Lecture 9 Infinite Impulse Response Filters Outline 9 Infinite Impulse Response Filters 9 First-Order Low-Pass Filter 93 IIR Filter Design 5 93 CT Butterworth filter design 5 93 Bilinear transform 7 9 Infinite Impulse Response Filters The class of causal infinite impulse response (IIR) filters can be captured by the difference equation M N y[n] = b k u[n k] a k y[n k], k= where there are M input coefficients b k R, and N output coefficients a k R The filter order is given by max(m, N ) and corresponds to the number of delay elements an implementation of the filter would require; it is also the size of the state in a state-space description of the system In contrast to FIR filters, the output of a causal IIR filter is dependent on both the filter s input and on previously generated outputs This dependency on previous outputs is described by the existence of one or more non-zero coefficients a k Dependence on previous output generally implies that the impulse response of an IIR filter has infinite length (hence the name) Another key difference between FIR and IIR filters is that IIR filters are not necessarily stable; the stability of an IIR filter depends on the coefficients a k The advantage of IIR filters, however, is that they generally meet a given set of filter specifications with a lower filter order (corresponding to lower computation and storage cost) compared to FIR filters The transfer function and frequency response are straightforward to calculate from the constant coefficient difference equation: k= H(z) = M k= b k z k + N a k z k k= z=e jω H(Ω) = M k= b k e jωk + N a k e jωk k= Updated: November 7, 7
The goal of IIR filter design is to find coefficients a k and b k such that the filter meets given specifications and is stable IIR filter design often employs established continuous-time (CT) filter design methods, for example Butterworth filter design, and then transforms the resulting CT filter into DT In this lecture, we introduce the concepts underlying IIR filters; how to design a CT Butterworth filter; and finally, how to convert a CT filter into DT using the bilinear transform 9 First-Order Low-Pass Filter To introduce the concepts of IIR filtering, we study the causal, first-order, low-pass IIR filter, which has the difference equation y[n] = αy[n ] + ( α)u[n], (9) where α < For α, this is an infinite impulse response filter The extremes of α provide some intuition about the filter behavior: If α = the output is equal to the input and no filtering occurs In contrast, as α, the output becomes increasingly constant The transfer function of the first-order low-pass filter is The filter has a single pole at z = α α < The frequency response is Properties H(z) = H (Ω) = Very low frequency signals remain unaltered since The magnitude response is: H(Ω = ) = α αz It immediately follows that the filter is stable if α αe jω α = αe j α H (Ω) = ( α cos Ω) + α sin Ω Furthermore, one can show that the magnitude is monotonically non-increasing: The phase is d H(Ω) dω ( H(Ω) = arctan, for Ω π Always negative {}}{ α sin Ω α cos Ω }{{} Always positive Therefore π < H(Ω) A plot of the magnitude and phase response follows: ), for Ω π
H(Ω) 5 α = 3 α = 5 α = 8 α = 9 π/ π Ω H(Ω) π/ π/ π Ω Design considerations A typical way to choose α is based on the decay time: the amount of time it takes the signal y to decay to the value e We later motivate using e (another choice is 5, the half-life) Assume that the initial output y[] =, and that the input is always zero, u[n] =, for n The output sequence of the filter is y[] = y[] = αy[] = α y[] = αy[] = α y[n] = α n = e therefore α = e n Given n, the required value of α can be determined To choose n, let T be the desired time for the continuous process to decay to e, and let T s be the sampling time Then n = T T s α = e n = e T, where we assume that T is an integer multiple of T s (if this is not the case, n may be rounded) For example, let the sampling time be T s = s and the decay time be T = s, then α = exp( ) 99 Usually, T is large relative to T s and we may use a first-order approximation to obtain α T s T 3
Connection to CT systems A connection to CT systems can be made: In CT, the transfer function of a first-order low-pass filter is given by H(s) = τs + The differential equation for the output is ẏ(t) = τ (y(t) u(t)) (9) Assuming u(t) = for t, we obtain the time-domain system response y(t) = y()e t τ Choosing y() =, we obtain the system response: e τ t Therefore, the time to reach e is τ Another connection to CT can be made by discretizing (9), assuming the input u is constant over a sample period T s, ie a zero-order hold device is used: [ẏ ] [ ] [ ] = τ τ y u u which we solve using the matrix exponential and obtain [ ] ([ y(t s ) T s u( = exp τ ) T s τ ]) [ ] [ y() = u() t < T s, e τ e τ ] [y() ] u() As discussed in Lecture, this solution is valid on any time interval because the system is time invariant Substituting the decay time T for the time constant τ, the resulting difference equation becomes y[n] = e T y[n ] + ( e T )u[n ] = αy[n ] + ( α)u[n ] which closely resembles the first-order, low-pass IIR filter (9), except that the input is delayed by one sample 4
93 IIR Filter Design FIR filter design only makes sense in DT In contrast, IIR filters are often designed in CT using an established method, for example a Butterworth or Chebyshev method, and are then converted to DT using the bilinear transform (sometimes also called Tustin method) As we will see in the following section, the bilinear transform has properties that make it a useful tool for the above design procedure 93 CT Butterworth filter design We now introduce the CT design of a Butterworth filter, which is a general purpose low-pass filter The starting point is the desired frequency response, with corner frequency rad/sec: where K is the order of the filter R(ω) = + ω K, R(ω) (db) 4 8 K = K = 3 K = 8 6 ω Properties This frequency response has two very desirable properties: it has no ripples, and is maximally flat We investigate these properties now First, let us calculate the derivative dr/dω By noting that R = ( + ω K ) and that we have, dr dω = dr dr dr dω = RdR dω dr dω = dr R dω, dr dω = ( + ωk ) Kω K = R 4 Kω K and therefore dr dω = KR3 ω K < for all ω > This means that the Butterworth filter has no ripples Furthermore, all derivatives of R up to K are at and the filter is said to be maximally flat 5
Transfer function Let H(s) be the transfer function of a filter with frequency response R(ω) We then have that H(jω) = R(ω) = ( + ω K ) The only stable transfer function that achieves this is H(s) =, K (s s k ) k= where s k = e j(k+k )π K, k =,, K Filters with a transfer function of this structure are known as Butterworth filters Example (Second-order Butterworth low-pass filter) Consider one of the most common Butterworth filters: a second-order (K = ) low-pass We have, s = e j3π/4 = + j, (35 ) s = e j5π/4 = j, (5 ) which result in the transfer function H(s) = (s + j )(s + +j ) = s + s + The poles of the CT filter lie on the unit circle in the s-plane (not to be confused with the z-plane) and are represented by the black crosses below The gray crosses represent the poles of H( s) and are useful to visualize the pole-placement pattern K = Im(s) 35 Re(s) 35 If, instead, a third-order low-pass filter is chosen, the pole plot looks as follows: 6
K = 3 Im(s) Re(s) Corner frequency specification The design method introduced above assumed a corner frequency of rad/sec other corner frequencies ω c can be chosen In that case, we have However, For example, the second order filter becomes H(s) = s s ω c ω c s + ω c s + ω c, which has the same response as a mass-spring-damper system with sub-critical damping / and natural frequency ω c 93 Bilinear transform Once a CT filter has been designed, the bilinear transform (also known as Tustin s method) can be used to convert it into a DT filter The bilinear transform uses the substitution that s = ( ) z, T s z + where T s is the sampling time Motivation Recall that z can be given the interpretation of a DT shift operator If Y (z) = zu(z), then y[n] = u[n + ] 7
Similarly, in CT, e s can be given the interpretation of a time shift operator If Y (s) = e s U(s) where Y (s) and U(s) are the Laplace transform of y(t) and u(t), respectively, then y(t) = u(t + T s ) Therefore, the two operators are equivalent: z = e s A rational approximation for the relation between z and s will map a rational CT transfer function to a rational DT transfer function This is equivalent to converting differential equations to difference equations We therefore use the approximation e s = which is valid if st s is small We call es s e + s, s the bilinear transform The inverse is z = + s s s = ( ) z T s z + and is straightforward to verify by substitution DT-CT frequency mapping We now evaluate the bilinear transform along the imaginary axis of the s-plane; that is, let s = jω Using the bilinear transform, this point maps to Note that z = z = + jω jω + jω jω = The bilinear transform therefore maps the imaginary axis of the s-plane to the unit circle in the z-plane We therefore write z = e jω = + jω, jω and calculate the mapping of a CT frequency ω to a DT frequency Ω as: e jω = ( + jω ) ( jω Ω = arctan(ω = arctan(ω ) 8 ) ) arctan( ω )
The frequency response of the CT system at ω (the CT transfer function evaluated on the imaginary axis at s = jω) directly corresponds to the frequency response of the resulting DT system at Ω = arctan(ω ) (the DT transfer function evaluated on the unit circle at z = e jω ) This is a desirable property, as we will later see For small ωt s, the DT frequency is approximately Ω (ω ) = ωt s This is also evident when we plot the mapping of CT frequencies to DT frequencies for z = e s and the bilinear transform: π Ω Ω = ωt s Ω = arctan(ωt s /) π/ /T s π/t s ω Note how Ω asymptotically converges to π as ω : The bilinear transform compresses the imaginary axis of the s-plane onto the unit circle 9
The underlying nonlinear relation between ω and Ω is called frequency warping A few common values are: ω = T s ω = Ω = ω = Ω = π ω = T s Ω = π s = jω z = e jω ω = ω = Ω ω = ω = T s Other points are: z = s = T s, z = s = T s Region of stability mapping Stable poles in the continuous domain are mapped to stable poles in the discrete domain: s-domain z-domain This is desirable, as it means that a stable CT filter is transformed into a stable DT filter You will prove this in the problem sets Summary The bilinear transform preserves stability and maps the imaginary axis in the s-plane to the unit circle in the z-plane by compressing the CT frequencies < ω < to DT frequencies π < Ω < π
Example (Converting a CT first-order low-pass filter) Consider the CT low-pass filter with time constant τ H(s) = τs + Using the bilinear transform, we obtain H(z) = ( ) = α + z + τ z αz T s z+ with α = τ + τ For small values of τ, α τ, as before Let us compare different discretization methods: Method Transfer function Filter parameter Direct H(z) = α αz Sample and Hold Bilinear H(z) = H(z) = ( α)z αz +z ( α)( ) αz α = e τ (decay time τ) α = e τ (time constant τ) α = τ + τ (time constant τ) A nice property of the bilinear transform, for this example, is that H(z = ) = This corresponds to z = = e jπ which is the highest possible DT frequency Observe that the CT first-order low-pass has H(s) = for s j Therefore, the frequency responses at the highest possible frequencies are the same for the CT and the DT system when using the bilinear transform It follows that their high-frequency behavior is similar, which is one of the advantages of the bilinear transform
This can be seen by looking at the frequency response plots of the resulting filters for T s = and τ = : H(Ω) 5 Continuous-time Bilinear transform ZOH Direct H(Ω) 9 9 π/ π Frequency (rad)