A new receiving scheme for self-mixing receivers is presented that overcomes the disadvantages of the self-heterodyne concept. Generally speaking, the self-mixing receiver offers immunity to phase noise and frequency offsets, especially at very high frequencies, since it does not require radio frequency local oscillators. Our proposed technique eliminates the drawbacks of the self-heterodyne transmission scheme, which are the poor power efficiency and the strong dependence on the continously transmitted carrier.

A nonlinear system of equations is constructed that describes a phase retrieval problem for the reconstruction of the original transmit signal before self-mixing. Two different solution strategies, with restrictions in time and frequency domain, are presented. As a consequence, the self-mixing equation system is shown to be solvable with some a-priori information about the transmit signal. With this novel approach, the transmitted information is distributed over the full available bandwidth, and there is no special dependence on a certain subcarrier for the down-conversion.

The general performance, regarding bit error ratio over signal to noise ratio, is improved by at least 2 dB as compared to the self-heterodyne transmission scheme. In the case of frequency selective channels, e.g. multi-path propagation, this improvement is shown to be much larger, because the presented approach is able to reconstruct the received subcarriers without the necessity of receiving all subcarriers.

For the fifth generation mobile communications standard and also for future
wireless local area network (WLAN) standards such as WiGig, mm-wave
frequencies are considered and even frequencies in the terahertz regime might
be of interest. Receive and transmit circuits at these frequencies are
challenging to design and fabricate, especially for the active components. As
an alternative to conventional super-heterodyne receivers, self-mixing
receivers, which employ a square-law detector for down-conversion from radio
frequency (RF) into the baseband, are investigated in particular due to their
low complexity. The main differences between these two receiver topologies
are shown in Fig.

Comparison of different receiver topologies.

Apart from these obvious savings in the receiver circuitry, self-mixing
receivers behave advantageously in several respects. First, they show a
beneficial behavior when employed in multi-antenna systems. The receive
signals from different antenna array elements are always superimposed in
phase. As a result, receiver sensitivity is improved in fading channels with
multi-path propagation

Down-conversion by selfmixing, as the working principle of the
self-heterodyne transmission technique.

However, it is clear that incoherent demodulation introduces critical signal
distortions as compared to conventional IQ-demodulation. This topic is also
active under research for single carrier acoustic underwater communication
systems

However, the self-heterodyne transmission scheme holds several disadvantages.
The bandwidth efficiency and the power efficiency are only 50 %, since
optimum performance is obtained with equal power distribution between the
carrier and the OFDM subcarriers

Another important drawback is the strong dependence of the self-heterodyne
concept on the transmitted carrier. This becomes crucial in frequency
selective channels, e.g. with fading and multi-path propagation. Some
solutions have been proposed, namely smart carrier positioning and subcarrier
grouping and pairing

In this paper, a novel technique for self-mixing signal recovery is presented that is capable of handling more general transmit signals. The necessity of a strong carrier signal for the down-conversion is dropped in favor of more bandwidth efficiency. In turn, the computational complexity of the signal reconstruction is increased. In this respect, this paper only aims at demonstrating a possibility for signal reconstruction, and not a solution ready for implementation in real time wireless applications.

In Sect. 2, we describe the general restrictions for the signal reconstruction of the transmit signal when utilizing a square-law detector for down-conversion. Based on this, two new transmission and reconstruction techniques are proposed in Sect. 3. Simulation results are discussed in Sect. 4 to demonstrate the superior behavior of the presented method over the self-heterodyne concept.

The down-conversion of the received (Rx) signal is performed by a square law detector in time domain. This down-conversion process will now be analyzed in detail, with a focus on OFDM-like transmit signals. This is not a really hard restriction, as any sampled and digitized signal is represented uniquely by a suitably sampled spectrum, which can be interpreted as an OFDM-like signal. More specific techniques such as pilot signals or specific implementation issues will not be discussed.

For the Rx signal analysis, a time domain transmit (Tx) signal is considered,
which is represented in a sampled version as the vector

Signal processing models for the self-mixing receiver including
channel and noise influence.

The representation of this model in frequency domain will be analyzed in the
following. The transmit signal is described by the frequency domain vector

This insight leads us to a much more convenient representation of the
self-mixing receiving mechanism. The described convolution can be computed
much easier, if just the complex baseband signal is considered, because
low-pass filtering is not necessary, as depicted in Fig.

One consequence of the self-mixing is that the resulting spectrum occupies
twice as much bandwidth as compared to the complex baseband signal.
Equation (

Therefore, Eq. (

Signal analysis shows that the self-mixed Rx signal contains further degrees
of freedom (DOFs) even if optimally measured (with a suitable oversampling)
and the underlying equation systems do not offer unique solutions. These
remaining DOFs can thus be represented by

The remaining task is to reconstruct the original receive signal

As a quite intuitive approach, the transmitted information is coded in the magnitude of the time domain signal. At each sampling point in time, the transmitted information is encoded in the envelope of the time domain signal with ASK modulation. This requires no reconstruction effort, as the data for each timestep is directly read from the receveid time domain signal. The extra DOFs in the Rx signal, i.e. the phases of the time domain signal, are simply ignored. As the separation of the transmitted information is achieved over time, this approach is called time-domain multiplexing (TDM).

Nevertheless, this approch works only well if the transmission channel, e.g.
represented by a diagonal matrix

A similar concept is investigated in frequency domain. An OFDM signal with
ASK for all the subcarriers is considered. Thereby, the phases of all OFDM
subcarriers

For the reconstruction of the original transmit signal, different solution
strategies can be applied, as discussed in Sect. 2. For the following
simulations, a solution is obtained with the help of the algorithm of

For the case of a frequency-dependent channel, the channel properties are
assumed to be known at the receiver. Therefore, the influence of the channel
on the receive signal can be corrected during the frequency-domain phase of
the Gerchberg-Saxton algorithm. In each iteration, the phase values are
corrected by the phases of the spectrum of the channel

To demonstrate the working principle of the proposed concepts, Monte Carlo
simulations have been carried out. Thereby, the BER was analyzed for
different SNRs. Thereby, the receiver model depicted in Fig.

For the sake of clarity, also the constellation diagrams for a lower
modulation order, 8-ASK and 64-QAM, are depicted in Fig.

Constellation diagram comparison for the normal OFDM versions with
QAM and ASK modulation, self-heterodyne OFDM with QAM and self-mixing with
ASK modulation for the same average transmit power per symbol.

For the two different modulation orders, the BER has been computed for
different receive SNR levels for an AWGN channel, without any further
influence on the received spectrum. This is shown in Figs.

BER simulation results for the comparison of standard ASK-OFDM, the self-heterodyne transmission scheme and the two proposed transmission schemes with AM and multiplexing in time and frequency domain.

BER simulation results for the comparison of standard ASK-OFDM, the self-heterodyne transmission scheme and the two proposed transmission schemes with AM and multiplexing in time and frequency domain, for the lower order modulation schemes.

A further example will demonstrate the advantages of the proposed method.
Now, a constant AWGN channel is considered, where, however, the frequency of
the transmitted carrier of the self-heterodyne scheme is damped by a factor
of

BER simulation results with a frequency selective channel, damping
the self-heterodyne carrier frequency by a factor of

As a final example, a two-path channel is considered. This channel exhibits
a direct line of sight connection with an amplitude of 1 and a slightly
attenuated reflected component with an amplitude of

BER simulation results for a two-path fading channel, with a direct
LOS connection with amplitude

A complex baseband description of the self-mixing receiver, or square law detector, has been proposed, which was shown to be very useful for the signal recovery. Two non-linear equation systems condensing the available information for the signal reconstruction have been obtained. Additionally, the general limitations of the self-mixing concept, as compared to a common super-heterodyne receiver, have been analyzed with these equation systems. It was found that parts of the transmit signal have to been known at the receiver for the possibility of a unique signal reconstruction after self-mixing and information transmission.

With the proposed method, the standard IQ demodulation as for super-heterodyne and self-heterodyne receivers has not been employed, but the signal has been reconstructed with the help of the presented equation systems. Thereby, the power-efficiency was increased as compared to the self-heterodyne transmission scheme and the strong dependence on one spectral component was eliminated, as verified in Monte Carlo simulations. Hence, the BER performance of the presented approach was shown to be much better than for the self-heterodyne transmission scheme, in particular for fading and frequency-selective channels. This comes at the cost of increased computational effort, while maintaining the known advantages of self-mixing receivers.

The underlying research data can be requested from the authors.

The authors declare that they have no conflict of interest.

This work was supported by the German Research Foundation (DFG) under grant EI 352/17-1.This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding programme Open Access Publishing. Edited by: Jens Anders Reviewed by: Emanuele Viterbo and one anonymous referee