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Low Complexity Transceiver Design for GFDM

Arman Farhang, Nicola Marchetti, Linda E. Doyle

arXiv:1501.02940v1cs.ITmath.IT

TL;DR

GFDM offers waveform properties attractive for 5G, but its transceiver operations can be computationally costly. This paper exploits modulation-matrix structure to design lower-complexity transmitter and unified MF, ZF, and MMSE receivers. The authors report substantial complexity reduction without loss of optimal receiver performance.

  • Problem

    GFDM’s modulation and detection require costly large-matrix operations, despite its attractive waveform properties for 5G.

    Source: §I · p. 1

  • Method

    The paper sparsifies the GFDM modulation matrix for the transmitter and uses its structure to build unified low-complexity MF, ZF, and MMSE receivers.

    Source: §I · p. 2, A. GFDM transmitter design · p. 4, C. Low complexity ZF receiver · p. 5

  • Results

    The proposed transmitter is more computationally efficient than the prior lowest-complexity GFDM transmitter, and the receivers substantially reduce complexity versus the low-complexity receiver in without performance loss.

    Source: §I · p. 2

  • Takeaways & Limitations

    The matrix-based approach also reduces memory requirements and enables parallel computations that can reduce processing delay.

    Source: §I · p. 2

Abstract

from arXiv · show

Due to its attractive properties, generalized frequency division multiplexing (GFDM) is recently being discussed as a candidate waveform for the fifth generation of wireless communication systems (5G). GFDM is introduced as a generalized form of the widely used orthogonal frequency division multiplexing (OFDM) modulation scheme and since it uses only one cyclic prefix (CP) for a group of symbols rather than a CP per symbol, it is more bandwidth efficient than OFDM. In this paper, we propose novel transceiver structures for GFDM by taking advantage of the particular structure in the modulation matrix. Our proposed transmitter is based on modulation matrix sparsification through application of fast Fourier transform (FFT) to reduce the implementation complexity. A unified receiver structure for matched filter (MF), zero forcing (ZF) and minimum mean square error (MMSE) receivers is also derived. The proposed receiver techniques harness the special block circulant property of the matrices involved in the demodulation stage to reduce the computational cost of the system implementation. We have derived the closed forms for the ZF and MMSE receiver filters. Additionally, our algorithms do not incur any performance loss as they maintain the optimal performance. The computational costs of our proposed techniques are analyzed in detail and are compared with the existing solutions that are known to have the lowest complexity. It is shown that through application of our transceiver structure a substantial amount of computational complexity reduction can be achieved.

I. INTRODUCTION

GFDM is presented as a circular-pulse-shaping waveform that addresses several OFDM limitations, but its non-orthogonality creates receiver complexity and BER trade-offs. The paper introduces a GFDM system model and low-complexity transceiver designs intended to retain optimal receiver performance.

I. INTRODUCTION

  • OFDM’s out-of-band emissions and synchronization sensitivity motivate waveforms with better time-frequency localization, but existing MUI remedies can increase receiver complexity.The added overhead and complexity are especially problematic for lightweight M2M applications.Source: §I · p. 1
  • GFDM is a circular-pulse-shaping FBMC waveform that uses one CP per block and offers low out-of-band radiation, reduced latency, and synchronization-error robustness.Its filtering limits ICI to adjacent subcarriers.Source: §I · p. 1
  • GFDM’s non-orthogonality raises BER relative to OFDM, while MF, ZF, and MMSE receivers trade interference suppression against noise enhancement and computational cost.MF cannot completely remove ICI; ZF removes ICI but can amplify noise, motivating MMSE.Source: §I · p. 2
  • The proposed transceiver exploits modulation-matrix structure to reduce transmitter and receiver complexity while retaining optimal MF, ZF, and MMSE performance.The receiver uses a unified structure whose filter coefficients can be changed or stored for different receiver choices.Source: §I · p. 2
  • The proposed matrix sparsification also reduces memory requirements and permits parallel computation that can reduce processing delay.The paper contrasts its direct receiver approach with iterative interference-cancellation methods.Source: §I · p. 2

II. SYSTEM MODEL FOR GFDM

  • The system model uses N subcarriers and M overlapping symbols per GFDM block, adds a CP longer than the channel delay spread, and applies frequency-domain equalization before MF, ZF, or MMSE detection.The receiver estimates transmitted data from the equalized signal; the transceiver is modeled as synthesis and analysis filter banks.Source: §II · pp. 2–4
  • Direct MN × MN matrix operations in modulation and detection can be too costly for practical systems, motivating lower-cost synthesis and analysis filter-bank implementations.The proposed techniques aim to reduce computational cost while maintaining optimal performance.Source: §II · p. 4

III. PROPOSED GFDM TRANSMITTER

This section introduces the proposed low-complexity GFDM transmitter and previews simplification of the synthesis filter bank.

III. PROPOSED GFDM TRANSMITTER

  • The transmitter design simplifies the GFDM synthesis filter bank to reduce its computational load.The section presents the design and implementation of the low-complexity transmitter.Source: §III · p. 4

A. GFDM transmitter design

The transmitter design reduces the cost of direct matrix modulation by transforming the data and exploiting sparse structure in the modulation matrix.

A. GFDM transmitter design

  • Direct multiplication of the modulation matrix by the data vector requires (MN)^2 complex multiplications, making transmitter complexity grow costly as N or M increases.The section therefore seeks a lower-complexity implementation.Source: A. GFDM transmitter design · p. 4
  • Applying the block DFT makes the transformed modulation matrix sparse, with nonzero entries tied to scaled prototype-filter coefficients.The block DFT is unitary, and the transformed data can be formed by M N-point DFT operations.Source: A. GFDM transmitter design · p. 4
  • The sparse transmitter computation uses M-point circular convolutions of transformed data with polyphase components of the prototype filter.The resulting sparse vectors have M nonzero elements at positions spaced N samples apart.Source: A. GFDM transmitter design · p. 4

B. GFDM transmitter implementation

The transmitter implementation uses FFT-based symbol transforms and circular convolutions. The proposed ZF and MMSE receivers exploit the structure of AHA for low-complexity designs without performance loss.

B. GFDM transmitter implementation

  • GFDM modulation is implemented as M N-point DFTs followed by N M-point circular convolutions.The DFTs can use FFT; the implementation cascades the corresponding blocks in Fig. 2.Source: B. GFDM transmitter implementation · p. 4
  • The transmitter’s commutators advance at different intervals: some after one sample, and one after sending M samples to each convolution block.The right-side commutators in Fig. 2(a) and (b) turn after one sample collection; the left-side commutator in (b) turns after M samples.Source: B. GFDM transmitter implementation · pp. 4–5

IV. PROPOSED GFDM RECEIVER

  • The proposed ZF and MMSE receivers use the special structure of AHA to reduce complexity without performance loss.The receiver section attributes the result to the structure of AHA in both receiver formulations.Source: §IV · p. 5

A. Block-diagonalization of the matrix AHA

AHA is block-circulant with M × M blocks, enabling block-diagonalization. The MF implementation uses a sparse transformed matrix and circular convolutions instead of direct matrix multiplication.

A. Block-diagonalization of the matrix AHA

  • AHA is block-circulant with M × M blocks, and its structure underlies the proposed receiver techniques.Its columns are circularly shifted relative to one another, and the matrix is expanded using a block-diagonal matrix D.Source: A. Block-diagonalization of the matrix… · p. 5
  • Each Di block is derived from prototype-filter polyphase components and is real and circulant.The paper defines each component using a circularly folded version of the corresponding polyphase component.Source: A. Block-diagonalization of the matrix… · p. 5

B. Low complexity MF receiver

  • Direct MF multiplication costs (MN)^2 complex multiplications, while applying the block DFT to AH yields a sparse matrix Γ.Γ contains scaled prototype-filter coefficients.Source: B. Low complexity MF receiver · p. 5
  • For MF, Γi y is computed as an M-point circular convolution, followed by M N-point IDFTs to obtain the symbol estimates.The convolution uses equidistant received-vector elements and a circularly folded polyphase component.Source: B. Low complexity MF receiver · p. 5

C. Low complexity ZF receiver

The ZF receiver uses the sparse transform Γ = F_b A^H and circulant-filter operations to reduce computation. Its symbol estimates are obtained through inverse DFT operations.

C. Low complexity ZF receiver

  • The ZF receiver’s A^H y multiplication costs (MN)^2, and using sparse Γ = F_b A^H reduces this computational burden.The passage introduces the sparse transform as the approach to reducing the first source of ZF computational burden.Source: C. Low complexity ZF receiver · p. 5
  • Each qκ is derived from the first column of the inverse circulant matrix circ(gκ), scaled by 1/N, and can be calculated offline.The passage also states that qκ is real when the prototype-filter coefficients are real.Source: C. Low complexity ZF receiver · p. 5
  • The ZF receiver computes N M-point circular convolutions, then uses M N-point IDFTs to obtain the symbol estimates.The IDFT operations produce the ZF estimates from the intermediate vector.Source: C. Low complexity ZF receiver · pp. 5–6
  • Fig. 3 presents the unified MF, ZF, and MMSE receiver implementation as a cascade of block diagrams (a) and (b).Source: Figure 3 · p. 6

D. Low complexity MMSE receiver

The MMSE derivation adds a noise-variance identity term to D, whose circulant blocks remain circulant. It uses frequency-domain fast convolution before producing MMSE symbol estimates.

D. Low complexity MMSE receiver

  • Because each Di is circulant, each D̄i is also circulant and can be expanded using the frequency-domain expression involving Φκ.The passage defines Φκ in terms of the Fourier transform of gκ.Source: D. Low complexity MMSE receiver · p. 6
  • Because the inverse depends on σν^2, the MMSE receiver uses frequency-domain fast convolution to obtain ˘y before estimating the transmitted symbols.Source: D. Low complexity MMSE receiver · p. 6

E. Receiver implementation

The receiver implementation uses a shared structure for MF, ZF, and MMSE, while the complexity analysis compares the proposed transmitter and receivers with existing low-complexity techniques. The reported savings vary by receiver and transmitter setting, with the proposed receivers maintaining optimal ZF and MMSE performance.

E. Receiver implementation

  • The unified receiver implementation cascades two blocks, using v_i for MF, q_i for ZF, and p_i for MMSE.The MF and ZF vectors can be calculated offline; MMSE vectors depend on SNR and are calculated in real time.Source: E. Receiver implementation · p. 6

V. COMPUTATIONAL COMPLEXITY

A. Transmitter complexity

  • The proposed transmitter is close to OFDM in complexity for small M, but its cost rises faster as M increases.For M up to 11, the transmitter in requires about twice as many complex multiplications; as M increases, the proposed cost approaches.Source: A. Transmitter complexity · p. 7

B. Receiver complexity

  • The proposed receivers use circular convolutions and IFFT operations; MMSE complexity differs from MF and ZF in circular-convolution implementation.MMSE vectors depend on SNR and are calculated in real time, while MF and ZF vectors are fixed and can be stored offline.Source: B. Receiver complexity · p. 7
  • At N = 1024 and M = 7, the proposed MF/ZF receiver is three orders of magnitude and MMSE receiver six orders simpler than direct methods in complex multiplications.The proposed ZF is around an order of magnitude simpler than SIC in, and MMSE has 2 to 3 times lower complexity than.Source: B. Receiver complexity · p. 7

VI. CONCLUSION

The paper presents low-complexity GFDM transmitter and receiver techniques that exploit matrix structure to reduce computation. The proposed techniques are reported to cost less than existing low-complexity methods, with the ZF receiver offering over an order of magnitude reduction relative to.

VI. CONCLUSION

  • The transmitter sparsifies the modulation matrix, while block diagonalization reduces receiver matrix inversion and multiplication costs.The paper derives unified MF, ZF, and MMSE receiver structures and closed-form ZF and MMSE filters.Source: §VI · p. 8
  • All proposed techniques have lower computational cost than existing low-complexity techniques,, with the ZF receiver reducing complexity by over an order of magnitude versus.The paper reports that these reductions do not incur a performance loss.Source: §VI · p. 8
  • The authors identify the reduced computation as making the transceiver structures attractive for real-time GFDM hardware implementation.The conclusion links the amount of computation involved to hardware implementation.Source: §VI · p. 8

APPENDIX A DERIVATION OF FbAH

The appendix derives the structure of Γ = FbAH from complex exponentials and the prototype filter. It shows that Γ is sparse and that its nonzero columns have a circular-shift pattern.

APPENDIX A DERIVATION OF FbAH

  • The derivation uses the zero inner product of complex exponentials at different frequencies to form Γ = FbAH.The Γ_i blocks are M × MN matrices, and their construction uses the prototype-filter vectors.Source: Appendix A · p. 8
  • The matrix Γ is real and consists of prototype-filter coefficients.The passage identifies the relevant filter vectors as circularly folded versions of ψκ.Source: Appendix A · p. 8
  • Γ is sparse, with Γ_i containing M^2 nonzero elements in circularly equidistant columns κ, κ+N, …, κ+(M−1)N.The nonzero columns are circularly shifted copies; the first nonzero column is a circularly folded version of a prototype-filter polyphase component.Source: Appendix A · p. 8

APPENDIX B CLOSED FORM DERIVATION OF D

The appendix derives a closed form for D by expressing it as ΓΓH. Orthogonality among the Γ_i blocks makes D block diagonal, and its diagonal blocks are circulant.

APPENDIX B CLOSED FORM DERIVATION OF D

  • The appendix defines the polyphase components of the prototype filter and describes Γ as sparse, with nonzero elements organized in its Γ_i blocks.The appendix states that Γ = FbAH and that its block rows contain circularly folded prototype-filter components.Source: Appendix B · pp. 8–9
  • D is calculated as ΓΓH, and orthogonality between distinct Γ_i blocks gives D a block-diagonal structure.The appendix states that Γ_iΓ_jH = 0 for i ≠ j.Source: Appendix B · p. 9
  • The diagonal blocks D_i = Γ_iΓ_iH are circulant matrices.The derivation uses the circularly shifted pattern of the nonzero columns of Γ_i.Source: Appendix B · p. 9
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