Lattice structures and canonical structures
The basic discussion begins with FIR and IIR
structures,furthermore some structures are canonical and some are non canonical
structures.We normally study and examine lattice and lattice ladder structures
and more specifically FIR lattice structures and IIR lattice structures,FIR
lattice ladder structures and IIR lattice ladder structures.We are not over
here to discuss all lattice and lattice ladder in detail our concern is
associated with FIR lattice structures only.
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| general building block of FIR lattice structure |
Lattice are canonical structures or not
Before exploring the idea lattice are canonical or not let us
learn what are canonical structure and what is special about canonical structures.The
structures where transfer function order and the number of delays in the block
diagram are same are called canonical structures.The condition for canonical
structures is obvious from the definition that delays in the block diagram and
the order of the transfer function must be the same.On the other hand direct
form structures categorized as direct form I and direct form II are canonical
structures as the condition required for structures to be canonical is present
in direct form of structures as well.One more reason why direct form structures
are canonical this is due to their requirement of only N multiplications for
their implementation.
If we talk about the lattice structures specifically then after
observing these structures we come to know that order of transfer function and
the number of delays in the block diagram are not equal in strength and FIR
lattice structures require 2N multiplications for their implementation.So
lattice structures are not canonical structures.
Advantage of lattice structures
The big advantage of lattice structures is their elementary and
fundamental usage in synthesis and analysis of speech and in adaptive
applications like linear prediction.Furthermore,they are used for modeling of
auto regressive signals.
FIR lattice structure simple definition
A system with two input and two outputs respectively of specific
nature and requirements is called FIR lattice structure.Just explanatory and
theoretical aspects are being discussed here.The lattice structures are completely
described and parameterized by their respective coefficients.The coefficients
of lattice structures are named differently including PARCOR coefficients and
reflection coefficients too.
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is an electronic circuit operating on continuous-time analog signals.
A digital filter system usually consists of an analog-to-digital converter to sample the input signal, followed by a microprocessor and some peripheral components such as memory to store data and filter coefficients etc. Finally a digital-to-analog converter to complete the output stage. Program Instructions (software) running on the microprocessor implement the digital filter by performing the necessary mathematical operations on the numbers received from the ADC. In some high performance applications, an FPGA or ASIC is used instead of a general purpose microprocessor, or a specialized DSP with specific paralleled architecture for expediting operations such as filtering.
Digital filters may be more expensive than an equivalent analog filter due to their increased complexity, but they make practical many designs that are impractical or impossible as analog filters. When used in the context of real-time analog systems, digital filters sometimes have problematic latency (the difference in time between the input and the response) due to the associated analog-to-digital and digital-to-analog conversions and anti-aliasing filters, or due to other delays in their implementation.
Digital filters are commonplace and an essential element of everyday electronics such as radios, cellphones, and AV receivers.
The transfer function for a linear, time-invariant, digital filter can be expressed as a transfer function in the Z-domain; if it is causal, then it has the form:
{\displaystyle H(z)={\frac {B(z)}{A(z)}}={\frac {b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots +b_{N}z^{-N}}{1+a_{1}z^{-1}+a_{2}z^{-2}+\cdots +a_{M}z^{-M}}}} H(z)={\frac {B(z)}{A(z)}}={\frac {{b_{{0}}+b_{{1}}z^{{-1}}+b_{{2}}z^{{-2}}+\cdots +b_{{N}}z^{{-N}}}}{{1+a_{{1}}z^{{-1}}+a_{{2}}z^{{-2}}+\cdots +a_{{M}}z^{{-M}}}}}
where the order of the filter is the greater of N or M. See Z-transform's LCCD equation for further discussion of this transfer function.
This is the form for a recursive filter, which typically leads to an IIR infinite impulse response behaviour, but if the denominator is made equal to unity i.e. no feedback, then this becomes an FIR or finite impulse response filter.
Another form of a digital filter is that of a state-space model. A well used state-space filter is the Kalman filter published by Rudolf Kalman in 1960.
Traditional linear filters are usually based on attenuation. Alternatively nonlinear filters can be designed, including energy transfer filters which allow the user to move energy in a designed way. So that unwanted noise or effects can be moved to new frequency bands either lower or higher in frequency, spread over a range of frequencies, split, or focused. Energy transfer filters complement traditional filter designs and introduce many more degrees of freedom in filter design. Digital energy transfer filters are relatively easy to design and to implement and exploit nonlinear dynamics.Courtesy of wikipedia...
Digital filter
A general finite impulse response filter with n stages, each with an independent delay, di, and amplification gain, ai.In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is an electronic circuit operating on continuous-time analog signals.
A digital filter system usually consists of an analog-to-digital converter to sample the input signal, followed by a microprocessor and some peripheral components such as memory to store data and filter coefficients etc. Finally a digital-to-analog converter to complete the output stage. Program Instructions (software) running on the microprocessor implement the digital filter by performing the necessary mathematical operations on the numbers received from the ADC. In some high performance applications, an FPGA or ASIC is used instead of a general purpose microprocessor, or a specialized DSP with specific paralleled architecture for expediting operations such as filtering.
Digital filters may be more expensive than an equivalent analog filter due to their increased complexity, but they make practical many designs that are impractical or impossible as analog filters. When used in the context of real-time analog systems, digital filters sometimes have problematic latency (the difference in time between the input and the response) due to the associated analog-to-digital and digital-to-analog conversions and anti-aliasing filters, or due to other delays in their implementation.
Digital filters are commonplace and an essential element of everyday electronics such as radios, cellphones, and AV receivers.
Characterization
A digital filter is characterized by its transfer function, or equivalently, its difference equation. Mathematical analysis of the transfer function can describe how it will respond to any input. As such, designing a filter consists of developing specifications appropriate to the problem (for example, a second-order low pass filter with a specific cut-off frequency), and then producing a transfer function which meets the specifications.The transfer function for a linear, time-invariant, digital filter can be expressed as a transfer function in the Z-domain; if it is causal, then it has the form:
{\displaystyle H(z)={\frac {B(z)}{A(z)}}={\frac {b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots +b_{N}z^{-N}}{1+a_{1}z^{-1}+a_{2}z^{-2}+\cdots +a_{M}z^{-M}}}} H(z)={\frac {B(z)}{A(z)}}={\frac {{b_{{0}}+b_{{1}}z^{{-1}}+b_{{2}}z^{{-2}}+\cdots +b_{{N}}z^{{-N}}}}{{1+a_{{1}}z^{{-1}}+a_{{2}}z^{{-2}}+\cdots +a_{{M}}z^{{-M}}}}}
where the order of the filter is the greater of N or M. See Z-transform's LCCD equation for further discussion of this transfer function.
This is the form for a recursive filter, which typically leads to an IIR infinite impulse response behaviour, but if the denominator is made equal to unity i.e. no feedback, then this becomes an FIR or finite impulse response filter.
Types of digital filters
Many digital filters are based on the fast Fourier transform, a mathematical algorithm that quickly extracts the frequency spectrum of a signal, allowing the spectrum to be manipulated (such as to create band-pass filters) before converting the modified spectrum back into a time-series signal.Another form of a digital filter is that of a state-space model. A well used state-space filter is the Kalman filter published by Rudolf Kalman in 1960.
Traditional linear filters are usually based on attenuation. Alternatively nonlinear filters can be designed, including energy transfer filters which allow the user to move energy in a designed way. So that unwanted noise or effects can be moved to new frequency bands either lower or higher in frequency, spread over a range of frequencies, split, or focused. Energy transfer filters complement traditional filter designs and introduce many more degrees of freedom in filter design. Digital energy transfer filters are relatively easy to design and to implement and exploit nonlinear dynamics.Courtesy of wikipedia...
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