Carlos, there is an entry for Ellettaria cardamom in the Plant Files, and the picture here http://davesgarden.com/guides/pf/showimage/41579/ looks just like yours with similar leaves and white flowers. But in the comments area under the picture a couple of people say that is not the right plant.
I'd guess the only way you're going to find out for sure is if your plant sets some seeds and they smell/taste like cardamom. So, if it does get as far as seed pods, put a net bag or some protection over them so the birds don't steal them, and let them ripen on the plant. Then smell a crushed pod. The scent is very distinctive as I'm sure you know.
After further reading in the comments below, on the Plant Files page for this,
"The differences are obvious and easy to see:
>Alpinia nutans: Glossy, aromatic leaves on 3' or smaller plants. Rarely blooms, but flowers are on the end of a leafy stalk. Easy-going plant.
>Elettaria cardamomum: Fuzzy, up to 10' tall plants with flowers on short stalks along the ground. Fairly rare in ornamental cultivation."
There was also an entry from someone who did indeed get seeds from their plant, ground them and used them in cooking. Said they tasted exactly like cardamom . .. maybe both plants give the seeds you want??