Multidimensional digital pre-distortion (MDDPD), often referred to as multiband digital pre-distortion (MBDPD), is a subset of digital predistortion (DPD) that enables DPD to be applied to signals (channels) that cannot or do not pass through the same digital pre-distorter but do concurrently pass through the same nonlinear system. 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. The first paper in which it found application was in 1991 as seen here. 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).
Motivation Although MDDPD enables the use of DPD in multi source systems, there is another advantage from implementing MDDPD over DPD which is the prime motivation of the initial studies. 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. MDDPD does just this.
Advantages Just as the digital pre-distortion in MDDPD is applied to the channels independently, the feedback sampling of the channels may also be done independently. In addition, as was mentioned previously, MDDPD allows the pre-distortion to be applied to channels that are generated independently. This enables the application of and thereby benefit of predistortion in systems which would not traditionally be able to benefit from one dimensional DPD.
Disadvantages 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. LOs and synthesizers are not trivial components in designs. Also, as will be seen later, the number of coefficients that must be solved for is much larger than the number of coefficients that would need to be solved for in one dimensional DPD. Finally, there must be a high-speed channel between the different channel sources as in order to adapt the digital pre-distorter and apply the pre-distortion as each source must have the channel information from each and every one of the other sources as will be shown in the derivation and approaches sections.
Applications The two markets that currently make use of MDDPD are the handset and the satellite communications market. 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. Multi-dimensional DPD allows the application of DPD in multi-source systems and therefore enables the transmitter to be kept closer to saturation power in multi-modem installations.
Derivation and differentiation of two-dimensional DPD from one-dimensional DPD 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)). The signals are orthogonal because they are frequency translated by ω1 and ω2 which are selected in a manner that guarantees channel orthogonality.
where
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 other words, there are no simple first, third, and fifth order components now but rather first, third, and fifth order interband and intraband coefficients instead. Equations ((13)) and ((14)) are those in-band terms in summation form.
The aesthetic difference between 1DDPD and MDDPD can be seen from a comparison of ((3)) and ((11)) and ((4)) and ((12)) and the result of these mathematical differences in a multichannel application can be seen by comparing the two graphs below.
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