Infinite Impulse Response Filters (IIR)

Infinite impulse response (IIR) filters are another class of traditional filters. A $K^{th}$ order IIR filter is described by the difference equation,

$$y(n)= -\sum^{K}_{k=1}a_ky(n-k) + \sum^{K}_{k=0}b_kx(n-k)$$ (4.6)

where each output sample depends on present and past input samples, as well as output samples. A direct form II realization of a general IIR filter is shown in Fig. 4.3 [PM92]. The recursive part of the filter causes the response to an impulse input to endure (theoretically) forever. This is why such filters are called IIR filters [BTK02a]. A distinct advantage of IIR filters over FIR filters is that IIR filters can yield an equivalent magnitude response using a much lower filter order. A drawback of IIR filters is that they have non-linear phase responses, which can cause severe problems for phase sensitive velocity estimation such as the autocorrelation technique. However, several investigations showed [BTK02a], [Bec93] that IIR filters can be used for clutter rejection.

Figure 4.3: Direct form II realization of a general IIR filter

Figure 4.4: An illustration of an ideal and practical magnitude specification for high-pass filter. The practical curve shows an elliptical filter design.

There are a number of well established algorithms for designing IIR filter types based on the steady-state magnitude response, e.g: Butterworth filters, Chebyshev type I and II and elliptic filters. Butterworth coefficients exhibit a flat pass-band and, for a high pass filter, zeros at zero frequency [OS99]. Chebyshev type I filters have an equi-ripple pass-band and a monotonic stop-band. The filter is optimal in the sense that among all all-pole filters of order $N$, this filter has the smallest pass-band ripples $d_p$ for fixed stop-band frequency $w_s$,and pass-band frequency $w_p$. (see Fig. 4.4). Chebyshev type II filters have an equi-ripple stop-band with stop-band ripples $d_s$ and a monotonic pass-band. Elliptic filters are equi-ripple both in the pass-band and in the stop-band [OS99]. An output of an IIR filter consists of a transient component and a steady-state component:

$$y(n)=y_{tr}(n)+y_{ss}(n).$$ (4.7)

Assuming a stable filter, the first term vanishes as $n$ $\rightarrow \infty$. [BTK02a]. CRFs must operate with a finite number of samples.$n$ does not go to $\infty$, so the transient response of the filter becomes important. Initialization of the inner states of the filter can yield a suppression of the transient response and can reduce the ``ring-down time". The ``ring-down time" is referred to as the time the transient response dominates the output of the IIR filter.

State Space Formulation

To investigate different techniques for reducing the transient response of an IIR-filter, a state space formulation [BTK02a] is more convenient than the difference equation formulation given in Eq. 4.6.

Figure 4.5: State space diagram for an IIR filter [BTK02a]. The thick lines represent matrix operations and the thin ones vector operations.

In Fig. 4.3, a direct form II realization of a general IIR filter is shown. The (inner) state vector is defined by $v(n)=[v_1(n)\ v_2(n) \ldots v_K(n)]^T$. The resulting state space filter is then:

$$\mathbf{v}(n+1) = \mathbf{Fv}(n)+\mathbf{q}x(n)$$ (4.8)

$$y(n) = \mathbf{g}^T\mathbf{v}(n)+dx(n)$$ (4.9)

where $\mathbf{F}$, $\mathbf{q}$, $\mathbf{g}$ and $d$ are:

$$\mathbf{F} = \left[ \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0\\ 0 & 0 & 1 &... ...{ q} = \left[ \begin{array}{c} 0\\ 0\\ \vdots\\ 0\\ 1\\ \end{array}\right]$$

$$\mathbf{g} = \left[ \begin{array}{c} b_K-b_0a_n\\ b_{K-1}-b_0a_{K-1}\\ \vdots\\ b_1-b_0a_1\\ \end{array} \right] , d=b_0$$ (4.10)

The state space description of the filter is illustrated in 4.5. With initial state $\mathbf{v}(0)$, the state at $n$>0 is given by

$$\mathbf{v(n)}= \mathbf{F}^n\mathbf{v}(0) + \sum^{n-1}_{k=0}\mathbf{F}^{n-1-k}\mathbf{q}x(k)$$ (4.11)

and the filter output is given by

$$y(n)= \left\{ \begin{array}{ll} \mathbf{g}^T\mathbf{v}(0... ...-k}\mathbf{q}x(k)+d\mathit{x}(n), & n>0 \end{array} \right.$$ (4.12)

The input and output sequences are viewed as $N\times1$ vectors, $x(n)=[x(0)\ x(1) \ldots x(N-1)]^T$, and $y(n)=[y(0)\ y(1) \ldots y(N-1)]^T$. Eq. 4.12 can be written as the matrix vector equation,

$$\mathbf{y}=\mathbf{B}\mathbf{v}(0)+\mathbf{C}\mathbf{x}$$ (4.13)

where

$$\mathbf{B} = \left[ \begin{array}{c} \mathbf{g}^T\\ \mathbf{g}^T\mathbf{F}\\ \vdots\\ \mathbf{g}^T\mathbf{F}^{N-1}\\ \end{array} \right]$$

$$\mathbf{C} = \left[ \begin{array}{ccccc} d & 0 & \ldots & 0 & 0\\ \mathbf{g}^... ...{F}^{N-3}\mathbf{q} & \ldots & \mathbf{g}^T\mathbf{q} &d\\ \end{array} \right]$$ (4.14)

The state space formulation can now be used to investigate different ways for v(0) for minimizing the transient filter response.

Zero initialization

The zero initialization sets the inner state $\mathbf{v}(0)$ equal to zero. This is equivalent to assuming that the input signal is identical to zero for $n$ <0. The filtering operation is equal to

$$\mathbf{y}=\mathbf{C}\mathbf{x}=\mathbf{A}_{zero}\ \mathbf{x},$$ (4.15)

where $\mathbf{C}$ is given in Eq. 4.14 and $\mathbf{A}_{zero}$ gives the zero-initialized IIR filter matrix. This initialization technique shows no damping of the transient response due to the leading edge of the data window.

Step initialization

Step initialization, first proposed by [FB72] sets the value of the initial state vector by assuming that the first complex sample value had existed since time sample n=$-\infty$. This is equivalent to projecting the first sample of the input data vector back to time sample n=$-\infty$, and running the filter on the resulting vector. For a stable filter, the transient dies out with time, and for a step input, the filter registers converge to constant values. The transient can thus be suppressed by setting the initial state equal to the state at infinitely long time after the step is applied at the input. This initial filter state is found by utilizing the final value theorem of the one sided $\mathcal{Z}$-transform [PM92]. Transformation of Eq. 4.8 gives

$$\mathbf{V}^+(z)=z(z\mathbf{I}-\mathbf{F})^{-1}\mathbf{v}(0)+z(z\mathbf{I}-\mathbf{F})^{-1}\mathbf{q}X^+(z)$$ (4.16)

where $\mathbf{I}$ is the identity matrix. Inserting the $\mathcal{Z}$-transform of the assumed step input gives:

$$\mathbf{v}_{step}(0) = \lim_{z\rightarrow 1} (z-1)\mathbf{V}^+$$

$$= \lim_{z\rightarrow 1} (z-1)\bigg(z(z\mathbf{I}-\mathbf{F})^{-1}\mathbf{v}(0)+z(z\mathbf{I}-\mathbf{F})^{-1}\mathbf{q}\frac{x(0)z}{z-1}\bigg)$$

$$= x(0)(\mathbf{I}-\mathbf{F})^{-1}\mathbf{q}.$$ (4.17)

Inserting this in Eq. 4.13 the filter with step initialization would be given by

$$\mathbf{y} = x(0)\mathbf{B}(\mathbf{I}-\mathbf{F})^{-1}\mathbf{q}+\mathbf{C}\mathbf{x}$$

$$= (\mathbf{B}(\mathbf{I}-\mathbf{F})^{-1}\mathbf{q}\mathbf{1}^T+\mathbf{C})\mathbf{x}$$

$$= \mathbf{A}_{step}\ \mathbf{x}$$ (4.18)

where $\mathbf{1}^T$ is the $1\times N$ vector [1 0 ...0$]$. Transients can be partially suppressed by using a priori knowledge that the input signal is dominated by high amplitude, low Doppler frequency (nearly stationary) clutter.

Projection initialization

In [Cho92] the projection-initialized method was proposed to minimize the transient response. The transient part of the output signal is of the same form as the response with only zero as the input signal [Cho92]. Setting $x$ equal to 0 in Eq. 4.13 shows that the transient is in the subspace spanned by the columns of matrix $\mathbf{B}$. The projection matrix $\mathbf{P_B}=\mathbf{B}(\mathbf{B}^T\mathbf{B})^{-1}\mathbf{B}^T$ is the projection into this transient subspace [Cho92]. The transient component can be removed by forcing $\mathbf{P_B}y=0$. This can be obtained with the following initial state vector:

$$\mathbf{v}(0)=-(\mathbf{B}^T\mathbf{B})^{-1}\mathbf{B}^T\mathbf{C}\mathbf{x}.$$ (4.19)

Inserting this in Eq. 4.13, the filter operation, with projection initialization, is given by:

$$\mathbf{y}=(\mathbf{I}-\mathbf{B}(\mathbf{B}^T\mathbf{B})^{-1}\mathbf{B}^T)\mathbf{C}\mathbf{x}=\mathbf{A_p}\mathbf{x}$$ (4.20)

Comparison of different filter characteristics and initialization techniques

Figure 4.6: Frequency responses for $2^{nd}$ order Butterworth, Chebyshev, and elliptical filters depending on the initialization technique. The bottom right plot shows the steady-state response of the filters, where Butterworth and Chebyshev I characteristics overdraw one another. ($N$=8)

In Fig. 4.6, the frequency responses of Butterworth, Chebyshev I, and elliptic filters are shown with different initialization techniques. The frequency responses are produced using Eq. 4.5 for zero-, step- and projection-initialized IIR filters. Additionally the steady-state response of the filter is plotted to indicate the result for an infinite number of samples. It can be seen for all designed filters, that no initialization technique yields the result of the steady-state response. Increased cut-off frequencies and wider transition-bands are markable for all initialization techniques. Zero initialization results in insufficient stop-band attenuation. The step-initialized filters have a higher attenuation at zero frequency, but the stop-band itself is very narrow. The projection-initialized filters have a steady-state equal stop-band, but a wider transition region. Among the filter characteristics for IIR filters, the Chebyshev is a good choice. It provides a sufficient stop-band attenuation for step- and projection-initialized filters with an additionally low cut-off frequency. The Chebyshev I filter design is therefore chosen for this Thesis. If the desired stop-band attenuation must be increased above -80 dB, choosing a higher cut-off frequency is the wrong way. Increasing the cut-off frequency results in an undesirable widening of the transition region. A better approach is to increase the filter order, while keeping the cut-off frequency low (Fig. 4.7). Varying the pass-band-ripples for the steady-state response showed no significant improvement in the frequency response character of the filters. A peak-to-peak ripple of 0.2 dB was therfore used throughout this Thesis for IIR filter designs.

Figure 4.7: Projection-initialized responses of Chebyshev filters as a function of (steady-state) cut-off frequency and filter order for $N$=8.

Figure 4.8: Frequency response for $3^{rd}$ order Chebyshev IIR filters, with packet size 8 and 16.

The transition region of the filters can be narrowed by increasing $N$, which is illustrated in Fig. 4.8. For the step- and projection-initialized filter, the attenuation at zero frequency is not influenced by increasing the packet size. It can be clearly seen that the cut-off frequencies for the initialized filters are closer to the desired cut-off frequency for the steady-state response. This results in a narrower transition region.