Open the the file that carries the.new image. Then open the image file carrying the signature.
With the signature showing, choose Select > All or make a rectangular selection around the signature and follow that with Edit > Copy
Now move to the image file and choose Edit > Paste. The result will look l like this
Last, in the Layers panel change Bllelnding Mode Normal to Multiply. This is the result.
You can now use the Move tool to change its position and/or Edit > Transform > Scale to change the signature size.
Save the file as a psd so that it may be edited later, if desired. Then save as jpg, for example, to flatten the file