How to calculate signal which is not changed by a filter?
If a filter with transfer function $H(f)$ passes a signal $x(t)$ with Fourier transform $X(f)$ unchanged, then $H(f)X(f) = X(f)$ for all $f$, and 
so $H(f)$ must have the property that
$$H(f) = 1 ~ \text{for all frequencies}~ f~ \text{for which}~ X(f) \neq 0.$$
Thus, an arbitrary filter will not enjoy this property for very many
signals, perhaps for no signal at all.  Indeed, if $H(f_0) = H(-f_0) = 1$ for
some $f_0$, then the filter will pass the sinusoid 
$A\cos(2\pi f_0t + \theta), -\infty &lt; t &lt; \infty$ unchanged, but it will
not pass a sinusoidal pulse such as
$$x(t) = \begin{cases} A\cos(2\pi f_0t + \theta), 
&amp; -T &lt; t &lt; T,\\ 0, &amp;\text{otherwise,}\end{cases}$$
without distorting it. Of course, it is possible that $H(f) \neq 1$ for
all $f$ and so even passing a monochromatic sinusoid is not possible
(cf. @hotpaw's comment).
In short, the subspace referred to by @AtulIngle might well be
the trivial subspace of zero dimension containing only the
zero signal.
It is also worth noting that if $H(f)$ has constant nonzero value
over an interval, then the filter is physically unrealizable for
essentially the same reason that an ideal low-pass filter is
unrealizable.

So, the OP's question is likely to have no answer at all because the
signals $S^\prime$ that he seeks to find may not exist at all for
the given filter. Perhaps something else was intended to be asked?