Is this a kind of aliasing error and how to make correction? I use a Discrete Fourier Transform to have a representation in the frequency domain:



Notice that, both the real and the imaginary line are softly reaching zero on the left, from 0.5 Hz to 0 Hz. This is a right function.

Subsequently, I wish to multiply the above complex frequency function with another frequency function: DFT*TF, where TF is in the form 



The result appears an undesired shift of the imaginary line (and of the module, as a consequence) instead of the previously observed reduction to zero of the above mentioned lines. This shift is a problem because of causing wrong results to the following signal processes.


How could we give a theoretical explanation what kind of problem it is and what have been so far explained about such cases? 


We may say that it happens due to the boundary conditions of zero out of the sampling period (the interval I apply the DFT is from 0 to 2047 sampling points, I plot only the part from 0 to 3.5 Hz below) and drives to this shift (see on the left) which is an aliasing error. 




Have you ever heard something about such a case?


A practical explanation I could give to this is that, we cannot avoid this phenomenon when multiplying two complex numbers (functions herein) where the imaginary part of (at least) one of them is reduced to zero and is low but not lower than the real part reduction to zero. For example, see below I attach indicative arithmetical values for which the first seven imaginary parts, being a little larger than the real ones, causes a great problem in the following methodology.



The correction I thought is to process the imaginary parts and reduce them to values at least equal to the real ones. Then all the following results works right !
Well, I 've tried this process and works but I cannot be sure myself before asking the experience of others.


What do you think? Could you please indicate me something from books or a useful link?
Complex multiplication dictates that even a complex series with a disappearing imaginary when multiplied with a complex number:
let

$$\begin{align}
z_n &amp;= a_n + i\cdot b_n,\quad a_n, b_n \in \mathbb{R}\\
&amp;\text{then}\\
 z_1 \cdot z_2 &amp;= a_1\cdot a_2+a_1\cdot ib_2+ ib_1 \cdot ia_2 + ib_1\cdot ib_2\\
&amp;=(a_1a_2-b_1b_2) + (a_1b_2+a_2b_1)i\\
&amp;\text{let }b_1\equiv0\\
&amp;=(a_1a_2-b_1b_2) + ({a_1b_2})i
\end{align}$$

and hence, the imaginary part of $z_1$ multiplied with a complex number is not inherently zero, just because the imaginary part of $z_1$ is zero.