I am confused about this concept. This theorem calculates the energy of a signal. For a discrete time signal x[n], what does energy mean?
To elaborate, from what I understand that samples in discrete time are usually represented using discrete values on the y-axis which are obtained from quantisation of the levels pf input signal which is usually a quantity represented by volts as an input to a signal processing system. From basic electrical theory, we can calculate the energy of this analogue signal in Joules but what does energy mean in this theorem for Discrete Time signals(sequences)? Surely, not in Joules!!

 

I am confused about this concept. This theorem calculates the energy of a signal. For a discrete time signal x[n], what does energy mean?
To elaborate, from what I understand that samples in discrete time are usually represented using discrete values on the y-axis which are obtained from quantisation of the levels pf input signal which is usually a quantity represented by volts as an input to a signal processing system. From basic electrical theory, we can calculate the energy of this analogue signal in Joules but what does energy mean in this theorem for Discrete Time signals(sequences)? Surely, not in Joules!!

First, a note on terminology. A discrete-time signal has been quantized in time (discrete steps along the horizontal axis), which is distinct from quantization applied to amplitude (discrete steps along the vertical axis). A sampled signal is discrete in time; a quantized signal is discrete in amplitude; and a digital signal is discrete in both time and amplitude. A sampled signal need not be quantized, e.g., analog multiplexing. Likewise, a quantized signal need not be sampled, e.g., LED bar graph being driven by an analog signal.

Now, to your question. At any instant of time, the power of a continuous-time signal is proportional to the square of its magnitude. This follows from the definition of power and treating the signal as a voltage or current. For certain continuous-time signals, we can calculate its energy E by integrating its power over time:

\(E = \int_{-\infty}^\infty |f(t)|^2 \, dt\)

We take E to be in Joules by assuming that f(t) is the voltage across a resistor of 1 Ω. Then, |f(t)|² is the power of f, in units J/s, and so E has units of (J/s) * s = J.

What about the discrete-time case? If we sample a continuous-time signal x(t) with a sampling interval of T seconds, we get a discrete-time signal x[n] = x(nT), where n ∈ ℤ is the sample index. The key transformation here is t → nT, with n the independent variable in units of samples, and T a constant with units of seconds per sample. Using dimensional analysis, we find that

\(\small{[nT] = [n][T] = (\text{samples}) (\frac{\text{seconds}}{\text{samples}}) = \text{seconds} = [t]}\)

and so we claim that the power -- and thus energy -- of a sampled signal has equivalent units as its continuous version. Of course, integrals in continuous time become sums in discrete time.

Note, however, that in both the continuous- and discrete-time case, signal energy is defined only for a certain class of signals, e.g., aperiodic signals of finite duration, or signals that approach 0 as t → ∞. For instance, the energy of f(t) = sin(t) is not well-defined as the integral does not converge.

That's the signals and systems view from an electronics perspective, but to better understand Parseval's theorem, we turn to a mathematical analysis, which abstracts away the representational details. This is appropriate because Parseval was a mathematician, not an electrical engineer.

There's a more general term for functions f(t) such that E -- as defined above -- exists. These are called square-integrable functions. The space of all such functions is L², aka, Hilbert space. While we can think of this space as the space of processes that have finite energy, it is fundamentally a mathematical object; indeed, we use the same object to represent quantum states in physics.

Formally, Parseval's theorem says that

\(\int_{-\infty}^\infty |f(t)|^2 \, dt = \frac{1}{2\pi} \int_{-\infty}^\infty |F(\omega)|^2 \, d\omega\)

which tells us that square-integrable functions and their Fourier transforms represent the same point in L² space. This means that the Fourier transform is an isomorphism to itself in Hilbert space, and this is the mathematical property that makes Fourier transforms so incredibly useful: they are norm-preserving isometries in Hilbert space.

So, if we think of the underlying Hilbert space in terms of energy, the isometry means that the Fourier transform preserves energy (your interpretation above of Parseval's theorem). But if we think in terms of quantum states, the isometry means that Fourier transform preserves state. Notice that, regardless of the interpretation (the model applied), the math is the same precisely because it is a mathematical relationship. That we can apply various physical interpretations is extremely convenient and, I believe, a clear indication of a deep connection.

FYI, the analogous case for discrete-time functions f:ℕ → ℝ is the set of square-summable functions that live in l² (little-el squared). Parseval's theorem applies here, too, which gives us the mathematical explanation for why the DFT works.

 

Thanks @bogosort for the explanation and for correcting my misuse of terminology. This totally makes sense now. I tried searching for a definition that links Parseval's theorem to a definition of energy of signal. Your explanation was perfect.

 

One thing to watch out for is the temptation to treat the energy of a discrete time signal as being an approximation of the energy of the continuous time signal that was sampled to produce it. The degree to which you can make this assumption depends on the sampling rate as compared to the frequency content of the continuous time signal. If you are sampling far below Nyquist, then it is actually quite easy to produce signals whose energy is essentially unrelated, either larger or smaller, than that of their sampled discrete-time signals.