DIGITAL SIGNAL PROCESSING UNIT III INFINITE IMPULSE RESPONSE DIGITAL FILTERS Contents: 3.1 Introduction IIR Filters 3.2 Transformation Function Derivation 3.3 Review of Analog IIR Filters 3.3.1 Butterworth Filter 3.3.2 Chebyshev Filter 3.4 Design of IIR Digital Filters 3.4.1 Impulse Invariant Technique 3.4.2 Bilinear Transformation 3.5 Pre Warping 3.6 Design of Digital Filter using Digital to Digital Transformation 3.7 Realization of IIR Filters 3.1 Introduction: IIR FILTERS Infinite impulse response (IIR) is a property of signal processing systems. Systems with this property are known as IIR systems or, when dealing with electronic filter systems, as IIR filters. IIR systems have an impulse response function that is non-zero over an infinite length of time. This is in contrast to finite impulse response filters (FIR), which have fixed-duration impulse responses. The simplest analog IIR filter is an RC filter made up of a single resistor (R) feeding into 1
a node shared with a single capacitor (C). This filter has an exponential impulse response characterized by an RC time constant. IIR filters may be implemented as either analog or digital filters. In digital IIR filters, the output feedback is immediately apparent in the equations defining the output. Design of digital IIR filters is heavily dependent on that of their analog counterparts because there are plenty of resources, works and straightforward design methods concerning analog feedback filter design while there are hardly any for digital IIR filters. As a result, usually, when a digital IIR filter is going to be implemented, an analog filter (e.g. Chebyshev filter, Butterworth filter, Elliptic filter) is first designed and then is converted to a digital filter by applying discretization techniques such as Bilinear transform or Impulse invariance. Example for IIR filters: Chebyshev filter, Butterworth filter, and the Bessel filter. An infinite impulse response filter (IIR) is one whose impulse response goes on forever. Recursive filters are closely associated with IIR filters but are not exclusively IIR. The transfer function of an IIR filter is shown below. For it to be IIR, at least one of the a's in the denominator must be non zero. M, the number of zeros should be less than N, the number of poles. To obtain the frequency response of the filter, substitute the following relation between z and omega. T is the sampling time. This results in a Frequency response with the general form. 2
To obtain the time sequence of the output or the impulse response use the following equivalent difference equation. The Linear Predictive model is a special case of an IIR filter and shown in figure The general IIR filter is given by: If p = 0 then the system represents a finite impulse response (FIR) filter. If p is not zero, then the system is an infinite impulse response (IIR) filter. Review Questions: 1. What is an Infinite Impulse Response filter? 3
2. Give examples for IIR filter. 3. Define Transfer function of IIR filter. 3.2 TRANSFER FUNCTION DERIVATION Digitals filters are often described and implemented in terms of the difference equation that defines how the output signal is related to the input signal: where: is the feedforward filter order are the feedforward filter coefficients is the feedback filter order are the feedback filter coefficients is the input signal is the output signal. A more condensed form of the difference equation is: which, when rearranged, becomes: 4
To find the transfer function of the filter, we first take the Z-transform of each side of the above equation, where we use the time-shift property to obtain: We define the transfer function to be: Considering that in most IIR filter designs coefficient is 1, the IIR filter transfer function takes the more traditional form: Stability The transfer function allows us to judge whether or not a system is bounded-input, bounded-output (BIBO) stable. To be specific, the BIBO stability criteria requires the ROC of the system include the unit circle. For example, for a causal system, all poles of the transfer function have to have an absolute value smaller than one. In other words, all poles must be located within a unit circle in the z-plane. The poles are defined as the values of z which make the denominator of H(z) equal to 0: 3.3 Review of Analog IIR Filters 3.3.1BUTTERWORTH FILTER: The Butterworth filter is one type of electronic filter design. It is designed to have a frequency response which is as flat as mathematically possible in the passband. Another name for it is maximally flat magnitude filter. 5
The transfer function Plot of the gain of Butterworth low-pass filters of orders 1 through 5. Note that the slope is 20n db/decade where n is the filter order. Like all filters, the typical prototype is the low-pass filter, which can be modified into a high-pass filter, or placed in series with others to form band-pass and bandstop filters, and higher order versions of these. The gain G(ω) of an n-order Butterworth low pass filter is given in terms of the transfer function H(s) as: where n = order of filter ω c = cutoff frequency (approximately the -3dB frequency) G 0 is the DC gain (gain at zero frequency) It can be seen that as n approaches infinity, the gain becomes a rectangle function and frequencies below ω c will be passed with gain G 0, while frequencies above ω c will be suppressed. For smaller values of n, the cutoff will be less sharp. We wish to determine the transfer function H(s) where s = σ + jω. Since H(s)H(-s) evaluated at s = jω is simply equal to H(jω) 2, it follows that: The poles of this expression occur on a circle of radius ω c at equally spaced points. The transfer function itself will be specified by just the poles in the negative real half-plane of s. The k-th pole is specified by: 6
The transfer function may be written in terms of these poles as: The denominator is a Butterworth polynomial in s. Normalized Butterworth polynomials The Butterworth polynomials may be written in complex form as above, but are usually written with real coefficients by multiplying pole pairs which are complex conjugates, such as s 1 and s n. The polynomials are normalized by setting ω c = 1. The normalized Butterworth polynomials then have the general form: 7
Digital implementation Digital implementations of Butterworth filters often use bilinear transform or matched z-transform to discretize an analog filter. For higher orders, they are sensitive to quantization errors. Comparison with other linear filters Here is an image showing the gain of a discrete-time Butterworth filter next to other common filter types. All of these filters are fifth-order. The Butterworth filter rolls off more slowly around the cutoff frequency than the others, but shows no ripples. 3.3.2 CHEBYSHEV FILTER Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized filter characteristic and the actual over the range of the filter, but with ripples in the passband. This type of filter is named in honor of Pafnuty Chebyshev 8
because their mathematical characteristics are derived from Chebyshev polynomials. Because of the passband ripple inherent in Chebyshev filters, filters which have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications. Type I Chebyshev filters The frequency response of a fourth-order type I Chebyshev low-pass filter with These are the most common Chebyshev filters. The gain (or amplitude) response as a function of angular frequency ω of the nth order low pass filter is where ε is the ripple factor, ω 0 is the cutoff frequency and T n () is a Chebyshev polynomial of the nth order. The passband exhibits equiripple behavior, with the ripple determined by the ripple factor ε. In the passband, the Chebyshev polynomial alternates between 0 and 1 so the filter gain will alternate between maxima at G = 1 and minima at. At the cutoff frequency ω 0 the gain again has the value but continues to drop into the stop band as the frequency increases. The ripple is often given in db: Ripple in db = so that a ripple amplitude of 3 db results from Type II Chebyshev filters The frequency response of a fifth-order type II Chebyshev low-pass filter with 9
Also known as inverse Chebyshev, this type is less common because it does not roll off as fast as type I, and requires more components. It has no ripple in the passband, but does have equiripple in the stopband. The gain is: In the stop band, the Chebyshev polynomial will oscillate between 0 and 1 so that the gain will oscillate between zero and and the smallest frequency at which this maximum is attained will be the cutoff frequency ω 0. The parameter ε is thus related to the stopband attenuation γ in decibels by: Review Questions: 1. What is an analog Butterworth filter? 2. Define the transfer function of Butterworth filter in terms of its gain. 3. Mention the types of Chebyshev filter. 4. Compare the gain of Butterworth filter with other types. 3.4 DESIGN OF IIR DIGITAL FILTERS: Methods: i) Impulse Invariant Technique ii) Bilinear Transformation 10
3.4.1 i) IMPULSE INVARIANT TECHNIQUE Output from a digital filter is made up from previous inputs and previous outputs, using the operation of convolution: Two convolutions are involved: one with the previous inputs, and one with the previous outputs. In each case the convolving function is called the filter coefficients. If such a filter is subjected to an impulse (a signal consisting of one value followed by zeroes) then its output need not necessarily become zero after the impulse has run through the summation. So the impulse response of such a filter can be infinite in duration. Such a filter is called an Infinite Impulse Response filter or IIR filter. Note that the impulse response need not necessarily be infinite: if it were, the filter would be unstable. In fact for most practical filters, the impulse response will die away to a negligibly small level. One might argue that mathematically the response can go on for ever, getting smaller and smaller: but in a digital world once a level gets below one bit it might as well be zero. The Infinite Impulse Response refers to the ability of the filter to have an infinite impulse response and does not imply that it necessarily will have one: it serves as a warning that this type of filter is prone to feedback and instability. 11
The filter can be drawn as a block diagram: The filter diagram can show what hardware elements will be required when implementing the filter: The left hand side of the diagram shows the direct path, involving previous inputs: the right hand side shows the feedback path, operating upon previous outputs. If the phase response is not specified, one prefers to use IIR digital filter. In case of an IIR filter design, the most common practice is to convert the digital filter specifications to analog low pass prototype filter specifications, to determine the analog low pass transfer function meeting these specifications, and then to transform it into desired digital filter transfer function. This methods is used for the following reasons: 1. Analog filter approximation techniques are highly advanced. 2. They usually yield closed form solutions. 3. Extensive tables are available for analog-design. 4. Many applications require the digital solutions of analog filters. 12
The transformations generally have two properties (1) the imaginary axis of the s-plane maps into unit circle of the z-plane and (2) a stable continuous time filter is transformed to a stable discrete time filter. Filter design by impulse invariance Here is a recipe for designing an IIR digital filter: decide upon the desired frequency response design an appropriate analogue filter calculate the impulse response of this analogue filter sample the analogue filter's impulse response use the result as the filter coefficients This process is called the method of impulse invariance. The method of impulse invariance seems simple: but it is complicated by all the problems inherent in dealing with sampled data systems. In particular the method is subject to problems of aliasing and frequency resolution. In the impulse variance design procedure the impulse response of the impulse response of the discrete time system is proportional to equally spaced samples of the continues time filter, i.e., where T d represents a sampling interval, since the specifications of the filter are given in discrete time domain, it turns out that T d has no role to play in design of 13
the filter. From the sampling theorem we know that the frequency response of the discrete time filter is given by Since any practical continuous time filter is not strictly bandlimited there issome aliasing. However, if the continuous time filter approaches zero at high frequencies, the aliasing may be negligible. Then the frequency response of the discrete time filter is We first convert digital filter specifications to continuous time filter specifications. Neglecting aliasing, we get specification by applying the relation where is transferred to the designed filter H(z), we again use equation (9.2) and the parameter T d cancels out. Let us assume that the poles of the continuous time filter are simple, then The corresponding impulse response is Then The system function for this is 14
We see that a pole at in the s-plane is transformed to a pole at T d in the z-plane. If the continuous time filter is stable, that is, then the magnitude of will be less than 1, so the pole will be inside unit circle. Thus the causal discrete time filter is stable. The mapping of zeros is not so straight forward. 3.4.2 II) BILINEAR TRANSFORMATION The method of filter design by impulse invariance suffers from aliasing. The aliasing will be a problem if the analogue filter prototype's frequency response has significant components at or beyond the Nyquist frequency. The problem of aliasing arises because we wrap an infinitely long, straight frequency axis around a circle. So the frequency axis wraps around and around, and any components above the Nyquist frequency get wrapped back on top of other components. The bilinear transform is a method of squashing the infinite, straight analogue frequency axis so that it becomes finite. To avoid squashing the filter's desired frequency response too much, the bilinear transform squashes the far ends of the frequency axis the most - leaving the middle portion relatively unsquashed: 15
The infinite, straight analogue frequency axis is squashed so that it becomes finite - in fact just long enough to wrap around the unit circle once only. This is also sometimes called frequency warping If is the continues time transfer function the discrete time transfer function is detained by replacing s with Rearranging terms in equation (9.3) we obtain. Substituting, we get If, it is then magnitude of the real part in denominator is more than that of the numerator and so. Similarly if, than for all. Thus poles in the left half of the s-plane will get mapped to the poles inside the unit circle in z-plane. If then So,, writing we get rearranging we get 16
or The compression of frequency axis represented by (9.5) is nonlinear. This is illustrated in figure 9.4. 17
Because of the nonlinear compression of the frequency axis, there is considerable phase distortion in the bilinear transformation. Review Questions : 1. What is meant by impulse invariant technique? 2. List out the steps to design a digital IIR filter using impulse invariant technique. 3. What is meant by Bilinear Transformation? 4. List out the steps to design a digital IIR filter using Bilinear Transformation. 3.5 PRE WARPING Frequency warping does change the shape of the desired filter frequency response. In particular, it changes the shape of the transition bands. This is a pity, since we went to a lot of trouble designing an analogue filter prototype that gave us the desired frequency response and transition band shapes. One way around this is to warp the analogue filter design before transforming it to the sampled data z plane Argand diagram: this warping being designed so that it will be exactly undone by the frequency warping later on. This is called prewarping. This technique avoids the problem of aliasing by mapping one revaluation of the unit circle in the z-plane. axis in the s-plane to Some frequently used analog filters In the previous two examples we have used Butterworth filter. The Butterworth filter of order n is described by the magnitude square frequency response of It has the following properties 18
is monotonically decreasing function of As n gets larger, approaches an ideal low pass filter are zero at is called maximally flat at origin, since all order derivative exist and they The poles of a Butterworth filter lie on circle of radius in s-plane. There are two types of Chebyshev filters, one containing ripples in the passband (type I) and the other containing a ripple in the stopband (type II). A Type I low pass normalizer Chebyshev filter has the magnitude squared frequency response. where is n th order Chebyshev polynomial. We have the relationship with Chebyshev filters have the following properties 1. The magnitude squared frequency response oscillates between 1 and within the passband, the so called equiripple and has a value of at, the normalized cut off frequency. 2. The magnitude response is monotonic outside the passband including transitionand stopband. 3. The poles of the Chebysher filter lie on an ellipse in s-plane. An elliptic filter has ripples both in passband and in stopband. The square magnitude frequency response is given by 19
where is Chebyshev rational function of O determined from specified ripple characteristics. An n th order Chebyshev filter has sharper cutoff than a Butterworth filter, that is, has a narrower transition bandwidth. Elliptic filter provides the smallest transition width. 3.6 DESIGN OF DIGITAL FILTER USING DIGITAL TO DIGITAL TRANSFORMATION There exists a set of transformation that takes a low pass digital filter and turn into highpass, bandpass, bandstop or another lowpass digital filter. These transformations are given in table 9.1. The transformations all take the form of replacing the in by some function of. Type From To Transformation Design Formula Low pass cutoff Low pass cutoff LPF HPF LPF BPF LPF BSF 20
Starting with a set of digital specifications and using the inverse of the design equation given in table 9.1, a set of lowpass digital requirements can be established. A LPF digital prototype filter is then selected to satisfy these requirements and the proper digital to digital transformation is applied to give the desired. Review Questions: 1. What is aliasing? 2. What do you mean by prewarping? 3. How a digital LPF can be transformed into HPF,BPF or BSF? 3.7 REALIZATION OF IIR FILTERS So far, we have studied the IIR filters by analysis of transfer functions or impulse responses. In this section we want to face the problem of implementing these filters as computational structures that can be directly coded using sound processing languages or real-time sound processing environments. i) Direct Form 1 ii) Direct Form 2 or Transposed form iii) Parallel Form iv) Cascade Form i) Direct form 1 So called because it can be drawn direct from the filter equation. Consider a second-order filter with two poles and two zeros, which is rep- Resented by the transfer function { δ(n) =1 ; n=0 }with N = M = 2. This can be realized by the signal flowgraph of fig., where the nodes having converging edges are considered as points of addition, and the nodes having diverging edges are considered as branching points. Such a realization is called Direct Form I. fig.1 21
Signal flowgraphs can be manipulated in several ways, thus leading to alternative realizations having different numerical properties and, possibly, more computationally efficient. For instance, if we want to implement a filter as a cascade of second-order cells such as that of fig.1 we can share, between two contiguous cells, the unit delays that are on the output stage of the first cell, with the unit delays that are on the input stage of the second cell, thus saving a number of memory accesses. This particular diagram is called the direct form 1 because the diagram can be drawn directly from the filter equation. The filter diagram can show what hardware elements will be required when implementing the filter: The left hand side of the diagram shows the direct path, involving previous inputs: the right hand side shows the feedback path, operating upon previous outputs. ii) Direct form 2 so called because it can be derived by changing the diagram of direct form 1 22
A first transformation comes from the observation that the structure of fig.1 is formed by the cascade of two blocks, each being linear and time invariant. Therefore, the two blocks can be commuted without altering the input-output behavior. Moreover, from the block exchange we get a flowgraph with two sideto-side stages of pure delays, and these stages can be combined in one only. The realization of these transformations is shown in fig. and it is called Direct Form II. fig.2 The filter diagram for direct form 1 can be drawn direct from the filter equation: The block diagram is in two halves: and since the results from each half are simply added together it does not matter in which order they are calculated. So the order of the halves can be swapped: 23
Now, note that the result after each delay is the same for both branches. So the delays down the centre can be combined: This is called direct form 2. Its advantage is that it needs less delay elements. And since delay elements require hardware (for example, processor registers) the direct form 2 requires less hardware and so is more efficient than direct form I. direct form 2 is also called canonic, which simply means 'having the minimum number of delay elements'. The transposition theorem says that if we take a filter diagram and reverse all the elements - swapping the order of execution for every element, and reversing the direction of all the flow arrows - then the result is the same: if everything is turned back to front, it all works just the same This means that the direct form 1 diagram can be obtained by transposition of the direct form 2 diagram: 24
For this reason, direct form 1 is often called transposed direct form 2. Transposed so called because it is obtained by transposition of direct form 2 - but really, this is just direct form 1 Another transformation that can be done on a signal flow graph without altering its input-output behavior is the transposition The transposition of a signal flow graph is done with the following operations: Inversion of the direction of all the edges Transformation of the nodes of addition into branching nodes, and vice versa The transposition of a realization in Direct Form II leads to the Transposed Form II, which is shown in fig.2 Similarly, the Transposed Form I is obtained by transposition of the Direct Form I. fig.3 25
Canonic so called because it has the minimum number of delay elements - but really, it is just direct form 2 iii) Parallel Form: Because IIR filters are very sensitive to quantisation errors, they are usually implemented as second order sections. The parallel form is simple: The outputs from each second order section are simply added together. If scaling is required, this is done separately for each section. It is possible to scale each section appropriately, and by a different scale factor, to minimise quantisation error. In this case another extra multiplier is required for each section, to scale the individual section outputs back to the same common scale factor before adding them. The order in which parallel sections are calculated does not matter, since the outputs are simply added together at the end. iv) Cascade form: In the cascade form, the output of one section forms the input to the next: 26
Mathematically, it does not matter in which order the sections are placed - the result will be the same. This assumes that there are no errors. In practice, the propagation of errors is crucial to the success of an IIR filter so the order of the sections in the cascade form is vital. In the cascade form, the output of one section forms the input to the next: In practice, the propagation of errors is crucial to the success of an IIR filter so the order of the sections in the cascade, and the selection of which filter coefficients to group in each section, is vital: Sections with high gain are undesirable because they increase the need for scaling and so increase quantisation errors It is desirable to arrange sections to avoid excessive scaling Review Questions: 1. Mention the types of realizing a digital IIR filter. 2. How a digital filter can be realized using cascade form? 27