Page 233 - DSP Integrated Circuits

218 Chapter 5 Finite Word Length Effects

5.10 INTERPOLATOR, CONT.

The interpolator filter will be implemented by multiplexing four processing ele-
ments (PEs). Each PE will therefore execute eight adaptor operations. Only 12
unique coefficients are needed since the two interpolator stages are identical. The
coefficient word lengths can be optimized for each PE in order to minimize the
cost. However, for the sake of simplicity we will here assume that the coefficient
word lengths are the same and in the range of 12 to 14 bits.
Since we do not have a specific application for the interpolator in mind, we
will arbitrarily select to use a data word length of 20 bits, which will be more than
sufficient for most applications.

5.11 FFT PROCESSOR, CONT.

To determine the required word length in the scaled FFT we first compute the
noise at the output of the FFT. Generalizing from Figure 5.19 we obtain the vari-
ance at the outputs:

where M = log 2UV). The first factor in the preceding sum corresponds to the num-
ber of noise sources in stage s, and the second factor corresponds to their noise
gains. For large FFTs we have

The variance of the output signal is related to the input signal variance
according to

The signal-to-noise ratio is

Hence the SNR will be about the same at the input and output of the FFT if
we use a 0.5 log2(8AO = 6.5 bits longer word length inside the FFT than for the
input signal. We select a data word length of 23 bits.
The coefficient word length will affect the cost of the butterfly PEs and the
cost of storing the coefficients, WP. The potential reduction in chip area and power
consumption that can be achieved by optimizing the coefficient word length is rel-
atively small. We will therefore select a coefficient word length of 14 bits [26].

5.12 DCT PROCESSOR, CONT.

The required data and coefficient word lengths required for the two-dimensional
DCT have been determined experimentally to be in the ranges of 10 to 14 and 8 to