Digital filter
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 digital signal processor (DSP) with specific paralleled architecture for expediting operations such as filtering.
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.
A digital filter is characterized by its transfer function, or equivalently, its difference equation.
This is the form for a recursive filter, which typically leads to an infinite impulse response (IIR) behaviour, but if the denominator is made equal to unity, i.e. no feedback, then this becomes a finite impulse response (FIR) filter.
A variety of mathematical techniques may be employed to analyze the behavior of a given digital filter.
Many of these analysis techniques may also be employed in designs, and often form the basis of a filter specification.
Typically, one characterizes filters by calculating how they will respond to a simple input such as an impulse.
In discrete-time systems, the digital filter is often implemented by converting the transfer function to a linear constant-coefficient difference equation (LCCD) via the Z-transform.
For example: This is expanded: and to make the corresponding filter causal, the numerator and denominator are divided by the highest order of
The resultant linear difference equation is: or, for the example above: rearranging terms: then by taking the inverse z-transform: and finally, by solving for
In plain terms, for example, as used by a computer programmer implementing the above equation in code, it can be described as follows:
= the 2nd-to-last raw input value Although filters are easily understood and calculated, the practical challenges of their design and implementation are significant and are the subject of much advanced research.
Specifically, some realizations are more efficient in terms of the number of operations or storage elements required for their implementation, and others provide advantages such as improved numerical stability and reduced round-off error.
This form is practical for small filters, but may be inefficient and impractical (numerically unstable) for complex designs.
[6] In general, this form requires 2N delay elements (for both input and output signals) for a filter of order N.
This structure is obtained by reversing the order of the numerator and denominator sections of Direct Form I, since they are in fact two linear systems, and the commutativity property applies.
) that tap off the center net, and these can be combined since they are redundant, yielding the implementation as shown below.
The disadvantage is that direct form II increases the possibility of arithmetic overflow for filters of high Q or resonance.
[8] This is because, conceptually, the signal is first passed through an all-pole filter (which normally boosts gain at the resonant frequencies) before the result of that is saturated, then passed through an all-zero filter (which often attenuates much of what the all-pole half amplifies).
Analog filters consist of imperfect electronic components, whose values may also change with temperature and drift with time.
In digital filters, the coefficient values are stored in computer memory, making them far more stable and predictable.
Analog filters that perform the same function are often significantly more complicated, as they would require many delay elements.
Digital filters rely less on analog circuitry, potentially allowing for a better signal-to-noise ratio.
In digital systems, latency is introduced not only by delay elements in the digital signal path, but also by analog-to-digital and digital-to-analog converters that are required for a system to process analog signals.
When making an electrical circuit on a PCB it is generally easier to use a digital solution, because the processing units are highly optimized over the years.
Some 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 very high order band-pass filters) before converting the modified spectrum back into a time-series signal with an inverse FFT operation.
Alternatively nonlinear filters can be designed, including energy transfer filters,[14] 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.
Digital energy transfer filters are relatively easy to design and to implement and exploit nonlinear dynamics.
