Signal Representation/Direct Current Signal - Limit Case of a Periodic Signal

1. Signal Representation
2. Periodic Signals
3. Direct Current Signal - Limit Case of a Periodic Signal

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Time signal representation

$\text{Definition:}$  A  »direct current (DC) signal«   is a deterministic signal whose instantaneous values are constant for all times  $t$  from  $-\infty$  to  $+\infty$.  Such a signal is the borderline case of a  »harmonic oscillation«,  where the period duration  $T_{0}$  has an infinitely large value.

DC signal in time domain

According to this definition a DC signal always ranges from  $t = -\infty$  to  $t = +\infty$.  If the constant signal is only switched on at the time  $t = 0$  there is no DC signal.

• A direct signal can never be a carrier of information in a communication system,&nbsp but transmitted signals can possess a  »direct signal component«. 

• All statements made in the following for the direct current signal apply in the same way also to such a direct signal component.

$\text{Definition:}$  For the  »DC signal component«  $A_{0}$  of any signal  $x(t)$  applies:

$$A_0 = \lim_{T_{\rm M}\to\infty}\,\frac{1}{T_{\rm M} }\cdot\int^{T_{\rm M}/2}_{-T_{\rm M}/2}x(t)\,{\rm d} t. $$

• The measurement duration  $T_{\rm M}$  should always be selected as large as possible  $($infinite in borderline cases$)$.

• The given equation is only valid if  $T_{\rm M}$  lies symmetrical about the time  $t=0$.

Random signal with DC componentl

$\text{Example 1:}$  The graph shows a random signal  $x(t)$.

• The DC component is here  $A_{0} = 2\ \rm V$.

• In the sense of statistics,  $A_{0}$  corresponds to the linear mean.

Spectral representation

We now look at the situation in the frequency domain.  From the time function it is already obvious,  that it contains – spectrally speaking – only one single  $($physical$)$  frequency,  namely the frequency  $f=0$.

⇒   This result shall now be derived mathematically.  In anticipation of the chapter  »Fourier Transform«  the connection between the time signal  $x(t)$  and the corresponding spectrum  $X(f)$  is already given here:

$$X(f)= \hspace{0.05cm}\int_{-\infty} ^{{+}\infty} x(t) \, \cdot \, { \rm e}^{-\rm j 2\pi \it ft} \,{\rm d}t.$$

The spectral function  $X(f)$  is called after the French mathematician  $\text{Jean Baptiste Joseph Fourier}$  the  »Fourier transform«  of the signal  $x(t)$  and the short labeling for this functional relation is

$$X(f)\ \bullet\!\!-\!\!\!-\!\!\circ\,\ x(t).$$

For example,  if  $x(t)$  describes a voltage curve,  so  $X(f)$  has the unit  "V/Hz".

⇒  Applying the Fourier transform to the DC signal  $x(t)=A_{0}$  yields the spectral function

$$X(f)= A_0 \cdot \int_{-\infty} ^{+\hspace{0.01cm}\infty}\rm e \it ^{-\rm {j 2\pi} \it ft} \,{\rm d}t$$

with the following properties:

• The integral diverges for  $f=0$,  i.e. it returns an infinitely large value  $($integration over the constant value  $1)$.

• For a frequency  $f\ne 0$,  on the other hand,  the integral is zero;  the corresponding proof,  however,  is not trivial  $($see next section$)$.

$\text{Definition:}$  The searched spectral function  $X(f)$  is compactly expressed by the following equation:

$$X(f) = A_0 \, \cdot \, \rm \delta(\it f).$$

1.  $\delta(f)$  is denoted as the  »Dirac delta function«,  also known as  »distribution«.
2.  $\delta(f)$  is a mathematically complicated function; the derivation can be found in the next section.

DC signal and its spectral function

$\text{Example 2:}$  The graphic shows the functional connection

• between an DC signal  $x(t)=A_{0}$  and

• its corresponding spectral function  $X(f)=A_{0} \cdot \delta(f)$.

The Dirac delta function at frequency  $f=0$  is represented by an arrow with weight  $A_{0}$.

Dirac (delta) function in frequency domain

$\text{Definition:}$  The  »Dirac delta function«   ⇒   short:  »Dirac function«  has the following properties:

• The Dirac delta function is infinitely narrow,  i.e. it is  $\delta(f)=0$  for  $f \neq 0$.

• The Dirac delta function  $\delta(f)$  is infinitely high at the frequency  $f = 0$ .

• The Dirac delta weight  $($area of the Dirac function$)$  yields a finite value, namely  $1$:

$$\int_\limits{-\infty} ^{+\infty} \delta( f)\,{\rm d}f =1.$$

• It follows from this last property that  $\delta(f)$  has the unit  ${\rm Hz}^{-1} = {\rm s}$ .

The derivation of the Dirac delta function

$\text{Proof:}$  For the mathematical derivation of these properties we assume a dimensionless DC signal  $x(t)$.

• To force the convergence of the Fourier integral,  the non-energy-limited signal  $x(t)$  is multiplied by a bilateral declining exponential function.

• The graph shows the signal  $x(t)=1$  and the energy-limited signal

$$x_{\varepsilon} (t) = \rm e^{\it -\varepsilon \hspace{0.05cm} \cdot \hspace{0.05cm} \vert \hspace{0.01cm} t \hspace{0.01cm}\vert}{.}$$

It applies  $\varepsilon > 0$.  At the limit   $\varepsilon \to 0$ ,   $x_{\varepsilon}(t)$  passes to  $x(t)=1$.

• The spectral representation is obtained by applying the Fourier integral given above:

$$X_\varepsilon (f)=\int_{-\infty}^{0} {\rm e}^{\varepsilon{t} }\, {\cdot}\, {\rm e}^{-\rm j 2\pi \it ft} \,{\rm d}t \hspace{0.2cm}+ \hspace{0.2cm} \int_{0}^{+\infty} {\rm e}^{-\varepsilon t} \,{\cdot}\, { \rm e}^{-\rm j 2\pi \it ft} \,{\rm d}t.$$

• After integration and combination of both parts we obtain the purely real spectral function of the energy-limited signal  $x_{\varepsilon}(t)$:

$$X_\varepsilon (f)=\frac{1}{\varepsilon -\rm j \cdot 2\pi \it f} + \frac{1}{\varepsilon+\rm j \cdot 2\pi \it f} = \frac{2\varepsilon}{\varepsilon^2 + (\rm 2\pi {\it f}\hspace{0.05cm} ) \rm ^2} \, .$$

• The area under the  $X_\varepsilon (f)$  curve is independent of the parameter  $\varepsilon$  equals  $1$. The smaller  $ε$  is selected,  the narrower and higher the function becomes,  as the  $($German language$)$  learning video  »Herleitung und Visualisierung der Diracfunktion«   ⇒   "Derivation and visualization of the Dirac delta function"  shows.

• The limit for  $\varepsilon \to 0$  returns the Dirac delta function with weight  $1$:

$$\lim_{\varepsilon \hspace{0.05cm} \to \hspace{0.05cm} 0}X_\varepsilon (f)= \delta(f).$$

Exercises for the chapter

Exercise 2.2: Direct Current Component of Signals

Exercise 2.2Z: Non–Linearities