Monday, December 22, 2008

Peak to Average Power Ratio II: An Introduction

A Signal Processing Perspective

The statistic properties of the PAPR of multi-carrier signals can be described by CCDF (Complementary Cumulative Distribution Function). If we assume the frequency-domain symbol is complex Gaussian distributed. When the number of subcarriers, L, become large, the instantaneous power of each multi-carrier signal chip can be modeled by a Chi-distributed signal with two degree of freedom.
CCDF( γ ) = Pr( PAPR > γ ) = 1- Pr( PAPR <= γ ) = 1 – [ Pr( p <= γ ) ]L >~ 1 – ( 1 – e )L 
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Figure 1.  A statistic modeling of PAPR

A Coding Perspective

Given a code of length n, coding rate R, what is the achievable region of triplets (R, d, PMEPR) ?  What is the relationship between PMEPR and the minimum Euclid distance (mED) d* ? All these questions belong to a Sphere Packing Problem. Given a codeset of q-ary code c with length n, what is the relationship between the size of the codeset and minimum Hamming distance (mHD) D ? This is a Sphere Packing Problem again.
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Figure 2.  A geometric modeling of PAPR

An Implementation Perspective

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Figure 3. Rapp’s SSPA model
 
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Figure 4. AM/AM characteristics of the Rapp SSPA model, P = 2. 

Popular PAPR Reduction Approaches

Clipping: In-band distortion mostly is negligible. But out-of-band distortion is more serious.
Filtering and Signal Processing
  • time-invariant linear filter results in less peak regrowth and lower PAPR than DFT filter in general, if there is no spectral masking.
  • Partial Transmit Signaling (PTS): divide/Group into clusters and each of them is done with a smaller IFFT. [Muller and Huber, 97]
  • Tone Reservation (TR): inserting anti-peak signals in unused or reserved subcarriers. The objective is to find the time-domain signal to be added into the original time-domain signal such that PAPR is reduced. [Tellado, 00]
Coding: The idea is to select a codeword with less PAPR. it still is an open problem to construct codes with both low PAPR and short Hamming distance.
Selected Mapping (SLM): it is based on selecting one of the transformed blocks for each data block, which has the lowest PAPR. [Bauml, Fisher and Huber, 96]
Constellation Optimization
  • Tone Injection (TI): the basic idea is to increase the constellation size so that each of the points in the original basic constellation can be mapped into several equivalent points in the expanded constellation.
  • Active constellation extension (ACE): similar to TI. [Krongold and Jones, 03]

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