Page 372 - Analog and Digital Filter Design

Introduction to Digital Filters 369

Now the fundamentals have been explained, we must examine some
arithmetic.

Adding Two’s Complement Numbers
Addition is the most basic of processes. Samples can be averaged by adding them
together and then dividing by the number of samples. Filtering processes need
addition (as well as multiplication).

Adding two’s complement numbers is as simple as adding decimal numbers; the
advantage is that the signs are automatically taken care of. A simple equation
using two positive numbers will be given first.

Number 0111,0011,1011,1011 (73BBh or 29627)
Add 0011,1010,1001,1010 (3A9Ah or 15002)

Result 1010,1110,0101,0101 (AE55h or -51ABh or -20907)
So, what went wrong? The problem here was not taking into account the size
of the numbers. The maximum number that can be used in 16-bit two’s com-
plement arithmetic is 32767, or 2” - 1. The addition of decimal 29627 and I5002
results in 44629, which is beyond the allowed range.

Adding a negative and a positive number together gives the correct result. This
is shown in the following example.
Number 01 1 1,001 1,101 1,101 1 (73BBh or 29627)

Add 101 1,1010,0001,1000 (BA18h or -45E8h or -17896)
Result (I), 0010,1101,1101,0011 (2DD3h or 11731)

Notice that there has been a carry, which is ignored.

In DSPs that use Q15 format numbers, the accumulator has a 430 structure. A
Q15 number must be converted into a Q30 number before addition. This can
be done by shifting the number left by 15 places in the 32-bit accumulator.
This means that, after addition, the two most significant bits comprise the sign
and carry information. The accumulator’s result is then shifted left by one
bit, so the carry falls off the end. The most significant bit in the accumulator
is now the sign bit, and the upper half of the accumulator can be stored in a
16-bit register.