Page 370 - Analog and Digital Filter Design

Introduction to Digital Filters 3

Many DSP specific programs such as FIR filters and Fast Fourier Transforms
need multiplication and addition, and possibly subtraction, so before the
budding DSP engineer can have programs running he should understand the
basics of signal-processing mathematics.

Many devices (particularly low-cost ones) use fixed point numbers, and the
mathematics requires some thought. Fixed point DSP devices that have 16-bit
busses usually use the notation of Q15: the Q represents the sign and the 15
represents the number of binary characters after the decimal point; thus in this
notation all numbers have a magnitude of less than one.

Binary and Hexadecimal
Each data sample, or filter coefficient, is a binary word of 16 or more bits. A
long series of ones and zeroes is difficult to describe. In the computing world,
multiples of 3 bits are grouped together. Each group is then converted into a
hexadecimal number and thus represents a decimal number between 0 and 16.
TQ signify a hexadecimal number a lower case h is appended. Decimal numbers
0 to 9 are the same in hexadecimal, but decimal numbers 10 to 16 are repre-
sented by A to F in hexadecimal. For the whole range, binary 0000 becomes
Oh and binary 11 11 becomes Fh. Many processors have a 16-bit bus; in 16-bit
format 1100,101 1,0110,flOl becomes CB6Dh.

How do we handle negative numbers? For these there is a different system where
the most significant bit represents the sign. The sign bit is a 0 for positive
numbers and a 1 for negative numbers. In 16-bit hexadecimal, 7FFFh is the
highest positive number (which is 32767 in decimal). Numbers 8000 h to FFFFh
are negative, with a maximum value of -32768. It would be possible for us to
simply change the most significant bit, depending on whether the number is
positive or negative. Unfortunately this makes the mathematics difficult, so an
alternative method is used where all the bits are inverted and then one is added.
This system is known as two’s complement. To see where the name comes from,
and to explain further, I will have to use binary notation.

Two’s Complement
The complement of a binary number is found by inverting each bit in turn. This
is known as one’s complement:

Binary Number 100 1 ,0 101,110 1,lO 1 1

One’s Complement 0 1 10,lO 10,0010,0 100
A two’s-complement number is found by taking the one’s complement and then
adding a binary one to it. In simple terms this can be remembered as one plus