System Design Flow and Fixed-point Arithmetic - Digital Design of Signal Processing Systems: A Practical Approach - page 115

e = 0 e = 3 e = 2 e = 1

x[n]

……..

n

Figure 3.22 Applying block floating-point format on data input in blocks to an algorithm

the next block. Every stage of the FFT causes bit growth, but the block floating-point implementation
can also cater for this growth.
The first two stages of the Radix-2 FFTalgorithm incur a growth by a factor of two. Before feeding
the data to these stages, two redundant sign bits are left in the representation to accommodate this
growth. The block floating-point computation, while scaling the data, keeps this factor in mind.
For the remainder of the stages in the FFTalgorithm, a bit growth by a factor of four is expected. For
this, three redundant sign bits are left in the block floating-point implementation. The number of
shifts these blocks of data go through from the first stage to the last is accumulated by adding the
block exponent of each stage. The output values are then readjusted by a right shift by this amount to
get the correct values, if required. The block floating point format improves precision as more valid
bits take part in computation.
Figures 3.23(a) and (b) illustrate the first stage of a block floating-point implementation of an 8-
point radix-2 FFT algorithm that caters for the potential of bit growth across the stages. While
observing the minimum redundant sign bits (four in the figure) and keeping in consideration the bit
growth in the first stage, the block of data is moved to the left by a factor of two and then converted to
the requisite 8-bit precision. This makes the block exponent of the input stage to be 2. The output of
the first stage in 8-bit precision is shown. Keeping in consideration the potential bit growth in the
second stage, the block is shifted to the right by a factor of two. This shift makes the block exponent
equal to zero. Not shown in the figure are the rest of the stages where the algorithm caters for bit
growth and also observes the redundant sign bits to keep adjusting the block floating-point exponent.

3.8 Forms of Digital Filter
This section briefly describes different forms of FIR and IIR digital filters.

3.8.1 Infinite Impulse Response Filter
The transfer function and its corresponding difference equation for an IIR filter are given in (3.12)
and (3.13), respectively:

X N
k

0 b k z k

Yz ðÞ

¼

Hz ðÞ¼

Þ ¼

ð 3 : 12 Þ

X M
k

X ð z

1 þ

1 a k z k

¼

X X

N M

y ½¼ b k x ½ n k a k y ½ n k ð 3 : 13 Þ

k ¼ 0 k ¼ 1