Mutual information of $\infty$-PAM

I'm trying to compute using matlab the mutual information for an $\infty$-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck.

I'm working with a real-valued scalar Gaussian channel with input power constrained to unity and standard Gaussian noise $N \sim (0, 1)$. The output of the channel is given by, $$y = \sqrt{ \gamma } x + n$$

I know that under the power input constraint the input is uniformly distributed on $\left[ -\sqrt{ 3 }, \sqrt{ 3 } \right]$.

I found that in HIGH-SNR regime, there exists a gap between the Shannon capacity and the capacity achieved with $\infty$-PAM around 0.254 bits. I know this is called the shaping loss and represents the loss of using an uniform input rather than a Gaussian input.

I know how to compute that. But, I got stuck trying to compute the capacity for a range of snr or $\gamma$. I tried using the formula, $$I(x;y) = h(y) - h(n)$$ with $f_{y}(y) = \int f_{x}(x)f_{y|x}(y|x) dx$ and I got $$f_{y} (y) = \frac{ 1 }{ 2 \sqrt{ 3 \gamma } } \frac{ 1 }{ 2 } \int_{ \frac{ y - \sqrt{3} }{ \sqrt{2} } }^{ \frac{ y + \sqrt{3} }{ \sqrt{2} } } \frac{ 2 }{ \sqrt{ \pi } } e^{ -t^{2} } dt \\ = \frac{ 1 }{ 2 \sqrt{ 3 \gamma } } \frac{ 1 }{ 2 } \left[ \text{erf} \left( \frac{ y + \sqrt{ 3 \gamma } }{ \sqrt{ 2 } } \right) - \text{erf} \left( \frac{ y - \sqrt{ 3 \gamma } }{ \sqrt{ 2 } } \right) \right]$$

I tried to compute $h(y) = \int_{-\infty}^{\infty} f_{y} (y) \log_{2} f_{y} (y)$ and the integral does not converge. I tried with int$(\cdot)$ and integral$(\cdot)$ methods in matlab.

I also tried approximating the $\text{erf}(\cdot)$ function with Taylor series expansion $$erf(x) = \frac{ 2 }{ \sqrt{ \pi } } \sum_{ n = 0 }^{ \infty } \frac{ (-1)^{n} x^{ 2n + 1 } }{ n!(2n + 1) }$$ but it didn't work.

How can I obtain a more manageable integral? Is my approach correct?

I appreciate any help!! Thanks.

• Do you have any good references for $\infty$-PAM? I had never encountered it before. – MBaz Dec 1 '15 at 13:52
• Yes, Lozano, A., Tulino, A. M., & Verdú, S. Optimum power allocation for parallel Gaussian channels with arbitrary input distributions. In page nº 6. They comment three types of $\infty$-ary modulations. I decided to start with the $\infty$-PAM. I found in Richard E. Blahut, Principles and practices of information theory how to compute the shaping loss for $\infty$-PSK in page 276. I found, in different publications, a lot of mentions on the shaping loss in high-snr regime, but no how to compute the mutual information for a continuous uniform distributed input. – SNR Dec 1 '15 at 18:07
• Here, Sun, F. W., & van Tilborg, H. C. Approaching capacity by equiprobable signaling on the Gaussian channel. I found bounds on this quantity.I don't know, I think either there would be another approach or I'm doing something wrong with matlab I'm using the int and integral functions. – SNR Dec 1 '15 at 18:15
• OK, thanks for the pointers. I'm interested in this so I'll take a look, but it may be a while before I have the time. – MBaz Dec 1 '15 at 18:58

The $\text{erf}(x)$ function approximates to 1 with an error less than 0.0001 for $x \geq 3$. I plotted $f_{y} (y)$ and I saw that the most of probability is within a band denpending on $\sqrt{ 3 \gamma }$. Moreover, taking the limit $\lim_{ y\to\infty } g(y,\gamma) \log g(y,\gamma) = 0$. Then, I computed
$$h(y) = \frac{1}{2} \log \left( 24 \gamma \right) - \frac{ 1 }{ 2 \sqrt{ 3\gamma} } \frac{1}{2} \int_{ - \sqrt{ 3\gamma } - a }^{ \sqrt{ 3\gamma } + a } g(y,\gamma) \log_{2}( g(y,\gamma) ) dy$$
$$g (y,\gamma) = \text{erf} \left( \frac{ y + \sqrt{3\gamma} }{ \sqrt{2} } \right) - \text{erf} \left( \frac{ y - \sqrt{3\gamma} }{ \sqrt{2} } \right)$$
using $quadgk(\cdot)$ function in matlab.