Complexity: Number of MAC operations on a DSP Suppose a complex $(N \times N)$ matrix $\mathbf{T}$, multiplying a complex vector $\mathbf{x}$ with length $N$.

On a given application that I'm studying, which is irrelevant for this discussion, I can get the same results of $\mathbf{Tx}$ by the following operation:

\begin{align}
\mathbf{W_1} \Re\{ \mathbf{y} \} + \mathbf{W_2} \Im\{ \mathbf{y} \}
\end{align}
where are $\mathbf{W_1}$ and $\mathbf{W_2}$ are complex $(\frac{N}{2} \times \frac{N}{2})$ matrices, $\mathbf{y}$ is a complex vector of length $\frac{N}{2}$, $\Re$ denote the real part and $\Im$ the imaginary part.

I'm trying to asses which one requires more operations in a real implementation.

My analysis is that the first ($\mathbf{Tx}$) requires $N^2$ MACs, while the second requires

$ 2(\frac{N}{2})^2 = \frac{N^2}{2}$ MACs, plus $\frac{N}{2}$ sums.

I'm not completely sure about this, neither about how a DSP deals with complex numbers.
Am I making any mistake?

I'd appreciate your help.
It appears to me that your analysis is basically correct, except that in the second case there are $\frac{N}{2}$ sums, not $N$ sums.

I'm not sure what you are looking for when you ask "how a DSP deals with complex numbers", but generally they are represented in one of two ways: as separate real and imaginary values, or as a "complex" entity (structure/object/whatever- depends on the language) that contains real and imaginary members.