Can you convert 1080i into 1080p?

The real reason to convert 1080i to 1080p is for archival purposes. In most cases, 1080i video is a program or film recorded off a television broadcast and "trapped" in your DVR (until a full hard drive forces deletion). However, those who still have a working Windows Media Center (hint: if you upgraded to Windows 10, you DON'T), are able to capture live 1080i broadcast video as playable files (*.WTV). These files can then be converted to the format of your choice. Herein lies the problem. Most (if not all) conversion software out there is unable to successfully deinterlace 1080i broadcast video. Certainly not in the most popular file formats (*.mp4, *.mkv, *.avi, *.mov, etc.) The result is noticeable motion blur. Hopefully one day the technology exists to do this without Final Cut Pro, After Effects and a whole lot of man hours at the professional edit bay. It will remain an issue until 1080p is adopted by the cable and satellite providers.
1080i delivers a picture in two fields and together they make up a full frame. 1080p delivers a full frame in a single pass. There are converters that will convert between all HD formats, including 1080i to 1080p. However, modern HD televisions, using LCD and plasma displays are progressive displays. They will all convert 1080i (interlaced signals) into a progressive image for display. Delivering a converted 1080p signal will not provide any better results in many cases and indeed may reduce the quality of the image. A true 1080p signal carries twice as much information as a 1080i signal. Converting a 1080i signal to a 1080p signal does not in any way increase the amount of information in the image. Before investing money in converters, one needs to see the end results to make sure there is a real reason to make the conversion.