Arithmetic - Signal Processing Systems - page 584

extend the sign by copying the sign-bit at the MSB side of the vector. On the LSB
side it is enough to extend with zeros . 1
The operation of shifting, i.e., multiplying or dividing by a power of two, is in
fact the same as increasing the word length. Hence, shifting a two's complement
number two positions to the right (dividing by 4) requires copying of the sign-bit to
the two introduced positions.
For carry-save representation the extension of the word length on the MSB side
require that the sign-bits of the carry and sum vectors are corrected to avoid what is
referred to as carry overflow. The origin of this effect is that once the carry and sum
vectors are added there is an overflow in the addition. As the resulting carry bit from
the MSB position is neglected the result still remains valid, in fact this is a required
property for two's complement representation to work. However, if the vectors are
shifted once before the final addition, the effect of the overflow will occur in an
earlier position.

1.5 Negation

Negation is a useful operation in itself, but also in a subtraction, Z = X − Y ,which

can be seen as an addition of the negated value, Z .

To negate a two's complement number one inverts all the bits and add a one
to the LSB position. As will be seen later on, this does not necessarily have to be
performed explicitly, but can rather be integrated with an addition.

= X +( − Y )

1.6 Finite Word Length Effects

In recursive algorithms it is necessary to use non-linear operations to maintain a
finite word length. This introduces small re-quantization errors, so called granularity
errors. In addition, very large overflow errors will occur if the finite number range
is exceeded. These errors will not only cause distortion, but may also be the cause
of parasitic oscillations in recursive algorithms [ 5 , 13 , 46 ] .
The effect of these errors depends on many factors, for example, type of
quantization, algorithm, type of arithmetic, representation of negative numbers, and
properties of the input signal [ 31 ] . The analysis of the influence of round-off errors
in floating-point arithmetic is very complicated [ 32 , 45 , 57 ] , because quantization
of products as well as addition in floating-point arithmetic and subtraction causes
errors that depend on the signal values. Quantization errors in fixed-point arithmetic
are with few exceptions independent of the signal values and can therefore be
analyzed independently. Lastly, the high dynamic range provided by floating-point

1 It is worth noticing that for one's complement the sign-bits are inserted at the LSB side.