Multidimensional digital pre-distortion

Its ability to do so comes from the portion of multidimensional signal theory that deals with one dimensional discrete time vector input - 1-D discrete time vector output systems.

[2] None of the applications of MDDPD are able to make use of the linear shift invariant (LSI) system properties as by definition they are nonlinear and not shift-invariant although they are often approximated as shift-invariant (memoryless).

[3] In one dimensional polynomial-based memory (or memoryless) DPD, in order to solve for the digital pre-distorter polynomials coefficients and minimize the mean squared error (MSE), the distorted output of the nonlinear system must be over-sampled at a rate that enables the capture of the nonlinear products of the order of the digital pre-distorter.

In systems where there is considerable spacing between carriers or the channel bandwidths are very wide, this leads to a significant increase in the minimum acceptable sampling rate of the analog-to-digital converter (ADC) used for feedback sampling over that of systems that are single channel or have tightly spaced carriers.

As ADCs are more expensive and harder to design than the digital-to-analog converter (DAC) used to generate the channels and ADCs get very expensive when the sampling rate approaches 1 Gs/s and higher, it is highly desirable to reduce the sampling rate of the ADC required to perform DPD.

In addition, as was mentioned previously, MDDPD allows the pre-distortion to be applied to channels that are generated independently.

In order to take advantage of the ability to reduce the ADC sampling rate, groups of channels must have their own down-conversion to baseband for sampling, thereby increasing the number of mixers and local oscillators (LO) or synthesizers.

In handsets it is important to keep power consumption low and size minimal which is what brought about the initial investigations into MDDPD as the reduction of the feedback sampling rate means a reduction in power and size of the ADC portion of the IC being used.

In satellite communications it is important to run transmitter power amplifier as close to its saturation power as possible in order to minimize operational expenditure and capital expenditure but often more than one modem is being used in conjunction with the same transmitter.

A fifth odd-only order nonlinear one dimensional memory (or memoryless) polynomial is taken ((1)) but in place of a single input signal used in the traditional derivation of 1DDPD the input to the nonlinear system is replaced with the summation of two orthogonal signals ((2)).

Equations ((3)) and ((4)) are the in-band terms that come from the expansion of the polynomials when done in the traditional one dimensional DPD manner, meaning, the first, third, and fifth order coefficients are considered coupled or non-orthogonal and equal to that of their value in the polynomial presented in ((1)).

Equations ((5)),((6)),((7)),((8)), ((9)), and ((10)) are the out-of-band terms that come from the polynomial expansion also done in the traditional 1D DPD manner.

Equations ((11)) and ((12)) are the in-band terms that come from the expansion of the polynomials when done in the MDDPD manner, meaning, the first, third, and fifth order coefficients are considered uncoupled or orthogonal and not equal to that of their value in the polynomial presented in ((1)).

In MDDPD the independent impulse responses are replaced with independent coefficients but represent a same concept that each input has a unique relation to each output and can be called a single sample impulse response.

This process can be done for any number of independent sources m in order to obtain the general forms of the equations for MDDPD.

However, this approach is a sub-set of the MIMO Volterra series, for the application of complex-valued equivalent time signals [5] One can choose to ignore harmonics if one considers ones systems representable by a "baseband" model, a model in which the system is considered to be accurately represented only by energy content within the frequency range that can be generated the systems DACs and measured by the systems ADCs, or one can choose to include the harmonics in the solving algorithm if ones system does not adhere to the baseband model but application of MDDPD to a non-baseband model is somewhat counterintuitive as it will increase the necessary sampling rate to capture the harmonic information and somewhat defeat one of the two prime advantages of MDDPD.

That is to say, that if it is known that a baseband model is adequate for a given multi-signal system, then MDDPD should be considered.

The approaches seen in,[6][7][8][9] and [10] attempt to break the problem into two orthogonal problems and deal with each separately in order to reduce the feedback sampling bandwidth over that of 1D DPD (hopefully to that of MDDPD).

They break the application of the pre-distortion and model extraction into in-band and interband systems.

It is stated that correction of interband inter-modulation distortion (IMD) generates inband IMD and that if the fully orthogonal polynomials are applied properly this will no longer be the case.

It appears that this approach in essence is trying to make ((3)) and ((4)) into ((11)) and ((12)) as the orthogonality of the inband and interband coefficients is guaranteed if the polynomials are properly derived and applied as in ((13)) and ((14)).

The approaches seen in,[11][12][13][14] are focused on the proper derivation and application of the MDDPD memory polynomial in multiband systems.

The disadvantages of the previous approaches is that they solely consider certain terms in the MIMO Volterra kernels as defined in [15][16] or described in its complex-valued time equivalent form in.

[17] That is, the models and compensation schemes are pruned forms of the MIMO Volterra series.

However, this series suffer from high dimensionality hidering[check spelling] of its practical application.

Thus, using the described pruned approaches sound solutions and models can be found for relatively general cases.

The approach seen in [18] attempts to further simplify the pre-distorter feedback system by applying subsampling in order to eliminate a down conversion stage.

This reference focuses on the subsampling portion of the system and characterizing the ranges of valid sampling frequencies based on carrier location and spacing.

The disadvantage of this approach is the restriction of the carrier location and spacing that is inherent to achieving proper subsampling.

The approach seen in [20] uses principal component analysis (PCA) to reduce the number of coefficients necessary to achieve similar adjacent channel power (ACP).