Optimization of Number Representations - Signal Processing Systems - page 1285

a b

Fig. 6 Fixed-point filters with reduced complexity. ( a ) Fixed-point filter without shift, ( b )Fixed-
point filter with a 16 bit adder

the range of x is de-normalized and the

upper 3 bits next to the sign bit are unused. Since the IWL's of x

[ n ] is 1.0. This means that the input x [ n ]

are the
same, there is no need of inserting a shifter. Figure 6 a shows the resulting hardware.
The SQNR (Signal to Quantization Noise Ratio) of the input is obviously lowered
by employing the de-normalized scheme. Another fixed-point implementation in
Fig. 6 b shows the hardware using a 16-bit adder. In this case, the quantizer (Q) is
moved to the output of the multiplier. Note that the SQNR of this scheme is even
lower than that of Fig. 6 a .
In the above example, the range of 1 is assumed to the input x

[ n ] and z [ n ]

, which is from
the floating-point design. However, assuming the range of 2 as for the input x

[ n ]

]
does not change the resultant hardware because the output range should be doubled
in this case.

[ n

3 Range Estimation for Integer Word-Length Determination

The floating-point to fixed-point conversion examples in the previous section shows
that estimating the ranges of all of variables is most crucial for this conversion
process. There are two different approaches for range estimation. One is to calculate
the L1-norm of the system and the other is using the simulation results of floating-
point systems [ 12 , 17 ] .

3.1 L1-Norm Based Range Estimation

The L1-norm of a linear shift-invariant system is the maximum value of the output
when the absolute value of the input is bounded to 1. If the unit-pulse response of a
system is h

[ n ] ,where n = 0 , 1 , 2 , 3 ,··· ∞ , the L1-norm of this system is defined as:

∞
n = 0 | h [ n ] |

L 1 ¯ norm ( h [ n ]) = (3)