I don't think that whiteness is of any importance here, and in particular, no digital modulation scheme is going to create a signal whose power spectrum is anywhere close to that of band-limited white noise -- flat in bandwidth $B$ and $0$ outside the band. The result the OP is asking about is based on 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, perhaps only at an intuitive level, 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 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 approximately $M = 2WT$.

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 approximately $2WT$ orthogonal signals with this property.

I don't think that whiteness is of any importance here, and in particular, no digital modulation scheme is going to create a signal whose power spectrum is anywhere close to that of band-limited white noise -- flat in bandwidth $B$ and $0$ outside the band. The result the OP is asking about is based on 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, perhaps only at an intuitive level, 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 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 approximately $M = 2WT$.

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 approximately $2WT$ orthogonal signals with this property.

I don't think that whiteness is of any importance here, and in particular, no digital modulation scheme is going to create a signal whose power spectrum is anywhere close to that of band-limited white noise -- flat in bandwidth $B$ and $0$ outside the band. The result referred to is based on 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, perhaps only at an intuitive level, 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 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 approximately $M = 2WT$.

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 approximately $2WT$ orthogonal signals with this property.

I don't think that whiteness is of any importance here, and in particular, no digital modulation scheme is going to create a signal whose power spectrum is anywhere close to that of band-limited white noise -- flat in bandwidth $B$ and $0$ outside the band. The result the OP is asking about is based on 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, perhaps only at an intuitive level, 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 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 approximately $M = 2WT$.

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 approximately $2WT$ orthogonal signals with this property.