User:Jsochacki/Multidimensional Digital Pre-distortion

Multidimensional Digital Pre-distortion (MDDPD), often referred to as multiband digital pre-distortion (MBDPD), is a subset of digital pre-distortion (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. It's ability to do so comes from the portion multidimensional signal theory that deals with 1D discrete time vector input - 1D discrete time vector output systems as defined in . The first paper in which it found application was in 1991 as seen here . It is important to note that none of the applications of MDDPD are able to make use of the linear shift invariant (LSI) system properties as by definition they are non linear and not shift invariant although they are often approximated as shift invariant (memoryless).

16APSK 2/3 2D DPD Comparison Using MDDPD

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 the minimize 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 ADC's are more expensive and harder to design than the digital to analog converters (DAC) used to generate the channels and ADC's get very expensive when the sampling rate approaches 1Gsps 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 pre-distortion 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 channel must have their own down conversion to baseband for sampling thereby increasing the number of mixers and local oscillators (LO) or synthesizers. LO's 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.

Derivation and Differentiation From One Dimensional DPD

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

$y=as+bs\left\vert {s}\right\vert ^{2}+cs\left\vert {s}\right\vert ^{4}$

(1)

$s=x_{1}e^{-j{\omega }_{1}t}+x_{1}e^{-j{\omega }_{2}t}$

(2)

Equations ((3)) and ((4)) are the in band terms that come from the expansion of the polynomials when done in the traditional 1D DPD manner and 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.

$y_{\omega _{1}}=(ax_{1}+bx_{1}\left\vert {x_{1}}\right\vert ^{2}+bx_{1}\left\vert {x_{2}}\right\vert ^{2}+cx_{1}\left\vert {x_{1}}\right\vert ^{4}+cx_{1}\left\vert {x_{2}}\right\vert ^{4}+4cx_{1}\left\vert {x_{1}}\right\vert ^{2}\left\vert {x_{2}}\right\vert ^{2})e^{-j{\omega }_{1}t}$

(3)

$y_{\omega _{2}}=(ax_{2}+bx_{2}\left\vert {x_{2}}\right\vert ^{2}+bx_{2}\left\vert {x_{1}}\right\vert ^{2}+cx_{2}\left\vert {x_{2}}\right\vert ^{4}+cx_{2}\left\vert {x_{1}}\right\vert ^{4}+4cx_{2}\left\vert {x_{2}}\right\vert ^{2}\left\vert {x_{1}}\right\vert ^{2})e^{-j{\omega }_{2}t}$

(4)

$y_{3{\omega _{1}}}=(bx_{1}^{2}x_{2}^{\ast }+2cx_{1}^{2}\left\vert {x_{1}}\right\vert ^{2}x_{2}^{\ast }+2cx_{1}^{2}\left\vert {x_{2}}\right\vert ^{2}x_{2}^{\ast })e^{-j3{\omega }_{1}t}$

(5)

$y_{3{\omega _{2}}}=(bx_{2}^{2}x_{1}^{\ast }+2cx_{2}^{2}\left\vert {x_{2}}\right\vert ^{2}x_{1}^{\ast }+2cx_{2}^{2}\left\vert {x_{1}}\right\vert ^{2}x_{1}^{\ast })e^{-j3{\omega }_{2}t}$

(6)

$y_{5{\omega _{1}}}=(cx_{1}^{3}x_{2}^{2\ast })e^{-j5{\omega }_{1}t}$

(7)

$y_{5{\omega _{2}}}=(cx_{2}^{3}x_{1}^{2\ast })e^{-j5{\omega }_{2}t}$

(8)

$y_{\omega _{2}-2{\omega }_{1}}=(bx_{1}\left\vert {x_{2}}\right\vert ^{2}+2cx_{1}\left\vert {x_{2}}\right\vert ^{2}\left\vert {x_{1}}\right\vert ^{2}+2cx_{1}\left\vert {x_{2}}\right\vert ^{4})e^{-j(\omega _{2}-2{\omega }_{1})t}$

(9)

$y_{\omega _{1}-2{\omega }_{2}}=(bx_{2}\left\vert {x_{1}}\right\vert ^{2}+2cx_{2}\left\vert {x_{1}}\right\vert ^{2}\left\vert {x_{2}}\right\vert ^{2}+2cx_{2}\left\vert {x_{1}}\right\vert ^{4})e^{-j(\omega _{1}-2{\omega }_{2})t}$

(10)

Equations ((11)) and ((12)) are the in band terms that come from the expansion of the polynomials when done in the MDDPD manner and ((13)) and ((14)) are those in band terms in summation form.

$y_{\omega _{1}}=x_{1}(c_{0,0}^{(1)}+c_{2,0}^{(1)}\left\vert {x_{1}}\right\vert ^{2}+c_{2,2}^{(1)}\left\vert {x_{2}}\right\vert ^{2}+c_{4,0}^{(1)}\left\vert {x_{1}}\right\vert ^{4}+c_{4,4}^{(1)}\left\vert {x_{2}}\right\vert ^{4}+c_{4,2}^{(1)}\left\vert {x_{1}}\right\vert ^{2}\left\vert {x_{2}}\right\vert ^{2})e^{-j{\omega }_{1}t}$

(11)

$y_{\omega _{2}}=x_{2}(c_{0,0}^{(2)}+c_{2,0}^{(2)}\left\vert {x_{2}}\right\vert ^{2}+c_{2,2}^{(2)}\left\vert {x_{1}}\right\vert ^{2}+c_{4,0}^{(2)}\left\vert {x_{2}}\right\vert ^{4}+c_{4,4}^{(2)}\left\vert {x_{1}}\right\vert ^{4}+c_{4,2}^{(2)}\left\vert {x_{2}}\right\vert ^{2}\left\vert {x_{1}}\right\vert ^{2})e^{-j{\omega }_{2}t}$

(12)

$y_{1}(n)=\sum _{m=0}^{M-1}\sum _{k=0}^{N}\sum _{j=0}^{k}c_{k,j,m}^{(1)}x_{1}(n-m)\times \left\vert {x_{1}(n-m)}\right\vert ^{k-j}\left\vert {x_{2}(n-m)}\right\vert ^{j}$

(13)

$y_{2}(n)=\sum _{m=0}^{M-1}\sum _{k=0}^{N}\sum _{j=0}^{k}c_{k,j,m}^{(2)}x_{2}(n-m)\times \left\vert {x_{2}(n-m)}\right\vert ^{k-j}\left\vert {x_{1}(n-m)}\right\vert ^{j}$

(14)

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 mathematic differences in an multichannel application can be seen by comparing the two graphs below.

16APSK 2/3 2D 2D DPD Comparison Using Correct MultiDimensional Math Where: The blue line is the non pre-distorted waveform at the input to the nonlinear system The red line is the non pre-distorted waveform at the output of the nonlinear system The black line is the pre-distorted waveform at the output of the nonlinear system when 1D DPD is applied to the system where both waveforms came from the same modulator and pre-distorter and the full oversampling rate was used The magenta line is the pre-distorted waveform at the output of the nonlinear system when MDDPD is applied properly to the system where each waveforms came from a different modulator and pre-distorter and the reduced oversampling rate was used

QPSK 2D DPD Comparison Using Incorrect MultiDimensional Math Where: The blue line is the non pre-distorted waveform at the input to the nonlinear system The red line is the non pre-distorted waveform at the output of the nonlinear system The black line is the pre-distorted waveform at the output of the nonlinear system when 1D DPD is applied to the system where both waveforms came from the same modulator and pre-distorter and the full oversampling rate was used The magenta line is the pre-distorted waveform at the output of the nonlinear system when MDDPD is applied improperly to the system where each waveforms came from a different modulator and pre-distorter and the reduced oversampling rate was used

As defined in multidimensional signal theory for 1D discrete time vector input - 1D discrete time vector output systems, if all inputs but one are set to zero and the one non zero input is an impulse, there will be an independent impulse response from that input to each independent output. This is true of each input in that system. This is why ((3)) and ((4)) are wrong in the end and need to be modified to ((11)) and ((12)) as they are 1D equations still and are not M dimensional until this is done.

Additional Considerations

One can choose to ignore harmonics if they consider there system representable by a "baseband" model or they can choose to include the harmonics in the solving algorithm if their system does not adhere to the baseband model but it should be noted that 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. This is to say that if it know that a baseband model is adequate for a given multi signal system then MDDPD should be considered but if this is not known then it is important to determine this first before going forward.

Approaches

Orthogonal Polynomial

The approachs seen in , , , , and 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 inband and interband systems. It is stated that correction of interband 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)). It appears that these approaches are the result of the improper application of the math such as that which was demonstrated in the figure comparison above.

2D (Dual-Band), 3D (Tri-Band), and MD Digital Pre-distortion

The approachs seen in , , , are focused on the proper derivation and application of the MDDPD memory polynomial in multiband systems. There are no disadvantages to the approaches in these references as they approach the problem properly and produce a sound solution and method. They lay the groundwork for some very interesting other approaches however.

MDDPD Using Subsampling Feedback

The approach seen in 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 advantage of this approach is the obvious advantage of the elimination of a mix stage. The disadvantage of this approach is the restriction of the carrier location and spacing that is inherent to achieving proper subsampling.

MDDPD Using Augmented Hammerstein

The approach seen in formulates the augmented Hammerstein model so that it is tractable for use with the 2D nonlinear polynomial model. The augmented Hammerstein model is used to implement memory while maintaining a memoryless polynomial model. The model as a whole becomes a memory model but the polynomial model it's self remains memoryless. This reduces the complexity of the polynomial model and has a net reduction on the overall complexity of the composite system.

MDDPD Coefficient Order Reduction Using PCA

The approach seen in uses principle component analysis (PCA) to reduce the number of coefficients necessary to achieve similar adjacent channel power (ACP). Although the normalized mean square error (NMSE) is significantly degraded the ACP is only degraded by ~3.5dB for a 87% reduction in the number of coefficients.

Additional References

Some additional papers can be seen here: ; ; ; ; ; ; ; ; ; ;

External links

• A list of papers from Shahin Gheitanchi and others. A few of which are on multiband digital pre-distortion