Digital Signal Quantized Processor Complexity In “traditional” digital signal processing, the complexity is computed as a number of multiplication the operation requires, e.g. the computation of the N-point DFT via the decimation-in-frequency FFT requires (N/2)log_2(N) complex multiplications (and the same number of additions, but in the DSP the additions are assumed to require less time and their contribution is traditionally omitted, unless specifically required). However, we typically do not define which numbers we multiply. My question is how we define the complexity in the DSP sense, if we apply some quantization rule for the numbers to be multiplied: e.g. we proceed from multiplying two real numbers in all operations of the processing devise to multiplying the pairs of int8, float 16, etc. The latter operations are evidently “simpler” that the multiplication of two reals, but how can we quantify and express this complexity reduction for the DSP element?
First, if there's a more formal way to do this than what I'm going to show, I don't know it.

Call out the MAC operations along with data path widths.  I.e., &quot;the FFT algorithm used takes $4 n \log n$ operations with 24-bit wide data paths&quot;

A corollary to this would be that sometimes you may have different precision in your coefficients and your actual data.  So you may call out &quot;24 bit signal path, 16 bit coefficients&quot;.


Call out the LUTs used (in an FPGA).  This isn't direct, and has a lot to do with the cleverness of the implementation, but within a specific FPGA technology it has meaning.

Note that these days, unless you're doing full-custom ASIC design, data paths come in chunks - processors give you multiples of their native word width, and FPGAs come with &quot;DSP blocks&quot; that come in fixed widths.  I'm only guessing, but I'd bet money that if you go to design an ASIC then at the first level of design complexity there will be signal processing libraries with their own fixed-width data paths, and that libraries with variable-width data paths would take more engineering resources to use.