15.1: Signal Measures

The signals of interest to us are defined as maps from a time set into \(\mathbb{R}^{n}\). A continuous-time signal is a map from \(\mathbb{R} \rightarrow \mathbb{R}^{n}\), and a discrete-time signal is a map from \(\mathbb{Z} \rightarrow \mathbb{R}^{n}\). If \(n = 1\) we have a scalar signal, otherwise we have a vector-valued signal. It is helpful, in understanding the various signal measures defined below, to visualize a discrete-time signal \(w(k)\) as just a vector of infinite (or, if our signal is defined only for non-negative time, then a vector of semi-infinite) length or dimension, concretely representing it as the array

\[\left(\begin{array}{c}
\vdots \\
w(0) \\
w(1) \\
\vdots
\end{array}\right) \quad \text { or } \quad\left(\begin{array}{c}
w(0) \\
w(1) \\
\vdots
\end{array}\right) \ \tag{15.1}\]

Three of the most commonly used DT signal measures are then natural generalizations of the finite-dimensional vector norms (\(\infty\)-, 2- and 1-norms) that we have already encountered in earlier chapters, generalized to such infinite-dimensional vectors. We shall examine these three measures, and a fourth that is related to the 2-norm, but is not quite a norm. We shall also define CT signal measures that are natural counterparts of the DT measures.

The signal measures that we study below are:

1. peak magnitude (or \(\infty\)-norm);
2. energy (whose square root is the 2-norm);
3. power (or mean energy, whose square root is the "rms" or root-mean-square value);
4. "action" (or 1-norm).

Peak Magnitude: The \(\infty\)-Norm

The \(\infty\)-norm \(\|w\|_{\infty}\) of a signal is its peak magnitude, evaluated over all signal components and all times :

\[\begin{aligned}
\|w\|_{\infty} & \triangleq \text { max magnitude of } w \\
& \triangleq \sup _{k} \max _{i}\left|w_{i}(k)\right|=\sup _{k}\|w(k)\|_{\infty} \ (\text{ for DT systems}) \ (15.2) \\
& \triangleq \sup _{t} \max _{i}\left|w_{i}(t)\right|=\sup _{t}\|w(t)\|_{\infty} \ (\text{ for CT systems}) \ (15.3)
\end{aligned}\nonumber\]

where \(w_{i}(k)\) indicates the \(i\)-th component of the signal vector \(w(k)\). Note that \(\|w(k)\|_{\infty}\) denotes the \(\infty\)-norm of the signal value at time \(k\), i.e. the familiar 1 norm of an n-vector, namely the maximum magnitude among its components. On the other hand, the notation \(\|w\|_{\infty}\) denotes the \(\infty\)-norm of the entire signal. The "sup" denotes the supremum or least upper bound, the value that is approached arbitrarily closely but never (i.e., at any finite time) exceeded. We use "sup" instead of "max" because over an infinite time set the signal magnitude may not have a maximum, i.e. a peak value that is actually attained | consider, for instance, the simple case of the signal

\[1-\frac{1}{1+|k|}\nonumber\]

which does not attain its supremum value of 1 for any finite \(k\).

Note that the DT definition is the natural generalization of the standard \(\infty\)-norm for finite-dimensional vectors to the case of our infinite vector in (15.1), while the CT definition is the natural counterpart of the DT definition. This pattern is typical for all the signal norms we deal with, and we shall not comment on it explicitly again.

Example 15.1

Some bounded signals:

(a) \(\begin{array}{l}
\text { For } w(t)=1, t \in \mathbb{R}, t \geq 0 \\
\|w\|_{\infty}=1
\end{array}\)

(b) \(\begin{array}{l}
\text { For } w(t)=a^{t}, t \in \mathbb{Z}: \\
\|w\|_{\infty}=\infty \text{ if } |a| \neq 1 \text { and } \|w\|_{\infty}=1 \text{ otherwise}
\end{array}\)

The space of all signals with finite \(\infty\)-norm are generally denoted by \(l_{\infty}\) and \(\mathcal{L}_{\infty}\) for DT and CT signals respectively. For vector-valued signals, the size of the vector may be explicitly added to the symbol, e.g., \(l_{\infty}^{n}\). These form normed-vector spaces.

Energy and the 2-Norm

The 2-norm of a signal is the square root of its "energy", which is in turn defined as the sum (in DT) or integral (in CT) of the squares of all components over the entire time set:

\[\begin{aligned}
\|w\|_{2} & \triangleq \text { square-root of energy in } w \\
& \triangleq\left[\sum_{k} w^{T}(k) w(k)\right]^{\frac{1}{2}}=\left[\sum_{k}\|w(k)\|_{2}^{2}\right]^{\frac{1}{2}} & \text { (for DT systems) } \ (15.4) \\
& \triangleq\left[\int w^{T}(t) w(t) d t\right]^{\frac{1}{2}}=\left[\int_{t}\|w(t)\|_{2}^{2} d t\right]^{\frac{1}{2}} & \text { (for CT systems) } \ (15.5)
\end{aligned}\nonumber\]

Example 15.2

Some examples:

(a) \(\begin{array}{l}
\text { For } w(t)= e^{-at} \text{ and time set } t \geq 0, \text{ with } a>0: \\
\|w\|_{2}=\frac{1}{\sqrt{2a}}< \infty \end{array}\)

(b) \(\begin{array}{l}
\text { For } w(t)=1 \text{ and time set } t \geq 0: \\
\|w\|_{2}=\infty \end{array}\)

(c) \(\begin{array}{l}
\text { For } w(t)=\cos \omega_{o}t \text{ and time set } t \geq 0: \\
\|w\|_{2}=\infty \end{array}\)

These examples suggest that bounded-energy signals go to zero as time progresses. For discrete-time signals, this expectation holds up: if \(\|w\|_{2}<\infty\), then \(\|w(k)\| \longrightarrow 0 \text { as } k \longrightarrow \infty\). However, for continuous-time signals, the property of having bounded energy does not imply that \(\|w(t)\| \longrightarrow 0 \text { as } t \longrightarrow \infty\), unless additional assumptions are made. This is because continuous-time bounded energy signals can still have arbitrarily large excursions in amplitude, provided these excursions occur over sufficiently narrow intervals of time that the integral of the square remains finite - consider, for instance, a CT signal that is zero everywhere, except for a triangular pulse of height \(k\) and base \(1/k^{4}\) centered at every nonzero integer value \(k\). If the continuous-time signal \(w(t)\) is differentiable and both \(w\) and its derivative \(\dot{w}\) have bounded energy (which is not the case for the preceding triangular-pulse example), then it is true that \(\|w(t)\| \longrightarrow 0 \text { as } t \longrightarrow \infty\). The reader may wish to verify this fact.

It is not hard to show that DT or CT signals with finite 2-norms form a vector space. On the vector space `\(l_{2}\) (respectively \(\mathcal{L}_{2}\)) of DT (respectively CT) signals with finite 2-norm, one can define a natural inner product as follows, between signals \(x\) and \(y\):

\[\langle x, y\rangle \triangleq\left[\sum_{k} x^{T}(k) y(k)\right] \quad \text { (for DT systems) } \ \tag{15.6}\]

\[\triangleq\left[\int x^{T}(t) y(t) d t\right] \quad \text { (for CT systems) } \ \tag{15.7}\]

(The 2-norm is then just the square root of the inner product of a signal with itself.) These particular infinite-dimensional inner-product vector spaces are of great importance in applications, and are the prime examples of what are known as Hilbert spaces.

Power and RMS Value

Another signal measure of interest is the "power" or mean energy of the signal. One also often deals with the square root of the power, which is commonly termed the "root-mean-square" (or "rms") value. For a signal \(w\) for which the following limits exist, we define the power by

\[\begin{aligned}
P_{w} \triangleq \lim _{N \rightarrow \infty}\left[\frac{1}{2 N} \sum_{k=-(N-1)}^{N-1} w^{T}(k) w(k)\right] \quad \text { (for discrete }-\text { time systems }) \ (15.8)\\
\triangleq \lim _{L \rightarrow \infty}\left[\frac{1}{2 L} \int_{-L}^{L} w^{T}(t) w(t) d t\right] \quad \text { (for continuous - time systems) } \ (15.9)
\end{aligned}\nonumber\]

(The above definitions assume that the time set is the entire time axis, but the necessary modifications for other choices of time set should be obvious.) We shall use the symbol \(\rho_{w}\) to denote the rms value, namely \(\sqrt{P_{w}}\). The reason that \(\rho_{w}\) is not a norm, according to the technical definition of a norm, is that \(\rho_{w}=0\) does not imply that \(w = 0\).

Example 15.3

Some finite-power signals:

(a) \(\begin{array}{l}
\text { For } w(t)=1 \\
\rho_{w}=1
\end{array}\)

(b) \(\begin{array}{l}
\text { For } w(t)=1 \text { such that} \|w_{2}\| < \infty\\
\rho_{w}=1
\end{array}\)

(c) \(\begin{aligned}
&\text { For } w(t)=\cos \omega_{0} t(\text { with } t \in \mathbb{R} \text { or } t \in \mathbb{Z})\\
&\rho_{w}=\frac{1}{\sqrt{2}}
\end{aligned}\)

Example c) points out an important difference between bounded power and bounded energy signals: unlike bounded energy signals, if \(\rho_{w}<\infty\), the signal doesn't necessarily decay to zero

As a final comment on the definition of the power of a signal, we elaborate on the hint in the preamble to our definition that the limit required by the definition may not exist for certain signals. The limit of a sequence or function (in our case, the sequence or function is the set of finite-interval rms values, considered over intervals of increasing length) may not exist even if the sequence or function stays bounded, as when it oscillates between two different finite values. The following signal is an example of a CT signal that is bounded but does not have a well-defined power, because the required limit does not exist:

\[w(t)=\left\{\begin{array}{ll}
1 & \text { if } t \in\left[2^{2 k}, 2^{2 k+1}\right], \text { for } k=0,1,2, \ldots \\
0 & \text { otherwise }
\end{array}\right.\nonumber\]

Also note that the desired limit may exist, but not be finite. For instance, the limit of a sequence is \(+\infty\) if the values of the sequence remain above any chosen finite positive number for sufficiently large values of the index.

Action: The 1-Norm

The 1-norm of a signal is also sometimes termed the "action" of the signal, which is in turn defined as the sum (in DT) or integral (in CT) of the 1-norm of the signal value at each time, taken over the entire time set:

\[\begin{aligned}
\|w\|_{1} & \triangleq \text {action of } w \\
& \triangleq\left[\sum_{k}\|w(k)\|_{1}\right] \quad(\text { for discrete }-\text { time systems }) \ (15.10) \\
& \triangleq\left[\int\|w(t)\|_{1} d t\right] \quad \text { (for continuous - time systems) } \ (15.11)
\end{aligned}\nonumber\]

Recall that \(\|w(k)\|\) for the \(n\)-vector \(w(k)\) denotes the sum of magnitudes of its components.

The space of all signals with finite 1-norm are generally denoted by `\(l_{1}\) and \(\mathcal{L}_{1}\) for DT and CT signals respectively. These form normed-vector spaces.

We leave you to construct examples that show familiar signals of finite and infinite 1- norm.

Relationships Among Signal Measures

a) If w is a discrete-time sequence, then

\[\|w\|_{2}<\infty \Longrightarrow\|w\|_{\infty}<\infty \ \tag{15.12}\]

but

\[\|w\|_{2}<\infty \not \Longleftarrow\|w\|_{\infty}<\infty \ \tag{15.15}\]

b) If \(w\) is a continuous-time signal, then

\[\|w\|_{2}<\infty \not \Longrightarrow\|w\|_{\infty}<\infty \ \tag{15.14}\]

and

\[\|w\|_{2}<\infty \not \Longleftarrow\|w\|_{\infty}<\infty \ \tag{15.15}\]

c) If \(\|w\|_{\infty}<\infty\), then (when \(\rho_{w}\) exists)

\[\rho_{w} \leq\|w\|_{\infty}\nonumber\]

Item a) is true because of the relationship between energy and magnitude for discrete-time signals. Since the energy of a DT signal is the sum of squared magnitudes, if the energy is bounded, then the magnitude must be bounded. However, the converse is not true -take for example, the signal \(w(k) = 1\). As item b) indicates, though, bounded energy implies nothing about the boundedness of magnitude for continuous time signals.

(Many more relationships of the above form can be stated.)