Processing Signals on Arbitrary Lattices

Figure 1: Block diagram of linear shift-invariant filtering input signal $x$ to produce output signal $y$.

• Almost any signal processing operation that can be performed on rectangularly sampled signals can be performed on arbitrarily sampled ones.
  • Filtering
  • Spectrum Analysis
  • Interpolation

• The response of the LSI system $h({\bf n})$ shown in Fig. 1 is
  
  $$y({\bf n}) = \sum_{\bf k} x({\bf k}) h({\bf n} - {\bf k}) = \sum_{\bf k} h({\bf k}) x({\bf n} - {\bf k})$$
  
  

• To compute the system frequency response, start with an analog continuous-space complex sinusoid with fixed frequency ${\bf\Omega}$, sample it with sampling matrix ${\bf V}$, and take the discrete-time Fourier transform:
  
  
  $$x_a(t)= e^{j {\bf\Omega}^T {\bf t}}$$
  
  
  
  
  $$x({\bf n}) = e^{j {\bf\Omega}^T {\bf V} {\bf n}}$$
  
  
  
  
  $$y({\bf n}) = \sum_{\bf k} h({\bf k}) x({\bf n} - {\bf k})$$
  
  
  
  
  $$y({\bf n}) = e^{j {\bf\Omega}^T {\bf V} {\bf n}} \underbrace... ...\right] }_{{\rm Frequency Response } H({\bf V}^T {\bf\Omega})}$$
  
  
  The relationship between the discrete-time and continuous-time Fourier domains is ${\bf\omega}^T = {\bf\Omega}^T {\bf V}$. Hence, ${\bf\omega} = {\bf V}^T {\bf\Omega}$.