Polynomial regression filters

Regression filters operate on the assumption that the slowly varying clutter component in the signal can be approximated by a polynomial [HdVD$^+$95]. The least-squares fit to the low-frequency clutter component in the echo signal is then subtracted from the signal. Regression filters therefore work on a different concept compared to FIR or IIR filters, which are based on theories that signals are the superposition of sinusiods. Thus, the regression filter design is not based upon commonly known impulse or frequency response concepts. In order to compare the regression filters to traditional filters (IIR, FIR), a frequency response can be calculated with Eq. 4.5. The polynomials can be chosen to form an orthonormal basis for a $K$-dimensional clutter subspace of the $N$-dimensional signal space. The least$­$ squares clutter fit is the projection of the signal into the clutter subspace. Fig. 4.9 illustrates the projection of the signal into the clutter space.

Figure 4.9: Interpretation of how the signal space spanned by $b_1,b_2,b_3$ can be projected into the clutter space spanned by $b_1,b_2$. The signal vector can be projected into the clutter subspace, and from that, the clutter signal can be obtained. Subtracting the clutter signal from the input signal gives the output signal.

A linear filtering operation can generally be expressed as

$$\mathbf{y}=\mathbf{A_{reg}}\mathbf{x},$$ (4.21)

where $\mathbf{x}$ is the complex input signal vector (slow-time samples), $\mathbf{y}$ is the complex output vector, both of dimension $N\times1$, and $\mathbf{A_{reg}}$ is an $N\times N$ dimensional filter matrix which is given by

$$\mathbf{A_{reg}}=\mathbf{I} - \sum_{k=1}^Kb_kb_k^H.$$ (4.22)

where $k$ is the set of basis vectors $b_k$, for orthonormal bases often Legendre or Chebyshev polynomials. $(\bullet)^H$ is the Hermitian transposition. The frequency response of the filter can be calculated by

$$H(e^{j\omega t})=1 - \frac{1}{N}\sum_{k=1}^K\vert B_k(e^{j\omega t})\vert^2$$ (4.23)

where $B_k(e^{j\omega t})$ is the Fourier transform of the basis vector $b_k$ [Tor97]. In order to design HP filters, $K$ must be small compared to $N$. Regression filters are adaptive in the sense that the polynomial coefficients vary depending on the data. Polynomial regression filters with real valued basis functions have a symmetric frequency response, for instance the Legendre polynomials [Tor97]. The Legendre polynomials can be obtained by applying the Gram-Schmidt orthonormalization process to the series of polynomials $\{1,n,n^2,\ldots,n^K\}$ [Tor97]. Although straightforward regression analysis is computationally demanding, an efficient implementation based on a matrix approach can be provided [AL95]. Further investigations in this thesis use the Legendre polynomials as basis vectors and the results are called polynomial regression filters. The mentioned matrix approach for regression filtering is briefly reviewed in the following. Rewriting Eq. 4.21 gives:

$$c(n)= \sum_{k=0}^{K-1}a_kn^k ,\quad n=1,2,\ldots,N$$

$$y(n)= x(n)-c(n)$$ (4.24)

where $c(n)$ is the approximated low-frequency signal, $a_k$ is the regression model coefficient and $K$ is the polynomial degree. Fitting a polynomial of degree $K$ to a set of data points $\{n,x(n)\}, n=1,2\ldots N$ involves finding a set of coefficients $a_k$, $k=0,1,\ldots K$, such that the sum of the squared differences between the actual data and the model

$$\chi^2=\sum_{n=1}^N\bigg\{x(n) - \sum_{k=0}^{K-1}a_kn^k\bigg\}^2$$ (4.25)

is minimized, which is equivalent to

$$\frac{\partial\chi^2}{\partial a_i}=0$$

$$\frac{\partial^2\chi^2}{\partial a_i^2}>0 , \quad i=0,1,\ldots K.$$ (4.26)

Explicit evaluation of Eq. 4.26 results in the following set of linear equations:

$$\sum_{n=1}^N\bigg\{x(n) - \sum_{k=0}^{K-1}(a_kn^k)\bigg\}n^i ,\quad i=0,1,\ldots K$$ (4.27)

which can be rearranged to give

$$\sum_{k=0}^{K-1}\sum_{n=1}^Nn^in^ka_k= \sum_{n=1}^{N}n^ix(n) , \quad i=0,1,\ldots K.$$ (4.28)

By introducing the matrix vector notation

$$\mathbf{N}=\left[ \begin{array}{cccc} 1^0 & 1^1 & \ldots & 1^K\\ ... ...\vdots & \vdots & & \vdots\\ N^0 & N^1 & \ldots & N^K\\ \end{array} \right] , \mathbf{a}=\left[ \begin{array}{c} a_0\\ a_1\\ \vdots\\ a_k\\ \end{array} \right] , \mathbf{x}=\left[ \begin{array}{c} x(0)\\ x(1)\\ \vdots\\ x(N)\\ \end{array} \right]$$

one can write the above equation as

$$(\mathbf{N}^T\mathbf{N})\mathbf{a}=\mathbf{N}^T\mathbf{x}.$$ (4.29)

This can the be rearranged to

$$\mathbf{a} = \mathbf{P} \mathbf{x}$$

$$\mathbf{P} = (\mathbf{N}^T\mathbf{N})^{-1}\mathbf{N}^T=\left[ \begin{array}{c} \mathbf{p}_1\\ \mathbf{p}_2\\ \vdots\\ \mathbf{p}_K\\ \end{array} \right].$$ (4.30)

Here, $\mathbf{P}$ is the projection operator and represents a matrix of size $K\times N$ and $\mathbf{p_k}$ are row vectors of the projection matrix. The evaluation of matrix $\mathbf{P}$ is computationally demanding, but is totally independent of the input sequence since its elements are only determined by $K$ and $N$. Inserting Eq. 4.30 into Eq. 4.24, with respect that the value $n^k$ is placed in ^4.1 $\mathbf{N}(n,k)$. The second precalculation for fixed $K$ and $N$ can be obtained by evaluation of Eq. 4.24:

$$\mathbf{A}=\left[\begin{array}{cccccccc} \mathbf{p}_1\mathbf{N}(1... ...2\mathbf{N}(N,2)&+&\ldots &+&\mathbf{p}_K\mathbf{N}(N,K)\\ \end{array} \right]$$ (4.31)

With the precalculated matrix $\mathbf{A}$, Eq. 4.22 can be redefined to

$$\mathbf{A_{reg}}=\mathbf{I} - \mathbf{A}.$$ (4.32)

This gives an efficient way of implementing the regression filters for color flow imaging. The entire filter matrix can be precalculated, if the order $K$ and the packet size $N$ is fixed. It must be noted here that the second precalculation is not given in [AL95]. This precalculation was developed for optimization purposes during implementation of the filters for this Thesis.

Figure 4.10: Frequency responses for polynomial regression filters with different orders. In (a) $N$=8 and in (b) $N$=16.

(a)	(b)

The frequency response of polynomial regression filters changes in discrete steps with clutter space dimension as seen in Fig. 4.10. The response also varies with $N$. To obtain the same stop-band width with larger packet size, the clutter dimension must be increased. Increasing the clutter space dimension results in an increase of the 0 Hz attenuation but also in a wider transition band. To suppress low frequencies sufficiently and keep the pass-band region large, a large $N$ must be used.

Figure 4.11: Signal approximations for different orders of Legendre-polynomials. To find the corresponding attenuation in the filter frequency responses in Fig. 4.10(a) the signal frequency ( $f_{\rm{signal}}$) is given in terms of units from the $x$-axis of the graph. $N$=8.

$f_{\rm {signal}}=0.05$	$f_{\rm {signal}}=0.1$
$f_{\rm {signal}}=0.2$	$f_{\rm {signal}}=0.3$

To illustrate the polynomial, a signal with $\mathit{\sin}(2\pi (n-1) f_{\rm {signal}}$ where $n=1,2,\ldots,8$ was approximated. Fig. 4.11 presents different approximation orders for low frequency signals (compared to the sampling frequency). The signal frequency ( $f_{\rm {signal}}$) is given in units of the sampling frequency where 0.5 corresponds to half of the sampling frequency. If the signal approximation is of order 1, only a linear slope can be fitted. Second-order approximations can fit up to parabolic form (see Fig. 4.11 (a)). Higher order polynomials can give hyperbolic approximations of the signals. According to Fig. 4.11, low frequency signals can be approximated with a low order of polynomials. The higher the signal frequency, the more the order must be increased to get reasonable approximations. For example, for $f_{\rm {signal}}$=0.1 a good approximation can be achieved for order 4, whereas $f_{\rm {signal}}$=0.2 order 6 is needed. This shows that low frequency signals can be sufficiently approximated with low-order Legendre-polynomials.