I am reading a electrical engineering book about digital signal processing, in the process, those fourier transform and discrete time fourier transform of constant and the exponential function lead to the delta function. I understand how to manipulate them formally, but I have serious trouble with them in terms of their mathematical foundation, such as why the DTFT "exist" (which definitely doesn't converge in the usual sense) and result in the delta function. I was told I need the theory of generalized function, which I have no background. I want to find some books or online lecture notes or resources to study them myself. Can anyone point me to the easiest way to study them rigorously without too big a detour to other area of mathematics? I have undergraduate analysis as well as some measure theory background. Please let me know what else I need to understand generalized functions, and what books would be best for this purpose. Thank you.