Page 227 - DSP Integrated Circuits

212 Chapters Finite Word Length Effects

2
noise gain is = 1. The noise variance at an output node is (JV-l)aB . The noise in
the safely scaled FFT is determined in Example 5.9.

EXAMPLE 5.9
Determine the signal-to-noise ratio for the safely scaled, 16-point FFT.
The quantization errors are divided by two in each stage. The output of the
last stage is multiplied by two. Hence the variance is divided by four. The noise
2
1
gains from stages 1 to 4 to the outputs are (1/4) , (1/4) ,1, and 4, respectively. The
number of noise sources are 8, 4, 2, and 1, respectively. Hence, the noise variance
at the output is

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

Hence, the SNR will be about the same at the input and output of the FFT if
we use a log2(Vl20) « 3.45 bits longer word length inside the FFT than for the
input signal. Note that is a stringent requirement and a somewhat shorter word
length may be used in practice.

5.6.2 Error Spectrum Shaping
Error spectrum shaping or error feedback is a general method that can be used
to reduce the errors inherent in quantization [13, 18, 29, 30]. The technique is
useful for both fixed-point and floating-point arithmetic. The reduction in
round-off noise is especially dramatic in narrow-band and wide-band lowpass
filters, which have poles close to the unit circle. The data word length can often
be reduced by several bits since the error feedback reduces the round-off noise.
This is important both if the algorithm is to be implemented in ASIC, since
memory is expensive, and in standard signal processors with short data word
lengths.
Figure 5.20 shows a quantizer with error feedback followed by a filter. The
transfer function from the quantizer to the output of the filter is