Blackboard Bold i

Here, I use dafrick's answer at Double-struck zero and one to use the boondox-ds versions of bb fonts, designated here as mymathbb{}. I used j in one location so that you can see it is not pixelated.

The fonts are installed via the boondox-dx package, which is not invoked below, so as not to overwrite the native mathbb implementation otherwise available through amssymb.

documentclass{article}
usepackage{amsmath}
DeclareMathAlphabet{mymathbb}{U}{BOONDOX-ds}{m}{n}
begin{document}
    [
      x_i(t) = mathop{mathrm{Re}}(A_i e^{8000pimymathbb{j}t}c^{phi_imymathbb{i}}) text{for $i = 1, 2, 3$}
    ]
end{document}

---

FOLLOW UP

In comments, the OP asks for the mathbb{i} from TG Pagella Math (OpenType format), but for use in pdflatex, which is not available. Since it is only a single glyph that is being requested, here is a kludge to obtain it:

Create the document TGbbi.tex as follows, and compile in Xelatex:

documentclass{standalone}
usepackage{unicode-math}
setmathfont{TeX Gyre Pagella Math}
begin{document}
$mathbb{i}$
end{document}

It creates the output TGbbi.pdf containing only the Pagella version of mathbb{i}. Now, reverting back to pdflatex, we are going to call upon that graphic for use in a macro named bbi defined as

newcommandbbi{ThisStyle{%
  setbox0=hbox{$SavedStylemathbb{i}$}includegraphics[height=ht0]{TGbbi}}}

This macro requires the graphicx package (to import the graphic) and the scalerel package (to auto-scale it to the proper math size, taken as the vertical height of the regular mathbb{i}). Thus, the implementation (showing use in several different math sizes) is

documentclass{article}
usepackage{amssymb,amsmath,scalerel,graphicx}
newcommandbbi{ThisStyle{%
  setbox0=hbox{$SavedStylemathbb{i}$}includegraphics[height=ht0]{TGbbi}}}
begin{document}
    [
      x_i(t) = mathop{bbimathrm{Re}}(A_i e^{8000pibbi t}c^{phi_ibbi}) text{for $i = 1, 2, 3$}
    ]
end{document}

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