x}}(t)$ is a linear combination: ${\hat{x}}(t)\approx 1.27 w_1(t) +0.36 w_2(t) -7.64 w_3(t)$ and the mean square error is $MSE=0.029$. In this case, $w_1=v_1(t)$, $w_2=v_2(t)- \frac{2}{3} v_1(t)$, $w_3=v_3(t)- \frac{6}{5} v_2(t)+ \frac{3}{10} v_1(t)$, $v_1(t)=\exp (-t) u(t)$, $v_2(t)=\exp (-2 t) u(t)$ and $v_3(t)=\exp (-3 t) u(t)$. Whatever the approximation (via $v_k(t)$ or $w_k(t)$ ), the mean square error from the original signal is significant ($MSE=0.029$). This deviation could be reduced by increasing the number K of signals used or by changing the base with a more adapted generative signal.

1.3 Appendix A.3. Derivative-Based Method

Let us show $\langle v_e,{\dot{v}}_e \rangle =0$. If the real signal $v(t)=v_e(t)$ is even (or odd $v(t)=v_o(t)$), then the derivative of the signal $y(t)={\dot{v}}_e(t)$ is orthogonal: $\langle v_e,{\dot{v}}_e \rangle =0$. To prove it, one could just show that y(t) is odd, i.e. $-y(t)=y(-t)$:

$$\begin{aligned} y(t)={\dot{v}}_e(t) =\lim _{h \rightarrow 0} \frac{v_e(t+h)-v_e(t)}{h}, \end{aligned}$$

(A.1)

$$\begin{aligned} y({-t})=\lim _{h \rightarrow 0} \frac{v_e({-t}+h)-v_e({-t})}{h}. \end{aligned}$$

As v(t) is even, we verify that $v_e(t)=v_e(-t)$ and $v_e(t-h)=v_e(-t+h)$. Hence,

$$\begin{aligned} y(-t)= \lim _{h \rightarrow 0} \frac{v_e(t-h)-v_e(t)}{h}, \end{aligned}$$

and if we impose $\tau =t-h$, this yields:

$$\begin{aligned} \lim _{h \rightarrow 0} \frac{v_e(\tau )-v_e(\tau +h)}{h}=\lim _{h \rightarrow 0} -\frac{v_e(\tau +h)-v_e(\tau )}{h}=-y(\tau ). \end{aligned}$$

By replacing $t\rightarrow \tau$, it comes

$$\begin{aligned} y(t)=\lim _{h \rightarrow 0} -\frac{v_e(t+h)-v_e(t)}{h}=-y(t). \end{aligned}$$

(A.2)

Consequently, as $y(t)={\dot{v}}_e(t)$ is odd and $v_e(t)$ is even, then $\langle v_e,{\dot{v}}_e \rangle =0$. QED (quod erat demonstrandum).

1.4 Appendix A.4. Hilbert Transform

• (a) Let us show $HT[HT[v_1(t)]]=HT[{\tilde{v}}_1(t)]=-v_1(t)$ where $HT[ \bullet ]$ refers to the Hilbert Transform. Let us consider $v_1(t)$ and ${\tilde{v}}_1(t):=HT[v_1(t)]$. By definition, the spectrum of $v_1(t)$ is: $FT[{\tilde{v}}_1(t)]={\tilde{V}}_1(f)$ ($FT[ \bullet ]$ refers to the Fourier transform) and the spectrum of ${\tilde{v}}_1(t)$ is:

  $$\begin{aligned} {\tilde{V}}_1(f):={(-j) \cdot \, \text{ sgn }(f) V_1(f)}. \end{aligned}$$

  By multiplying the right and left terms by ${(-j) \text{ sgn }(f)}$, it comes:

  $$\begin{aligned} {(-j) \, \text{ sgn }(f)} \times {\tilde{V}}_1(f)={(-j) \, \text{ sgn }(f)} (-j) \, \text{ sgn }(f) V_1(f)=-V_1(f). \end{aligned}$$

  With $\text{ sgn}^2(f)=1$, it comes:

  $$\begin{aligned}{} & {} FT^{-1}[(-j) \, \text{ sgn }(f) {\tilde{V}}_1(f)]=FT^{-1}[-V_1(f)] \\{} & {} HT[{\tilde{v}}_1(t)]=-v_1(t). \end{aligned}$$

  QED.

• (b) Let us show $v_1(t) \perp {\tilde{v}}_1(t)$, i.e. $<v_1,{\tilde{v}}_1>_t=0$. Let us consider $v_1(t)$ is real. By virtue of the product theorem, it comes:

  $$\begin{aligned}{} & {} <v_1,{\tilde{v}}_1>_t=<V_1,{\tilde{V}}_1>_f \\{} & {} <V_1,{\tilde{V}}_1>_f=\int V_1(f) \times {\tilde{V}}_1(f) df=\int V_1(f) \times {(-j) \, sgn(f) V_1(f)} df \\{} & {} <V_1,{\tilde{V}}_1>_f=(- j) \int V^2_1(f) sgn(f)df. \end{aligned}$$

  As $v_1(t)$ is real, $V_1(f)$ is even. As $\text{ sgn }(f)$ is odd then $V_1(f) \times \text{ sgn }(f)$ is odd too and $\int V^2_1(f) \text{ sgn }(f)df=0$ that implies $<V_1,{\tilde{V}}_1>_f=0<v_1,{\tilde{v}}_1>_t$, QED.

1.5 Appendix A.5. Matlab Code of Example from Fig. 11

1.6 Appendix A.6. Some Definitions

• $\text{ Rect}_T(t)= {\left\{ \begin{array}{ll} 0 &{} \text {if } |t|>T/2 \\ 1/2 &{} \text {if } |t|=T/2 \\ 1 &{} \text {if } |t|<T/2. \\ \end{array}\right. }$;

• ${{\,\textrm{Tri}\,}}_T(t)= {\left\{ \begin{array}{ll} 1-|t/T| &{} \text {for } |t| \le T \\ 0 &{} \text {otherwise} \end{array}\right. }$

• $\text{ sgn }(t)= {\left\{ \begin{array}{ll} -1 &{} \text {if } t<0 \\ 0 &{} \text {if } t=0 \\ 1 &{} \text {if } t>1. \\ \end{array}\right. }$

• $\textrm{III}_{T}(t)=\sum _{k=-\infty }^{+\infty }\delta (t-kT)$;

• $\delta (t)= {\left\{ \begin{array}{ll} 1 &{} \text {if } t=0 \\ 0 &{} \text {if } t\ne 0. \\ \end{array}\right. }$

• $u(t)={\left\{ \begin{array}{ll} 0 &{} \text {if } t<0 \\ 1/2 &{} \text {if } t=0 \\ 1 &{} \text {if } t>0. \\ \end{array}\right. }$

• ${{\,\textrm{sinc}\,}}(t)=\frac{\sin (\pi t)}{(\pi t)}$ where ${{\,\textrm{sinc}\,}}(n)=0$ $\forall n \in \mathbb {Z}^*$;

• $x(t)*y(t)=\int x(\tau ) y(\tau -t) d\tau$;

• Scalar product of a signal of finite energy: $\langle x, y \rangle _t =\int _{t_1}^{t_2} x(t) y(t) \text{ d }t$;

• Scalar product of a periodic signal of period T: $\langle x, y \rangle _t =\frac{1}{T}\int _{t_1}^{t_1+T} x(t) y(t) \text{ d }t$.

• $HT[x(t)]=\frac{1}{\pi }\int _{-\infty }^{+\infty } \frac{x(\tau )}{(t-\tau )} d\tau$;

• $FT[x(t)]=\int _{-\infty }^{+\infty } x(t) \text{ e}^{-2 \pi j f t} dt$;

• Weighted Hermite polynomials: $(-1)^n\frac{\text{ d}^n\text{ e}^{-t^2}}{\text{ d }t^n}$;

• orthogonal Haar decomposition: $\psi _{m,k}(t)=\psi (2^{m/2}t + k)$ with $\psi (t)=1 \, \forall \, 0 \leqslant t \leqslant 1/2$, $\psi (t)=-1 \, \forall \, 1/2 \leqslant t< 1$ and $\psi (t)=0$ otherwise;

• Rademacher orthogonal decomposition: $\psi _{k}(t)=\text{ sgn }\left( \sin (2^k \pi t /T) \right)$.

1.7 Appendix A.7

See Table 1.

Table 1 First group (G1): finite energy signals with compact support. Second group (G2): even signals of finite energy. Third group (G3): causal signals of finite energy. Fourth group (G4): signals at finite average power (periodic). For all groups, solutions with singularities: 1–2, 8–9, 17–19