Deterministic signal modeling

Deterministic signal modeling is based upon the resolution of signals into elementary or basic signals. This way of modeling is convenient because many information processing systems are intended to be linear and decomposition of arbitrary signals into elementary or basic signals facilitates the performance analysis of those systems using superposition. If an arbitrary signal can be written as a finite or infinite sum of basic signals that all have the same shape, the system response to an arbitrary signal can then easily be derived from the response to the basic signal. Basic signals that are often used for this purpose are:

1. Unit impulse and unit step functions

   In section Time-domain modeling of signals, we will discuss the resolution of signals into a continuum of time-shifted impulse functions or of step functions. These signals have special interest for studying the performance of linear stationary dynamic systems. If the response to a single impulse or step is known,

   For example, by measurement or analysis.

   the system response to an arbitrary signal can easily be related to it. Step functions are often used as test signals for those systems. The popular square wave test signal can be constituted from two step functions. System analysis using impulses or step functions as test signals is also referred to as*\ time domain analysis*.

2. Imaginary exponentials and sinusoidal signals

   In section Frequency-domain modeling of signals, we will discuss the resolution of signals into a discrete series, or into a continuum of imaginary exponentials. Periodic signals can be resolved into a discrete series of imaginary exponentials. Such a resolution is described by the Fourier series, and by the sine or cosine transform. Non-periodic signals can, under certain conditions, be resolved into a continuum of imaginary exponential functions. The Fourier transform describes such a resolution. The Fourier transform only applies for signals that are absolute integrable. For a signal \(x(t)\), this means that the integral of the absolute value of the signal over all time is finite:

   \[\int_{-\infty}^{\infty}\left\vert x(t)\right\vert dt<\infty.\]

   Practically, this means that the signal values only differ from zero over a limited time interval. Signals that meet this condition are called energy signals or pulse signals.

   Energy signals have a finite nonzero energy. Their average power over all time is zero.

   Since exponential signals and sinusoidal signals retain their shape under differentiation and integration, they are often used for characterization of linear dynamic systems. Any deviation of the shape of the response from a sinusoid, indicates system nonlinearity. The amount of nonlinearity of a system can thus be evaluated by measuring the distortion in the system’s response to a sinusoidal signal. Sinusoidal test signals can be constructed from two complex conjugated imaginary exponentials. System analysis with the aid of imaginary exponentials is referred to as frequency domain analysis.

3. Complex exponentials

   In section Complex frequency domain modeling, we will discuss the resolution of signals into a continuum of complex exponentials. This resolution is described by the Laplace transform. Application of the Laplace transform is not limited to energy signals; it can also be applied for signals that are not absolutely integrable. Theoretically, these signals may occur in unstable systems: systems that have an unbounded response to a bounded excitation.

Above, we have introduced the concepts of power signals and energy signals. Before we continue with signal modeling, we will give the definitions of power signals and energy signals.

Power signals and energy signals

The average power that can be extracted from a signal \(x(t)\) is proportional to the mean square value \(\overline{x(t)^{2}}\) of that signal. The mean square value is defined as

\[\overline{x(t)^{2}}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2} ^{T/2}\{x(t)\}^{2}dt.\]

A signal with a nonzero mean square value is called a power signal. If the mean square value of a signal equals zero, it is called an energy signal. The energy \(W\{x(t)\}\) of a signal \(x(t)\) is defined as

\[W\{x(t)\}=\lim_{T\rightarrow\infty}\int_{-T/2}^{T/2}\{x(t)\}^{2}dt.\]

The energy of a power signal is (theoretically) unlimited. In the real world, all signals are energy signals, but signal modeling uses abstractions for which it is useful to make a distinction between power signals and energy signals.

Fig. 525 Three examples of pulse signals with unit energy and unit width (\(\tau=1\)).

Fig. 525 shows some examples of energy signals with their energy normalized to unity.

The unit gate pulse

The unit gate pulse signal \(x(t,\tau)\) is defined as:

\[\begin{split}x(t,\tau) & =1,~\left\vert t\right\vert <\frac{\tau}{2},\\ x(t,\tau) & =0,~\left\vert t\right\vert \geq\frac{\tau}{2}.\end{split}\]

The unit Gaussian pulse

A unit Gaussian pulse signal \(x(t,\tau)\) is defined as

\[x(t,\tau)=\frac{1}{\tau\sqrt{2\pi}}\exp\left( \frac{-t^{2}}{2\tau^{2} }\right) .\]

The unit sinc pulse

A unit sinc pulse signal \(x(t,\tau)\) is defined as

(181)\[x(t,\tau)=\frac{\sin(\pi t/\tau)}{\pi t}. \label{sinc-pulse}\]

The unit impulse \(\delta(t)\)

The unit impulse function can be obtained as \(\lim_{\tau\rightarrow0}\) of all three above pulse signals. The area (strength) of the unit impulse function is unity

\[\int_{-\infty}^{\infty}\delta(t)=1,\]

but the value at time instants other than \(t=0\) equals zero:

\[\delta(t)=0~\text{for~}t\neq0.\]

Time-domain modeling of signals

In this section, we will discuss the resolution of arbitrary time functions \(x(t)\) into unit impulse and unit step functions. We will see that an arbitrary time function can be considered to be constituted from a linear superposition of time delayed unit impulse- or unit step functions. As a consequence, the behavior of linear systems can be written as a linear superposition of the responses to these unit impulses or unit steps. This superposition, which is known as the convolution integral, is often used for graphical determination of the response of a linear system to an arbitrary input signal. Linear dynamic systems can thus be characterized by their so-called unit impulse response or unit step response. Step functions are often used as test signals for the verification of the behavior of linear dynamic systems.

Resolution in unit impulse functions

A signal \(x(t)\) can be resolved into a continuum of time-shifted unit impulses. We will do this for a signal that has nonzero values between \(t=0\) and \(t=T\). The signal value at time instant \(0<\tau<T\) is then obtained as the strength of a unit impulse at \(t=\tau\) with the scalar \(x(\tau)\):

(182)\[x(t)=\int_{0}^{T}x(\tau)\delta(t-\tau)d\tau. \label{resolution-unit-impulses}\]

Resolution in unit step functions

The resolution of a signal in unit step functions \(\mu(t)\) can be found in a similar way:

(183)\[x(t)=\int_{0}^{T}\dot{x}(\tau)\mu(t-\tau)d\tau, \label{resolution-unit-steps}\]

where \(\dot{x}(t)=\frac{d}{dt}\{x(t)\}\).

Frequency-domain modeling of signals

Periodic signals can be modeled as a linear superposition of undamped sinusoidal signals. These signals play an important role in the verification of dynamic systems. This is due to the property that linear dynamic systems with sinusoidal excitations give sinusoidal responses.

The Fourier series description is based upon the resolution of periodic signals into a series of imaginary exponentials. It can be shown that an arbitrary signal \(x(t)\), within a limited interval \(-\frac{T}{2}<t<\frac{T}{2}\), under certain conditions may be resolved into a series of imaginary exponentials. It will be clear that only if \(x(t)\) itself is periodic with period \(T\) does this decomposition hold for all values of \(t\). The set of complex amplitudes \(X_{i}\) of the imaginary exponentials constituting this periodic signal \(x(t)\) is called the Fourier series for \(x(t):\)

(184)\[x(t)=\sum_{n=-\infty}^{n=\infty}X_{n}\exp(jn\omega_{0}t). \label{inv-Fourier-series}\]

The complex amplitudes \(X_{n}\) are obtained as

(185)\[X_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)\exp(-jn\omega_{0}t)dt. \label{Fourier-series}\]

Fig. 526 An imaginary exponential \(A_{i}\exp(\omega_{i}t+\phi_{i} )\) represented by a rotating phasor.

Phasor representation

A single exponential component can be depicted as a phasor (rotating arrow) by plotting the imaginary part along the \(y\)-axis and the real part along the \(x\)-axis. This is shown in Fig. 526. Any signal that can be resolved into complex exponentials can be written as a superposition of phasors. Phasor representations of signals are often used for modulated signals. Fig. 527 shows a real sinusoidal signal constructed from two complex conjugated phasors.

Fig. 527 Two phasors constitute a real sinusoidal signal.

The complex amplitude \(X_{n}\) of an element of the Fourier series can be represented by a phasor with amplitude \(|X_{n}|\) and initial phase \(\varphi_{n}\) rotating with an angular speed \(n\omega_{0}\). \(|X_{n}|\) and \(\varphi_{n}\) are known as the discrete magnitude and phase spectrum of the signal \(x(t)\), respectively.

Power spectrum, Parseval’s theorem

The mean square value of a periodic signal \(x(t)\) can be obtained from the complex amplitudes of the elementary sinusoids from which it is constituted. Parseval’s theorem asserts that the mean square value of a periodic signal equals the sum of the squares of the amplitudes of the harmonics that compose that signal:

(186)\[\overline{\{x(t)\}^{2}}=\sum_{n=-\infty}^{n=\infty}X_{n}.X_{n}^{\ast} =\sum_{n=-\infty}^{n=\infty}|X_{n}|^{2}. \label{parseval-power}\]

The graphical representation of \(|X_{n}|^{2}\) is called the power spectrum of the periodic signal \(x(t)\).

AC and DC signals

The mean value, the time-average value, or the DC value of a signal \(x(t)\) over a time interval \(T\), is defined as

(187)\[x_{DC}=\overline{\{x(t)\}}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)dt. \label{XDC}\]

The AC signal is the signal that remains after subtraction of the DC value from the original signal:

\[x_{AC}(t)=x(t)-x_{DC}.\]

The power of a signal can be written in terms of its DC value and the power of the AC signal:

\[\overline{\{x(t)\}^{2}}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2} ^{T/2}\{x_{AC}(t)+x_{DC}\}^{2}dt\]

Since the average value of \(x_{AC}(t)\) equals zero, we obtain

\[\overline{\{x(t)\}^{2}}=\overline{\{x_{AC}(t)\}^{2}}+x_{DC}^{2}.\]

The RMS (root mean square) value \(x_{\text{rms}}\) of a signal \(x(t)\) is defined as

(188)\[x_{\text{rms}}=\sqrt{\overline{\{x(t)\}^{2}}}=\sqrt{x_{AC_{rms}}^{2} +x_{DC}^{2}}. \label{Xeff}\]

A signal with an RMS value \(x_{\text{rms}}\) delivers the same power to a load as a signal having a constant value of \(x_{DC}=x_{\text{rms}}\).

Cosine transformation

A periodic signal can also be resolved in a series of cosine functions. This resolution can be written as

\[x(t)=x_{DC}+\sum_{n=1}^{\infty}C_{n}\cos(n\omega_{0}t-\varphi_{n}).\]

in which \(C_{n}=\sqrt{A_{n}^{2}+B_{n}^{2}},\) and \(\varphi_{n}=\arctan \frac{B_{n}}{A_{n}}\)with

\[\begin{split}A_{n} & =\frac{2}{T}\int_{-T/2}^{T/2}x(t)\cos(n\omega_{0}t)\ dt,\\ B_{n} & =\frac{2}{T}\int_{-T/2}^{T/2}x(t)\sin(n\omega_{0}t)\ dt.\end{split}\]

This follows directly from the Fourier series description of a periodic signal. With the aid of Parseval’s theorem for periodic signals, the mean square value of a periodic signal \(x(t)\) can thus be written as

(189)\[\overline{\{x(t)\}^{2}}=x_{DC}^{2}+\frac{1}{2}\sum_{n=0}^{n=\infty}C_{n}^{2}. \label{Parseval-cosine}\]

Fourier transform

Non-periodic signals cannot be modeled by a discrete Fourier series. Under certain conditions, however, these signals can be resolved into a continuum of imaginary exponentials. An important condition is that the signals are absolute integrable:

(190)\[\int_{-\infty}^{\infty}|x(t)|dt<\infty. \label{abs-integrable}\]

For real-world pulse signals, the conditions are always satisfied. We can then define an energy spectrum by letting the period \(T\) approaching infinity. It is apparent that such a resolution will then have frequency components at all frequencies, but the amplitudes of these components will all approach zero. For this reason, we speak of a frequency density spectrum. The resolution of a signal into a continuum of imaginary exponentials is known as the inverse Fourier transform:

(191)\[x(t)=\mathcal{F}^{-1}\left\{ X(j\omega)\right\} =\frac{1}{2\pi}\int _{-\infty}^{\infty}X(j\omega)\exp(j\omega t)d\omega. \label{inv-Fourier-transform}\]

The complex amplitude function \(X(j\omega)\) is known as the Fourier transform of \(x(t)\):

(192)\[X(j\omega)=\int_{-\infty}^{\infty}x(t)\exp(-j\omega t)dt. \label{Fourier-Transform}\]

The functions \(x(t)\) and \(X(j\omega)\) form a so-called Fourier pair for which we will use the short hand notation:

\[X(j\omega)=\mathcal{F}\left\{ x(t)\right\} .\]

Some frequently used Fourier pairs are listed in Table 26.

Table 26 Fourier pairs

time description	time function	frequency function	frequency description
impulse at \(t=0\)	\(\delta\left( t\right) \)	\(1\)	flat spectrum
impulse at \(t=t_{0}\)	\(\delta\left( t-t_{0}\right) \)	\(\exp\left( -j\omega t_{0}\right) \)	imaginary exponential
impulse sequence	\(\sum_{-\infty}^{\infty}\delta\left( t-nT\right) \)	\(\frac{2\pi}{T}\sum_{-\infty}^{\infty}\delta\left( \omega-\frac{2\pi n}{T}\right) \)	impulse sequence
real sinusoid	\(\cos\left( \omega_{0}t\right) \)	\(\pi\delta\left( \omega\pm\omega_{0}\right) \)	impulse pair
Gaussian pulse	\(\exp\left( -at^{2}\right) \)	\(\sqrt{\frac{\pi}{a}} \exp\left( \frac{-\omega^{2}}{4a}\right) \)	Gaussian pulse
unit gate pulse	\(G_{\Delta\tau}\left( t\right) \)	\(\tau\frac{\sin\left( \omega\tau/2\right) }{\omega\tau/2}\)	sinc function

Important properties of the Fourier transformation are shown in table Table 27.

Table 27 Properties of the Fourier Transform

property	time function	frequency function
linearity	\(a~x_{1}\left( t\right) +b~x_{2}\left( t\right) \)	\(a~X_{1}\left( j\omega\right) +b~ X_{2}\left( j\omega\right) \)
scaling	\(x(at)\)	\(\frac{1}{\left\vert a\right\vert }X\left( \frac{j\omega}{a}\right) \)
time shift	\(x\left( t-\tau\right) \)	\(X\left( j\omega\right) \exp\left( -j\omega\tau\right) \)
frequency shift	\(x\left( t\right) \exp\left( j\omega _{0}t\right) \)	\(X\left( j\omega-j\omega\tau\right) \)
duality	\(X\left( -t\right) \)	\(2\pi x\left( j\omega\right) \)
time derivative	\(\frac{d^{n}}{dt^{n}}\left\{ x\left( t\right) \right\} \)	\(\left( j\omega\right) ^{n}X\left( j\omega\right) \)
frequency derivative	\(\left( -jt\right) ^{n}x\left( t\right) \)	\(\frac{d^{n}}{d\omega^{n}}X\left( j\omega\right) \)
time integration	\(\int_{-\infty}^{t}x\left( \tau\right)d\tau\)	\(\frac{1}{j\omega}X\left( j\omega\right)+\pi X\left( 0\right) \delta\left( \omega\right) \)
average power	\(\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}\left\{ x\left( t\right) \right\} ^{2}dt\)	\(\lim_{T\rightarrow\infty }\frac{1}{2\pi T}\int_{-\infty}^{\infty}\left\vert X\left( j\omega\right) \right\vert ^{2}d\omega\)
total energy	\(\lim_{T\rightarrow\infty}\int_{-T/2}^{T/2}\left\{ x\left( t\right) \right\} ^{2}dt\)	\(\frac{1}{2\pi}\int_{-\infty}^{\infty }\left\vert X\left( j\omega\right) \right\vert ^{2}d\omega\)
time-domain convolution	\(x\left( t\right) \ast h\left( t\right) \)	\(X\left( j\omega\right) H\left( j\omega\right) \)
time domain multiplication	\(x\left( t\right) g(t)\)	\(\frac{1}{2\pi}X\left( j\omega\right) \ast G\left( j\omega\right) \)
time domain correlation	\(\lim_{T\rightarrow\infty}\frac{1}{T}\int _{-T/2}^{T/2}x\left( t\right) x\left( t+\tau\right)dt\)	\(\lim_{T\rightarrow\infty}\frac{1}{2\pi T}\left\vert X\left( j\omega\right) \right\vert ^{2}\)

Energy density spectrum and Parseval’s theorem

The energy of a pulse signal \(x(t)\) may as well be obtained from its frequency domain description \(X(j\omega)\). This relation is given by Parseval’s theorem, given in Parseval-energy:

(193)\[W\left\{ x\left( t\right) \right\} =\lim_{T\rightarrow\infty}\int _{-T/2}^{T/2}\{x(t)\}^{2}dt=\frac{1}{2\pi}\int_{-\infty}^{\infty} |X(j\omega)|^{2}d\omega, \label{Parseval-energy}\]

where \(|X(j\omega)|^{2}\) is called the energy density spectrum of \(x(t)\).

Fourier transform of periodic signals

The Fourier transform gives the spectral density of a signal. Since periodic signals have a discrete amplitude spectrum, the density must be infinite at multiples of the fundamental frequency. From this, we expect the Fourier transform of a periodic signal to be written as a series of frequency impulses occurring at \(\omega=n\omega_{0}\). Moreover, we expect the strength of the impulses to be \(2\pi X_{n}\). Without exact mathematical derivation, it follows that the Fourier transform of a periodic signal can be written as

\[X(j\omega)=\mathcal{F}\left\{ \sum_{n=-\infty}^{n=\infty}X_{n}\exp (jn\omega_{0}t)\right\} =\sum_{n=-\infty}^{n=\infty}2\pi X_{n}\delta (\omega-n\omega_{0}).\]

Complex frequency domain modeling

As discussed earlier, periodic signals and absolute integrable pulse signals can be resolved into undamped sinusoids. Many theoretical signals have no power limitation (unbounded signals) and are not absolutely integrable. These signals can be decomposed into complex exponentials. The inverse Laplace transform of \(X(s)\), denoted as \(\mathcal{L}^{-1}\left\{ X(s)\right\} \), writes a time function \(x(t)\) as a continuum of complex exponentials:

\[x(t)=\mathcal{L}^{-1}\left\{ X(s)\right\} =\frac{1}{2\pi j}\oint _{\sigma-j\omega}^{\sigma+j\omega}{}X(s)\exp(st)ds.\]

The continuum of the complex coefficients \(X(s)\) is called the Laplace transform of \(x(t)\):

(194)\[X(s)=\mathcal{L}\left\{ x(t)\right\} =\int_{-\infty}^{\infty}x(t)\exp (-st)dt. \label{Laplace-transform}\]

The time function \(x(t)\) and its Laplace transform \(X(s)\) form a so-called Laplace pair for which we will use the short hand notation:

\[X(s)=\mathcal{L}\left\{ x(t)\right\} .\]

Some often used Laplace pairs are given in Table 28.

Table 28 Laplace pairs

time description	time function	frequency function
unit impulse at \(t=0\)	\(\delta\left( t\right) \)	\(1\)
unit step at \(t=t_{0}\)	\(\mu\left( t\right) \)	\(\frac{1}{s}\)
power function	\(\frac{t^{n-1}}{\left( n-1\right) !}\)	\(\frac{1}{s^{n}}\)
real exponential	\(\exp\left( -at\right) \)	\(\frac{1}{s+a}\)
cosine function	\(\cos\left( \omega t\right) \)	\(\frac{s}{s^{2}+\omega ^{2}}\)
sine function	\(\sin(\omega t)\)	\(\frac{\omega}{s^{2}+\omega^{2}}\)

The function \(X(s)\) from expression Laplace-transform is called the two-sided Laplace transform of \(x(t)\). Most problems can be formulated in a manner that permits all signals to be zero for \(t<0.\) The Laplace transform then reduces to the one-sided Laplace transform:

\[\mathcal{L}\left\{ x(t)\right\} =\int_{0}^{\infty}x\left( t\right) \exp(-st)dt.\]

For analysis of linear dynamic systems, the state of the system before \(t=0\) can be described by means of so-called initial conditions.

Some important properties of the Laplace transform are listed in Table 29.

Table 29 Properties of the Laplace transform

property	time function	Laplace transform
linearity	\(a\left\{ x_{1}\left( t\right) \right\}+b\left\{ x_{2}\left( t\right) \right\} \)	\(a\left\{ X_{1}\left( s\right) \right\} +b\left\{ X_{2}\left( s\right) \right\} \)
scaling	\(x\left( \frac{t}{a}\right) \)	\(aX\left( as\right) \)
time-domain shift	\(x\left( t-\tau\right) \mu\left( t-\tau\right) \)	\(X\left( s\right) \exp\left( -s\tau\right) \)
s-domain shift	\(x\left( t\right) \exp\left( -at\right) \)	\(X\left( s+a\right) \)
time derivative	\(\frac{d^{n-1}}{dt^{n-1}}\left\{ x\left( t\right) \}\right\vert _{t=0}\)	\(s^{n}X(s)-s^{n-1}x(0)-... \)
		\(...-s^{0}\frac{d^{n-1}}{dt^{n-1}}\left\{ x\left( t\right) \}\right\vert _{t=0}\)
s-domain derivative	\(tx\left( t\right) \)	\(\frac{d}{ds}X\left( s\right) \)
time-domain integration	\(\int_{-\infty}^{t}x\left( \tau\right)d\tau\)	\(\frac{1}{s}X\left( s\right) +\frac{1}{s}\int_{-\infty}^{t}x\left( \tau)d\tau\right\vert _{t=0}\)
time-domain convolution	\(x\left( t\right) \ast h\left( t\right) \)	\(X\left( s\right) H\left( s\right) \)
time-domain multiplication	\(x\left( t\right) g(t)\)	\(X\left( s\right) \ast G\left( s\right) \)
initial value theorem	\(x\left( 0^{+}\right)=\lim_{s\rightarrow\infty}sX\left( s\right) \)
final value theorem	\(\lim_{t\rightarrow\infty}x\left( t\right) =\lim_{s\rightarrow0}sX\left( s\right) \)