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Given a finite duration of discrete time domain samples as $x[n]$, the square of each sample $x[n]^2$ would have both units of energy and units of power, and specifically it can be described as "<_____> Power" given it is the energy at that instant, over the time duration until the next sample.

Using SI units for clarity or explanatory analogy, if we provide the time duration of one sample as $T$ seconds, and $x[n]^2$ as energy in Joules, then we would have the <_____> power in each sample as $x[n]^2/T$ Joules/sec = Watts. I'll simplify this to the case where $T=1$ seconds such that, in this case, $x[n]^2$ is both the energy of one sample and the power over one sample duration.

If we sum all the samples for $x[n]^2$, we get the total energy:

$$E_T = \sum_n x[n]^2$$

$$P_T = \frac{1}{N}\sum_n x[n]^2$$

Similarly, given a finite duration of discrete frequency domain samples as $X[k]$, the square of each sample would have both units of power and units of energy, and specifically it can be described as "<_____> Energy" given it is the power at that location in frequency over the frequency span until the next sample. The result of this is a power spectral density, as the power over some unit of BW.

Using SI units in this case, if we provide the frequency spacing of index $k$ as $B$ Hz, and $X[k]^2$ as power in Watts, then we would have the "<_____> Energy" in each bin as $X[k]^2/B$ Watts/Hz = Joules. I'll simplify this to the case of $B=1$ Hz, such that, in this case, $X[k]^2$ is both power of one sample and then energy over the frequency span of one bin.

If we sum all the samples for $X[k]^2$, we get the total power:

$$P_T = \sum_n X[k]^2$$

$$E_T = N\sum_n X[k]^2$$

$$E_T = N\sum_n X[k]^2 = N/2 \text{ Joules}$$

Given a finite duration of discrete time domain samples as $x[n]$, the conjugate product of each sample $|x[n]|^2$ would have both units of energy and units of power, and specifically it can be described as "<_____> Power" given it is the energy at that instant, over the time duration until the next sample.

Using SI units for clarity or explanatory analogy, if we provide the time duration of one sample as $T$ seconds, and $|x[n]|^2$ as energy in Joules, then we would have the <_____> power in each sample as $|x[n]|^2/T$ Joules/sec = Watts. I'll simplify this to the case where $T=1$ seconds such that, in this case, $|x[n]|^2$ is both the energy of one sample and the power over one sample duration.

If we sum all the samples for $|x[n]|^2$, we get the total energy:

$$E_T = \sum_n |x[n]|^2$$

$$P_T = \frac{1}{N}\sum_n |x[n]|^2$$

Similarly, given a finite duration of discrete frequency domain samples as $X[k]$, the conjugate product of each sample would have both units of power and units of energy, and specifically it can be described as "<_____> Energy" given it is the power at that location in frequency over the frequency span until the next sample. The result of this is a power spectral density, as the power over some unit of BW.

Using SI units in this case, if we provide the frequency spacing of index $k$ as $B$ Hz, and $|X[k]|^2$ as power in Watts, then we would have the "<_____> Energy" in each bin as $X[k]^2/B$ Watts/Hz = Joules. I'll simplify this to the case of $B=1$ Hz, such that, in this case, $|X[k]|^2$ is both power of one sample and then energy over the frequency span of one bin.

If we sum all the samples for $|X[k]|^2$, we get the total power:

$$P_T = \sum_n |X[k]|^2$$

$$E_T = N\sum_n |X[k]|^2$$

$$E_T = N\sum_n |X[k]|^2 = N/2 \text{ Joules}$$

Specific to discrete time systems, and preferably without bringing in continuous time equivalents, I would like to confirm my understanding of the following. I'm looking both for confirmation or correction, and if I am correct I would be interested in other perhaps clearer or alternate ways to describe this or other related insights. Note that there is an interesting Time Frequency duality with Power and Energy. After that I will ask a related question:

To help focus answers: The following is specific to “Energy” and “Power” as used for signal processing, not physical units of energy and power (then to ask can units of “Watts” and “Joules” be used, and if not- what then are the units that could be used that would make more sense). Further, this is specific to ergodic, stationary (if random) or otherwise deterministic discrete time signals with a finite duration, as well as discrete in frequency (via the DFT).

Specific to discrete time systems, and preferably without bringing in continuous time equivalents, I would like to confirm my understanding of the following. I'm looking both for confirmation or correction, and if I am correct I would be interested in other perhaps clearer or alternate ways to describe this or other related insights. Note that there is an interesting Time Frequency duality with Power and Energy. I summarize my questions at the end:

• Are my statements accurate?
  
• Are there alternate, more concise and clearer ways to explain this (ideally without continuous time analogies)?
  
• What would the best terms be for the <_____> positions above in time and frequency. In the time domain, I have seen $x[n]^2$ for each sample referred to as "instantaneous power", although it is really instantaneous energy and then the power over one sample duration (which is nearly instantaneous, and approaches the continuous time instantaneous power as $T \rightarrow 0$). I'm fine with referring to it as "instantaneous power", but then what do we refer to the frequency domain counterpart to instantaneous referring to something which occurs at one frequency?

• Are my statements accurate?
  
• Are there alternate, more concise and clearer ways to explain this (ideally without continuous time analogies)?
  
• What would the best terms be for the <_____> positions above in time and frequency. In the time domain, I have seen $x[n]^2$ for each sample referred to as "instantaneous power", although it is really instantaneous energy and then the power over one sample duration (which is nearly instantaneous, and approaches the continuous time instantaneous power as $T \rightarrow 0$). I'm fine with referring to it as "instantaneous power", but then what do we refer to the frequency domain counterpart to "instantaneous", as referring to something which occurs at one frequency?