I was reading a chapter of my book (I am not putting it as reference since it is not in english) about the quantization noise problem in digital filters. Specifically, it writes about a filter that performs multiplication operations:
"If the input, the output and the multiplicative coefficients of the filter are numbers that can be represented with n bits, maintaining this accuracy in the calculations requires greater precision since, typically, multiplying two numbers of n bits produces a number that can be represented with 2n bits. The need to round the intermediate results therefore produces an error called truncation noise".
To understand exactly what happens I would need a practical example. For example I don't understand if the truncation refers to integers (for example 180 becomes 200) or only for decimal numbers (for example 180.5 becomes 180).
Take for example the numbers 200 and 4, each expressed with 4 bits, that is with 16 levels (therefore 200 will have a resolution of 200/16 = 12.5, and 4 will have a resolution of 4/16 = 0.25). Multiplying them you get 800. Representing it with 4 bits you get a resolution of 800/16 = 50. Is this what is meant by truncation? What is the relationship between the accuracy of the result and that of the operands?