Bidirectional filtering

[Adel]

Hello.

I have a question concerning a bidirectional filtering using IIR
filter.
The procedure consists of 2 steps:
1. Filtering a time domain signal x[i] from left to right using IIR.
2. The result signal of 1st step is filtered from the right to the
left
with the same filter.
Result of the second step gives me a final result y[i].

Now I look at this whole procedure in the frequency domain:
X[i] - discrete spectra of original signal,
H[i] - discrete spectral response of IIR (truncated)
Y[i] - discrete spectra of result.
T[i] - temporary array

First step:
T[i] = X[i]H[i] - spectrums multiplication (first filtering)

Second step:
T[i] = T[i]* - complex conjugate
(reverse result of first filtering),
Y[i] = T[i]H[i] - spectrum multiplication (second filtering)

Now my question:
It seems to me that I should complex conjugate Y[i] to get a final
result or in the time domain - reverse signal. Or to get the result
faster I first need to filter the reversed original signal and then
filter it in the forward direction. I am right?

Adel