Circular Shift Time Reversal

If instead of time reversed channel impulse response (CIR), a circularly shifted time reversed CIR is used as a transmitter pre-filter, the resulting scheme can be called as Circular Shift Time Reversal (CSTR). CSTR can be further classified into left CSTR and right CSTR, depending upon the direction of the CS operation applied on signal

consider a CIR,h(t), withN multi-path taps, represented as:

h(t) = XN

i=1

β_iδ(t−τ_i) (5.1)

where β_i and τ_i are the amplitude and the delay of the CIR coefficient associated to the i_th tap. The time reversed version of the CIR is written as:

h(−t) = XN

i=1

a_iδ(t−τ_i) (5.2)

where a_i = β_N−i+1. If h(−t) is circularly shifted to right by l taps, the resulting function is written as:

circshif t(h(−t), l(right)) = Xl

i=1

aN−l+iδ(t−τ_i) + XN i=l+1

ai−lδ(t−τ_i) (5.3) The equation for the left circular shift can be written as:

circshif t(h(−t), l(lef t)) = XN−l

i=1

ai+lδ(t−τi) + XN i=N−l+1

ai+l−Nδ(t−τi) (5.4) where l can have integer values: 1 ≤l≤N −1

Fig. 5.1 shows a time reversed CIR without any circular shift, with right and left circular shift (CS) of l taps (l = 3). In a multiuser scenario, the interference at the peak increases with the number of simultaneous transmissions. Simultaneous transmission with the simple TR scheme means that the maximal peaks of the time reversed impulse responses intended for different users are aligned with each other in order to be transmitted. This results in the creation of the interference which is equal to the magnitude of the CIRs cross-correlations (R^cross_jk ). This interference will increase to the sum of M −1 cross-correlations for M simultaneous users. If CSTR is used instead of simple TR, the effects of the interference can be greatly reduced [63]. If symbols for more than one user are transmitted simultaneously with CSTR, the maximal peaks of the time reversed impulse responses will no longer be aligned with each other. The taps in the propagating channel containing more energy are multiplied by the taps of the transmitted signal with less energy and vice versa. This results in greatly reduced interference with CSTR [63].

The received signal for the right CSTR while neglecting the noise is written as:

R_x,right^{CST R} =circshif t(h(−t), l(right))? h(t) (5.5)

a0

Figure 5.1: Time Reversed CIR a) without CS b) with right CS of 3 taps c) with left CS of 3 taps

Applying (5.3), the equation becomes:

R^{CST R}_x,right =X^l The received signal for a right CSTR consists of two parts. The first part is the convolution of the first l taps of the right CSTR transmitted signal (i.e. the last l taps of the time reversed CIR moved to the start) with the CIR and the second part is the convolution of the rest of the taps with CIR. These two convolutions result in two signals with two significant peaks. We call the latter asSignal and the former as Image. The position of the Signalpeak also moves from its position toward right by l taps. The distance between the position of the Signal and Image peaks is always equal to N taps. The received signal for left CSTR can be constructed in the same manner. As the number of shifted taps increases, the amplitude of the Image peak also increases depending upon the energy contained in the shifted taps.

Thus, the amplitude of the Signal and Image peaks depends directly on the energy of the taps responsible for their creation. To summarize, ifl is the number of taps to be shifted then, the firstl taps of the circularly shifted time reversed CIR are responsible for the creation ofImagein case of right CSTR. In case of left CSTR, the

0 500 1000 1500 2000 2500 3000

−1

−0.5 0 0.5 1

Number of samples

Amplitude (V)

Image Signal Signal Peak Image Peak

Figure 5.2: Received Signal and Image with right circular shift of 0.25N taps

last l taps of the circularly shifted time reversed CIR are responsible for the creation of anImage.

Thus, the reduction in the interference with CSTR has its cost. On one hand, CSTR improves the performance of the system by reducing the interference caused by the simultaneous transmissions, but on the other hand, it results in the degradation of the system performance as the received signal peak amplitude is reduced with the increase of circular shift. This behavior of the CSTR was not elaborated by the authors in [63]. Thus, CSTR has its limits and we can get a better performance, only if these are taken into account. Otherwise, CSTR could lead to a very poor performance.

5.2.1 Effects of CSTR on Received Signal Peak

We have found that the received signal peak is reduced with CSTR depending upon the amount of the shift. The lost power appears as anImageand is thus lost. These Image and Signal are represented mathematically in (5.6). The amplitude of the Image peak becomes comparable to the amplitude of the Signal peak when the energy contained in the circularly shifted taps approaches to the energy contained in the rest of the taps.

Fig. 5.2 shows the received signal for a right CS of 0.25N taps where N is the total number of taps in the transmitted signal. Signal and Image peaks can easily be distinguished.

Fig. 5.3 shows the variation of the received Signal peak power (normalized to the received peak power without CS) with CSTR using the 35 measured channels.

The bold line shows the average variation over multiple measurements. The dashed line is our defined threshold that is considered as -3 dB. This threshold should be selected according to the application and received power requirements. As shown, the received Signal peak power reduces as the percentage of the CS (either left or right) increases. The graph is plotted till left circular shift of 0.70N taps and right circular shift of 0.30N taps instead of more symmetric 0.50N taps shift on both sides.

These percentages are chosen because energy contained in the shifted taps for right

−70 −60 −50 −40 −30 −20 −10 0 10 20 30

−8

−6

−4

−2 0

Left cir. sh. % Right cir. sh. %

Peak value of the signal (dB)

Peak Signal Variation for different channels Avg. variation 3dB threshold

Figure 5.3: Variation of the received signal peak values normalized to the received signal peak without circular shift with different values of circular shift

CSTR is higher than energy contained in the shifted taps for left CSTR, resulting in a greater power loss with the right circular shift (see Fig. 5.1). Therefore, the allowable shift margin obtained from the threshold increases considerably with our choice of CS percentages.

When the sum of Signal and Image peak amplitudes is taken, it comes out to be equal to the amplitude of the received signal peak without any shift, ascertaining that the received signal peak is decomposed into a Signal peak and an Image peak after CS.

5.3 Effects of CSTR on Signal to Interference