Multirate Processing of Digital Signals: Fundamentals

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Multirate Processing ofDigital Signals
Fundamentals

• For NTUEE VLSI Signal Processing Course
• Instructor ?????

Updated on 4/4/2001
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Outline

• Introduction
• Sampling Rate Conversion
• Multistage Implementation
• Practice Structure
• Polyphase Implementation

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Motivation

• Definition
• More than one sampling rate (clock) are used in a
  system

Module 1
Module 2
?
clock 1
clock 2
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Conversion Approach

• Analog approach
• Digital approach (multirate DSP system)

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Analog Approach
Advantages Simple Straightforward Arbitrary
sampling rate Disadvantages D/A A/D converter
are needed Ideal (near perfect) lowpass filter is
needed Introduced noise and distortion
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Digital Approach

• Sampling rate conversion
• Interpolation
• Increase the sampling rate
• Decimation
• Decrease the sampling rate

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Sampling Theory

• If the highest frequency component in a signal is
  fmax, then the signal should be sampled at the
  rate of at least 2fmax for the samples to
  describe the signal completely, i.e.,

For Fs lt 2fmax, alias occurs in the sampling
process.? Alias Distortion (aliasing)
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Aliasing
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Interpolation by L
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Interpolation by L
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Decimation by M
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Decimation by M
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Conversion by a Rational Factor M/L
Cascade of two process
?L
h1(m)
h2(m)
?M
Interpolation by L
Decimation by M
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Conversion by a Rational Factor M/L
A more efficiency implementation
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Multistage Implementation
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Multistage Implementation
Advantages Reduce the complexity Reduce storage
devices (registers) Simplify (relax) filter
design problem Reduce the finite wordlength
effect Disadvantages Increase the control
circuit Difficulty in choosing I and best Lj for
1? i ? I
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Interpolated FIR (IFIR) Approach
Nothing to do with interpolation and
decimation Conceptually similar Suitable for
narrowband FIR filter design LPF HPF BPF
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Application Interpolated FIR (IFIR)
Desired narrowband response Assume required
filter order is N.
Stretched filter Required filter order is reduced
to N/2.
Desired
Undesired
Interpolated version of stretched filter Required
filter order is still N/2.
Image suppresser Required filter order is
M. Order (N/2M) is needed to implement! (N/2M)
ltlt N for small M
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Interpolated FIR (IFIR)
(a) G(z)
(a) G(z2)
(a) G(z2)I(z)
(b) I(z)
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Interpolated FIR (IFIR)
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Some Useful Operations
Duality and Transposition A dual system is that
performs a complementary operation to that of an
original system, and it can be constructed form
the original system through the process of
transposition. The transposition operation is one
in which the direction of all branches in the
network are reversed, and the roles of the input
and output of the network are interchanged.
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Duality and Transposition
transposition
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Duality and Transposition
They are not true in time-varying system, but can
be treated as sampling rate reverse process.
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Practical Structure
Decimation
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Practical Structure
Interpolation
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Application Polyphase FIR Filter
Polyphase decomposition
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Polyphase FIR Filter
Noble identity
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Polyphase FIR Filter
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Polyphase FIR Filter
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Structure Comparison
Polyphase implementation
Direct implementation