# Non-orthogonal frequency-division multiplexing

(Redirected from N-OFDM)

Non-orthogonal frequency-division multiplexing (N-OFDM) is a method of encoding digital data on multiple carrier frequencies with non-orthogonal intervals between frequency of sub-carriers[1][2][3]. N-OFDM signals can be used in communication and radar systems.

## Subcarriers system

Subcarriers system of N-OFDM signals after FFT

The low-pass equivalent N-OFDM signal is expressed as:[3][2]

${\displaystyle \ \nu (t)=\sum _{k=0}^{N-1}X_{k}e^{j2\pi \alpha kt/T},\quad 0\leq t

where ${\displaystyle {X_{k}}}$ are the data symbols, ${\displaystyle N}$ is the number of sub-carriers, and ${\displaystyle T}$ is the N-OFDM symbol time. The sub-carrier spacing ${\displaystyle \alpha /T}$ for ${\displaystyle \alpha <1}$ makes them non-orthogonal over each symbol period.

## History

The history of N-OFDM signals theory was started in 1992 from the Patent of Russian Federation No. 2054684[1]. In this patent, Vadym Slyusar proposed the 1st method of optimal processing for N-OFDM signals after Fast Fourier transform (FFT).

In this regard need to say that W. Kozek and A. F. Molisch wrote in 1998 about N-OFDM signals with ${\displaystyle \alpha <1}$ that "it is not possible to recover the information from the received signal, even in the case of an ideal channel." [4]

In 2001 V. Slyusar proposed non-orthogonal frequency digital modulation (N-OFDM) as an alternative of OFDM for communications systems [5].

The next publication about this method has priority in July 2002[2] before the conference paper regarding SEFDM of I. Darwazeh and M.R.D. Rodrigues (September, 2003)[6].

Despite the increased complexity of demodulating N-OFDM signals compared to OFDM, the transition to non-orthogonal subcarrier frequency arrangement provides several advantages:

1. higher spectral efficiency, which allows to reduce the frequency band occupied by the signal and improve the electromagnetic compatibility of many terminals;
2. adaptive detuning from interference concentrated in frequency by changing the nominal frequencies of the subcarriers[7];
3. an ability to take into account Doppler frequency shifts of subcarriers when working with subscribers moving at high speeds;
4. reduction of the peak factor of the multi-frequency signal mixture.

## Idealized system model

This section describes a simple idealized N-OFDM system model suitable for a time-invariant AWGN channel[8]

## Transmitter N-OFDM signals

An N-OFDM carrier signal is the sum of a number of not-orthogonal subcarriers, with baseband data on each subcarrier being independently modulated commonly using some type of quadrature amplitude modulation (QAM) or phase-shift keying (PSK). This composite baseband signal is typically used to modulate a main RF carrier.

${\displaystyle \scriptstyle s[n]}$ is a serial stream of binary digits. By inverse multiplexing, these are first demultiplexed into ${\displaystyle \scriptstyle N}$ parallel streams, and each one mapped to a (possibly complex) symbol stream using some modulation constellation (QAM, PSK, etc.). Note that the constellations may be different, so some streams may carry a higher bit-rate than others.

A Digital Signal Processor (DSP) is computed on each set of symbols, giving a set of complex time-domain samples. These samples are then quadrature-mixed to passband in the standard way. The real and imaginary components are first converted to the analogue domain using digital-to-analogue converters (DACs); the analogue signals are then used to modulate cosine and sine waves at the carrier frequency, ${\displaystyle \scriptstyle f_{c}}$, respectively. These signals are then summed to give the transmission signal, ${\displaystyle \scriptstyle s(t)}$.

## Demodulation

The receiver picks up the signal ${\displaystyle \scriptstyle r(t)}$, which is then quadrature-mixed down to baseband using cosine and sine waves at the carrier frequency. This also creates signals centered on ${\displaystyle \scriptstyle 2f_{c}}$, so low-pass filters are used to reject these. The baseband signals are then sampled and digitised using analog-to-digital converters (ADCs), and a forward FFT is used to convert back to the frequency domain.

This returns ${\displaystyle \scriptstyle N}$ parallel streams, which use in appropriate symbol detector.

## Demodulation after FFT

The 1st method of optimal processing for N-OFDM signals after FFT was proposed in 1992[1]

## Demodulation without FFT

### Demodulation by using of ADC samples

The method of optimal processing for N-OFDM signals without FFT was proposed in October 2003[3][9]. In this case can be used ADC samples.

## N-OFDM+MIMO

N-OFDM+MIMO system model

The combination N-OFDM and MIMO technology is similar to OFDM. To the building of MIMO system can be used digital antenna array as transmitter and receiver of N-OFDM signals.

## Fast-OFDM

Fast-OFDM[10][11][12] method was proposed in 2002.[13]

## FBMC

FBMC is Filter-Bank Multi-Carrier Modulation[14][15][16]. As example of FBMC can consider Wavelet N-OFDM.

### Wavelet N-OFDM

N-OFDM has become a technique for power line communications (PLC). In this area of research, a wavelet transform is introduced to replace the DFT as the method of creating non-orthogonal frequencies. This is due to the advantages wavelets offer, which are particularly useful on noisy power lines.[17]

To create the sender signal the wavelet N-OFDM uses a synthesis bank consisting of a ${\displaystyle \textstyle N}$-band transmultiplexer followed by the transform function

${\displaystyle F_{n}(z)=\sum _{k=0}^{L-1}f_{n}(k)z^{-k},\quad 0\leq n

On the receiver side, an analysis bank is used to demodulate the signal again. This bank contains an inverse transform

${\displaystyle G_{n}(z)=\sum _{k=0}^{L-1}g_{n}(k)z^{-k},\quad 0\leq n

followed by another ${\displaystyle \textstyle N}$-band transmultiplexer.\ The relationship between both transform functions is

${\displaystyle f_{n}(k)=g_{n}(L-1-k)}$
${\displaystyle F_{n}(z)=z^{-(L-1)}G_{n}*(z-1)}$

## Spectrally efficient FDM (SEFDM)

N-OFDM is a spectrally efficient method.[6][18] All SEFDM methods are similar to N-OFDM.[6][19][20][21][22][23][24]

## GFDM

GFDM is generalized frequency division multiplexing.

## References

1. ^ a b c RU2054684 (C1) G01R 23/16. Amplitude-frequency response measurement technique// Slyusar V. – Appl. Number SU 19925055759, Priority Data: 19920722. – Official Publication Data: 1996-02-20 [1]
2. ^ a b c Slyusar, V. I. Smolyar, V. G. Multifrequency operation of communication channels based on super-Rayleigh resolution of signals// Radio electronics and communications systems c/c of Izvestiia- vysshie uchebnye zavedeniia radioelektronika.. – 2003, volume 46; part 7, pages 22–27. – Allerton press Inc. (USA)[2]
3. ^ a b c Slyusar, V. I. Smolyar, V. G. The method of nonorthogonal frequency-discrete modulation of signals for narrow-band communication channels// Radio electronics and communications systems c/c of Izvestiia- vysshie uchebnye zavedeniia radioelektronika. – 2004, volume 47; part 4, pages 40–44. – Allerton press Inc. (USA)[3]
4. ^ W. Kozek and A. F. Molisch.“Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1579–1589, Oct. 1998.
5. ^ Pat. of Ukraine № 47835 A. IPС8 H04J1/00, H04L5/00. Method of frequency-division multiplexing of narrow-band information channels// Sliusar Vadym Іvanovych, Smoliar Viktor Hryhorovych. – Appl. № 2001106761, Priority Data 03.10.2001. – Official Publication Data 15.07.2002, Official Bulletin № 7/2002
6. ^ a b c M. R. D. Rodrigues and I. Darwazeh. A Spectrally Efficient Frequency Division Multiplexing Based Communications System.// InOWo'03, 8th International OFDM-Workshop, Proceedings, Hamburg, DE, September 24–25, 2003. - https://www.researchgate.net/publication/309373002
7. ^ Vasilii A. Maystrenko, Vladimir V. Maystrenko, Alexander Lyubchenko. Interference Immunity Analysis of an Optimal Demodulator Under Peak Multiplexing of N-OFDM Spectrum.//Conference Paper of 2017 International Siberian Conference on Control and Communications (SIBCON).· June 2017. - DOI: 10.1109/SIBCON.2017.7998458
8. ^ Vasilii A. Maystrenko, Vladimir V. Maystrenko, Evgeny Y. Kopytov, Alexander Lyubche. Analysis of Operation Algorithms of N-OFDM Modem in Channels with AWGN.// Conference Paper of 2017 Dynamics of Systems, Mechanisms and Machines (Dynamics). · November 2017. - DOI: 10.1109/Dynamics.2017.8239486
9. ^ Maystrenko, V. A., & Maystrenko, V. V. (2014). The modified method of demodulation N-OFDM signals. 2014 12th International Conference on Actual Problems of Electronics Instrument Engineering (APEIE). doi:10.1109/apeie.2014.7040919
10. ^ Dimitrios Karampatsis, M.R.D. Rodrigues and Izzat Darwazeh. Implications of linear phase dispersion on OFDM and Fast-OFDM systems.// London Communications Symposium 2002. - http://www.ee.ucl.ac.uk/lcs/previous/LCS2002/LCS112.pdf.
11. ^ D. Karampatsis and I. Darwazeh. Performance Comparison of OFDM and FOFDM Communication Systems in Typical GSM Multipath Environments. // London Communications Symposium 2003 (LCS2003), London, UK, Pp. 360 – 372. - http://www.ee.ucl.ac.uk/lcs/previous/LCS2003/94.pdf.
12. ^ K. Li and I. Darwazeh. System performance comparison of Fast-OFDM system and overlapping Multi-carrier DS-CDMA scheme.// London Communications Symposium 2006. - http://www.ee.ucl.ac.uk/lcs/previous/LCS2006/54.pdf.
13. ^ M.R.D. Rodrigues, Izzat Darwazeh. Fast OFDM: A Proposal for Doubling the Data Rate of OFDM Schemes.// International Conference on Communications, ICT 2002, Beijing, China, June 2002. - Pp. 484 – 487
14. ^ Bellanger M.G. FBMC physical layer: a primer / M.G. Bellanger et al. - January 2010.
15. ^ Farhang-Boroujeny B. OFDM Versus Filter Bank Multicarrier//IEEE Signal Processing M agazine.— 2011.— Vol. 28, № 3.— P. 92— 112.
16. ^ Behrouz Farhang-Boroujeny. Filter Bank Multicarrier for Next Generation of Communication Systems.//Virginia Tech Symposium on Wireless Personal Communications. — June 2–4, 2010.
17. ^ S. Galli; H. Koga; N. Nodokama (May 2008). Advanced Signal Processing for PLCs: Wavelet-OFDM. 2008 IEEE International Symposium on Power Line Communications and Its Applications. pp. 187–192. doi:10.1109/ISPLC.2008.4510421. ISBN 978-1-4244-1975-3.
18. ^ Safa Isam A Ahmed. Spectrally Efficient FDM Communication Signals and Transceivers: Design, Mathematical Modelling and System Optimization.//A thesis submitted for the degree of PhD. — Communications and Information Systems Research Group Department of Electronic and Electrical Engineering University College London. — October 2011.- http://discovery.ucl.ac.uk/1335609/1/1335609.pdf
19. ^ Masanori Hamamura, Shinichi Tachikawa. Bandwidth efficiency improvement for multi-carrier systems. //15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, vol. 1, Sept. 2004, pp. 48 — 52.
20. ^ Li. D. B. A high spectral efficiency technology and method for overlapped frequency division multiplexing [P]. 2006, PCT/CN2006/002012 (in Chinese)
21. ^ Xing Yang, Wenbao Ait, Tianping Shuait, Daoben Li. A Fast Decoding Algorithm for Non-orthogonal Frequency Division Multiplexing Signals // Communications and Networking in China, 2007. CHINACOM '07. — 22-24 Aug. 2007, P. 595—598.
22. ^ I. Kanaras, A. Chorti, M. Rodrigues, and I. Darwazeh, "A combined MMSE-ML detection for a spectrally efficient non orthogonal FDM signal, " in Broadband Communications, Networks and Systems, 2008. BROADNETS 2008. 5th International Conference on, Sept. 2008, pp. 421 −425.
23. ^ I. Kanaras, A. Chorti, M. Rodrigues, and I. Darwazeh, "Spectrally efficient FDM signals: Bandwidth gain at the expense of receiver complexity, " in IEEE International Conference on Communications, 2009. ICC ’09., June 2009, pp. 1 −6.
24. ^ Bharadwaj, S., Nithin Krishna, B.M.; Sutharshun, V.; Sudheesh, P.; Jayakumar, M. Low Complexity Detection Scheme for NOFDM Systems Based on ML Detection over Hyperspheres. 2011 International Conference on Devices and Communications, ICDeCom 2011 - Proceedings, Mesra, 24-25 February 2011, Pp. 1-5.