At that price, I am guessing it is a passive adapter.
Some hardware is designed with both digital and analog in mine. In fact DVI-I ports for instance are build to do either digital or analog(It would be the same as having a VGA port on the card) so any after image issue is the screen or its cable. As for displayport to HDMI, it depends on the hardware. Many of these devices can use a passive DP to HDMI adapter, but AMD when they first started to add DisplayPort to cards said active adapters are needed if you want to use 3 monitors on a card(talking 5xxx cards when they started this).
DVI -> HDMI can be done with a simple passive cable upto 1920 x 1200 @ 60hz on almost any hardware and higher if both sides can handle the increased link speed. Even audio can be passed from DVI on some hardware.
DVI-D -> HDMI with audio.