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I am new to GIS and I want to fit a DEM given as an xyz file into an existing map.

My problem is that when I open the xyz in qgis the tile is not at the correct position.

Changing the CRS to the one given in the documentation of the tile results in a white raster image.

Here are the xyz tiles (they are quite large):

https://www.opengeodata.nrw.de/produkte/geobasis/hm/dgm1_xyz/dgm1_05314000_Bonn_EPSG4647_XYZ.zip

For the tiles EPSG:25832 is given in the documentation, here is a link:

https://www.opengeodata.nrw.de/produkte/geobasis/hm/dgm1_xyz/dgm1_meta.zip

Here is the gdalinfo for one of the tiles:

Driver: XYZ/ASCII Gridded XYZ
Files: dgm1_32370_5614_2_nw.xyz
Size is 2000, 2000
Coordinate System is `'
Origin = (32369999.500000000000000,5613999.500000000000000)
Pixel Size = (1.000000000000000,1.000000000000000)
Corner Coordinates:
Upper Left  (32369999.500, 5613999.500) 
Lower Left  (32369999.500, 5615999.500) 
Upper Right (32371999.500, 5613999.500) 
Lower Right (32371999.500, 5615999.500) 
Center      (32370999.500, 5614999.500) 
Band 1 Block=2000x1 Type=Float32, ColorInterp=Undefined
  Min=46.540 Max=169.160 

Ideally I would like to save the tiles all as one georeferenced raster with the correct CRS. The solution does not have to involve QGIS, if there is another alternative.

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I assume, that you used the EPSG 25832 because it says ETRS89_UTM32 in the field Koordinatenreferenzsystem in the metadata file. However, there are several different variants of this particular projection, one is indeed coded as 25832 by EPSG. The one, that should be used here is probably EPSG 4647, as hinted in the file name of the download link (which is a surprising exception. Hint: ALWAYS write the EPSG-code in your filenames, you never know where the stuff might end up).

When in doubt about the CRS this website is often very helpfull: https://spatialreference.org/

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