Fixed Point Multiplication in Verilog or Quartus II

Thanks a lot fvm, for such an enlightening post.

@amilcar : Not that I couldn't understand how an overflow would occur in the case of multiplying two most negative numbers; what confused me was the term "two sign bits".

But, now I completely understand what FvM had to say. Even I cross-checked my logic, and here's my elaborate proof:

Consider this - All we need is to multiply 2 numbers A & B (which are 16-bit signed).

We assume that (we have checked that) whatever range of values A & B will take (fractional or integer) - the integer-part of our "result_axb" is small enough to be represented by less than or equal to 15-bits.

(i.e. -32768 <= result_axb <= 32767)

Which means, that we have already checked that no overflow would occur with our result_AxB ( 'overflow' w.r.t. 16-bits signed representation )

now, in 32-bit signed representation:

Any positive no. less than +32767: has all it 17 msb's [31:15] = 0 (= sign bit)

Any negative no. greater than -32768: has all it 17 msb's [31:15] = 1 ( = sign bit)

I am attempting to generate a 32-bit signed result: mult_out32 = k*l*m

( refer my 2nd post: A*B = K*L*(M/N) ). This product (K*L*M) may exceed the 16-bit signed range.

But, since i'm sure that later on, dividing it by N (i.e. stripping-off its P LSBs) would bring the result back in the 16-bit signed range

... and in this resulting form, all its [31-p:15] msbs would be equal ) ...

therefore, before stripping-off, these same bits are at the position [31:P+15], and hence these are necessarily equal !

(We can collectively call all these equal bits as 'The sign bit' !)

So, summing up -

I can rest assured that, iff I've checked the 16-bit overflow condition at the start - then mult_out32[31] & mult_out32[30:p+15] are all equal !

Moreover,

result_axb = {mult_32out[31], mult_32out[p + 14 : p]};

will give me the correct 16-bit signed result, in the end !

Hence proved ! :-)

p.s.: Thanks again FvM, for helping me arrive at this proof; and also giving an Overflow-condition-checker, in case we wish to make a more generic module !