The output of ADC is at very high sampling rate as compared
to output data of the filter. In such a
case it is more efficient to change the sampling rate in stages
by means of a sequence of filters
than it is to do so all at once with a single filter.
HENCE THE FILTER COMPRISES OF THE FOLLOWING TWO STAGES:
1. FIRST STAGE FIR FILTER
2. SECOND STAGE FIR FILTER
The objective of each stage is to filter and to decimate the data.
FIRST STAGE FIR FILTER
It is also known asCascaded integrator-comb-filter(CIC)
THE TRANSFER FUNCTION IS GIVEN BY
Nth Order Sinc Transfer Function
The objectives of CIC are:
1. High value of Decimation ( order M=16,
32, 64...)
2. Removal of high Frequency Quantisation
noise.
The first stage of CIC filter can be configured
in two ways:
1. Recursive Algorithm
2. Non Recursive Algorithm
Recursive Algorithm
An N-stage CIC decimation filter consists of a cascade of N integrators
operating at the input
sampling rate Fs, an M-fold decimator, and a cascade of N comb filters
operating at the reduced
sampling rate fs/M. The integrator and comb sections are separated
by a decimator with a decimation
factor N.
Transfer Function of Cascaded Integrators
H(z) = ( 1 ) / ( 1-z^-1)^N
Transfer Function of Cascaded Combs
H(z) = ( 1- z^-1)^N
The throughput rate of the CIC decimation filter is determined by
the speed of the integrators since
they operate at the higher input sampling rate. The comb section
operates at a much lower rate and
is less of an issue in its implementation.
"Comb-Filter" reduces the sampling rate to an intermediate frequency.
The decimation done in first stage is of very high value M=64, 128
etc...
The intermediate frequency is decided by the Decimation factor M
=Fs/Fd.
Fd= Frequecy where First zero occours.
Fs= Sampling Frequency.
M is also known as the Length of Comb-Filter.
Drawbacks of Recursive Scheme
1. The output of sigma-delta modulator usually has
a short word length, say m bits ( 1 bit for single
stage modulators or 4 ~ 6 bits in case of cascaded modulators),
and a very high word rate, i.e. oversampling
rate. To maintain accuracy the data word inside the IIR part must
be m+N*log2(M) bits (N is filter order).
M is usually large( 16,32,64,etc.), hence IIR part
a) has a long word length,
b) has to operate at the very high ovesampling
frequency
c) long word additions performed at high sampling
frequency result in large power consumption.
2. As log2(M) increases the power consumption increases almost linearly
for large M .
3. The highest working frequency of recursive algorithm decreases
near linearly with Log2(M).
Therefore IIR part limits the applicability of the recursive structure
to a very high oversampling frequency
application.
Nonrecursive Algorithm
By applying the commutative rule, the non-recursive algorithm as shown in figure is obtained.
The switches in the figure indicate the reduction in sampling rate
by a factor of 2. Every stage is a low
order filter with a different sampling rate.
Advantages of Nonrecursive Algorithm
1. The word length increases through every stage by N bits ( N is
filter order ) and the sampling rate
decreases through every stage by a factor of
2.
2. The word length is short when the sampling rate is high; and
when the word length increases the
sampling rate decreases.
3. Results in highly regular structure which makes layout easier.
4. The frequency limitation is relaxed.
5. As log2(M) increases the power consumption increases only slightly
for large M .
6. The highest working frequency of non-recursive algorithm is at
a higher constant value when log2(M)
increases.
Thus, Non-recursive algorithm is suitable for the low power and
high speed decimator designs for large
decimator designs.
Change in the sampling rate implies change in the Nyquist Frequency.
So aliasing noise appears.
To avoid that, the signal must be low pass filtered before decimation.The
denominator of the Sinc
function is indicative of the low pass filtering.
For both Recursive and Non-recursive Filetrs all the filter coefficients
are unity. Hence, a multiplier is not
required, resulting in simple hardware.
This comb-filter operation is equivalent to a rectangular window
FIR filter, resulting in narrowband
low pass filtering. The rectangular window has poor stopband
performance but gives a sharp cutoff
(narrow transition band).
Limitations of CIC Filters
1. A single comb-filter is unable to effectively remove the out
of band quantization noise.
For M=64:
i. Sinc Comb Filter
: The first sidelobe peak is at -15dB
ii. Sinc^2 Comb Filter : The First
sidelobe peak is at - 25dB
iii. Sinc^3 Comb Filter : The First sidelobe
peak is at -40db
Hence, a cascade of comb-filters are used, which is sin^2,
sinc^3, sinc^4, etc. operation.
This gives enough stop-band attenuation. The First stgae
FIR Filter is concerned more with stopband
response than with passband response.
On cascading comb-filters, in frequency domain
we are multiplying the Impluse response of two
identical comb-filters. In time domain
it is equivalent to convolving the Impluse response of the
two identical comb-filters. Which essentially
increases the length of the impulse response.
2. Causes magnitude drooping at the upper region of the baseband.
Hence comb-filters are used in conjuction with
additional stages of FIR Filters to compensate for
this drooping.
3. Power consumption of the decimation filters will be very large
at high oversampling frequency. Hence,
decimation is done in two stages
I. PLOTS OF SINC, SINC^2, SINC^3, SINC^4, SINC^5 FOR VARIOUS DECIMATION FACTORS
FIGURE OF SINC FUNCTIONS FOR DECIMATION=64
FIGURE OF SINC FUNCTIONS FOR DECIMATION=32
FIGURE OF SINC FUNCTIONS FOR DECIMATION=16
ZOOMED VIEW OF SINC FUNCTIONS N=64
From plot of Noise Transfer function it is clear as we increase the
order of modulation, the quantization
noise in the baseband gets reduced thus increasing the resolution.
II. PLOTS OF SINC^3 FOR VARIOUS DECIMATION FACTORS
FIGURE OF SINC3 FUNCTION FOR VARIOUS DECIMATION FACTORS
It can be seen from the plots that the zeros of infinite rejection
(at fs/M, 2fs/M, 3fs/M,etc..) can be
strategically placed by selecting fs and the number of samples averaged,N.
Increase in decimation causes decrease in the width of main lobe.
Hence, a higher value of Decimation
is desirable. All the decimation can't be done in one stage as it
leads to a complex circuitry to design.
Hence a part of decimation is done in second Stage of FIR Filter.
The Decimation factor indicates how many samples are used to calculate
the weighted average. Hence,
M is indicative of the impluse response of the comb-filter. Comb-Filter
being an averaging filter ignores
the information contained in the relationship between various samples,
as it gives equal weightage to
all the samples.
Result: The Comb-Filter provides maximum attenuation
only on the higher frequency
components which will be aliased into
the band of interest after decimation. It also yields
very high reduction in decimation rate.
OVERALL IMPLEMENTATION OF SINC^3 (MISTUBISHI)
SECOND STAGE FILTER
The Second stage can be either IIR or FIR Filter.
It provides
sharp attenuation to low frequency components near the base band.
IMPLEMENTATION OF FIR FILTER
The Comb-Filter is followed by FIR Filter, with symmetric coefficients
to provide a linear phase response
Provides:
1. Decimation, of the order of 2 or 4.
2. Magnitude Compensation for the magnitude change (droop) from
the comb-filter output.
3. Sharp low pass filtering
The second Stage Filter is concerned with passband(droop compensation) and also stopband(high attenuation)
The choice of window function is determined mainly by the stopband
attenuation requirement.
A 16 bit Sigma delta AD Converter requires around 80dB stopband
attenuation. Blackman window
can provide atleast 74dB Stopband attenuation. Hence, Blackman window
is used.
The order of filter is decided by the transition bandwidth.
OVERALL BLOCK DIAGRAM OF FILTER (FROM MOTOROLA 7-4)
