Advanced: Power and Energy in Discrete-Time

1.12 Advanced: Power and Energy in Discrete-Time

The concepts of power and energy are better defined for continuous-time than for discrete-time signals and vectors. Dealing with the concept of power of a discrete-time signal $x [n]$ requires some caution because the discrete-time $n$ does not have the dimension of time as the continuous-time t, which can be given in seconds, for instance. Sometimes one has a set of values, but the index that helps locating each individual value cannot be interpreted as discrete-time. In such cases, to distinguish from $x [n]$ (with $n$ representing discrete-time), the notation $x_{i}$ is used. The key concept here is that $i$ simply identifies the $i$ -th element of the set and is not interpreted as “time”. Hence, as will be discussed in this section, taking an average over $| x_{i} |^{2}$ values will not be interpreted as power (energy divided by time).

1.12.1 Power and energy of discrete-time signals

The following definition of average power will be adopted in this text

P ≜ lim_{N \to \infty} \frac{1}{2 N + 1} \sum_{n = - N}^{N} | x [n] |^{2} .

(1.63)

and interpreted as Watts given that $x [n]$ is assumed to be in volts. This is a sensible definition, as will be discussed in Section 1.12.5.

Accordingly, the energy $E$ of a discrete-time signal is

E ≜ \sum_{n = - \infty}^{\infty} | x [n] |^{2} .

(1.64)

such that $P = E ∕ N$ when the support of $x [n]$ are $N$ samples.

A similar situation occurs when dealing with vectors.

1.12.2 Power and energy of signals represented as vectors

Here, $x [n]$ denotes the $n$ -th element of a vector $x$ . The order of these elements is assumed to correspond to a time evolution, such that a finite-dimension vector is equivalent to a finite-duration discrete-time sequence. Hence, the energy $E$ of vector $x$ is its squared Euclidean norm:

E = \sum_{n = 1}^{N} | x [n] |^{2} = | | x | |^{2},

(1.65)

where $N$ is the dimension of $x$ . Accordingly, the power of such finite-dimension vector is

P = \frac{E}{N} = \frac{1}{N} \sum_{n = 1}^{N} | x [n] |^{2} .

(1.66)

1.12.3 Advanced: Power and energy of vectors whose elements are not time-ordered

In contrast to the previous equations, there are cases in which the vector elements are not indexed according to a time evolution.

It should be noticed that the average of the squared norms of several vectors should be interpreted as their average energy, not power. For example, assuming there are $M$ vectors $x_{1}, \dots, x_{M}$ of dimension $N$ , an average energy is obtained with

\bar{E} = \frac{1}{M} \sum_{i = 1}^{M} E_{i} = \frac{1}{M} \sum_{i = 1}^{M} | | x_{i} | |^{2},

(1.67)

and interpreted in joules.

Eq. (1.66) and Eq. (1.67) are similar due to the connection between vectors and finite-duration discrete-time signals. The distinction that allows to observe their results as “power” or “energy” relies on interpreting the index $i$ as “time” or not. In Eq. (1.66), an average of instantaneous power values $| x [n] |^{2}$ along the discrete-time $n$ leads to an estimate of power. But when taking an average $\bar{E}$ over vectors of energy $E_{i}$ in Eq. (1.67), the result is (average) energy.

1.12.4 Power and energy of discrete-time random signals

If $x [n]$ represents a random signal, with samples $x [n_{0}]$ corresponding to outcomes of a random variable $X$ , an alternate definition is

P ≜ 𝔼 [| X |^{2}] = 𝔼 [| x [n] |^{2}],

where $𝔼 [\cdot]$ is the expectation operator.

The power of a random signal $x [n]$ (or $x (t)$ ) can be decomposed into two parcels as follows:

P = 𝔼 [X^{2}] = σ_{x}^{2} + μ_{x}^{2},

(1.68)

where the variance $σ_{x}^{2}$ and the squared-mean $μ_{x}^{2}$ correspond to the powers of the AC and DC components of a real $x [n]$ , respectively. The proof is derived in Eq. (A.67).

Most of the signals in telecommunications and other applications have zero mean ( $μ = 0$ ). In this case, Eq. (1.68) shows that the power $P = σ_{x}^{2}$ coincides with the variance of the random signal and the standard deviation $σ_{x}$ with its RMS value.

1.12.5 Advanced: Relating Power in Continuous and Discrete-Time

The goal here is to relate the power of a discrete-time $x [n]$ to the power of a continuous-time $x (t)$ where these signals are related by an A/D or D/A conversion. A possible processing chain relating these signals with their associated power in parenthesis is:

\underset{(P_{c})}{x (t)} \to sampling \to x_{s} (t) \to S ∕ D \to \underset{(P_{d})}{x [n]} = x (n T_{s}) .

(1.69)

Another processing chain of interest is the reconstruction of a continuous-time signal:

x (n T_{s}) = \underset{(P_{d})}{x [n]} \to D ∕ S \to x_{s} (t) \to h (t) \to \underset{(P_{c})}{x (t)}

(1.70)

which will be discussed in Section 3.5.7.

Special interest lies on systems that have equivalence between power in discrete and continuous-time, such that

P_{d} = P_{c},

(1.71)

where $P_{d}$ and $P_{c}$ are given by Eq. (1.63) and Eq. (1.23), respectively.¹⁸

Eq. (1.71) is further discussed in Section 3.5.3, but here it is derived¹⁹ using the rectangle method to approximate the integral of $p (t) = | x (t) |^{2}$ as a sum of rectangles with bases $T_{s}$ and heights $| x [n] |^{2}$ , as follows:

\begin{array}{l} P_{c} & = lim_{N \to \infty} [\frac{1}{(2 N + 1) T_{s}} \int_{- N T_{s}}^{N T_{s}} | x (t) |^{2} d t] \\ \approx lim_{N \to \infty} [\frac{1}{(2 N + 1) T_{s}} \sum_{n = - N}^{N} T_{s} | x (n T_{s}) |^{2}] \\ = lim_{N \to \infty} [\frac{1}{(2 N + 1)} \sum_{n = - N}^{N} | x [n] |^{2}] \\ = P_{d} . & (1.72) \end{array}

In summary, when the sampling theorem is obeyed, the signal processing chains (filtering, amplification, etc.) associated to the A/D and D/A processes are assumed here not to alter the power of $x (t)$ and $x [n]$ , such that Eq. (1.71) holds.

¹⁸ This is not the same, but similar to the energy-conservation property of Fourier transforms (e. g., Eq. (A.51), or its version for periodic signals Eq. (A.52)).

¹⁹ Eq. (1.71) can also be obtained by assuming the zero-order hold (ZOH) reconstruction of Figure 1.35.