Finite Impulse Response (FIR) Digital Filters

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Finite Impulse Response (FIR) Digital Filters

• Digital filters are rapidly replacing classic
  analog filters.
• Programmable DSP with MAC can be used to
  implement digital filters.
• For high-bandwidth signal processing
  applications, FPGA technology can provide
  multiple MACs to achieve the desired thoughput.

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• FIR Theory
• An FIR with constant coefficient is an Linear
  Time-Invariant (LTI) filter.
• The ouput of an FIR of order (or length) L, to an
  input time-series xn, is given by a finite
  version of the convolution sum
• yn fnxn S fk xn-k
• where f0 ? 0 through fL-1 ? 0 are the
  filters L coefficients. They also correspond to
  the FIRs impulse response.

L-1
k0
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• LTI system expressed in the z-domain
• Y(z) F(z) X(z)
• where F(z) is the FIRs transfer function defined
  in z-domain by
• F(z) S fk zk
• The roots of polynomial F(z) define the zeros of
  the filter. FIRs are also called all zero filters.

L-1
k0
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Direct Form FIR Filter
Tapped delay line
Fig. 3.1.
Tapped weight
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FIR Filter with Transposed Structure

• A variation of the direct FIR model is called
  the transposed FIR filter. It can be constructed
  from the direct form FIR filter by
• Exchanging the input and output
• Inverting the direction of signal flow
• Substituting an adder by a fork, and vice versa

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FIR Filter in the Transposed Structure
Fig. 3.3. See Example 3.1 Programmable FIR
Filter
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Comparison of the two forms of the FIR filter

• The direct form FIR filter needs extra pipeline
  registers between the adders to reduce the delay
  of the adder tree and to achieve high throughput.
  
• The FIR filter with transposed structure has
  registers between the adders and can achieve
  high throughput without adding any extra pineline
  registers.

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Symmetry in FIR Filters

• Define the center point of an odd-order FIRs
  impulse response as the 0th sample
• F(z) S fk zk
• (An even-order FIR can be similarly defined.)
• Linear-phase FIR filter
• Linear-phase is achieved if the filter is
  symmetric or antisymmetric.
• See Table 3.1.

k(L-1)/2
k-(L-1)/2
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Linear-phase even-order filter with reduced
number of multipliers
fL-1
fL-2
Fig. 3.5. L L/2, where L is an even number.
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Constant Coefficient FIR Design

• There are only a few applications, e.g., adaptive
  filters, where we need a general programmable
  filter.
• In many applications, the filters are Linear Time
  Invariant (LTI) and the coefficients do not
  change over time.
• The hardware effort can be reduced for constant
  coefficient FIR.

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Direct FIR implementation

• In a practical situation, the FIR coefficients
  are obtained from a computer design tool and
  presented to the designer as floating point
  numbers.
• The performance of a fixed-point FIR, based on
  the floating-point coefficients, should be
  verified using simulation or algebraic analysis
  to ensure that design specifications remain
  satisfied.

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• Dynamic range overflow should be avoided.
• The worst-case dynamic range growth G of an Lth-
  order FIR is
• G ? log2(S fk)
• See Example 3.2 Four-tap direct FIR filter.

L-1
k0
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Improve the direct FIR design

• Realize each filter coefficient with an optimal
  CSD code.
• Increase effective multiplier speed by
  pipelining.
• For FIR with symmetric coefficients, the number
  of multipliers can be reduced.
• See Table 3.3 and Example 3.3.

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Rephasing pipelined multiplier in FIR filter
Add a positive delay fnz d fn z -d
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FIR Filter with Transposed Structure

• If the transposed filter has constant
  coefficients, two improved designs should be
  considered
• Multiple use of the repeated coefficients using
  the reduced adder graph (RAG) algorithm
• Pipeline adders

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Reduced Adder Graph

• Algorithm 3.4 Reduced Adder Graph
• Remove the sign of the coefficient.
• Remove all coefficients and factors that are a
  power of two.
• Realize all cost 1 coefficients.
• Use cost 1 coefficients in building the
  multiplier of higher cost. (Use Table 2.3.)
• See Example 3.5.

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Reduced Adder Graph for F6 Half-band Filter
Fig. 3.11. Realization of F6 using RAG algorithm
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FIR Filter Using Distributed Arithmetic

• Distributed Arithmetic Using Logic Cells
• See Example 3.6 Distributed Arithmetic Filter
  as State Machine.
• Logic cells are used to implement small look-up
  tables for low-order filters.
• The outputs of a collection of low-order filters
  can be added together to define the output of a
  high-order FIR.
• See Example 3.7 Five-input DA Table.

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• DA Using Embedded Array Blocks
• It is not economical to use the 2-kbit EABs for a
  short FIR filter, mainly because the number of
  available EABs is limited.
• The maximum registered speed of an EAB is 76 MHz,
  and an LC table implementation may be faster for
  a short FIR filter.
• For long filters, EABs have registered throughput
  at a constant 76 MHz and routing effort is
  reduced.
• See Example 3.8 Distributed Arithmetic Filter
  using EABs.

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Example 3.10 Loop Unrolling for DA FIR Filter
Fig. 3.16 Parallel implementation of a
distributed arithmetic FIR filter