# Maxium value of discrete convolution

I'm trying to calculate the maximum possible short-term energy $E[n]$ of a sampled signal $s$ in terms of $N$ and $\text{bitdepth}$.

$$E[n] =\sum_{m=-\infty}^{\infty} s^2[n]w[n-m]$$

where

$$w(n) = 0.54 - 0.46\; \cos \left ( \frac{2\pi n}{N-1} \right) \quad \text{(Hamming window)}$$

and $$-2^{bitdepth} \le s[n] \le 2^{bitdepth}$$ hope this makes sense

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8234259486198425, "perplexity_flag": "middle"}