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Amplitude ModulationAn Optimal Filter

4 Filter Analysis and Design

Filtering, the ability to selectively suppress or enhance particular parts of a signal, is perhaps the most important tool for signal processing. Signals and Systems meets this need by providing representations of rational pole-zero filters in both the continuous (analog) and discrete (digital) domains. Routines to transform a filter from analog to discrete-time are provided, as are the most common analog filter prototypes.

While general nonlinear filters or design tools for multidimensional filters are not yet directly supported by these routines, there are some special routines for designing two-dimensional decimation systems. They do not yet assist in the design of the filter itself, but they can produce optimal decimation systems for a given passband specification with rectangular sampling.

As per usual convention, discrete-time filter design is also referred to as digital filter design in this chapter and in Signals and Systems in general.

4.1 Analog Filter Design

The most common representation for a rational filter is in terms of the poles and zeros of the system function. Signals and Systems employs this in the AnalogFilter object.

Representing analog filters.

The AnalogFilter object is the basic object in this application for representing an analog filter. Other tools are described later on that allow you to derive an analog filter in terms of standard design classes (e.g., Bessel filters, elliptic filters, etc.). The filter object acts as a Signals and Systems operator; to determine the impulse response, you apply the filter to an impulse, and use the analysis tools to convert the system into a functional or graphical form.

First, make the functions available for use.

In[1]:=Needs["SignalProcessing`"]

Here is a stable analog filter. Remember that the poles of the filter's transfer function are given first, then the zeros.

In[2]:=filt = AnalogFilter[
{-.3, -.2 + .7 I, -.2 - .7 I},
{0, .2, -.2 },
t
];

The filter object normally acts in the same fashion as other operators. To determine the response of the filter to a signal, the filter object should be applied to the signal and evaluated. Here is the impulse response of our sample filter.

In[3]:=EvaluateOperators[
filt[DiracDelta[t]]
]

Out[3]=

We can plot the computed response.

In[4]:=MagnitudePhasePlot[
%, {t, 0, 20}
];

Unlike other operators, certain functions can act solely on the filter object. For instance, here is a Laplace transform of the object, which returns the system transfer function with information about the region of convergence.

In[5]:=LaplaceTransform[filt, t, s]

Out[5]=

A pole-zero plot can be generated from the transform.

In[6]:=PoleZeroPlot[%]

Out[6]=

All of the filter objects can be used to represent multidimensional filters. At this point in time, computations and design procedures based on multidimensional filters are not supported, however. To represent an -dimensional pole-zero filter, the lists of poles and zeros become matrices, while the time variable becomes a list of variables. This may be useful for custom system rewrite code, or for export to Ptolemy (see Chapter 7.2).

A number of tools are provided for filter design. They first require that you specify the characteristics of the filter.

Objects for specifying the magnitude response of a filter.

The FilterSpecification object represents the magnitude response of a filter that is being designed. It is given in terms of Passband and Stopband objects. The passbands and stopbands take as arguments the allowable variation in magnitude and the frequency range over which the band is defined (in radians/second).

Here is a specification for a lowpass analog filter. Note in particular that the final stopband extends to Infinity.

In[7]:=spec = FilterSpecification[
Passband[0.1, {0, 8000}],
Stopband[0.18, {12000, Infinity}]
];

Here is one way to visualize the specification. Note that the Infinity needed to be transformed to a finite value so that a nice graph could be generated.

In[8]:=graph = Show[Graphics[
{GrayLevel[0.8],
Map[If[Head[#] === Stopband,
Rectangle[{#[[2,1]], 0},
{#[[2,2]], #[[1]]}
],
Rectangle[{#[[2,1]], 1 - #[[1]]},
{#[[2,2]], 1}
]
]&,
List @@ spec/.Infinity -> 20000]},
Frame -> True, PlotRange -> All
]]

Out[8]=

You can specify lowpass, highpass, bandpass, and bandstop filters with the FilterSpecification object. General multiband filters can be given in principle, but the current design routines do not handle those cases.

Be aware that the FilterSpecification object has the attribute Orderless. This means that Mathematica sorts the arguments in canonical (alphanumeric) order. They will not be arranged by frequency range. Remember this particularly when writing programs based on the FilterSpecification object.

This bandpass specification demonstrates the automatic reordering of FilterSpecification arguments.

In[9]:=FilterSpecification[
Stopband[0.2, {0, 4000}],
Passband[0.14, {6000, 8000}],
Stopband[0.35, {9000, Infinity}]
]

Out[9]=

The filter specification is passed to a design procedure to create a filter object. Currently, the only analog design function is DesignAnalogFilter, which is used to create standard analog filter types. The resulting analog filters may be used as prototypes for digital filter design.

Designing an analog filter.

DesignAnalogFilter enables automated design of a filter from an input filter specification and a given filter type. If the filter paramaters are for a type of filter other than lowpass, the specification will first be transformed to a lowpass type, then the resulting filter will be inversely processed by transformation of the poles and zeros (see AnalogFilterTransformation, below).

Here is a lowpass elliptic filter designed with the specification we entered above. The result is a third-order filter.

In[10]:=filt = DesignAnalogFilter[
Elliptic, t, spec
]

Out[10]=

We can determine the frequency response of our new filter.

In[11]:=fresp = FourierTransform[filt, t, w]

Out[11]=

The equation may be less than completely informative. It is often useful to visualize the result. We compare the magnitude response here with the graph of the specification generated earlier. Remember that the frequency is given in radians per second!

In[12]:=Show[ graph,
First[MagnitudePhasePlot[
fresp, {w, 0, 20000},
MagnitudeScale -> Linear,
DisplayFunction -> Identity
]]
]

Out[12]=

Allowed types of filters for analog filter design.

The usual classical analog infinite impulse response filter types are used by DesignAnalogFilter. Both the Bessel and Butterworth filters have a smooth magnitude response that decreases monotonically with frequency. (The Bessel type of filter has a nearly constant group delay throughout the passband, giving it low overshoot in its step response; but this means that it does not have the linear phase response favored in digital filter design.) The type I Chebyshev filter (ChebyshevI) is equiripple in the passband, while the type II, or inverse, Chebysev filter (ChebyshevII) is equiripple in the stopband. The Elliptic filter is equiripple in both the passband and the stopband.

Here is a specification for a highpass filter, where the stopband has an attenuation of at least 10 decibels at 60 hertz, while the passband is at worst 2 decibels down at 100 hertz.

In[13]:=hispec = FilterSpecification[
Stopband[ 10^(-10/20), {0, 2 Pi 60}],
Passband[ 1 - 10^(-2/20), {2 Pi 100, Infinity}]
]//N

Out[13]=

When numeric computation with complex numbers is performed, rounding errors can cause small complex values to appear. If you do not expect small values in your output, you may eliminate them by judicious use of Chop.

This is a type I Chebyshev filter derived from the specification.

In[14]:=DesignAnalogFilter[ChebyshevI, t, hispec]//Chop

Out[14]=

Here is the frequency response of the filter.

In[15]:=FourierTransform[%, t, w]

Out[15]=

We can check that the response has the desired characteristics. Here, we verify that the stopband attenuation is at least 10 decibels.

In[16]:=20 Log[10, Abs[Normal[%]/.w -> 2 Pi 60]]//N

Out[16]=

Option to DesignAnalogFilter.

In some cases, the order of a filter that meets the given specification cannot be determined automatically by DesignAnalogFilter, or you may wish to design the filter with a particular order. In these cases, you can explicitly specify an order for the filter by the option FilterOrder. The order should be a positive integer. (The default value is the setting Automatic, which tells DesignAnalogFilter to use the standard algorithms for determining the minimum filter order which meets the specification.)

Using the lowpass filter specification employed in the first example, we attempt to design a Bessel filter. The filter order could not be determined automatically, so a twelfth-order filter was used instead.

In[17]:=DesignAnalogFilter[Bessel, t, spec]

Out[17]=

We can explicitly request that a fifth-order filter be used instead.

In[18]:=DesignAnalogFilter[Bessel, t, spec,
FilterOrder -> 5
]

Out[18]=

The Bessel filters are the most likely to have the problem of being unable to automatically determine the filter order. The algorithm employed involves an iterative search that can be quite time consuming; it automatically terminates at the twelfth order to reduce unnecessary computation. High-order Bessel filters can be somewhat unstable numerically, so it is often ineffective to employ filters above the twelfth order. It is still possible to request a higher order by way of the FilterOrder option, however.

Since Mathematica is a symbolic system, purely symbolic filter designs are also possible.

Here is a symbolic design for a lowpass Butterworth filter. The message that is generated is harmless in this context. Because the exact list of roots cannot be determined with a symbolic order parameter, Table is called with a symbolic range. If you substitute a numeric parameter for the symbolic iteration limit, the Table will be able to evaluate.

In[19]:=DesignAnalogFilter[
Butterworth,
t,
FilterSpecification[
Passband[pd, {0, wpc}],
Stopband[sd, {wsc, Infinity}]
],
FilterOrder -> a
]

Out[19]=

Symbolic design can lead to complex expressions, but it allows a more flexible approach to varying the design of a filter.

Information derived from a filter specification.

The automated filter order determination can be performed as a separate step. Enter the filter type and specification in the same fashion as you would when using the filter design function.

Here we determine what order of a type II Chebyshev filter would be needed to meet the lowpass filter specification we defined in the first example.

In[20]:=AnalogFilterOrder[ChebyshevII, spec]

Out[20]=

A filter specification is often described in terms of several other parameters. Some of these parameters can be derived from a FilterSpecification object by the AnalogFilterParameters function. In particular, it returns a list , which consists of the normalized passband ripple , the normalized stopband ripple , the percent deviation from a constant passband response , the stopband attenuation (equivalent to ), the stopband attenuation in decibels , the ripple control parameter (found in Chebyshev and elliptic filters), the ratio between the transition frequencies , and finally the parameter used in computing filter orders. These eight values will always be returned in the order listed.

Here are the filter parameters for the highpass filter defined previously.

In[21]:=AnalogFilterParameters[hispec]

Out[21]=

Frequency transformations of an analog filter object.

LowpassPrototypeSpecification takes a filter specification object and converts it to the design lowpass filter specification. This routine is used internally by DesignAnalogFilter, so it usually is not necessary to do this by hand.

Here is the lowpass specification used in the design of the highpass filter defined earlier.

In[22]:=LowpassPrototypeSpecification[hispec]

Out[22]=

The AnalogFilterTransformation function performs arbitrary mappings of the poles and zeros of a filter object. These mappings are used to convert a lowpass filter to the requested highpass, bandpass, bandstop, etc. filter by DesignAnalogFilter. However, an arbitrary mapping can be employed. The first argument is the filter, while the second is given in the form of a transformation rule. The rule should be a mapping of the state-space variable to some other expression (any symbol can be used for the state-space variable).

Some common mappings for use with AnalogFilterTransformation.

The parameters involved in the transformation take on different meanings depending on the design procedure involved.

Here is a normalized third-order lowpass Butterworth filter. Note that the poles are the roots of the third-order Butterworth polynomial. We will use this filter to design a highpass filter.

In[23]:=filt = AnalogFilter[
GetRootList[(s + 1)(s^2 + s + 1), s],
{},
t
]

Out[23]=

Here is a highpass filter with a decibel passband cutoff at 100 hertz.

In[24]:=AnalogFilterTransformation[
filt,
s -> (2 Pi 100)/s
]//N//Chop

Out[24]=

We can check the resulting frequency response at the cutoff frequency.

In[25]:=20 Log[10, Abs[
Normal[FourierTransform[%, t, w]]/.
w -> 2 Pi 100//N
]]

Out[25]=

Finding the DC gain.

The DC gain (gain at constant input) of a filter can be found by the DCGain function. It accepts an analog or digital filter object as input.

Here is the DC gain of an elliptic filter designed from the highpass specification used in previous examples.

In[26]:=DCGain[
DesignAnalogFilter[
Elliptic, t, hispec
]
]

Out[26]=

Representing an analog tapped delay line.

The closest thing to a finite impulse response (FIR) filter in the analog domain is the tapped delay line. Its syntax is similar to the AnalogFilter object. In this case, the first argument is the list of tap values and the second argument is the delay between the taps.

Here is a tapped delay line.

In[27]:=dline = TappedDelayLine[
{1, -0.2, 0.3, -0.4},
0.5, t
];

The impulse response can be found in the same fashion as that used for an analog filter object.

In[28]:=EvaluateOperators[
dline[DiracDelta[t]]
]

Out[28]=

We can plot the impulse response.

In[29]:=SignalPlot[%, {t, 0, 2.5}]

Out[29]=

4.2 Digital Filter Design

Signals and Systems naturally provides representations for working with filters in the discrete-time domain. In particular, general pole-zero filters (usually infinite impulse response (IIR)) can be manipulated by way of DigitalFilter, while all-zero finite impulse response (FIR) filters are represented with DigitalFIRFilter.

Representing a digital IIR filter.

The DigitalFilter object takes a list of poles and a list of zeros of the transfer function to specify a particular digital filter. This notation is particularly useful for design techniques based on pole-zero placement. The DigitalFilter object is a Signals and Systems operator. Like AnalogFilter, it is used in operator form to determine the output of a signal passing through the filter. The filter object can be passed directly to transform functions to find out about various filter characteristics, such as the transfer function or the frequency response.

This is a third-order digital IIR filter.

In[30]:=filt = DigitalFilter[
{0.2 + 0.3 I, 0.2 - 0.3 I, 0.3},
{-0.2, 0.2},
n
];

The impulse response can be determined by passing a discrete impulse through the filter, and forcing evaluation of the operator.

In[31]:=EvaluateOperators[
filt[DiscreteDelta[n]]
]

Out[31]=

We can plot the impulse response.

In[32]:=DiscreteSignalPlot[%, {n, 0, 10}]

Out[32]=

Note that with the current design, care must be taken when using the just described syntax. If filt contains a gain term, the impulse would not be attached to the filter object itself (since filt will then be a product of the gain term and the filter object). You can instead use various manipulation functions to place the impulse; one possibility is

EvaluateOperators[filt/.d:DigitalFilter[___] :> d[DiscreteDelta[n]]]

Of course, the appropriate filter type and time-domain variable should be used.

The system transfer function can be determined by taking the Z transform of the filter object. Here, a pole-zero plot is generated from the transfer function.

In[33]:=PoleZeroPlot[
ZTransform[filt, n, z]
]

Out[33]=

The frequency response can also be found by transforming the filter.

In[34]:=DiscreteTimeFourierTransform[filt, n, w]

Out[34]=

Here is a plot of the frequency response.

In[35]:=MagnitudePhasePlot[%, {w, 0, Pi}];

Methods for converting an analog prototype filter into a digital filter.

Classical design techniques for digital IIR filters involve prototyping the filter with an analog design, then transforming it to a digital filter. The functions ImpulseInvariance and BilinearTransformation perform the conversion of analog filters to digital filters.

In signal processing applications, design by impulse invariance usually works directly from a digital specification. In Signals and Systems, however, the design is implemented as a transformation of an analog filter object. Hence, the design sampling period must be input to the transformation routine. (Note that is not necessarily the frequency at which the filter operates.) When an analog filter is being designed specifically for a digital application, it is useful to specify the corner frequencies in the normalized digital frequency space (between and ). If this is done, you can set to without any problems.

Be aware that this procedure is only appropriate for bandlimited prototypes. Highpass and bandstop filters, for instance, will show severe aliasing when designed by impulse invariance.

This is a lowpass Butterworth filter in which the passband ends at decibels and the stopband begins at decibels attenuation. The frequency ranges are for the normalized digital frequencies.

In[36]:=anfilt = DesignAnalogFilter[Butterworth, t,
FilterSpecification[
Passband[1 - 10^(-1/20), {0, .2 Pi}],
Stopband[10^(-15/20), {0.3 Pi, Infinity}]
]
]//N

Out[36]=

Here is a digital filter designed by the method of impulse invariance.

In[37]:=dfilt = ImpulseInvariance[anfilt, 1, t, n]

Out[37]=

The magnitude of the frequency response of the digital filter is easily displayed. In this case, the analog filter was reasonably bandlimited, so the potential aliasing effects of impulse invariance are not noticeable.

In[38]:=MagnitudePhasePlot[
DiscreteTimeFourierTransform[
dfilt, n, w
],
{w, 0, Pi},
PhaseScale -> None,
MagnitudeScale -> Log,
Frame -> True, Axes -> False
]

Out[38]=

To compensate for aliasing effects, the design procedure based on the bilinear transform is also available. This transformation maps the entire continuous-time frequency axis to the unit circle, avoiding the aliasing problem. It will, however, introduce warping of the frequency response, unlike impulse invariance, which is a linear transformation. To counteract the distortion, prewarping of the analog frequency specification should be performed by where is the digital frequency.

Here is a digital filter derived from the analog filter for which prewarping of the frequencies was not done.

In[39]:=BilinearTransformation[anfilt, 1, t, n]

Out[39]=

We see that the magnitude response doesn't match our desired output, where the passband would end at .

In[40]:=MagnitudePhasePlot[
DiscreteTimeFourierTransform[
%, n, w
],
{w, 0, Pi},
MagnitudeScale -> Linear,
PhaseScale -> None
]

Out[40]=

Here, we redesign the analog filter with prewarped frequency bands.

In[41]:=DesignAnalogFilter[Butterworth, t,
FilterSpecification[
Passband[1 - 10^(-1/20),
{0, Tan[.2 Pi/2]}],
Stopband[10^(-15/20),
{Tan[0.3 Pi/2], Infinity}]
]
]//N

Out[41]=

This is the new digital filter derived from the redesigned analog filter.

In[42]:=bilinfilt =
BilinearTransformation[%, 1, t, n]

Out[42]=

This output matches what we want. A close check at the corner frequencies will reveal that the passband is better than the specification, while the stopband design is almost exactly met.

In[43]:=MagnitudePhasePlot[
DiscreteTimeFourierTransform[
bilinfilt, n, w
],
{w, 0, Pi},
MagnitudeScale -> Linear,
PhaseScale -> None
]

Out[43]=

The gain term in the bilinear transform can also be set for specific purposes, such as removing a pole from the transfer function. See the section of this document entitled Interlude-A Quick Start for more information.

Representing a digital FIR filter.

A discrete-time all-zero finite impulse response filter is often designed in terms of its impulse response, which is equivalent to the sequence of coefficients of its transfer function. The DigitalFIRFilter object can be used when this is the case.

Perhaps the simplest technique for designing an FIR filter is to sample the ideal frequency response. The most common expression of the ideal response is for it to be in the passband and elsewhere. Remembering the symmetry of the discrete-time Fourier transform, we can write the ideal response as a sum of pulses. The samples can be multiplied by an additional exponential term to generate a linear phase response.

This small function will generate the ideal response for a linear phase FIR lowpass filter, with the normalized cutoff frequency given as the first argument in radians per second, and the filter length given as the second argument.

In[44]:=idealResponse[cutoff_, length_] :=
Drop[Table[ Exp[-I t (length - 1)/2]
(ContinuousPulse[2 cutoff, t + cutoff] +
ContinuousPulse[2 cutoff,
t - 2 Pi + cutoff]),
{t, 0, 2 Pi, 2 Pi/length}], -1]//N

We can conveniently use the special vector input form of the InverseDiscreteFourierTransform function.

Here is the truncated impulse response of the ideal filter. This is equivalent to the coefficients of the filter's transfer function.

In[45]:=impresp = InverseDiscreteFourierTransform[
idealResponse[0.4 Pi, 17]
]//Chop

Out[45]=

An FIR filter object is easily constructed from the impulse response given as a vector.

In[46]:=dfilt = DigitalFIRFilter[impresp, n];

We see several familiar phenomena, such as the Gibb's effect from the simple truncation of the response and the linear phase resulting from the exponential term in the response.

In[47]:=MagnitudePhasePlot[
DiscreteTimeFourierTransform[dfilt, n, w],
{w, 0, Pi}
];

Note that the response of the filter exactly interpolates the sample points, marked by the grid lines.

In[48]:=Show[First[%],
Frame-> True,
Axes -> False,
GridLines -> {Range[0, 2 Pi, 2 Pi/17], {0,1}}
]

Out[48]=

Here is a pole-zero plot of the filter's transfer function.

In[49]:=PoleZeroPlot[ZTransform[
dfilt, n, z
]]

Out[49]=

The customary way of improving the frequency response of a filter designed by sampling the ideal response is to window the impulse response. This can be done in a variety of ways, but given the technique just used for generating the impulse response, the best method is to sample the windowing function, and multiply the impulse response vector by the sampled window.

This samples a discrete Hamming window.

In[50]:=window = N[Table[
DiscreteWindow[Hamming, 17, n],
{n, 0, 16}
]];

The filter object is created by multiplying the window vector and the impulse response computed above.

In[51]:=dfilt2 = DigitalFIRFilter[impresp window, n];

Here is the frequency response of the windowed filter.

In[52]:=MagnitudePhasePlot[
DiscreteTimeFourierTransform[dfilt2, n, w],
{w, 0, Pi}
];

4.3 Design of Two-Dimensional Decimation Systems

In some applications, such as seismic and video signal processing, it is useful to resample a signal close to its Nyquist rate to reduce the amount of information needed to represent the signal. The resampling system, also known as a decimation system, is fairly straightforward to design in one dimension, but becomes much more complex in higher dimensions. Signals and Systems provides some tools for decimation system design in two dimensions. These are currently limited to determining the resampling matrices for optimal resampling on a rectangular grid; determination of the lowpass filter is not yet supported. Also, functions for examining possible aliasing in two dimensions can assist in the design process.

Functions for assisting in the design of two-dimensional decimation systems.

A decimation system is defined as a cascade of a phase shifter (which centers the passband about the origin), an upsampler, a lowpass filter, and a downsampler. The passband is specified as a two-dimensional polygon in the frequency domain, normalized to the fundamental frequency tile, using the syntax of the Mathematica Polygon graphics primitive. The decimator design will be based on circumscribing the convex hull of the passband with a minimal parallelogram. The parallelogram will be aligned on the grid points of the fundamental frequency tile, determined by the GridPoints option. A set of resampling matrices and the shift vector that can be used to map the parallelogram to the fundamental frequency tile will be returned. Alternately, a cascade of Signals and Systems operators that perform this computation can be generated.

Here is a polygon specifying a particular passband.

In[53]:=poly = Polygon[{{-1.95, -1.94},
{-1.01, -1.64}, {-0.29, -1.05},
{0.29, -0.47}, {0.99, 0.47},
{1.54, 1.16}, {1.92, 2.07},
{1.20, 1.97}, {0.57, 1.66},
{0.07, 1.16}, {-0.51, 0.66},
{-1.17, -0.06}, {-1.70, -0.93}}
];

Here is the decimation system determined to best sample this passband given a rectangular sampling grid and the current setting for the GridPoints option.

In[54]:=DesignDecimationSystem2D[poly]

Out[54]=

This is the system expressed as operators.

In[55]:=DesignDecimationSystem2D[
poly, {n1, n2}
]

Out[55]=

Options for DesignDecimationSystem2D.

A good deal of useful information can be determined during this design process, such as the theoretical best and the actual achieved compression ratios, assuming an initial rectangular sampling grid. Visualization of the design process can also be performed. This information can be generated in the form of a report by use of the Justification option.

Here is a report of the design procedure.

In[56]:=DesignDecimationSystem2D[
poly, Justification -> All
]

Out[56]=

The GridPoints option specifies the grid superimposed on the fundamental frequency tile. If given as a single integer n, the tile is sampled by 2 n + 1 points in each dimension; if given as a pair of integers and the grid has 2 + 1 points on the horizontal axis and 2 + 1 points on the vertical axis.

Note how the resampling changes when GridPoints is changed from the default of to .

In[57]:=DesignDecimationSystem2D[
poly, Justification -> All,
GridPoints -> 7
]

Out[57]=

The DownsamplingAliasing function analyzes possible aliasing in a downsampling operation. Downsampling in two dimensions produces images of the original baseband in the fundamental frequency tile. If these images overlap, aliasing occurs. The DownsamplingAliasing function accepts the downsampling matrix as the first argument, and the baseband polygon (or list of polygons) as the second argument. It returns a list of the image bands produced in the downsampling process. With the default option of Justification -> All an informative report, including visualization of the possible aliasing, is generated. You can also supply options controlling the graphics of the fundamental frequency tile.

This is an example of quincunx downsampling that creates aliasing.

In[58]:=DownsamplingAliasing[
{{1, 1}, {2, -1}},
Polygon[{{-Pi, 0}, {0, Pi}, {Pi,0}, {0,-Pi}}]
]

Out[58]=

Amplitude ModulationAn Optimal Filter



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