In Analog-to-Digital Converters(ADC), the signal is first sampled at a rate higher than or equal to the Nyquist, then quantized and encoded. In Analog-to-Informations (AIC), the sampling and compression are combined in one step, mathematically:
$$ y = \Phi \cdot x $$
where $y$ is compressed signal, $\Phi$ is the sensing matrix, and $x$ is the input signal. My questions are: * Simulating the above equation using MATLAB is downsampling, First the input signal is read to obtain a vector of say $N$ samples representing the signal, and then multiplied by the matrix $\Phi$.
Does this mean that we sample the signal to obtain the vector $N$? If so, where is the benefit? How sampling and compression are combined in one step? How can the sampling and compression be combined to perform compressive sensing in a practical way?