# Mathematical model of a signal in Compressive Sampling

Currently I am reading a paper on Compressive Sampling, and trying to understand each and every parts of it. When I came across the mathematical model of the signal in paper, I got confused. I have mentioned the part of paper for your convenience

Consider the following mathematical model for a class of discrete multitone signals. Let $W/2$ be a positive integer that exceeds the highest frequency present in the continuous-time signal $f$. Fix a number $K$ that represents the active tones. The model contain each signal of the form $$f(t)=\sum_{\omega \in \Omega} a_{\omega} e^{-2\pi i \omega t},\hspace{5mm} for \hspace{2mm} t \in [0,1)$$ Here, $\Omega$ is a set of $K$ integer-valued frequencies that satisfies $$\Omega \subset \left \lbrace 0, \pm 1 , \pm 2, \ldots , \pm (W/2-1),W/2\right \rbrace$$ and $$\left\lbrace a_{\omega}:\omega \in \Omega\right\rbrace$$ is a set of complex-valued amplitudes. We focus on the case where the number $K$ of active tones is much smaller than the available bandwidth $W$.

Now my questions are:

1. I think there should be $\pm W/2$ in the set $\Omega$.
2. Does the signal model consider both the positive and negative value of frequency in a single signal?
3. What will be the signal $f(t)$ when $\Omega = \left\lbrace 0, \pm1, \pm2\right\rbrace$?

Thanks.

## 1 Answer

The model is the Fourier series of a signal up to the W/2th component, with sparse coefficients. That is, most of the set $\{a_k\}_{k=-W/2}^{W/2}$ is zero.

1. It is because the signal is bandlimited, and thus for a real signal, $a_{W/2}$ will be real. However, if this constraint is not met it is better to use $\pm W/2$
2. For a real valued signal, the coefficient of the negative frequency is just the conjugate of the positive frequency coefficient. It really just is a mathematical artifact. You could also write the model as

$$f(t) = \sum_{\omega\in\Omega} b_w \cos(2\pi \omega + \phi_\omega)$$

where $b_\omega$ is real and $\phi_\omega \in [0,2\pi)$ is a real phase value.

That is, the model represents the signal as the sum of sinusoids of different amplitudes and phases.

1. It will be the Fourier series approximation of the signal using just the first two frequencies.

You should take a look at this paper and the example code:

Towards a mathematical theory of super-resolution

• Thanks for your answer. Can we represent the signal as $$f(t)=\sum_{\omega \in \Omega} a_{\omega} e^{2\pi i \omega t}, \hspace{5mm} \text{for} \hspace{2mm} t \in [0,1)$$ – J Cian Oct 7 '15 at 14:56
• Yes. Keep in mind that $a_\omega = \bar{a}_{-\omega}$, and for a real-valued signal you MUST have both $a_\omega$ and $a_{-\omega}$. – geometrikal Oct 8 '15 at 5:32