I read from a paper (Ozaktas, H.M., et al. (1996) Digital Computation of the Fractional Fourier Transform) that says

The time-bandwidth product can be crudely defined as the product of the temporal extent of the signal and its (double-sided) bandwidth. It is equal to the number of degrees of freedom and the number of complex numbers required to uniquely characterize the signal among others of the same time-bandwidth product.

I understand the first statement, but the second statement is hard to interpretate.

First, why is the time-bandwidth product equal to the number of degrees of freedom and what is the number of degrees of freedom of a signal? Does it hold for both the deterministic and random signal?

Second, why is there a (minimum) number of complex numbers required to characterize a signal? What if the number is not reached?

Hope you can help clarify these confusions.


A strictly time-limited signal $x(t)$, meaning that $x(t)$ is exactly $0$ outside an interval of duration $T$, must have a Fourier transform $X(f)$ whose support is $(-\infty, \infty)$. Similarly, if the Fourier transform $Y(f)$ of a real-valued signal $y(t)$ is strictly band-limited, meaning that $Y(f)$ is exactly $0$ outside a frequency band $[-W,W]$, then $y(t)$ must have support $(-\infty, \infty)$. More succinctly, strictly time-limited signals cannot be strictly band-limited, and strictly band-limited signals cannot be strictly time-limited. So, if strictness is demanded, the notion of a time-bandwidth product is meaningless; it is always infinite.

On the other hand, suppose that we are a little more lax in what we mean by time-limited and band-limited. Then it is possible to make some progress. One result of this type is what is called the Landau-Pollak theorem.

The late U.S. Supreme Court Justice Potter Stewart once rendered a decision in which he memorably said

``I may not know how to define it legally, but I know it when I see it.''

He was, of course, speaking of pornography, but the notion of bandwidth of a signal is very similar. Every engineer understands the notion of bandwidth, perhaps only at an intuitive level or maybe with more erudition and detail for a specific purpose, but a definition of bandwidth that is satisfactory for all purposes remains as elusive as ever.

A signal $s(t)$ is said to be strictly time-limited if $s(t)$ is 0 outside a time interval of finite length, e.g. $s(t) = 0$ if $t < 0$ or $t > T$. It is said to be strictly band-limited if its Fourier transform $S(f)$ is 0 outside a frequency interval of finite length, e.g. $S(f) = 0$ if $\vert f\vert > W$, or $S(f) = 0$ if $\vert f\vert > f_c+W/2$, or $\vert f \vert < f_c - W/2$ corresponding respectively to lowpass or bandpass signals of bandwidth $W$. Unfortunately, a signal cannot be both strictly time-limited and strictly band-limited. Now, let $s(t)$ be a unit-energy strictly time-limited function. In particular, suppose that $s(t) = 0$ if $t < 0$ or if $t > T$. Let $\eta$ denote a very small positive number, that is, $0 < \eta \ll 1$. Let $W$ be a number such that $$ \int_{-W}^W \vert S(f)\vert^2 \, df > 1 - \eta, $$ that is, almost all of the energy in $s(t)$ is in the frequency band $[-W, W]$. We say that $s(t)$ is essentially band-limited to $W$ Hz or an essentially low-pass signal of bandwidth $W$ Hz. Now, suppose that $\{s(t)\}$ denotes a collection of unit-energy signals that are all strictly time-limited to $[0, T]$ and all essentially band-limited to $[-W, W]$. How many mutually orthogonal signals does the set $\{s(t)\}$ contain? The answer is given by a result called the Landau-Pollak Theorem. The number of orthogonal signals is less than $2WT/(1 - \eta)$. Since $\eta$ is very close to 0, the denominator is just slightly less than 1, and so we conclude that the number of orthogonal signals strictly time-limited to $[0, T]$ and essentially band-limited to $W$ Hz is just a tad larger than $M = 2WT$.

Duality says that if the signal is strictly band-limited to $W$ Hz, then it is essentially time-limited to a duration of $T$ seconds (that is, ($1-\eta$) of the energy is in that duration) and there are just a few more than $2WT$ (real-valued) orthogonal signals with this property.

As one might expect, it is possible to ease the "strictly" restriction in one domain to essentially restricted and arrive at the result that

If $2WT$ is defined as the time-bandwidth product of the set of signals that are essentially time-limited to $T$ seconds of duration and also essentially band-limited to $[-W,W]$ Hz of bandwidth, then the set contains just a smidgen more than $2WT$ orthogonal signals.

Now, all the signals in the space of essentially time-limited and essentially band-limited signals can be expressed as a linear combination of the $2WT$ orthogonal complex-valued signals in the space, and the coefficients of the linear combination are the coordinates or representation of a given signal in the space with respect to the basis consisting of these $2WT$ orthogonal signals. So, we can represent (and reproduce if need be from this representation) any such signal from its coordinates with respect to the basis, that is, from $2WT$ complex-valued numbers.

A different notion of time duration and bandwidth of an arbitrary unit-energy signal $x(t)$ comes from noting that $|x(t)|^2$ is a pdf (note that $|x(t)|^2 \geq 0$ for all $t$, and $\int |x(t)|^2 dt = 1$ which are the two defining properties of pdfs). Similarly, $|X(f)|^2$ is also a pdf since $\int |X(f)|^2 df$ is the signal energy which has value $1$. Some folks, especially physicists, like to use the standard deviations of these pdfs as the (root-mean-square) measures $\Delta t$ and $\Delta f$ of the time duration and the bandwidth respectively. I don't like $\Delta t$ and $\Delta f$ because (for example) if these pdfs are Gaussian, then only $68\%$ of the signal energy is contained in the frequency band $[-\Delta f, \Delta f]$ which is a rather poor approximation compared to the (say) $99\%$ essential bandwidth described earlier (when $\eta = 0.01$) that us poor engineers are more concerned with, but ymmv. Anyway, to cut a long story short, the product $\Delta t\cdot\Delta f$ is bounded below )cf. this answer by Matt L. This is a minor variant of the Heisenberg Uncertainty Principle beloved of physicists and OverLordGoldDragons, but I say Meh! Essential notions are more useful for engineers than RMS notions.

  • $\begingroup$ I'm not following these answers; isn't "time-bandwidth" product referring to the Heisenberg relation? Search engines are strangely devoid of much information. $\endgroup$ – OverLordGoldDragon Feb 2 at 4:09
  • 2
    $\begingroup$ @OverLordGoldDragon, no, unless otherwise specified, the time bandwidth product is about effective support. Effective support is bounded by regions where the signal decays faster than an exponential function. If you want Heisenberg, you need to explicitly talk about the quadratic deviation of time or frequency. $\endgroup$ – Jazzmaniac Feb 2 at 11:48
  • $\begingroup$ @Jazzmaniac In that case my answer's non-applicable; thanks. Some detailed reference on the topic would help. $\endgroup$ – OverLordGoldDragon Feb 2 at 12:30
  • $\begingroup$ @DilipSarwate could you please give us some papers/discussions/applications showing the usefulness of considering unit-energy signals as pdf? Thanks $\endgroup$ – AlexTP Feb 3 at 10:09
  • $\begingroup$ @AlexTP There is no specific utility in considering signals in terms of pdfs. As the answer by MattL cited above says, "If we now define time and frequency widths as follows: $$\Delta_t^2=\int_{-\infty}^{\infty}t^2|f(t)|^2dt\\\Delta_{\omega}^2=\int_{-\infty}^{\infty}\omega^2|F(\omega)|^2d\\omega,$$ then..." _ The _definitions came first in the derivation of the Heisenberg Uncertainty Principle; and it was soon pointed out that effectively this meant that $\Delta_t$ and $\Delta_\omega$ were being defined as the standard deviations of the corresponding pdfs. $\endgroup$ – Dilip Sarwate Feb 3 at 15:35

This is not a rigorous answer.

First, why is the time-bandwidth product equal to the number of degrees of freedom and what is the number of degrees of freedom of a signal? Does it hold for both the deterministic and random signal?

The degree of freedom is the number of samples needed to represent a complex continuous-time signal. The Nyquist Shannon sampling theorem states that for any signal confined within $\left[-\frac{\Delta f}{2},+\frac{\Delta f}{2}\right]$, we need at least $\Delta f$ sample per time unit to perfectly reconstruct it.

If in the time domain, the signal can be approximated by its duration $\left[-\frac{\Delta t}{2},+\frac{\Delta t}{2}\right]$, the number of samples needed is thus $\Delta f \Delta t$, which is the time-bandwidth product defined in [Ozaktas et al.].

Second, why is there a (minimum) number of complex numbers required to characterize a signal? What if the number is not reached?

If the number is not reached, the discrete-time signal represented by the insufficient number of samples cannot be used to reconstruct the continuous-time signal. An aliasing will be observed, for example.

  • $\begingroup$ Authors define $\Delta f$ in terms of "sufficient energy concentration" rather than perfect reconstruction, the two are at odds. "samples / time" units are also neither physical nor discrete-time, I don't see how one'd compute frequency-domain spread that yields them. $\endgroup$ – OverLordGoldDragon Feb 1 at 15:55
  • $\begingroup$ @OverLordGoldDragon the authors of the cited paper tried to relate the time-bandwidth product to the degree of freedom. This does not mean their definition is rigorous. They did not redefine the degree of freedom which is typically defined for perfect reconstruction. "Sample" does imply discrete-time. About your comment "physical", this is philosophical lol; you have to give me some context on what you want to refer to. $\endgroup$ – AlexTP Feb 1 at 16:08
  • $\begingroup$ No philosophy, "samples / time" is sampling frequency units, physical is "cycles / time", and discrete-time "cycles / samples". That authors lumping reconstruction with "spread" or "energy concentration" is a mistake, not plain ambiguity. $\endgroup$ – OverLordGoldDragon Feb 2 at 0:57
  • $\begingroup$ @OverLordGoldDragon I don't see why samples/time cannot be associated with physical phenomena. "Energy concentration" is an assumption relaxing strict mathematical requirements. Please take a look at DilipSarwate's more thorough answer. $\endgroup$ – AlexTP Feb 2 at 11:00

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