Calculation of time-variant Magnitude Responses

Before start of introducing clutter rejection filters, a common way of calculating the frequency response of a non-causal time-variant filter is described. A general 1-D filter operation can be expressed by a matrix multiplication A with an input vector $\mathbf{x}$ = $[x(0),x(1),\ldots,x(N-1)]^T$. The input vector is an N-dimensional complex vector $\mathcal{C^N}$. Therefore, the output vector y is also complex of type $\mathcal{C^N}$. Mathematically it can be written as:

$$\mathbf{y}=\mathbf{A}\mathbf{x},$$ (4.1)

where A is an $M\times N$ matrix, and the output vector y = $[y(0),y(1),\ldots,y(M-1)]^T$ has dimension M. With the matrix elements in row n and k denoted by a(n,k), the elements of the output vector are given by:

$$\begin{array}{lll} y(n) = \displaystyle{\sum^{N-1}_{k=0}} a(n,k)x(k)& & n=0,\ldots,M-1. \end{array}$$ (4.2)

The filter is linear, but generally not time invariant. Therefore, it is not possible to define the frequency response as the Fourier transform of the impulse response [BTK02a]. However, for a general linear filter, the frequency response can be defined as the power of the output signal when the input is a complex harmonic signal with unit amplitude. As an input function, a discrete-time complex exponential function, defined by

$$\begin{array}{ll} x(k)=e^{jk\omega} & k=0,\ldots,N-1 \end{array}$$ (4.3)

where $\omega \in [-\pi,\pi]$ is the normalized angular frequency and $j$ is the imaginary unit can be used. With the input signal of Eq. 4.3 the output signal becomes

$$\begin{array}{ll} y_{\omega}(n) = \displaystyle{\sum^{N-1... ...ga}\equiv \mathbf{A}_n(-\omega) & n=0,\ldots,M-1 \end{array}$$ (4.4)

where $\mathbf{A}_n(\omega)$ is the Fourier transform of row n in $a(n,k)$. The frequency response then becomes

$$H(\omega)=\frac{1}{M}\displaystyle{\sum^{M-1}_{n=0}\vert y... ...isplaystyle{\sum^{M-1}_{n=0}\vert\mathbf{A}(-\omega)\vert^2}.$$ (4.5)

This general method allows the calculation of the magnitude response of every linear time-variant non-causal filter.