Page 190 - DSP Integrated Circuits

4.22 Decimation With A Factor M 175

oversampled by a factor that is a large power of two. The digital signal is then
bandlimited by a digital filter with a stopband angle slightly less than jt/2. The
sample frequency can therefore be reduced by a factor of two by simply dropping
every other sample. This process of bandlimiting and dropping every other sample
is repeated until the desired sample rate is obtained.
The relation between the Fourier transforms of the decimated signal and the
original signal is [3, 13, 29]

where M is the decimation factor. The Fourier transform of the decimated signal
consists of a sum of shifted replicas of the Fourier transform of the original signal.
Generally, aliasing takes place if the original signal bandwidth is larger than n/M.
Although the aliasing can be removed by a bandlimiting digital filter, the deci-
mated signal will no longer be the same.
The decimation filter can also be realized efficiently using wave digital filters.
It is efficient from a computational point of view to also do the decimation in steps
of two.

4.22.1 HSP43220™
The HSP43220™ from HARRIS Corp. is a two-stage linear-phase FIR filter for
decimation. The first stage, which is shown in Figure. 4.66, is a high-order decima-
tion filter that allows decimation by factors up to 1024. The maximum input sam-
ple frequency is 30 MHz. Input data word length is 16 bits using two's-complement
representation.
The second stage consists of a linear-phase direct-form FIR filter with up to
512 taps. This filter provides the proper frequency response, but can also be used
to decimate by factors up to 16. The first stage can be bypassed in order to use only
the linear-phase FIR filter or several chips can be cascaded to obtain higher deci-
mation ratios or longer FIR filters.
The filtering and decimation in the first stage is accomplished by connecting
up to five accumulators in cascade. The accumulators yield the transfer function

where n is the number of accumulators. The accumulators represent first-order
recursive filters. Next follows the decimation. The resulting transfer function
becomes

Finally, n first-order FIR filters, so-called comb filters, with the transfer function