Mixed Signal Techniques - Signal Processing Systems - page 802

Fig. 11 Multistage
decimation filter for
decimation by 32. The
downward array indicates
decimation, followed by the
decimation factor

In

H sinc ( z ) 8 H 1 ( z )

2

32 f s

Out

H 2 ( z ) H 3 ( z )

2

f s

2 -P

In

+ + + K

K f s

Z -1 Z -1 Z -1

Out

+ + +

f s

- - -

Z -1 Z -1 Z -1

Fig. 12 Implementation of the averaging filters for decimation by factor K

An efficient way to implement the sinc filter is shown in Fig. 12 . The filter has
the transfer function

z − K

1 −

2 − P

M + 1

H sinc ( z )= (

z − 1 )

(27)

−

1

where K is the decimation ratio of the sinc filter. The scaling factor 2 − P has to satisfy
the condition

2 − P

M + 1

≤ ( 1 / K ) . (28)

It should be noticed that although the structure of Fig. 12 may have occasional
overflows in the internal nodes, the result is still correct if proper scaling and two's
complement arithmetic is used.

4 Digital to Analog Converters

In principle, DAC operation could be achieved by generating a sequence of weighted
impulses from the digital samples and lowpass filtering to remove the image
frequencies. Such a filter is called a reconstruction filter. In practice, the DAC
output is usually taken from a S/H circuit. The sinc type response of the S/H circuit
causes a droop of about

− .

3 9 dB at the Nyquist frequency, as illustrated in Fig. 3 .