Designing a Digital Filter

This chapter describes the steps involved in designing a digital filter. It also includes reference and background information on the available options. The two basic steps for designing a digital filter using Digital Filter Designer are:

1. Define the filter design specifications.

This step consists of defining the filter type, design method, and the filter order.

2. Define the filter response specifications.

Choosing a Filter Response Type


Choose either an Infinite Impulse Response (IIR) filter or a Finite Impulse Response (FIR) filter based upon your performance, design, and implementation needs and limitations. For example, when linearity of phase is an issue, an FIR filter is a better choice because an IIR filter achieves its computational efficiency at the cost of nonlinear phase.

FIR

A Finite Impulse Response (FIR) type is an all-zero or moving average (MA) filter that uses past and present input samples to calculate the output value.

In general the FIR design problem is easier to control in a wide range of practical situations. An FIR filter is able to provide pure linear-phase characteristics. It also can accurately approximate arbitrary frequency-response characteristics.

Even though a non-recursive design typically needs a large number of coefficients, it is inherently stable because it does not involve feedback.

The following illustration depicts the difference equation and implementation structure for a transposed FIR filter.

IIR

An Infinite Impulse Response (IIR) type is a pole and zero or auto-regressive, moving average (ARMA) filter. Each output value in such a filter is calculated using previous outputs, as well as past and present input samples. The feedback in this system gives this filter an impulse response that is infinite in duration.

Simplicity of its design procedure is a major advantage of an IIR filter. A variety of classical frequency-selective filters can be designed using closed-form formulas. Another major advantage of an IIR filter is its computational economy with respect to the filter implementation. Filter responses with high frequency selectivity can be obtained using a few recursive coefficients.

An IIR filter is generally unable to provide a linear phase response and its implementation is relatively more complex. However, when phase considerations are not an issue, a given magnitude response specification can be implemented more efficiently with an IIR filter.

The following illustration depicts the difference equation and structure for an IIR filter.

Choosing a Frequency Response

The output spectrum of a linear, time-invariant filter is a result of the product of the input signal spectrum and the filter's frequency response. Specifying the frequency response of a filter thus determines which frequency bands of the input signal will be passed and which will be rejected.

Note: An ideal filter would do a perfect job of either passing or attenuating a region of the spectrum. Actual filters, however, have passbands that are not perfectly flat, stopbands that don't reject bands of frequencies completely, and transitions between passbands and stopbands that are not instantaneous.

Choose a frequency response to specify the desired passband and stopband response. In addition to the basic filter responses, more advanced filter responses are also supported within Digital Filter Designer. Any one of the following responses can be specified using Digital Filter Designer. The following table lists the amplitude responses Digital Filter Designer currently supports for FIR and IIR filters.

Amplitude Response FIR Filter IIR Filter
Lowpass †¢ †¢
Highpass †¢ †¢
Bandpass †¢ †¢
Bandstop †¢ †¢
Multipass †¢ †¢
Multistop †¢ †¢
Differentiator †¢
Hilbert Transformer †¢
Raised Cosine †¢
Root Raised Cosine †¢
Gaussian †¢
Edge †¢

Lowpass

A lowpass frequency response can be achieved with either an FIR or an IIR filter. A lowpass filter is made up of a passband and a stopband, where the lower frequencies of the input signal are passed through while the higher frequencies are attenuated. An idealized lowpass filter frequency response has the following shape.

Highpass

A highpass frequency response can be achieved with either an FIR or an IIR filter. A highpass filter is made up of a stopband and a passband where the lower frequencies of the input signal are attenuated while the higher frequencies are passed. An idealized highpass filter frequency response has the following shape.

Bandpass

A bandpass frequency response can be achieved with either an FIR or an IIR filter. A bandpass filter is made up of two stopbands and one passband so that the lower and higher frequencies of the input signal are attenuated while the intervening frequencies are passed. An idealized bandpass filter frequency response has the following shape.

Bandstop

A bandstop frequency response can be achieved with either an FIR or an IIR filter. A bandstop filter is made up of two passbands and one stopband so that the lower and higher frequencies of the input signal are passed while the intervening frequencies are attenuated. An idealized bandstop filter frequency response has the following shape.


Multipass

A multipass frequency response can be achieved with either an FIR or an IIR filter. A multipass filter begins with a stopband followed by more than one passband. By default, a multipass filter in Digital Filter Designer consists of three passbands and four stopbands. The frequencies of the input signal at the stopbands are attenuated while those at the passbands are passed. An idealized multipass filter frequency response has the following shape.

Multistop

A multistop frequency response can be achieved with either an FIR or an IIR filter. A multistop filter begins with a passband followed by more than one stopband. By default, a multistop filter in Digital Filter Designer consists of three passbands and two stopbands. The frequencies of the input signal at the passbands are passed while those at the stopbands are attenuated. An idealized multistop filter frequency response has the following shape.

Differentiator

A differentiator frequency response can be achieved only with an FIR filter. An ideal differentiator provides a purely imaginary, linear response across the entire spectrum with a slope of . An idealized differentiator filter frequency response has the following shape.

Hilbert Transformer

A Hilbert transformer frequency response can be achieved only from an FIR filter. An ideal Hilbert transformer filter provides a purely imaginary constant linear response across the entire spectrum.

Raised Cosine

A raised cosine frequency response can be achieved only from an FIR filter. A raised cosine filter provides a flat passband with a roll-off that has a sinusoidal form. This type of a filter is typically used in the design of pulse-amplitude modulation (PAM) digital communication systems. When used within such a system, it achieves a minimum probability of error by reducing the combined effects of intersymbol interference and noise.

Root Raised Cosine

A root raised cosine frequency response can be achieved only from an FIR filter. Typically, two root raised cosine filters are used together within a pulse-amplitude modulation (PAM) digital communication system, one in the transmitter and other in the receiver. When combined in this manner, they achieve the raised cosine characteristics.

Gaussian and Edge

Gaussian and Edge frequency responses can also be achieved for an FIR filter using AEL functions to generate their ideal coefficients. For details on generating these frequency responses, refer to Importing an AEL Ideal Coefficients Function. For details on using AEL to create functions that generate ideal coefficients for these frequency responses, refer to the Manipulating Arrays.

Note An AEL template that can be customized is included with the current installation of Digital Filter Designer in $HPEESOF_DIR/dfilter/function . You can modify this template to design symmetric FIR filters according to GSM specifications using the desired rolloff, symbol period, and tolerable order values. For details on using an AEL functions file to generate filter coefficients, refer to Importing an AEL Ideal Coefficients Function".

Choosing a Design Method

Digital Filter Designer's filter design methods include the classic algorithms for FIR and IIR filter design. Each method offers its own way of approximating the desired response, where, depending upon the purpose of the filter, one method may be better suited for a particular design.

FIR

The techniques for designing FIR filters are based on directly approximating the desired frequency response of the discrete-time system. For an FIR filter, Equiripple and Least Squares are among the most flexible methods. In addition, numerous windowing methods that truncate the ideal impulse response using a finite-length window are also available. The simplicity of windowing makes it an attractive alternative to the other more complicated optimizations.

Equiripple

The Equiripple method approaches filter design as an optimization problem in which the coefficient values are adjusted to create an optimal filter with ripples that are of equal height. This method uses the most efficient optimization procedure to minimize the transition width along with the stopband and passband ripple.

Hint Modify the filter order if an optimization ever fails to converge. Although rare, such a situation may occur occasionally in the design of a high-order filter.

The Equiripple method uses the Parks-McClellan algorithm to compute the filter such that its response
represents the best approximation to the ideal frequency response
in a manner that minimizes the maximum weighted approximation error (where Q() is the weighting function):

Note Specify a weighting function for the Least Squares method (and for the Equiripple method when the Auto Order feature is not used) to assign an appropriate relative cost for the deviation from a desired response. Frequency regions where more accurate approximations are required need larger weighting values than the less critical regions. A zero weighting is typically specified in the transitional frequency band between a passband and a stopband.

Least Squares

The Least Squares method computes the filter such that its response
represents the best approximation to the ideal frequency response
in a weighted least-squares sense (where Q() is the weighting function):

Windowing

Windowing is the quickest method for designing an FIR filter. It begins with an ideal desired frequency response that can be expressed in terms of the corresponding impulse response. A windowing function simply truncates the ideal impulse response to obtain a causal FIR approximation that is noncausal and infinitely long. Smoother window functions provide higher out-of band rejection in the filter response. However this smoothness comes at the cost of wider stopband transitions.

Selecting the appropriate windowing method for a particular application involves weighing trade-offs between ripple and transition width to achieve acceptable results. For the most part, a good windowing method attempts to minimize the width of the main lobe (peak) of the frequency response. In addition, it attempts to minimize the side lobes (ripple) of the frequency response. Some of the windowing methods can be used to make this trade-off by adjusting their windowing parameter.

Rectangular This is the most basic of windowing methods. It does not require any operations because its values are either 1 or 0. It creates an abrupt discontinuity that results in sharp roll-offs but large ripples.



The Rectangular windowing method is defined by the following equation:

Triangular This is the simplest windowing method that exhibits a nonnegative transform. The computational simplicity of this window, a simple convolution of two rectangle windows, and the lower sidelobes make it a viable alternative to the rectangular window.

The Triangular windowing method is defined by the following equation:

Kaiser This windowing method is designed to generate a sharp central peak. Although it is computationally involved, its coefficients are easy to generate largely as a result of helpful design rules that eliminate iterations typically needed to achieve desired ripple and transition specifications.

The Kaiser windowing method is defined by the following equation using the windowing parameter, α:

Note For details on the Kaiser windowing method, refer to J. F. Kaiser, "Design Methods for Sampled Data Filters," Proceedings First Allerton Conference on Circuit and System Theory, 221-236, November, 1963.

Hamming This windowing method generates a moderately sharp central peak. Its ability to generate a maximally flat response makes it convenient for speech processing filtering.

The Hamming windowing method is defined by the following equation using the windowing parameter, α:

Hanning This windowing method generates maximally flat filter designs as well. Its other advantages are an easy to generate set of coefficients and easily identified properties of the transform of the cosine function.

The Hanning windowing method is defined by the following equation using the windowing parameter, α:

Dolph Chebyshev This windowing method provides good sidelobe rejection but its sidelobe structure displays extreme sensitivity to coefficient errors. It is optimal for most instances that don't use a fixed-point implementation.

Note For details on the equations used by the Dolph Chebyshev windowing function, refer to "Digital Signal Processing" by Roberts & Mullis. pp 225-228.

Blackman This windowing method provides good sidelobe rejection although it results in a broad central peak.


The Blackman windowing method is defined by the following equation:

IIR

IIR filter design is based on transformations of continuous-time IIR systems into discrete-time IIR systems. For an IIR filter type, Butterworth and Chebyshev are among the better known design methods. In addition to the auto-selection option that picks the most suitable method automatically, the Elliptic method is also available.

Auto Selection

Choose this option to pick the most appropriate method automatically. This option evaluates the filter design using each of the available methods and picks the method that results in a filter with the smallest order. Typically this choice is used to obtain the most economical filter design that may be further refined using a different method, if desired.

Chebyshev I

The Chebyshev method of approximation provides an equiripple performance in the passband and it varies monotonically in the stopband. Although an increase in the filter order increases the number of passband ripples, it also leads to better stopband performance. In addition, the good magnitude characteristics of this method come at the cost of phase characteristics that depart considerably from the ideal linear-phase.

An nth order Chebyshev I lowpass filter with cutoff frequency
is specified using the following equation:

Chebyshev II

The Chebyshev II or Inverse Chebyshev method of approximation provides an equiripple performance in the stopband and it is monotonic in the passband. Although an increase in the filter order increases the number of stopband ripples, it also leads to better passband performance because it yields the smallest delay in the passband. In addition, the Chebyshev II method provides good phase and delay response by providing the widest region of the passband over which the group delay is constant.

An nth order Chebyshev II lowpass filter with cutoff frequency
is specified using the following equation:

Butterworth

The Butterworth method of approximation provides a monotonic performance in the passband and the stopband. As the filter order is increased, the passband and stopband performance improves, with the transition from passband to stopband becoming sharper. The Butterworth method is best suited for design problems that focus on controlling passband and stopband ripple.

An nth order Butterworth lowpass filter with cutoff frequency
is specified using the following equation:

Elliptic

The Elliptic method of approximation provides an equiripple response in both the passband and the stopband. This method uses the smallest filter order for a transition from passband to stopband, but it does so at the cost of some ripple in both bands. As a result, when phase linearity is not an issue, the elliptic method results in a filter with the lowest-order and therefore the least computation overhead.

An Elliptic filter is based on a mapping of the following function:

Defining the Filter Order


The most apparent measure of the complexity of a filter is its order. The order refers to the order of the filter's system function. Within Digital Filter Designer, the design process can be driven either by the filter order or by the filter specifications.

That is, given the ripple specifications, the minimum filter order can be calculated automatically in all cases except the Least Squares method. On the other hand, given a filter order constraint, the filter coefficients can also be generated without constraining the ripple.

For a FIR filter the order is defined as the number of zeros of its transfer function H(z), which is a finite polynomial. The number of filter taps (coefficients) will be the order+1.
For an IIR filter the order is defined as the number of poles of its transfer function H(z), which is a rational function.

Digital Filter Designer may be unable to achieve the desired response using certain window design methods, regardless of the filter order. In such cases, a feedback message is displayed in the Status window to let you know.

Be sure you verify the accuracy of the results obtained using the auto-order feature when designing a filter with a nonlinear passband amplitude, a raised cosine filter for example.
For best results in a filter design with passbands at the Nyquist frequency (frequency = sampling frequency/2) use an odd numbered filter order for an antisymmetric filter, and an even numbered filter order for a symmetric filter. For details on the symmetricity option used in a filter design, refer to Defining the Symmetricity.

Choosing the Frequency Unit

Choose a frequency unit for the specifications you enter. The default unit is MHz.

Specifying the Sampling Frequency

Enter the desired sampling frequency. The default frequency is 1 MHz. The sampling frequency you enter is the rate at which the signal is sampled ( ). The high frequency, FH, in the last band is automatically changed to
to avoid designing a filter that suffers from aliasing distortion.

Defining the Response Specifications

Desired response specifications define the parameters for the passbands, stopbands, and weightbands of a digital filter. Response specifications include the number of bands in the case of a multiband filter, the low and high cutoff frequencies, the amplitude, ripple or attenuation, and phase.

Entering the Number of Bands


The number of bands you enter defines the shape of the filter. Within Digital Filter Designer you can specify the number of passbands, stopbands, and weightbands. These values need to be entered for multipass and multistop filters; they are otherwise automatically set to the appropriate values based upon the specified amplitude response.

Entering Low and High Frequencies

Define the acceptable cutoffs for the passbands, stopbands, and weightbands. Enter the passband cutoff as its High Frequency value, and the stopband cutoff as its Low Frequency value. The region in the middle of the two values defines the transition band.

Note When you use the Equiripple or frequency-weighted Least Squares method, take care to not use very wide transition bands. Transition bands are treated as "don't care" regions and are left unconstrained in the attempt to minimize the approximation effort while using the minimum filter order. Consequently, unexpected peaks can occur in wide transition bands.

Entering the Amplitude

Define the amplitude for the passbands. For details on the format and conventions used for entering the amplitude value, refer to Selecting the Amplitude Format and Nonsymmetric.

Specifying the Ripple

Define the acceptable deviation in the passbands and rejection (attenuation) in the stopbands. This value is specified in dB.

Entering the Exponential Amplitudes

Define the desired exponential amplitude in the passbands. The exponential amplitude response field provides another representation for the desired amplitude response.
The overall desired frequency response will be the product of the response specified in the amplitude and exponential amplitude fields.

The response in the exponential amplitude field corresponds to

These values are entered in the exponential amplitude field as

For information on how to control the exponential amplitude specification of the filter design, refer to Nonsymmetric.

Specifying the Weighting


Weighting is an available option when the Equiripple or frequency-weighted Least Squares method is used to design an FIR filter. The approximation to the desired frequency response can be weighted using one or more weightbands, each with its own low and high frequency and magnitude specifications. The magnitude values you specify are relative. That is, a magnitude value of 1 is ten times greater than a magnitude value of 0.1.

Hint For numeric simplicity, scale the weightband magnitude to the range 0 to 1.0 when using the Equiripple method.

The weighting function assigns an appropriate relative cost for the deviation from a desired response. Frequency regions where more accurate approximations are required need larger weighting values than the less critical regions. A zero weighting is typically specified in the transitional frequency band between a passband and a stopband.

Specification of these bands is not required. However, regions that are not weighted are automatically assumed to have zero weighting.

Note The response in weighted regions can be improved by not weighting the response in regions where the response is not important. However, when an unweighted region becomes very wide, unexpected peaks may result.

Copying the Response Specifications

Click Copy Desired Bands to use the stopband and passband frequency specifications of the desired response for the weightbands as well. This feature provides a useful reminder of the passband edges specified in the Desired Response tab. As a result, you may wish to add more weightbands subsequently to weight the stopbands.


Selecting the Magnitude Format

For the case of weighting specifications, only magnitudes are considered. Therefore, all values in the magnitude column should be real values. Also, unlike the desired response, there is no exponential amplitude specification for the weighting response.

Selecting the Amplitude Format

The amplitude format used depends upon the desired shape of the passbands or weightbands. In both formats, the amplitude is essentially represented by a polynomial of some degree, zero or greater.

Digital Filter Designer automatically computes the actual inverse Fourier transform for the piecewise polynomial magnitude representation to provide superior optimization accuracy when compared to approximations that use FFTs.

Polynomial

This is the recommended default for most designs. This format is able to express both linear and non-linear polynomial response shapes. The overall desired frequency response will be the product of the response specified in the amplitude and phase fields.

Given a band in the range , the desired complex amplitude shape given values a, b, c, d is defined by the equation:

In addition, the polynomial format is best able to express a linear shape with a constant or zero slope. That is, this format is also the most convenient for expressing the complex amplitude given by a single value, a.

Within Digital Filter Designer, multiple polynomial values for a given complex amplitude are separated by commas, as follows: 2.5, 0.03125 or 0.5, 0, 1.5625e-5
where a=2.5, b=0.03125 or a=0.5, b=0.0, c=1.5625e-5, respectively.

Linear

This format is best able to express a first degree polynomial shape, that is a non-constant linear response using the two band-edge complex amplitude values. In such situations, it is a more convenient format for expressing complex amplitude values a, b . Within Digital Filter Designer, linear values are separated by commas, as follows: 0.5, 0.8 or 2, 2.5.
Given a band in the range , the linear format for a complex amplitude that is expressed by two values a and b so that
and
is best defined using the Linear complex amplitude format.

Note For example, the following passband shape is obtained by using amplitude values of 0.5 and 0.8 with the Linear format. To obtain the same passband shape using the Polynomial format would require that the amplitude values be specified as 0.5 and 1.5e-6.

 

Choosing the Filter Transform

A filter transform maps the filter's poles and zeros in the s plane to the z plane to convert the analog filter into an acceptable digital IIR filter.

Bilinear Z Transform

This transform avoids the problem of aliasing by using an algebraic transformation between the variables s and z that map the
axis in the s plane to one revolution of the unit circle in the z plane. However, neither the impulse response nor the phase response of the analog filter are preserved in the digital filter that is obtained.
The mapping of the continuous-time system to the discrete-time system is accomplished using the following equation:

Impulse Invariant

This transform is a method for obtaining a discrete-time system whose frequency response is determined by the frequency response of a continuous-time system. Because the impulse invariance relationship between continuous-time and discrete-time frequency is linear, the shape of the frequency response is preserved, except for aliasing effects. Filters with high frequency passband are not designed using this transform because of the threat of aliasing.
The system function for the discrete-time system is expressed by the following equation where:

Choosing the Compensation

Use Compensation when filtering a discrete-time signal prior to a D/A conversion. Compensation takes into account any attenuation that occurs due to a zero-order hold operation. It is used only when designing an FIR filter.

Sinc

Select this option to compensate for any magnitude attenuation in the designed filter.

Unity

This is the default option that is used in the absence of any magnitude attenuation. That is, no compensation is designed for the filter.

Defining the Symmetricity

Symmetricity is used only when designing an FIR filter. One of the reasons that the FIR filter is popular is the ease in which linear phase frequency responses can be achieved. Specifically, a linear phase response is obtained if the resulting unit pulse response is either symmetric or antisymmetric.

To select the symmetricity used for an FIR filter design within Digital Filter Designer, click the desired option within the Other Parameters tab.

Symmetric

This is the default option for most designs. Selecting this option results in an FIR filter design with a symmetric unit pulse phase response of

Antisymmetric

An antisymmetric sequence is used to design FIR filters with a purely imaginary frequency response (such as Differentiators and Hilbert Transformers). Selecting this option will result in an antisymmetric unit pulse response of

An antisymmetric design is not generally used for a filter with passbands starting at DC (frequency= 0 Hz) because of the poor response characteristics that result from the basic Fourier transform properties.

Note For best results in a filter design with passbands at the Nyquist frequency (frequency = sampling frequency/2) use an odd number for an antisymmetric filter with a manually specified order, and an even number for a symmetric filter with a manually specified order. For details on the filter order used in a filter design, refer to Defining the Filter Order".

Nonsymmetric

The Digital Filter Designer allows the user to control the phase by specifying a complex amplitude response if a nonlinear phase response is desired. In this case, the Nonsymmetric option must be chosen to withdraw the design constraint on symmetry. This is useful in applications that require phase compensation or low system delay.

Entering a Windowing Parameter

Enter a value for the windowing parameter, α. The value you enter as the windowing parameter is used in the equation for the windowing method you selected.

Raised Cosine Parameters

Response parameters specific to Raised Cosine and Root Raised Cosine filters are specified in a separate dialog box within the Digital Filter Designer.

The Finish Computing Desired [Root] Raised Cosine dialog box appears. The application automatically selects the appropriate amplitude, frequency, ripple, and exponential amplitude values based on the entries made by the user. The user can change the automatic selections by clicking Back , manually editing the field entries, and clicking Next .

If you are satisfied with the computed desired values, click the Finish button to dismiss the dialog box. Once the Finish Computing Desired [Root] Raised Cosine dialog box has been dismissed, it can be reopened by clicking the Root/Raised Cosine icon on the top toolbar.

Symbol Rate

Enter a value to specify the symbol rate in Hz.

When designing a Raised Cosine, or Root Raised Cosine filter a window pops up with fields: Oversampling Rate (Hz), Roll Off, Passband Ripple (dB), Stopband Ripple (dB). For these filter types, Ovesampling Rate (Hz) is the same as the Symbol Rate.

Roll Off

Enter a value to specify the desired roll-off or width of the transition from a passband to a stopband. The acceptable range for the roll-off value, , is .
leads to a zero-width transition band, while
leads to a transition band width that is equal to the symbol rate.

Passband Ripple

Enter a value to specify the acceptable ripple in the passband.

Stopband Ripple

Enter a value to specify the acceptable ripple or attenuation in the stopband.

Note For more information on pulse-amplitude modulation (PAM) digital communication systems and the use of Raised Cosine and Root Raised Cosine filters, refer to R. W. Lucky, Salz J., & Weldon Jr. E. J., "Principles of Data Communication," McGraw-Hill. New York, 1968. pp 45-51.

 

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