Frequency Selection Encoding

While existing approaches such as studied by Schuster et al. (2011) strive to reduce the crosstalk noise by devising phase encoding functions such that $\langle N_s^{*} N_q\rangle = 0$ for $s\neq q$ , this chapter relies on devising source spectra $W_{q}(\omega)$ 's in order to eliminate the noise term ${\bf {c}}$ defined in equation 2.15. To this end, I single out an arbitrary term in equation 2.15 and investigate how to make it zero. An example of such a term is expressed as

$${\bf {c}}_{sq} =\: \sum_{\omega} W^{*}_s(\omega) {\boldsymbol{\nu}}_{sq}(\omega) W_{q}(\omega) , \quad s\neq q$$ (2.17)

$$\noalign{where} {\boldsymbol{\nu}}_{sq}(\omega) \stackrel{\mathrm{def}}{=}N_s^{*} N_q \underline{{\bf {L}}}_s^{\dagger}\underline{{\bf {L}}}_q {\bf {m}}.$$ (2.18)

Because the dependence of ${\boldsymbol{\nu}}_{sq}(\omega)$ on $\omega$ is typically spatially varying and unknown, it is impossible to construct $W_{q}(\omega)$ 's that can suppress all elements of ${\bf {c}}_{sq}$ , unless the source spectra are non-overlapping. Non-overlapping source spectra ensure that

$$W^{*}_s(\omega)W_{q}(\omega) = 0, \quad \textrm{for}~ s\neq q, \,\, \forall \omega$$ (2.19)

and in turn reduce equation 2.15 to zero. I refer to this encoding scheme as frequency selection. The previous analysis contrasts the different roles that the phase encoder $N_s$ and frequency encoder $W_{s}(\omega)$ play. For notational economy, however, hereafter in the context of frequency selection I recast $N_s$ as a frequency encoder, on which $W_{s}(\omega)$ is predicated; in addition, omega is discretized, and is identified with a frequency index j running from 1 to nomega. The frequency encoder is given as a binary vector

$$N_s(j) \stackrel{\mathrm{def}}{=}\begin{cases}1 & \textrm{if ~~~t... ...textrm{~frequency belongs to source~} s, \\ 0 & \textrm{otherwise.} \end{cases}$$ (2.20)

Note that $N_s(j)$ 's are no longer of pure phase; this can be regarded as a form of amplitude encoding (Godwin and Sava, 2010). If no frequency index is shared by multiple sources, then equation 2.20 leads to

$$N_s(j) N_q(j) = 0, \textrm{~for~} s \neq q, ~\forall j=1,\ldots,n_\omega.$$ (2.21)

Thus equation 2.19 is guaranteed, in my new notation, by the choice

$$W_s(j) = N_s(j) W(j), ~\forall s=1,\ldots,S, ~\forall j,$$ (2.22)

where $W(j)$ is the intact source spectrum. In shorthand, equation 2.21 can be expressed as

$${\bf {N}}_s \odot {\bf {N}}_q = {\bf {0}}, \textrm{~for~} s \neq q,$$ (2.23)

where $\odot$ represents element-wise multiplication between two vectors. If, moreover, every frequency index is assigned to some source, then equation 2.20 leads to

$$\bigoplus_{s=1}^{S} {\bf {N}}_s = {\bf {1}},$$ (2.24)

where $\oplus$ represents element-wise addition. Accordingly, following equation 2.22, we have

$${\bf {W}}_s \odot {\bf {W}}_q = {\bf {0}}, \textrm{~for~} s \neq q,$$ (2.25)

$$\bigoplus_{s=1}^{S} {\bf {W}}_s = {\bf {W}}.$$ (2.26)

Given $S$ sources and $n_\omega$ frequency indices, frequency selection endeavors to evenly divide the latter among the former. That is, on average each source is assigned $n_\omega/S$ frequency indices.

I outline next how $n_\omega$ is determined. Suppose the maximal travel time between sources and their associated receivers is $T$ , the peak frequency of the source wavelet is $f_0$ , and the cutoff high frequency is at $f_{hi} = 2.5 f_0$ . The Nyquist sampling theorem dictates $dt < \frac{1}{2 f_{hi}} = \frac{1}{5 f_{0}}$ , and therefore the total number of time samples is $n_t = T/dt > 5 T f_0$ . For real signals devoid of dc, the number of independent angular frequencies is given by

$$n_\omega = 2.5 T f_0.$$ (2.27)

For example, the parameters chosen for my 2D and 3D simulations are: $n_\omega = 160$ as $T=2$ s and $f_0=32$ Hz, and $n_\omega = 360$ as $T=9$ s and $f_0=16$ Hz, respectively. Note that the effective number of independent frequencies is smaller than $n_\omega$ , because the source spectrum $W(j)$ is far from uniform.