The motivation for computing a fast Discrete Fourier Transform was the concern about verifying a nuclear arms treaty with the Soviet Union. A very much faster Fourier Transform was needed to plant sensors in the ground in countries surrounding the Soviet Union. After the invention of the FFT, sensors were planted which allowed locating nuclear explosions to within 15 kilometers of where they were occurring.
MIT OCW - Digital Signal Processing. This consists of 22 video lectures (20 lectures plus 2 demonstration videos) given by Prof. Alan V. Oppenheim, discussing the analysis and representation of discrete-time signal systems, digital filters, and computation of the discrete Fourier transform.
Z transform is used to convert discrete time domain into a complex frequency domain where, discrete time domain represents an order of complex or real numbers. It is generalize form of Fourier transform, which we get when we generalize Fourier transform and get z transform.