Multidimensional modulation

Definition/Basics

MD modulation is modifying or multiplying an MD signal (typically sinusoidal and referred to as the carrier signal) with another signal that carries some information or message. In the frequency domain, the signal is moved from one frequency index to another.

if

$x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$ 1

then

$e^{(j\theta _{1}n_{1}+,...,+\theta _{M}n_{M})}x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1}-\theta _{1},...,\omega _{M}-\theta _{M})$ 2

Typically the carrier signal is a sinusoidal signal and in various application, a real signal as such (with no imaginary component). The figures below illustrate a quick example of a 2-D modulation. The original signal from (3) is modulated with a sinusoidal signal to get (4). The equations (5) and (6) are the real and the imaginary components of the modulated signal.

$x(n_{1},n_{2}){=}\cos {(2\pi n_{1}+4\pi n_{2})}$ 3

$y(n_{1},n_{2}){=}e^{(j\pi n_{1}+\pi n_{2})}x(n_{1},n_{2})$ 4

$\mathbf {Re} \lbrace y(n_{1},n_{2})\rbrace {=}\cos {(\pi n_{1}+\pi n_{2})}x(n_{1},n_{2})$ 5

$\mathbf {Im} \lbrace (n_{1},n_{2})\rbrace {=}\sin {(\pi n_{1}+\pi n_{2})}x(n_{1},n_{2})$ 6

Original plot of MD signal

Real value plot of MD modulated signal

Absolute plot of MD modulated signal

Absolute value plot of MD modulated signal

Phase plot of MD modulated signal

Background/Motivation

The MD modulation is one of the properties of the Fourier Transform in the MD sense.
1.2.1 MD Fourier Transform (FT):
Fourier Transform (FT) of multi-dimensional (MD) signal or system is the transform of the MD signal or system that decomposes it into its frequency components. Essentially, it's the frequency response of the MD signal or system, so it depicts the frequency characteristics of the signal or system. A special case of the MD transform is the 1-D FT. Typically, the signals or systems in question are Linear (System) Space Invariant or LSI (LTI in cases where the independent variable is in time domain), and the transform is known to exist if the signal or system is absolutely summable or MD summable i.e.

$\sum _{n_{1}=-\infty }^{\infty }...\sum _{n_{M}=-\infty }^{\infty }|x(n_{1},...,n_{M})|<\mathbf {S}$
$\sum _{n_{1}=-\infty }^{\infty }...\sum _{n_{M}=-\infty }^{\infty }|x(n_{1},...,n_{M})|^{M}<\mathbf {S_{M}}$

The signal or system is then bounded in this sense. This categorizes the signal or system as stable (BIBO).

Computation

The formula for computing the Fourier transform of an MD signal is

$X(\omega _{1},\omega _{2},...,\omega _{M}){=}\sum _{n_{1}=-\infty }^{\infty }\sum _{n_{2}=-\infty }^{\infty }...\sum _{n_{M}=-\infty }^{\infty }x(n_{1},n_{2},...,n_{M})e^{-j(\omega _{1}n_{1}+\omega _{2}n_{2}+,...,+\omega _{M}n_{M})}$

If the transform to be studied, is a system, then the transform is performed on the impulse response of the system. The impulse response is normally assigned the variable h(n1,…,nM), so the formula used is

$H(\omega _{1},\omega _{2},...,\omega _{M}){=}\sum _{n_{1}=-\infty }^{\infty }\sum _{n_{2}=-\infty }^{\infty }...\sum _{n_{M}=-\infty }^{\infty }h(n_{1},n_{2},...,n_{M})e^{-j(\omega _{1}n_{1}+\omega _{2}n_{2}+,...,+\omega _{M}n_{M})}$

If the frequency response is given instead, then the inverse FT formula is used to derive the input signal or system. In this case the formula used is:

$x(n_{1},n_{2},...,n_{M}){=}{\frac {1}{(2\pi )^{M}}}\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }...\int \limits _{-\pi }^{\pi }X(\omega _{1},\omega _{2},...,\omega _{M})e^{j(\omega _{1}n_{1}+\omega _{2}n_{2}+,...,+\omega _{M}n_{M})}d\omega _{1}d\omega _{2}...d\omega _{M}$

Properties

Similar properties of the 1-D FT transform apply, but instead of the input parameter being just a single entry, it’s an M-array or M-vector. Hence, it’s x(n1,…,nM) instead of x(n).

Linearity

if $x_{1}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{1}(\omega _{1},...,\omega _{M})$ , and $x_{2}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{2}(\omega _{1},...,\omega _{M})$ then,

$ax_{1}(n_{1},...,n_{M})+bx_{2}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}aX_{1}(\omega _{1},...,\omega _{M})+bX_{2}(\omega _{1},...,\omega _{M})$

Shift Invariance

if $x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$, then
$x(n_{1}-a_{1},...,n_{M}-a_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}e^{-j(\omega _{1}a_{1}+,...,+\omega _{M}a_{M})}X(\omega _{1},...,\omega _{M})$

Modulation

if $x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$, then
$e^{j(\theta _{1}n_{1}+,...,+\theta _{M}n_{M})}x(n_{1}-a_{1},...,n_{M}-a_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1}-\theta _{1},...,\omega _{M}-\theta _{M})$

Multiplication

if $x_{1}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{1}(\omega _{1},...,\omega _{M})$, and $x_{2}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{2}(\omega _{1},...,\omega _{M})$
then,

$x_{1}(n_{1},...,n_{M})**x_{2}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}{\frac {1}{(2\pi )^{M}}}\int \limits _{-\pi }^{\pi }...\int \limits _{-\pi }^{\pi }X_{1}(\omega _{1}-\theta _{1},...,\omega _{M}-\theta _{M})X_{2}(\theta _{1},...,\theta _{M})d\theta _{1}...d\theta _{M}$ MD Convolution in Spatial Domain

or,

$x_{1}(n_{1},...,n_{M})**x_{2}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}{\frac {1}{(2\pi )^{M}}}\int \limits _{-\pi }^{\pi }...\int \limits _{-\pi }^{\pi }X_{2}(\omega _{1}-\theta _{1},...,\omega _{M}-\theta _{M})X_{1}(\theta _{1},...,\theta _{M})d\theta _{1}...d\theta _{M}$ MD Convolution in Spatial Domain

It can be seen from the Multiplication and Modulation properties above that the Modulation property is a special case of the Multiplication property which in the frequency domain is a convolution. In modulation however, the carrier signal typically a sinusoidal signal that is localized to the unit circle (or bi-unit circle in case of 2-D and tri-unit circle in case of 3-D signals, etc.). This means that the carrier signal has a magnitude of 1, regardless of the frequency or the phase of that signal.

Differentiation

if $x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$, then

$-jn_{1}x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}{\frac {\delta }{(\delta \omega _{1})}}X(\omega _{1},...,\omega _{M})$,
$-jn_{2}x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}{\frac {\delta }{(\delta \omega _{2})}}X(\omega _{1},...,\omega _{M})$,
$(-j)^{M}(n_{1}n_{2}...n_{M})x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}{\frac {(\delta )^{M}}{(\delta \omega _{1}\delta \omega _{2}...\delta \omega _{M})}}X(\omega _{1},...,\omega _{M})$,

Transposition

if $x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$, then

$x(n_{M},...,n_{1}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{M},...,\omega _{1})$

Reflection: if $x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$, then

$x(\pm n_{1},...,\pm n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\pm \omega _{1},...,\pm \omega _{M})$

Complex conjugation

if $x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$, then

$x^{*}(\pm n_{1},...,\pm n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X^{*}(-\omega _{1},...,-\omega _{M})$

Parseval's theorem (MD)

if $x_{1}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{1}(\omega _{1},...,\omega _{M})$ , and $x_{2}(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{2}(\omega _{1},...,\omega _{M})$ then,

$\sum _{n_{1}=-\infty }^{\infty }...\sum _{n_{M}=-\infty }^{\infty }x_{1}(n_{1},...,n_{M})x_{2}^{*}(n_{1},...,n_{M}){=}{\frac {1}{(2\pi )^{M}}}\int \limits _{-\pi }^{\pi }...\int \limits _{-\pi }^{\pi }X_{1}(\omega _{1},...,\omega _{M})X_{2}^{*}(\omega _{1},...,\omega _{M})d\omega _{1}...d\omega _{M}$

if $x_{1}(n_{1},...,n_{M}){=}x_{2}(n_{1},...,n_{M})$, then

$\sum _{n_{1}=-\infty }^{\infty }...\sum _{n_{M}=-\infty }^{\infty }|x_{1}(n_{1},...,n_{M})|^{2}{=}{\frac {1}{(2\pi )^{M}}}\int \limits _{-\pi }^{\pi }...\int \limits _{-\pi }^{\pi }|X_{1}(\omega _{1},...,\omega _{M})|^{2}d\omega _{1}...d\omega _{M}$ 7

A special case of the Parseval’s theorem is when the two multi-dimensional signals are the same. In this case, the theorem portrays the energy conservation of the signal and the term in the summation or integral is the energy-density of the signal.

Separability

One property is the separability property. A signal or system is said to be separable if it can be expressed as a product of 1-D functions with different independent variables. This phenomenon allows computing the FT transform as a product of 1-D FTs instead of multi-dimensional FT.

if $x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})$, $a(n_{1}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}A(\omega _{1})$, $b(n_{2}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}B(\omega _{2})$ ... $y(n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}Y(\omega _{M})$, and if
$x(n_{1},...,n_{M}){=}a(n_{1})b(n_{2})...y(n_{M})$, then

$X(\omega _{1},...,\omega _{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}x(n_{1},...,n_{M}){=}a(n_{1})b(n_{2})...y(n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}A(\omega _{1})B(\omega _{2})...Y(\omega _{M})$, so

$X(\omega _{1},...,\omega _{M}){=}A(\omega _{1})B(\omega _{2})...Y(\omega _{M})$

Approaches / Extended Applications of MD Modulation

From the special case of Parseval’s theorem (7), it is noted that the energy or power of a signal is directly proportional to the magnitude of the signal, and since most of the signals in this case are sinusoidal, it is well noted that the energy or power is directly proportional to the amplitude of the sinusoid. Since power management is always an issue, the modulation of the amplitude isn’t always the best solution when the application is trying to be power conscious. For such an application, the amplitude of the sinusoid is typically set to 1. This very well means that the information is normally carried in the phase or the frequency index of the signal, as can be seen from the Fourier Transform (or frequency response) of the signal in the equation (4) above

1-D Modulation

The most popular and more conversant form of signal modulation is the 1-D modulation e.g. amplitude modulation, frequency modulation. (Note: the conversion of a signal from analog to discrete can be accomplished with a sampling Matrix assuming Nyquist sampling for reproducibility).

Other extended applications

Speech analysis: The fundamental theory of speech analysis, one can say starts with the digitization and the frequency response representation of a 1-D time domain signal. Since the speech signal can be represented as a sum of sinusoids and a change to the amplitude of the speech signal during its processing also impact the information in the signal itself, frequency type modulation is what is normally employed. The signal can be moved to lower band frequencies for processing without actually changing the information embedded in the signal. Audio signal processing: The pitch shifting phenomenon in audio signal processing essentially moves a signal from one frequency to another (FM). Ring modulation is another application or extension of 1-D Modulation of signals. It is typically used in audio signal processing.

2-D Modulation

A two-D modulation comprises a space domain that has 2 independent variables e.g. x(n1,n2) with a corresponding frequency domain that also has 2 independent variables X(ω1, ω2).

===2-D AM and FM analysis=== Image texturization is an application of multidimensional AM-FM modulation. In this method, the image (2-D signal) is expressed into its special frequencies and amplitude estimates. The signal is represented as a product of 2 FM functions (using the independent frequencies), making is separable. Using the instantaneous frequency, the image can be represented topologically to illustrate its texture.

3-D Modulation

Similar to 2-D modulation, 3-D has 3 independent variables instead of 2. Some applications even though aren’t based exactly on modulation, are based on the sum of the modulation across its entire spatial domain e.g. 3-D audio. 3-D scanners: Some 3-D scanners used modulated light, typically amplitude modulation, together with a camera to scan an object. 3-D TOF cameras: They use modulated light sources and it’s reflection for depth perception computation. In essence, creating a 2-D image with depth perception helps separate the foreground and background. 3-D Audio: 3-D audio (processing) is the spatial domain application of sound waves. It is the phenomenon of transforming sound waves (using Head Related Transfer function, HTRF filters and cross talk cancellation techniques) to mimic natural sounds waves, which emanate from a point in a 3-D space. It allows trickery of the brain using the ears and auditory nerves, pretending to place different sounds in different 3-D locations upon hearing the sounds, even though the sounds may just be produced from just 2 speakers (dissimilar to surround sound). [Note that, transfer functions are functions that depict the relationship between an input and the output of a system. It’s a fraction representation with the denominator not equal to 1, if there’s feedback in the system. If the transfer functions e.g. z-transform are localized to the unit circle (or bit-unit circle or tri-unit circle for 2-D and 3-D cases respectively), it then becomes the MD FT or the frequency response of the system.]

References

• Dudgeon and Mersereau (1984). Multidimensional Digital Signal Processing. Upper Saddle River, NJ: Prentice-Hall. ISBN 978-0136049593.
• Li, Larry. "Time-of-Flight Camera - An introduction" (PDF). Texas Instruments.