Digital filter

A digital filter is a mathematical filter used to manipulate a signal such as blocking or letting a certain frequency range through. The difference to the analog filter lies in the implementation: Analog filters are built with passive electronic components such as capacitors , coils , resistors or actively with operational amplifiers . Digital filters are implemented with logic modules such as ASICs , FPGAs or in the form of a sequential program with a signal processor .

properties

Block structure of a digital IIR filter

Another essential feature of digital filters is that they do not process continuous signals, but only signals with discrete times and values. In the time-periodic sequence, a time-discrete signal consists only of individual pulses, which represent the signal curve over time, the respective sample values . The sample value is discrete in value , since the digital representation of numbers only offers a finite resolution.

The filter behavior of digital filters is easier to reproduce. Certain types of filters, such as the so-called FIR filters, can only be implemented as digital filters and not as an analog filter circuit. Digital filters in combination with analog-to-digital converters and digital-to-analog converters are also increasingly replacing filter structures that were previously implemented as purely analog. Digital filters represent the basis of digital signal processing and are used, for example, in communication technology .

Continuous filter transfer functions and analog filters formed from them such as Butterworth filters , Bessel filters , Chebyshev filters or elliptical filters can be simulated after adapting the filter transfer function to the finite, discrete spectrum in the form of digital IIR filters with suitably selected filter coefficients.

Mathematical definition

An abstract digital filter is an operator that assigns time-discrete digital signals to the same. Often, for the sake of simplicity of description, it is assumed that the signal has real numbers as values; H. the quantization of the samples (i.e. the rounding to one of the finitely many values of the bit representation) of the digital signal is not taken into account. A time-discrete signal x is a map that represents each point of the discrete, equidistant set

{\ displaystyle \ Gamma = \ {t_ {n}: = t_ {0} + n \ Delta t, n \ in \ mathbb {Z} \}}

assigns a number. It can also be through the consequence of its functional values

{\ displaystyle x [n] = x_ {n}: = x (t_ {n})}

can be specified. The notation with square brackets is preferred in computer science to that with index in mathematics.

The basic functionality of a (finite, non-recursive) filter operation is as follows: At each point in time, or point from the grid, a neighborhood of nearby points in time is fixed, e.g. B. two points before and after. The shape of this environment is constant over time. If the environment only contains points that preceded it in time, the filter is called causal .

Now the tuple of values is available in its environment at any point in time . The same function is always used on this tuple, e.g. B. Maximum formation, mean value formation, weighted mean values, ... If this function is linear, the filter is called linear , otherwise non-linear.

If one considers a family of signals that result from one another due to a time shift and generates the family of the signals transformed by the filter, then the filtered signals differ from one another by precisely the same time shift. The filter is time-invariant. Signal transformations with these characteristics are also called LTI systems referred, English for L inear T ime I nvariant. If one considers the discrete signal as a coefficient sequence of a Fourier series expansion , i. H. the signal values as Fourier integrals , then an LTI system is capable of the amplitudes | f (s) | of the individual frequencies and to rotate it in phase arg (f (s)) compared to the input signal . ${\ displaystyle \ textstyle x_ {n} = \ int _ {- 1/2} ^ {1/2} \; f (s) \ cdot e ^ {i2 \ pi sn} \, ds}$

Convolution operators as LTI systems

A convolution operator is given by a sequence f of coefficients, which acts on the discrete signal x by convolution :

{\ displaystyle x \ mapsto y: = f * x \;, \; y_ {n} = \ sum _ {k = - \ infty} ^ {\ infty} f_ {k} \ cdot x_ {nk}}

This sum is well defined in the following cases:

x is arbitrary and f is finite as a result, so that the sum is finite,
x is bounded, and f is absolutely summable ⇒ y is bounded,
x can be "summed by squares" and f has a limited frequency response ⇒ y can be "summed by squares",
x can be added up absolutely and f can be added up absolutely ⇒ y can be added up absolutely.

This means

x bounded if −K <x _n <K for some K and all n ∈ ℤ ,
x "sumable by squares " if the series of squares converges
${\ displaystyle E (x): = \ | x \ | _ {2} ^ {2}: = \ sum _ {n = - \ infty} ^ {\ infty} | x_ {n} | ^ {2} < \ infty}$ ,

f finite if there is a finite subset I of ℤ such that f _n ≠ 0 only holds for n ∈ I ,

f can be summed absolutely if the series of amounts converges

{\ displaystyle \ | f \ | _ {1}: = \ sum _ {n = - \ infty} ^ {\ infty} | f_ {n} | <\ infty}

,

f of limited frequency response when the Fourier series becomes f

${\ displaystyle {\ hat {f}} (\ xi): = \ sum _ {k = - \ infty} ^ {\ infty} f_ {k} e ^ {- ik \ xi}}$

converges almost everywhere and is (essentially) restricted.

As you can see, the impulse response of the convolution operator in all these cases is the sequence f .

For a finite filter , the set I is also called the carrier, the difference between the start and end point of the carrier is called the length of the filter . The elements of the carrier are often called taps , their number is one more than the length of the signal. Only this first, finite case corresponds to that described in the introduction. The set I defines the environment which is used to determine the filtered values, the terms of f define a linear function of the values of this environment.

The absolutely summable filter sequences f of the second case have not only a limited, but even a continuous frequency response. This is the amplitude change for the elementary vibrations with at. These are limited, therefore is defined and ${\ displaystyle e (\ omega) = (e_ {n} (\ omega): n \ in \ mathbb {Z})}$ ${\ displaystyle e_ {n} (\ omega): = \ exp (in \ omega) = \ cos (n \ omega) + i \ sin (n \ omega)}$ ${\ displaystyle f * e (\ omega)}$

{\ displaystyle [f * e (\ omega)] _ {n} = \ sum f_ {k} e_ {nk} (\ omega) = e_ {n} (\ omega) \ sum f_ {k} e ^ {- ik \ omega} = {\ hat {f}} (\ omega) e_ {n} (\ omega)}

.

Ideal frequency-selective filters only have the values 0 and 1 in their frequency response. The jumps that occur can only be approximated with difficulty with the constant frequency responses that are absolutely summable, and even worse with the polynomial frequency responses of finite filters.

For the Fourier series , which only exist in the third case (as L² functions), the relationship applies:

{\ displaystyle {\ hat {y}} (\ xi) = {\ hat {f}} (\ xi) \ cdot {\ hat {x}} (\ xi)}

.

The sum of squares E ( x ) is also known as the “energy” of the signal. Because of the Parseval identity

{\ displaystyle \ | x \ | _ {2} ^ {2} = {\ frac {1} {2 \ pi}} \ | {\ hat {x}} \ | _ {2} ^ {2}: = {\ frac {1} {2 \ pi}} \ int _ {- \ pi} ^ {\ pi} | {\ hat {x}} (\ xi) | ^ {2} \, d \ xi}

an orthogonal breakdown of the signal can be achieved using frequency-selective filters.

Finite special cases

If the carrier of the filter f finite length, so the filter as an FIR system is referred to, FIR for finite impulse response (English Finite Impulse Response ). These filters are also referred to as non-recursive or implementable without feedback .

If the carrier of the filter f no finite length, so the filter as an IIR system is referred to, IIR for infinite impulse response (English Infinite Impulse Response ). Among these there is a class of filters f , which are referred to as recursive or implementable with feedback, which can be represented as the quotient of finite filters, i.e. H. there are two finite sequences a and b such that a * f = b in the convolution product . Only such infinite filters can be implemented at all.

Advantages and disadvantages of digital filters

Digital filters play a major role in communication technology . Compared to analog filters, they have the important advantage that their technical data are precisely adhered to at all times .

advantages

no fluctuations due to the tolerance of the components
no aging of the components
no manual adjustment necessary in production, therefore faster final testing of devices
possible filter functions that are difficult or impossible to implement with analog filters, for example filters with linear phase.

Disadvantages of digital filters

limited frequency range (through periodic continuation of the spectrum)

limited range of values (through value quantization)

Due to internal rounding, truncating and limiting operations to limit the word length, digital filters show quantization noise and other non-linear effects in practice, which are particularly noticeable in recursive higher-order filters and finer quantization, use of floating point numbers, adapted filter structures such as the use of May require digital wave filters .

In the case of non-electrical input and output variables, additional effort for the conversion.

Classification of digital filters

Frequency linear filters

Based on the structure, two classes of digital filters can be distinguished:

Non-recursive filters: Filter without feedback
Recursive filters: Filter with feedback

A second distinction can be made using the impulse response:

FIR filter (Finite Impulse Response): Filter with a finite long impulse response . FIR filters usually do not contain any feedback. But there are also special FIR filter structures with feedback, one example of which are CIC filters .
IIR filter (Infinite Impulse Response): Filters with an infinitely long impulse response, these always have feedback branches.

FIR filters are basically stable, even those with recursive elements. This is due to the fact that the non-recursive forms only have zeros and trivial poles at the origin in the transfer function and the nontrivial poles of recursive forms of the FIR filter always lie on the unit circle. With regard to the stability criterion, zeros are not subject to any restriction in their position in the pole-zero diagram . If they are all within the unit circle , one speaks of a minimum-phase system; if at least one is outside, then it is a non- minimum-phase system. When designing an FIR filter, windowing is used in most cases to reduce the leakage effect .

IIR filters are only stable if all poles lie within the unit circle. If simple poles lie on the unit circle, the system is conditionally stable, i.e. H. depending on the input signal. As soon as two or more poles are on the same point of the unit circle or even only one pole is outside the unit circle, an unstable filter is present.

The advantage of IIR filters is that in the transfer function they also have poles in addition to the zeros and thus enable higher filter quality . The calculation of an IIR filter is more complex than that of an FIR filter and should also include a stability study of the quantized coefficients. The Prony method offers a reliable method for determining the coefficients of an IIR filter .

In practice, the coefficient determination is carried out with programs such as MATLAB .

Frequency distorted filters

(based on the low-pass-low-pass transformation )

A distinction between these filters is no longer possible on the basis of the impulse response.

WFIR filters warped FIR - are stable. These filters are based on an FIR filter, which is, however, frequency-distorted. They always have an infinite impulse response.
WIIR filters warped IIR - are also only stable if all poles lie within the unit circle. They also belong to the frequency-distorted filters. They cannot be implemented directly, since coefficient mapping is required to remove instantaneous loops.

Multi-rate filter

They are used to convert between different sampling rates and avoid the occurrence of mirror spectra or aliasing . Examples of multi -rate filters are CIC filters .

literature

Karl-Dirk Kammeyer, Kristian Kroschel: Digital signal processing . 6th edition. Teubner, Stuttgart 2006. ISBN 3-8351-0072-6